Bifurcation dynamics of the tempered fractional Langevin equation
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Zeng, Caibin, E-mail: macbzeng@scut.edu.cn; Yang, Qigui, E-mail: qgyang@scut.edu.cn [School of Mathematics, South China University of Technology, Guangzhou 510640 (China); Chen, YangQuan, E-mail: ychen53@ucmerced.edu [MESA LAB, School of Engineering, University of California, Merced, 5200 N. Lake Road, Merced, California 95343 (United States)
2016-08-15
Tempered fractional processes offer a useful extension for turbulence to include low frequencies. In this paper, we investigate the stochastic phenomenological bifurcation, or stochastic P-bifurcation, of the Langevin equation perturbed by tempered fractional Brownian motion. However, most standard tools from the well-studied framework of random dynamical systems cannot be applied to systems driven by non-Markovian noise, so it is desirable to construct possible approaches in a non-Markovian framework. We first derive the spectral density function of the considered system based on the generalized Parseval's formula and the Wiener-Khinchin theorem. Then we show that it enjoys interesting and diverse bifurcation phenomena exchanging between or among explosive-like, unimodal, and bimodal kurtosis. Therefore, our procedures in this paper are not merely comparable in scope to the existing theory of Markovian systems but also provide a possible approach to discern P-bifurcation dynamics in the non-Markovian settings.
Jeon, Jae-Hyung; Metzler, Ralf
2010-02-01
Motivated by subdiffusive motion of biomolecules observed in living cells, we study the stochastic properties of a non-Brownian particle whose motion is governed by either fractional Brownian motion or the fractional Langevin equation and restricted to a finite domain. We investigate by analytic calculations and simulations how time-averaged observables (e.g., the time-averaged mean-squared displacement and displacement correlation) are affected by spatial confinement and dimensionality. In particular, we study the degree of weak ergodicity breaking and scatter between different single trajectories for this confined motion in the subdiffusive domain. The general trend is that deviations from ergodicity are decreased with decreasing size of the movement volume and with increasing dimensionality. We define the displacement correlation function and find that this quantity shows distinct features for fractional Brownian motion, fractional Langevin equation, and continuous time subdiffusion, such that it appears an efficient measure to distinguish these different processes based on single-particle trajectory data.
International Nuclear Information System (INIS)
Lim, S C; Teo, L P
2009-01-01
Single-file diffusion behaves as normal diffusion at small time and as subdiffusion at large time. These properties can be described in terms of fractional Brownian motion with variable Hurst exponent or multifractional Brownian motion. We introduce a new stochastic process called Riemann–Liouville step fractional Brownian motion which can be regarded as a special case of multifractional Brownian motion with a step function type of Hurst exponent tailored for single-file diffusion. Such a step fractional Brownian motion can be obtained as a solution of the fractional Langevin equation with zero damping. Various kinds of fractional Langevin equations and their generalizations are then considered in order to decide whether their solutions provide the correct description of the long and short time behaviors of single-file diffusion. The cases where the dissipative memory kernel is a Dirac delta function, a power-law function and a combination of these functions are studied in detail. In addition to the case where the short time behavior of single-file diffusion behaves as normal diffusion, we also consider the possibility of a process that begins as ballistic motion
Generalised and Fractional Langevin Equations-Implications for Energy Balance Models
Watkins, N. W.; Chapman, S. C.; Chechkin, A.; Ford, I.; Klages, R.; Stainforth, D. A.
2017-12-01
Energy Balance Models (EBMs) have a long heritage in climate science, including their use in modelling anomalies in global mean temperature. Many types of EBM have now been studied, and this presentation concerns the stochastic EBMs, which allow direct treatment of climate fluctuations and noise. Some recent stochastic EBMs (e.g. [1]) map on to Langevin's original form of his equation, with temperature anomaly replacing velocity, and other corresponding replacements being made. Considerable sophistication has now been reached in the application of multivariate stochastic Langevin modelling in many areas of climate. Our work is complementary in intent and investigates the Mori-Kubo "Generalised Langevin Equation" (GLE) which incorporates non-Markovian noise and response in a univariate framework, as a tool for modelling GMT [2]. We show how, if it is present, long memory simplifies the GLE to a fractional Langevin equation (FLE). Evidence for long range memory in global temperature, and the success of fractional Gaussian noise in its prediction [5] has already motivated investigation of a power law response model [3,4,5]. We go beyond this work to ask whether an EBM of FLE-type exists, and what its solutions would be. [l] Padilla et al, J. Climate (2011); [2] Watkins, GRL (2013); [3] Rypdal, JGR (2012); [4] Rypdal and Rypdal, J. Climate (2014); [5] Lovejoy et al, ESDD (2015).
International Nuclear Information System (INIS)
Sandev, Trifce; Metzler, Ralf; Tomovski, Živorad
2014-01-01
We study generalized fractional Langevin equations in the presence of a harmonic potential. General expressions for the mean velocity and particle displacement, the mean squared displacement, position and velocity correlation functions, as well as normalized displacement correlation function are derived. We report exact results for the cases of internal and external friction, that is, when the driving noise is either internal and thus the fluctuation-dissipation relation is fulfilled or when the noise is external. The asymptotic behavior of the generalized stochastic oscillator is investigated, and the case of high viscous damping (overdamped limit) is considered. Additional behaviors of the normalized displacement correlation functions different from those for the regular damped harmonic oscillator are observed. In addition, the cases of a constant external force and the force free case are obtained. The validity of the generalized Einstein relation for this process is discussed. The considered fractional generalized Langevin equation may be used to model anomalous diffusive processes including single file-type diffusion
Pomeau, Yves; Piasecki, Jarosław
2017-11-01
The existence of atoms has been long predicted by philosophers and scientists. The development of thermodynamics and of the statistical interpretation of its concepts at the end of the nineteenth century and in the early years of the twentieth century made it possible to bridge the gap of scales between the macroscopic world and the world of atoms. Einstein and Smoluchowski showed in 1905 and 1906 that the Brownian motion of particles of measurable size is a manifestation of the motion of atoms in fluids. Their derivation was completely different from each other. Langevin showed in 1908 how to put in a coherent framework the subtle effect of the randomness of the atomic world, responsible for the fluctuating force driving the motion of the Brownian particle and the viscosity of the "macroscopic" flow taking place around the same Brownian particle. Whereas viscous forces were already well understood at this time, the "Langevin" force appears there for the first time: it represents the fluctuating part of the interaction between the Brownian particle and the surrounding fluid. We discuss the derivation by Einstein and Smoluchowski as well as a previous paper by Sutherland on the diffusion coefficient of large spheres. Next we present Langevin's short note and explain the fundamental splitting into a random force and a macroscopic viscous force. This brings us to discuss various points, like the kind of constraints on Langevin-like equations. We insist in particular on the one arising from the time-reversal symmetry of the equilibrium fluctuations. Moreover, we discuss another constraint, raised first by Lorentz, which implies that, if the Brownian particle is not very heavy, the viscous force cannot be taken as the standard Stokes drag on an object moving at uniform speed. Lastly, we examine the so-called Langevin-Heisenberg and/or Langevin-Schrödinger equation used in quantum mechanics.
Lorentz Covariance of Langevin Equation
International Nuclear Information System (INIS)
Koide, T.; Denicol, G.S.; Kodama, T.
2008-01-01
Relativistic covariance of a Langevin type equation is discussed. The requirement of Lorentz invariance generates an entanglement between the force and noise terms so that the noise itself should not be a covariant quantity. (author)
Lisý, Vladimír; Tóthová, Jana
2018-02-01
Nuclear magnetic resonance is often used to study random motion of spins in different systems. In the long-time limit the current mathematical description of the experiments allows proper interpretation of measurements of normal and anomalous diffusion. The shorter-time dynamics is however correctly considered only in a few works that do not go beyond the standard Langevin theory of the Brownian motion (BM). In the present work, the attenuation function S (t) for an ensemble of spins in a magnetic-field gradient, expressed in a form applicable for any kind of stationary stochastic dynamics of spins with or without a memory, is calculated in the frame of the model of fractional BM. The solution of the model for particles trapped in a harmonic potential is obtained in a simple way and used for the calculation of S (t). In the limit of free particles coupled to a fractal heat bath, the results compare favorably with experiments acquired in human neuronal tissues.
Nuclear fission with a Langevin equation
International Nuclear Information System (INIS)
Boilley, D.; Suraud, E.; Abe, Yasuhisa
1992-01-01
A microscopically derived Langevin equation is applied to thermally induced nuclear fission. An important memory effect is pointed out and discussed. A strong friction coefficient, estimated from microscopic quantities, tends to decrease the stationary limit of the fission rate and to increase the transient time. The calculations are performed with a collective mass depending on the collective variable and with a constant mass. Fission rates calculated at different temperatures are shown and compared with previous available results. (author) 23 refs.; 7 figs
Deriving Langevin equations in curved spacetime
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Ramos, Rudnei O.; Tavares, Romulo F.
2013-01-01
Full text: Warm inflation is an inflationary scenario where the interactions between the inflaton and other degrees of freedom are considered. The effective equation of motion for the inflaton is in general of the form of a Langevin equation, that includes both quantum and thermal effects and where these effects manifest in the form of dissipation and stochastic noise terms, which are related by a generalized fluctuation-dissipation relation. The dissipation term is related to the interactions of the inflaton with other degrees of freedom of the thermal bath that can be obtained from the appropriate Feynman propagators. As the inflaton evolves into an expanding metric, these effects have to be taken into account when calculating the Green functions and consequently the Feynman propagators. In this work we present the considerations that must be made to calculate the Green functions in curved space (expanding metric) and in the presence of radiation in order to proper derive the effective evolution of the inflaton in the warm-inflation scenario. (author)
Computing generalized Langevin equations and generalized Fokker-Planck equations.
Darve, Eric; Solomon, Jose; Kia, Amirali
2009-07-07
The Mori-Zwanzig formalism is an effective tool to derive differential equations describing the evolution of a small number of resolved variables. In this paper we present its application to the derivation of generalized Langevin equations and generalized non-Markovian Fokker-Planck equations. We show how long time scales rates and metastable basins can be extracted from these equations. Numerical algorithms are proposed to discretize these equations. An important aspect is the numerical solution of the orthogonal dynamics equation which is a partial differential equation in a high dimensional space. We propose efficient numerical methods to solve this orthogonal dynamics equation. In addition, we present a projection formalism of the Mori-Zwanzig type that is applicable to discrete maps. Numerical applications are presented from the field of Hamiltonian systems.
On the Langevin equation for stochastic quantization of gravity
International Nuclear Information System (INIS)
Nakazawa, Naohito.
1989-10-01
We study the Langevin equation for stochastic quantization of gravity. By introducing two independent variables with a second-class constraint for the gravitational field, we formulate a pair of the Langevin equations for gravity which couples with white noises. After eliminating the multiplier field for the second-class constraint, we show that the equations leads to stochastic quantization of gravity including an unique superspace metric. (author)
Simplified simulation of Boltzmann-Langevin equation
International Nuclear Information System (INIS)
Ayik, S.; Randrup, J.
1994-01-01
We briefly recall the Boltzmann-Langevin model of nuclear dynamics. We then summarize recent progress in deriving approximate analytical expressions for the associated transport coefficients and describe a numerical method for simulating the stochastic evolution of the phase-space density. (orig.)
Brownian motion of spins; generalized spin Langevin equation
International Nuclear Information System (INIS)
Jayannavar, A.M.
1990-03-01
We derive the Langevin equations for a spin interacting with a heat bath, starting from a fully dynamical treatment. The obtained equations are non-Markovian with multiplicative fluctuations and concomitant dissipative terms obeying the fluctuation-dissipation theorem. In the Markovian limit our equations reduce to the phenomenological equations proposed by Kubo and Hashitsume. The perturbative treatment on our equations lead to Landau-Lifshitz equations and to other known results in the literature. (author). 16 refs
Numerical integration of the Langevin equation: Monte Carlo simulation
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Ermak, D.L.; Buckholz, H.
1980-01-01
Monte Carlo simulation techniques are derived for solving the ordinary Langevin equation of motion for a Brownian particle in the presence of an external force. These methods allow considerable freedom in selecting the size of the time step, which is restricted only by the rate of change in the external force. This approach is extended to the generalized Langevin equation which uses a memory function in the friction force term. General simulation techniques are derived which are independent of the form of the memory function. A special method requiring less storage space is presented for the case of the exponential memory function
Localization and Ballistic Diffusion for the Tempered Fractional Brownian-Langevin Motion
Chen, Yao; Wang, Xudong; Deng, Weihua
2017-10-01
This paper discusses the tempered fractional Brownian motion (tfBm), its ergodicity, and the derivation of the corresponding Fokker-Planck equation. Then we introduce the generalized Langevin equation with the tempered fractional Gaussian noise for a free particle, called tempered fractional Langevin equation (tfLe). While the tfBm displays localization diffusion for the long time limit and for the short time its mean squared displacement (MSD) has the asymptotic form t^{2H}, we show that the asymptotic form of the MSD of the tfLe transits from t^2 (ballistic diffusion for short time) to t^{2-2H}, and then to t^2 (again ballistic diffusion for long time). On the other hand, the overdamped tfLe has the transition of the diffusion type from t^{2-2H} to t^2 (ballistic diffusion). The tfLe with harmonic potential is also considered.
Stochastic TDHF and the Boltzman-Langevin equation
International Nuclear Information System (INIS)
Suraud, E.; Reinhard, P.G.
1991-01-01
Outgoing from a time-dependent theory of correlations, we present a stochastic differential equation for the propagation of ensembles of Slater determinants, called Stochastic Time-Dependent Hartree-Fock (Stochastic TDHF). These ensembles are allowed to develop large fluctuations in the Hartree-Fock mean fields. An alternative stochastic differential equation, the Boltzmann-Langevin equation, can be derived from Stochastic TDHF by averaging over subensembles with small fluctuations
Applications of Boltzmann Langevin equation to nuclear collisions
International Nuclear Information System (INIS)
Suraud, E.; Belkacem, M.; Stryjewski, J.; Ayik, S.
1991-01-01
An approximate method for obtaining numerical solutions of the Boltzmann-Langevin equation is proposed. The method is applied to calculate the time evolution of the mean value and dispersion of the quadrupole and octupole moments of the momentum distribution in nucleus-nucleus collisions, and some consequences are discussed
Boltzmann-Langevin equation, dynamical instability and multifragmentation
International Nuclear Information System (INIS)
Feng-Shou Zhang
1993-02-01
By using simulations of the Boltzmann-Langevin equation which incorporates dynamical fluctuations beyond usual transport theories and by coupling it with a coalescence model, we obtain information on multifragmentation in heavy-ion collisions. From a calculation of the 40 Ca + 40 Ca system, we recover some trends of recent multifragmentation data
Numerical simulations of generalized Langevin equations with deeply asymptotic parameters
International Nuclear Information System (INIS)
Bao Jingdong; Li Rongwu; Wu Wei
2004-01-01
A unified algorithm for solving Langevin equations with deeply asymptotic parameters is proposed and tested. The method consists of identifying solvable linear friction and implementing the force evaluations by use of the Runge-Kutta method. We apply the present scheme to the periodic motion of an overdamped particle subjected to a multiplicative white noise. The accurate calculations for the temporal velocity of the particle and its correlation function can be realized by introducing an inertial term. It is shown that the fluctuation around the steady quantity increases with decreasing time step in the overdamped white-noise algorithm, however, a massive white-noise technique greatly reduces this spurious drift, and the result can converge to the correct value if the added inertia approaches zero. The other application is the simulation of generalized Langevin equation with an exponential memory friction, this allows us to treat a weak non-Markovian process
Langevin equations with multiplicative noise: application to domain growth
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Sancho, J.M.; Hernandez-Machado, A.; Ramirez-Piscina, L.; Lacasta, A.M.
1993-01-01
Langevin equations of Ginzburg-Landau form with multiplicative noise, are proposed to study the effects of fluctuations in domain growth. These equations are derived from a coarse-grained methodology. The Cahn-Hilliard-Cook linear stability analysis predicts some effects in the transitory regime. We also derive numerical algorithms for the computer simulation of these equations. The numerical results corroborate the analytical productions of the linear analysis. We also present simulation results for spinodal decomposition at large times. (author). 28 refs, 2 figs
An Analysis of Vehicular Traffic Flow Using Langevin Equation
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Çağlar Koşun
2015-08-01
Full Text Available Traffic flow data are stochastic in nature, and an abundance of literature exists thereof. One way to express stochastic data is the Langevin equation. Langevin equation consists of two parts. The first part is known as the deterministic drift term, the other as the stochastic diffusion term. Langevin equation does not only help derive the deterministic and random terms of the selected portion of the city of Istanbul traffic empirically, but also sheds light on the underlying dynamics of the flow. Drift diagrams have shown that slow lane tends to get congested faster when vehicle speeds attain a value of 25 km/h, and it is 20 km/h for the fast lane. Three or four distinct regimes may be discriminated again from the drift diagrams; congested, intermediate, and free-flow regimes. At places, even the intermediate regime may be divided in two, often with readiness to congestion. This has revealed the fact that for the selected portion of the highway, there are two main states of flow, namely, congestion and free-flow, with an intermediate state where the noise-driven traffic flow forces the flow into either of the distinct regimes.
On the non-stationary generalized Langevin equation
Meyer, Hugues; Voigtmann, Thomas; Schilling, Tanja
2017-12-01
In molecular dynamics simulations and single molecule experiments, observables are usually measured along dynamic trajectories and then averaged over an ensemble ("bundle") of trajectories. Under stationary conditions, the time-evolution of such averages is described by the generalized Langevin equation. By contrast, if the dynamics is not stationary, it is not a priori clear which form the equation of motion for an averaged observable has. We employ the formalism of time-dependent projection operator techniques to derive the equation of motion for a non-equilibrium trajectory-averaged observable as well as for its non-stationary auto-correlation function. The equation is similar in structure to the generalized Langevin equation but exhibits a time-dependent memory kernel as well as a fluctuating force that implicitly depends on the initial conditions of the process. We also derive a relation between this memory kernel and the autocorrelation function of the fluctuating force that has a structure similar to a fluctuation-dissipation relation. In addition, we show how the choice of the projection operator allows us to relate the Taylor expansion of the memory kernel to data that are accessible in MD simulations and experiments, thus allowing us to construct the equation of motion. As a numerical example, the procedure is applied to Brownian motion initialized in non-equilibrium conditions and is shown to be consistent with direct measurements from simulations.
Quantum Non-Markovian Langevin Equations and Transport Coefficients
International Nuclear Information System (INIS)
Sargsyan, V.V.; Antonenko, N.V.; Kanokov, Z.; Adamian, G.G.
2005-01-01
Quantum diffusion equations featuring explicitly time-dependent transport coefficients are derived from generalized non-Markovian Langevin equations. Generalized fluctuation-dissipation relations and analytic expressions for calculating the friction and diffusion coefficients in nuclear processes are obtained. The asymptotic behavior of the transport coefficients and correlation functions for a damped harmonic oscillator that is linearly coupled in momentum to a heat bath is studied. The coupling to a heat bath in momentum is responsible for the appearance of the diffusion coefficient in coordinate. The problem of regression of correlations in quantum dissipative systems is analyzed
Solving the generalized Langevin equation with the algebraically correlated noise
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Srokowski, T.; Ploszajczak, M.
1997-01-01
The Langevin equation with the memory kernel is solved. The stochastic force possesses algebraic correlations, proportional to 1/t. The velocity autocorrelation function and related quantities characterizing transport properties are calculated at the assumption that the system is in the thermal equilibrium. Stochastic trajectories are simulated numerically, using the kangaroo process as a noise generator. Results of this simulation resemble Levy walks with divergent moments of the velocity distribution. The motion of a Brownian particle is considered both without any external potential and in the harmonic oscillator field, in particular the escape from a potential well. The results are compared with memory-free calculations for the Brownian particle. (author)
Coffey, W T; Titov, S V
2003-01-01
A theory of orientational relaxation for the inertial rotational Brownian motion of a symmetric top molecule is developed using the Langevin equation rather than the Fokker-Planck equation. The infinite hierarchy of differential-recurrence relations for the orientational correlation functions for the relaxation behaviour is derived by averaging the corresponding Euler-Langevin equations. The solution of this hierarchy is obtained using matrix continued fractions allowing the calculation of the correlation times and the spectra of the orientational correlation functions for typical values of the model parameters.
Is the Langevin phase equation an efficient model for oscillating neurons?
Ota, Keisuke; Tsunoda, Takamasa; Omori, Toshiaki; Watanabe, Shigeo; Miyakawa, Hiroyoshi; Okada, Masato; Aonishi, Toru
2009-12-01
The Langevin phase model is an important canonical model for capturing coherent oscillations of neural populations. However, little attention has been given to verifying its applicability. In this paper, we demonstrate that the Langevin phase equation is an efficient model for neural oscillators by using the machine learning method in two steps: (a) Learning of the Langevin phase model. We estimated the parameters of the Langevin phase equation, i.e., a phase response curve and the intensity of white noise from physiological data measured in the hippocampal CA1 pyramidal neurons. (b) Test of the estimated model. We verified whether a Fokker-Planck equation derived from the Langevin phase equation with the estimated parameters could capture the stochastic oscillatory behavior of the same neurons disturbed by periodic perturbations. The estimated model could predict the neural behavior, so we can say that the Langevin phase equation is an efficient model for oscillating neurons.
Is the Langevin phase equation an efficient model for oscillating neurons?
International Nuclear Information System (INIS)
Ota, Keisuke; Tsunoda, Takamasa; Aonishi, Toru; Omori, Toshiaki; Okada, Masato; Watanabe, Shigeo; Miyakawa, Hiroyoshi
2009-01-01
The Langevin phase model is an important canonical model for capturing coherent oscillations of neural populations. However, little attention has been given to verifying its applicability. In this paper, we demonstrate that the Langevin phase equation is an efficient model for neural oscillators by using the machine learning method in two steps: (a) Learning of the Langevin phase model. We estimated the parameters of the Langevin phase equation, i.e., a phase response curve and the intensity of white noise from physiological data measured in the hippocampal CA1 pyramidal neurons. (b) Test of the estimated model. We verified whether a Fokker-Planck equation derived from the Langevin phase equation with the estimated parameters could capture the stochastic oscillatory behavior of the same neurons disturbed by periodic perturbations. The estimated model could predict the neural behavior, so we can say that the Langevin phase equation is an efficient model for oscillating neurons.
Mélykúti, Bence; Burrage, Kevin; Zygalakis, Konstantinos C.
2010-01-01
The Chemical Langevin Equation (CLE), which is a stochastic differential equation driven by a multidimensional Wiener process, acts as a bridge between the discrete stochastic simulation algorithm and the deterministic reaction rate equation when
Fractional Schroedinger equation
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Laskin, Nick
2002-01-01
Some properties of the fractional Schroedinger equation are studied. We prove the Hermiticity of the fractional Hamilton operator and establish the parity conservation law for fractional quantum mechanics. As physical applications of the fractional Schroedinger equation we find the energy spectra of a hydrogenlike atom (fractional 'Bohr atom') and of a fractional oscillator in the semiclassical approximation. An equation for the fractional probability current density is developed and discussed. We also discuss the relationships between the fractional and standard Schroedinger equations
Müller, Eike H; Scheichl, Rob; Shardlow, Tony
2015-04-08
This paper applies several well-known tricks from the numerical treatment of deterministic differential equations to improve the efficiency of the multilevel Monte Carlo (MLMC) method for stochastic differential equations (SDEs) and especially the Langevin equation. We use modified equations analysis as an alternative to strong-approximation theory for the integrator, and we apply this to introduce MLMC for Langevin-type equations with integrators based on operator splitting. We combine this with extrapolation and investigate the use of discrete random variables in place of the Gaussian increments, which is a well-known technique for the weak approximation of SDEs. We show that, for small-noise problems, discrete random variables can lead to an increase in efficiency of almost two orders of magnitude for practical levels of accuracy.
Superstatistical generalised Langevin equation: non-Gaussian viscoelastic anomalous diffusion
Ślęzak, Jakub; Metzler, Ralf; Magdziarz, Marcin
2018-02-01
Recent advances in single particle tracking and supercomputing techniques demonstrate the emergence of normal or anomalous, viscoelastic diffusion in conjunction with non-Gaussian distributions in soft, biological, and active matter systems. We here formulate a stochastic model based on a generalised Langevin equation in which non-Gaussian shapes of the probability density function and normal or anomalous diffusion have a common origin, namely a random parametrisation of the stochastic force. We perform a detailed analysis demonstrating how various types of parameter distributions for the memory kernel result in exponential, power law, or power-log law tails of the memory functions. The studied system is also shown to exhibit a further unusual property: the velocity has a Gaussian one point probability density but non-Gaussian joint distributions. This behaviour is reflected in the relaxation from a Gaussian to a non-Gaussian distribution observed for the position variable. We show that our theoretical results are in excellent agreement with stochastic simulations.
Langevin equation in systems with also negative temperatures
Baldovin, Marco; Puglisi, Andrea; Vulpiani, Angelo
2018-04-01
We discuss how to derive a Langevin equation (LE) in non standard systems, i.e. when the kinetic part of the Hamiltonian is not the usual quadratic function. This generalization allows to consider also cases with negative absolute temperature. We first give some phenomenological arguments suggesting the shape of the viscous drift, replacing the usual linear viscous damping, and its relation with the diffusion coefficient modulating the white noise term. As a second step, we implement a procedure to reconstruct the drift and the diffusion term of the LE from the time-series of the momentum of a heavy particle embedded in a large Hamiltonian system. The results of our reconstruction are in good agreement with the phenomenological arguments. Applying the method to systems with negative temperature, we can observe that also in this case there is a suitable LE, obtained with a precise protocol, able to reproduce in a proper way the statistical features of the slow variables. In other words, even in this context, systems with negative temperature do not show any pathology.
The Boltzmann-Langevin Equation derived from the real-time path formalism
International Nuclear Information System (INIS)
Suraud, E.; Reinhard, P.G.
1991-01-01
We derive the Boltzmann-Langevin equation using Green's functions techniques in the real-time path formalism. We start from the Martin-Schwinger hierarchy and close it approximately at the two-body level. A careful discussion of the initial conditions for the free two-body Green's function provides the flexibility to recover the discarded correlations as fluctuations leading to the Langevin force. The derivation is generalized to the T-matrix approach which allows to prove that one can use the same effective interaction in the mean-field as well as in the collision term and Langevin force
Schrödinger–Langevin equation with quantum trajectories for photodissociation dynamics
Energy Technology Data Exchange (ETDEWEB)
Chou, Chia-Chun, E-mail: ccchou@mx.nthu.edu.tw
2017-02-15
The Schrödinger–Langevin equation is integrated to study the wave packet dynamics of quantum systems subject to frictional effects by propagating an ensemble of quantum trajectories. The equations of motion for the complex action and quantum trajectories are derived from the Schrödinger–Langevin equation. The moving least squares approach is used to evaluate the spatial derivatives of the complex action required for the integration of the equations of motion. Computational results are presented and analyzed for the evolution of a free Gaussian wave packet, a two-dimensional barrier model, and the photodissociation dynamics of NOCl. The absorption spectrum of NOCl obtained from the Schrödinger–Langevin equation displays a redshift when frictional effects increase. This computational result agrees qualitatively with the experimental results in the solution-phase photochemistry of NOCl.
Langevin equation in effective theory of interacting QCD pomerons in the limit of large Nc
International Nuclear Information System (INIS)
Bondarenko, S.
2007-01-01
Effective field theory of interacting BFKL pomerons is investigated and Langevin equation for the theory, which arises after the introduction of additional auxiliary field, is obtained. The Langevin equations are considered for the case of interacting BFKL pomerons with both splitting and merging vertexes and for the interaction which includes additional 'toy' four pomeron interaction vertex. In the latest case an analogy with the Regge field theory in zero dimensions (RFT-0) was used in order to obtain this 'toy' vertex, which coincided with the four point function of two-dimensional conformal field theory obtained in [G.P. Korchemsky, Nucl. Phys. B 550 (1999) 397]. The comparison between the Langevin equations obtained in the frameworks of dipole and RFT approaches is performed, the interpretation of results is given and possible application of obtained equations is discussed
Treatment of constraints in the stochastic quantization method and covariantized Langevin equation
International Nuclear Information System (INIS)
Ikegami, Kenji; Kimura, Tadahiko; Mochizuki, Riuji
1993-01-01
We study the treatment of the constraints in the stochastic quantization method. We improve the treatment of the stochastic consistency condition proposed by Namiki et al. by suitably taking into account the Ito calculus. Then we obtain an improved Langevin equation and the Fokker-Planck equation which naturally leads to the correct path integral quantization of the constrained system as the stochastic equilibrium state. This treatment is applied to an O(N) non-linear σ model and it is shown that singular terms appearing in the improved Langevin equation cancel out the δ n (0) divergences in one loop order. We also ascertain that the above Langevin equation, rewritten in terms of independent variables, is actually equivalent to the one in the general-coordinate transformation covariant and vielbein-rotation invariant formalism. (orig.)
On the interpretations of Langevin stochastic equation in different coordinate systems
International Nuclear Information System (INIS)
Martinez, E.; Lopez-Diaz, L.; Torres, L.; Alejos, O.
2004-01-01
The stochastic Langevin Landau-Lifshitz equation is usually utilized in micromagnetics formalism to account for thermal effects. Commonly, two different interpretations of the stochastic integrals can be made: Ito and Stratonovich. In this work, the Langevin-Landau-Lifshitz (LLL) equation is written in both Cartesian and Spherical coordinates. If Spherical coordinates are employed, the noise is additive, and therefore, Ito and Stratonovich solutions are equal. This is not the case when (LLL) equation is written in Cartesian coordinates. In this case, the Langevin equation must be interpreted in the Stratonovich sense in order to reproduce correct statistical results. Nevertheless, the statistics of the numerical results obtained from Euler-Ito and Euler-Stratonovich schemes are equivalent due to the additional numerical constraint imposed in Cartesian system after each time step, which itself assures that the magnitude of the magnetization is preserved
Elizondo-Aguilera, L. F.; Zubieta Rico, P. F.; Ruiz-Estrada, H.; Alarcón-Waess, O.
2014-11-01
A self-consistent generalized Langevin-equation theory is proposed to describe the self- and collective dynamics of a liquid of linear Brownian particles. The equations of motion for the spherical harmonics projections of the collective and self-intermediate-scattering functions, Fl m ,l m(k ,t ) and Flm ,l m S(k ,t ) , are derived as a contraction of the description involving the stochastic equations of the corresponding tensorial one-particle density nl m(k ,t ) and the translational (α =T ) and rotational (α =R ) current densities jlm α(k ,t ) . Similar to the spherical case, these dynamic equations require as an external input the equilibrium structural properties of the system contained in the projections of the static structure factor, denoted by Sl m ,l m(k ) . Complementing these exact equations with simple (Vineyard-like) approximate relations for the collective and the self-memory functions we propose a closed self-consistent set of equations for the dynamic properties involved. In the long-time asymptotic limit, these equations become the so-called bifurcation equations, whose solutions (the nonergodicity parameters) can be written, extending the spherical case, in terms of one translational and one orientational scalar dynamic order parameter, γT and γR, which characterize the possible dynamical arrest transitions of the system. As a concrete illustrative application of this theory we determine the dynamic arrest diagram of the dipolar hard-sphere fluid. In qualitative agreement with mode coupling theory, the present self-consistent equations also predict three different regions in the state space spanned by the macroscopic control parameters η (volume fraction) and T* (scaled temperature): a region of fully ergodic states, a region of mixed states, in which the translational degrees of freedom become arrested while the orientational degrees of freedom remain ergodic, and a region of fully nonergodic states.
Elizondo-Aguilera, L F; Zubieta Rico, P F; Ruiz-Estrada, H; Alarcón-Waess, O
2014-11-01
A self-consistent generalized Langevin-equation theory is proposed to describe the self- and collective dynamics of a liquid of linear Brownian particles. The equations of motion for the spherical harmonics projections of the collective and self-intermediate-scattering functions, F_{lm,lm}(k,t) and F_{lm,lm}^{S}(k,t), are derived as a contraction of the description involving the stochastic equations of the corresponding tensorial one-particle density n_{lm}(k,t) and the translational (α=T) and rotational (α=R) current densities j_{lm}^{α}(k,t). Similar to the spherical case, these dynamic equations require as an external input the equilibrium structural properties of the system contained in the projections of the static structure factor, denoted by S_{lm,lm}(k). Complementing these exact equations with simple (Vineyard-like) approximate relations for the collective and the self-memory functions we propose a closed self-consistent set of equations for the dynamic properties involved. In the long-time asymptotic limit, these equations become the so-called bifurcation equations, whose solutions (the nonergodicity parameters) can be written, extending the spherical case, in terms of one translational and one orientational scalar dynamic order parameter, γ_{T} and γ_{R}, which characterize the possible dynamical arrest transitions of the system. As a concrete illustrative application of this theory we determine the dynamic arrest diagram of the dipolar hard-sphere fluid. In qualitative agreement with mode coupling theory, the present self-consistent equations also predict three different regions in the state space spanned by the macroscopic control parameters η (volume fraction) and T* (scaled temperature): a region of fully ergodic states, a region of mixed states, in which the translational degrees of freedom become arrested while the orientational degrees of freedom remain ergodic, and a region of fully nonergodic states.
Study of fission dynamics with the three-dimensional Langevin equations
Energy Technology Data Exchange (ETDEWEB)
Eslamizadeh, H. [Persian Gulf University, Department of Physics, Bushehr (Iran, Islamic Republic of)
2011-11-15
The dynamics of fission has been studied by solving one- and three-dimensional Langevin equations with dissipation generated through the chaos weighted wall and window friction formula. The average prescission neutron multiplicities, fission probabilities and the mean fission times have been calculated in a broad range of the excitation energy for compound nuclei {sup 210}Po and {sup 224}Th formed in the fusion-fission reactions {sup 4}He+{sup 206}Pb, {sup 16}O+{sup 208}Pb and results compared with the experimental data. The analysis of the results shows that the average prescission neutron multiplicities, fission probabilities and the mean fission times calculated by one- and three-dimensional Langevin equations are different from each other, and also the results obtained based on three-dimensional Langevin equations are in better agreement with the experimental data. (orig.)
On a class of quantum Langevin equations and the question of approach to equilibrium
International Nuclear Information System (INIS)
Maassen, J.D.M.
1982-01-01
This thesis is concerned with a very simple 'open' quantum system, i.e. being in contact with the outer world. It is asked whether the motion of this system shows frictional behaviour in that it tends to thermal equilibrium. A partial positive answer is given to this question, more precisely, to the question if the solution of the quantum mechanical Langevin equation that describes the Lamb-model (a harmonic oscillator damped by coupling with a string), approaches an equilibrium state. In two sections, the classical and quantum Langevin equations are treated analogously. (Auth.)
Hadamard-type fractional differential equations, inclusions and inequalities
Ahmad, Bashir; Ntouyas, Sotiris K; Tariboon, Jessada
2017-01-01
This book focuses on the recent development of fractional differential equations, integro-differential equations, and inclusions and inequalities involving the Hadamard derivative and integral. Through a comprehensive study based in part on their recent research, the authors address the issues related to initial and boundary value problems involving Hadamard type differential equations and inclusions as well as their functional counterparts. The book covers fundamental concepts of multivalued analysis and introduces a new class of mixed initial value problems involving the Hadamard derivative and Riemann-Liouville fractional integrals. In later chapters, the authors discuss nonlinear Langevin equations as well as coupled systems of Langevin equations with fractional integral conditions. Focused and thorough, this book is a useful resource for readers and researchers interested in the area of fractional calculus.
International Nuclear Information System (INIS)
Menezes, G.; Svaiter, N.F.
2006-04-01
We use the method of stochastic quantization in a topological field theory defined in an Euclidean space, assuming a Langevin equation with a memory kernel. We show that our procedure for the Abelian Chern-Simons theory converges regardless of the nature of the Chern-Simons coefficient. (author)
Langevin equation of a fluid particle in wall-induced turbulence
Brouwers, J.J.H.
2010-01-01
We derive the Langevin equation describing the stochastic process of fluid particle motion in wall-inducedturbulence (turbulent flow in pipes, channels, and boundary layers including the atmospheric surface layer).The analysis is based on the asymptotic behavior at a large Reynolds number. We use
Brownian motion of classical spins: Anomalous dissipation and generalized Langevin equation
Bandyopadhyay, Malay; Jayannavar, A. M.
2017-10-01
In this work, we derive the Langevin equation (LE) of a classical spin interacting with a heat bath through momentum variables, starting from the fully dynamical Hamiltonian description. The derived LE with anomalous dissipation is analyzed in detail. The obtained LE is non-Markovian with multiplicative noise terms. The concomitant dissipative terms obey the fluctuation-dissipation theorem. The Markovian limit correctly produces the Kubo and Hashitsume equation. The perturbative treatment of our equations produces the Landau-Lifshitz equation and the Seshadri-Lindenberg equation. Then we derive the Fokker-Planck equation corresponding to LE and the concept of equilibrium probability distribution is analyzed.
International Nuclear Information System (INIS)
Brett, Tobias; Galla, Tobias
2014-01-01
We present a heuristic derivation of Gaussian approximations for stochastic chemical reaction systems with distributed delay. In particular, we derive the corresponding chemical Langevin equation. Due to the non-Markovian character of the underlying dynamics, these equations are integro-differential equations, and the noise in the Gaussian approximation is coloured. Following on from the chemical Langevin equation, a further reduction leads to the linear-noise approximation. We apply the formalism to a delay variant of the celebrated Brusselator model, and show how it can be used to characterise noise-driven quasi-cycles, as well as noise-triggered spiking. We find surprisingly intricate dependence of the typical frequency of quasi-cycles on the delay period
Brett, Tobias; Galla, Tobias
2014-03-28
We present a heuristic derivation of Gaussian approximations for stochastic chemical reaction systems with distributed delay. In particular, we derive the corresponding chemical Langevin equation. Due to the non-Markovian character of the underlying dynamics, these equations are integro-differential equations, and the noise in the Gaussian approximation is coloured. Following on from the chemical Langevin equation, a further reduction leads to the linear-noise approximation. We apply the formalism to a delay variant of the celebrated Brusselator model, and show how it can be used to characterise noise-driven quasi-cycles, as well as noise-triggered spiking. We find surprisingly intricate dependence of the typical frequency of quasi-cycles on the delay period.
Remarks on the chemical Fokker-Planck and Langevin equations: Nonphysical currents at equilibrium.
Ceccato, Alessandro; Frezzato, Diego
2018-02-14
The chemical Langevin equation and the associated chemical Fokker-Planck equation are well-known continuous approximations of the discrete stochastic evolution of reaction networks. In this work, we show that these approximations suffer from a physical inconsistency, namely, the presence of nonphysical probability currents at the thermal equilibrium even for closed and fully detailed-balanced kinetic schemes. An illustration is given for a model case.
Critique of the Brownian approximation to the generalized Langevin equation in lattice dynamics
International Nuclear Information System (INIS)
Diestler, D.J.; Riley, M.E.
1985-01-01
We consider the classical motion of a harmonic lattice in which only those atoms in a certain subset of the lattice (primary zone) may interact with an external force. The formally exact generalized Langevin equation (GLE) for the primary zone is an appropriate description of the dynamics. We examine a previously proposed Brownian, or frictional damping, approximation that reduces the GLE to a set of coupled ordinary Langevin equations for the primary atoms. It is shown that the solution of these equations can contain undamped motion if there is more than one atom in the primary zone. Such motion is explicitly demonstrated for a model that has been used to describe energy transfer in atom--surface collisions. The inability of the standard Brownian approximation to yield an acceptable, physically meaningful result for primary zones comprising more than one atom suggests that the Brownian approximation may introduce other spurious dynamical effects. Further work on damping of correlated motion in lattices is needed
Langevin equation with the deterministic algebraically correlated noise
International Nuclear Information System (INIS)
Ploszajczak, M.; Srokowski, T.
1995-01-01
Stochastic differential equations with the deterministic, algebraically correlated noise are solved for a few model problems. The chaotic force with both exponential and algebraic temporal correlations is generated by the adjoined extended Sinai billiard with periodic boundary conditions. The correspondence between the autocorrelation function for the chaotic force and both the survival probability and the asymptotic energy distribution of escaping particles is found. (author)
Relations between the kinetic equation and the Langevin models in two-phase flow modelling
International Nuclear Information System (INIS)
Minier, J.P.; Pozorski, J.
1997-05-01
The purpose of this paper is to discuss PDF and stochastic models which are used in two-phase flow modelling. The aim of the present analysis is essentially to try to determine relations and consistency between different models. It is first recalled that different approaches actually correspond to PDF models written either in terms of the process trajectories or in terms of the PDF itself. The main difference lies in the choice of the independent variables which are retained. Two particular models are studied, the Kinetic Equation and the Langevin Equation model. The latter uses a Langevin equation to model the fluid velocities seen along particle trajectories. The Langevin model is more general since it contains an additional variable. It is shown that, in certain cases, this variable can be summed up exactly to retrieve the Kinetic Equation model as a marginal PDF. A joint fluid and solid particle PDF which includes the characteristics of both phases is proposed at the end of the paper. (author)
Langevin equation with the deterministic algebraically correlated noise
Energy Technology Data Exchange (ETDEWEB)
Ploszajczak, M. [Grand Accelerateur National d`Ions Lourds (GANIL), 14 - Caen (France); Srokowski, T. [Grand Accelerateur National d`Ions Lourds (GANIL), 14 - Caen (France)]|[Institute of Nuclear Physics, Cracow (Poland)
1995-12-31
Stochastic differential equations with the deterministic, algebraically correlated noise are solved for a few model problems. The chaotic force with both exponential and algebraic temporal correlations is generated by the adjoined extended Sinai billiard with periodic boundary conditions. The correspondence between the autocorrelation function for the chaotic force and both the survival probability and the asymptotic energy distribution of escaping particles is found. (author). 58 refs.
Hamiltonian models for the Madelung fluid and generalized Langevin equations
International Nuclear Information System (INIS)
Nonnenmacher, T.F.
1985-01-01
We present a Hamiltonian formulation of some type of an 'electromagnetic' Madelung fluid leading to a fluid mechanics interpretation of the Aharonov-Bohm effect and to a subsidary condition to be required in order to make the correspondence between Schroedinger's quantum mechanics and Madelung's fluid mechanics unique. Then we discuss some problems related with the Brownian oscillator. Our aim is to start out with a Hamiltonian for the composite system with surrounding heat bath) and to finally arrive at a stochastic differential equation with completely determined statistical properties. (orig./HSI)
Quantum qubit measurement by a quantum point contact with a quantum Langevin equation approach
International Nuclear Information System (INIS)
Dong, Bing; Lei, X.L.; Horing, N.J.M.; Cui, H.L.
2007-01-01
We employ a microscopic quantum Heisenberg-Langevin equation approach to establish a set of quantum Bloch equations for a two-level system (coupled quantum dots) capacitively coupled to a quantum point contact (QPC). The resulting Bloch equations facilitate our analysis of qubit relaxation and decoherence in coupled quantum dots induced by measurement processes at arbitrary bias-voltage and temperature. We also examine the noise spectrum of the meter output current for a symmetric qubit. These results help resolve a recent debate about a quantum oscillation peak in the noise spectrum. (copyright 2007 WILEY-VCH Verlag GmbH and Co. KGaA, Weinheim) (orig.)
On matrix fractional differential equations
Directory of Open Access Journals (Sweden)
Adem Kılıçman
2017-01-01
Full Text Available The aim of this article is to study the matrix fractional differential equations and to find the exact solution for system of matrix fractional differential equations in terms of Riemann–Liouville using Laplace transform method and convolution product to the Riemann–Liouville fractional of matrices. Also, we show the theorem of non-homogeneous matrix fractional partial differential equation with some illustrative examples to demonstrate the effectiveness of the new methodology. The main objective of this article is to discuss the Laplace transform method based on operational matrices of fractional derivatives for solving several kinds of linear fractional differential equations. Moreover, we present the operational matrices of fractional derivatives with Laplace transform in many applications of various engineering systems as control system. We present the analytical technique for solving fractional-order, multi-term fractional differential equation. In other words, we propose an efficient algorithm for solving fractional matrix equation.
International Nuclear Information System (INIS)
Kwok, Sau Fa
2012-01-01
A Langevin equation with multiplicative white noise and its corresponding Fokker–Planck equation are considered in this work. From the Fokker–Planck equation a transformation into the Wiener process is provided for different orders of prescription in discretization rule for the stochastic integrals. A few applications are also discussed. - Highlights: ► Fokker–Planck equation corresponding to the Langevin equation with mul- tiplicative white noise is presented. ► Transformation of diffusion processes into the Wiener process in different prescriptions is provided. ► The prescription parameter is associated with the growth rate for a Gompertz-type model.
Colmenares, Pedro J.
2018-05-01
This article has to do with the derivation and solution of the Fokker-Planck equation associated to the momentum-integrated Wigner function of a particle subjected to a harmonic external field in contact with an ohmic thermal bath of quantum harmonic oscillators. The strategy employed is a simplified version of the phenomenological approach of Schramm, Jung, and Grabert of interpreting the operators as c numbers to derive the quantum master equation arising from a twofold transformation of the Wigner function of the entire phase space. The statistical properties of the random noise comes from the integral functional theory of Grabert, Schramm, and Ingold. By means of a single Wigner transformation, a simpler equation than that mentioned before is found. The Wigner function reproduces the known results of the classical limit. This allowed us to rewrite the underdamped classical Langevin equation as a first-order stochastic differential equation with time-dependent drift and diffusion terms.
Fine tuning classical and quantum molecular dynamics using a generalized Langevin equation
Rossi, Mariana; Kapil, Venkat; Ceriotti, Michele
2018-03-01
Generalized Langevin Equation (GLE) thermostats have been used very effectively as a tool to manipulate and optimize the sampling of thermodynamic ensembles and the associated static properties. Here we show that a similar, exquisite level of control can be achieved for the dynamical properties computed from thermostatted trajectories. We develop quantitative measures of the disturbance induced by the GLE to the Hamiltonian dynamics of a harmonic oscillator, and show that these analytical results accurately predict the behavior of strongly anharmonic systems. We also show that it is possible to correct, to a significant extent, the effects of the GLE term onto the corresponding microcanonical dynamics, which puts on more solid grounds the use of non-equilibrium Langevin dynamics to approximate quantum nuclear effects and could help improve the prediction of dynamical quantities from techniques that use a Langevin term to stabilize dynamics. Finally we address the use of thermostats in the context of approximate path-integral-based models of quantum nuclear dynamics. We demonstrate that a custom-tailored GLE can alleviate some of the artifacts associated with these techniques, improving the quality of results for the modeling of vibrational dynamics of molecules, liquids, and solids.
On matrix fractional differential equations
Adem Kılıçman; Wasan Ajeel Ahmood
2017-01-01
The aim of this article is to study the matrix fractional differential equations and to find the exact solution for system of matrix fractional differential equations in terms of Riemann–Liouville using Laplace transform method and convolution product to the Riemann–Liouville fractional of matrices. Also, we show the theorem of non-homogeneous matrix fractional partial differential equation with some illustrative examples to demonstrate the effectiveness of the new methodology. The main objec...
On generalized fractional vibration equation
International Nuclear Information System (INIS)
Dai, Hongzhe; Zheng, Zhibao; Wang, Wei
2017-01-01
Highlights: • The paper presents a generalized fractional vibration equation for arbitrary viscoelastically damped system. • Some classical vibration equations can be derived from the developed equation. • The analytic solution of developed equation is derived under some special cases. • The generalized equation is particularly useful for developing new fractional equivalent linearization method. - Abstract: In this paper, a generalized fractional vibration equation with multi-terms of fractional dissipation is developed to describe the dynamical response of an arbitrary viscoelastically damped system. It is shown that many classical equations of motion, e.g., the Bagley–Torvik equation, can be derived from the developed equation. The Laplace transform is utilized to solve the generalized equation and the analytic solution under some special cases is derived. Example demonstrates the generalized transfer function of an arbitrary viscoelastic system.
Power-law Exponent in Multiplicative Langevin Equation with Temporally Correlated Noise
Morita, Satoru
2018-05-01
Power-law distributions are ubiquitous in nature. Random multiplicative processes are a basic model for the generation of power-law distributions. For discrete-time systems, the power-law exponent is known to decrease as the autocorrelation time of the multiplier increases. However, for continuous-time systems, it is not yet clear how the temporal correlation affects the power-law behavior. Herein, we analytically investigated a multiplicative Langevin equation with colored noise. We show that the power-law exponent depends on the details of the multiplicative noise, in contrast to the case of discrete-time systems.
Isospin dependent Boltzmann-langevin equation and the production cross section of 19Na
International Nuclear Information System (INIS)
Ming Zhaoyu; Zhang Fengshou; Chen Liewen; Zhu Zhiyuan; Zhang Wenlong; Guo Zhongyan; Xiao Guoqing
2000-01-01
A new transport model (isospin dependent Boltzmann-Langevin equation) is developed and it is shown that this model can regenerate the experimental data for reaction of 12 C + 12 C at 28.7 MeV/u. The production cross section of 19 Na is systematically studied for reactions of 17-20,22 Ne + 12 C at 28.7 MeV/u. It is found that a neutron deficient projectile has larger 19 Na cross section than a stable projectile
Model reduction of multiscale chemical langevin equations: a numerical case study.
Sotiropoulos, Vassilios; Contou-Carrere, Marie-Nathalie; Daoutidis, Prodromos; Kaznessis, Yiannis N
2009-01-01
Two very important characteristics of biological reaction networks need to be considered carefully when modeling these systems. First, models must account for the inherent probabilistic nature of systems far from the thermodynamic limit. Often, biological systems cannot be modeled with traditional continuous-deterministic models. Second, models must take into consideration the disparate spectrum of time scales observed in biological phenomena, such as slow transcription events and fast dimerization reactions. In the last decade, significant efforts have been expended on the development of stochastic chemical kinetics models to capture the dynamics of biomolecular systems, and on the development of robust multiscale algorithms, able to handle stiffness. In this paper, the focus is on the dynamics of reaction sets governed by stiff chemical Langevin equations, i.e., stiff stochastic differential equations. These are particularly challenging systems to model, requiring prohibitively small integration step sizes. We describe and illustrate the application of a semianalytical reduction framework for chemical Langevin equations that results in significant gains in computational cost.
Fractional vector calculus and fractional Maxwell's equations
International Nuclear Information System (INIS)
Tarasov, Vasily E.
2008-01-01
The theory of derivatives and integrals of non-integer order goes back to Leibniz, Liouville, Grunwald, Letnikov and Riemann. The history of fractional vector calculus (FVC) has only 10 years. The main approaches to formulate a FVC, which are used in the physics during the past few years, will be briefly described in this paper. We solve some problems of consistent formulations of FVC by using a fractional generalization of the Fundamental Theorem of Calculus. We define the differential and integral vector operations. The fractional Green's, Stokes' and Gauss's theorems are formulated. The proofs of these theorems are realized for simplest regions. A fractional generalization of exterior differential calculus of differential forms is discussed. Fractional nonlocal Maxwell's equations and the corresponding fractional wave equations are considered
Energy Technology Data Exchange (ETDEWEB)
Harko, Tiberiu [University College London, Department of Mathematics, London (United Kingdom); Leung, Chun Sing [Polytechnic University, Department of Applied Mathematics, Hong Kong (China); Mocanu, Gabriela [Babes-Bolyai University, Faculty of Physics, Cluj-Napoca (Romania)
2014-05-15
We consider a description of the stochastic oscillations of the general relativistic accretion disks around compact astrophysical objects interacting with their external medium based on a generalized Langevin equation with colored noise and on the fluctuation-dissipation theorems. The former accounts for the general memory and retarded effects of the frictional force. The presence of the memory effects influences the response of the disk to external random interactions, and it modifies the dynamical behavior of the disk, as well as the energy dissipation processes. The generalized Langevin equation of the motion of the disk in the vertical direction is studied numerically, and the vertical displacements, velocities, and luminosities of the stochastically perturbed disks are explicitly obtained for both the Schwarzschild and the Kerr cases. The power spectral distribution of the disk luminosity is also obtained. As a possible astrophysical application of the formalism we investigate the possibility that the intra-day variability of the active galactic nuclei may be due to the stochastic disk instabilities. The perturbations due to colored/nontrivially correlated noise induce a complicated disk dynamics, which could explain some astrophysical observational features related to disk variability. (orig.)
International Nuclear Information System (INIS)
Harko, Tiberiu; Leung, Chun Sing; Mocanu, Gabriela
2014-01-01
We consider a description of the stochastic oscillations of the general relativistic accretion disks around compact astrophysical objects interacting with their external medium based on a generalized Langevin equation with colored noise and on the fluctuation-dissipation theorems. The former accounts for the general memory and retarded effects of the frictional force. The presence of the memory effects influences the response of the disk to external random interactions, and it modifies the dynamical behavior of the disk, as well as the energy dissipation processes. The generalized Langevin equation of the motion of the disk in the vertical direction is studied numerically, and the vertical displacements, velocities, and luminosities of the stochastically perturbed disks are explicitly obtained for both the Schwarzschild and the Kerr cases. The power spectral distribution of the disk luminosity is also obtained. As a possible astrophysical application of the formalism we investigate the possibility that the intra-day variability of the active galactic nuclei may be due to the stochastic disk instabilities. The perturbations due to colored/nontrivially correlated noise induce a complicated disk dynamics, which could explain some astrophysical observational features related to disk variability. (orig.)
Harko, Tiberiu; Leung, Chun Sing; Mocanu, Gabriela
2014-05-01
We consider a description of the stochastic oscillations of the general relativistic accretion disks around compact astrophysical objects interacting with their external medium based on a generalized Langevin equation with colored noise and on the fluctuation-dissipation theorems. The former accounts for the general memory and retarded effects of the frictional force. The presence of the memory effects influences the response of the disk to external random interactions, and it modifies the dynamical behavior of the disk, as well as the energy dissipation processes. The generalized Langevin equation of the motion of the disk in the vertical direction is studied numerically, and the vertical displacements, velocities, and luminosities of the stochastically perturbed disks are explicitly obtained for both the Schwarzschild and the Kerr cases. The power spectral distribution of the disk luminosity is also obtained. As a possible astrophysical application of the formalism we investigate the possibility that the intra-day variability of the active galactic nuclei may be due to the stochastic disk instabilities. The perturbations due to colored/nontrivially correlated noise induce a complicated disk dynamics, which could explain some astrophysical observational features related to disk variability.
Numerical simulation of transmission coefficient using c-number Langevin equation
Barik, Debashis; Bag, Bidhan Chandra; Ray, Deb Shankar
2003-12-01
We numerically implement the reactive flux formalism on the basis of a recently proposed c-number Langevin equation [Barik et al., J. Chem. Phys. 119, 680 (2003); Banerjee et al., Phys. Rev. E 65, 021109 (2002)] to calculate transmission coefficient. The Kramers' turnover, the T2 enhancement of the rate at low temperatures and other related features of temporal behavior of the transmission coefficient over a range of temperature down to absolute zero, noise correlation, and friction are examined for a double well potential and compared with other known results. This simple method is based on canonical quantization and Wigner quasiclassical phase space function and takes care of quantum effects due to the system order by order.
Accelerating the convergence of path integral dynamics with a generalized Langevin equation
Ceriotti, Michele; Manolopoulos, David E.; Parrinello, Michele
2011-02-01
The quantum nature of nuclei plays an important role in the accurate modelling of light atoms such as hydrogen, but it is often neglected in simulations due to the high computational overhead involved. It has recently been shown that zero-point energy effects can be included comparatively cheaply in simulations of harmonic and quasiharmonic systems by augmenting classical molecular dynamics with a generalized Langevin equation (GLE). Here we describe how a similar approach can be used to accelerate the convergence of path integral (PI) molecular dynamics to the exact quantum mechanical result in more strongly anharmonic systems exhibiting both zero point energy and tunnelling effects. The resulting PI-GLE method is illustrated with applications to a double-well tunnelling problem and to liquid water.
Accelerating the convergence of path integral dynamics with a generalized Langevin equation.
Ceriotti, Michele; Manolopoulos, David E; Parrinello, Michele
2011-02-28
The quantum nature of nuclei plays an important role in the accurate modelling of light atoms such as hydrogen, but it is often neglected in simulations due to the high computational overhead involved. It has recently been shown that zero-point energy effects can be included comparatively cheaply in simulations of harmonic and quasiharmonic systems by augmenting classical molecular dynamics with a generalized Langevin equation (GLE). Here we describe how a similar approach can be used to accelerate the convergence of path integral (PI) molecular dynamics to the exact quantum mechanical result in more strongly anharmonic systems exhibiting both zero point energy and tunnelling effects. The resulting PI-GLE method is illustrated with applications to a double-well tunnelling problem and to liquid water.
Fractional Differential Equation
Directory of Open Access Journals (Sweden)
Moustafa El-Shahed
2007-01-01
where 2<α<3 is a real number and D0+α is the standard Riemann-Liouville fractional derivative. Our analysis relies on Krasnoselskiis fixed point theorem of cone preserving operators. An example is also given to illustrate the main results.
Mélykúti, Bence
2010-01-01
The Chemical Langevin Equation (CLE), which is a stochastic differential equation driven by a multidimensional Wiener process, acts as a bridge between the discrete stochastic simulation algorithm and the deterministic reaction rate equation when simulating (bio)chemical kinetics. The CLE model is valid in the regime where molecular populations are abundant enough to assume their concentrations change continuously, but stochastic fluctuations still play a major role. The contribution of this work is that we observe and explore that the CLE is not a single equation, but a parametric family of equations, all of which give the same finite-dimensional distribution of the variables. On the theoretical side, we prove that as many Wiener processes are sufficient to formulate the CLE as there are independent variables in the equation, which is just the rank of the stoichiometric matrix. On the practical side, we show that in the case where there are m1 pairs of reversible reactions and m2 irreversible reactions there is another, simple formulation of the CLE with only m1 + m2 Wiener processes, whereas the standard approach uses 2 m1 + m2. We demonstrate that there are considerable computational savings when using this latter formulation. Such transformations of the CLE do not cause a loss of accuracy and are therefore distinct from model reduction techniques. We illustrate our findings by considering alternative formulations of the CLE for a human ether a-go-go related gene ion channel model and the Goldbeter-Koshland switch. © 2010 American Institute of Physics.
NMR signals within the generalized Langevin model for fractional Brownian motion
Lisý, Vladimír; Tóthová, Jana
2018-03-01
The methods of Nuclear Magnetic Resonance belong to the best developed and often used tools for studying random motion of particles in different systems, including soft biological tissues. In the long-time limit the current mathematical description of the experiments allows proper interpretation of measurements of normal and anomalous diffusion. The shorter-time dynamics is however correctly considered only in a few works that do not go beyond the standard memoryless Langevin description of the Brownian motion (BM). In the present work, the attenuation function S (t) for an ensemble of spin-bearing particles in a magnetic-field gradient, expressed in a form applicable for any kind of stationary stochastic dynamics of spins with or without a memory, is calculated in the frame of the model of fractional BM. The solution of the model for particles trapped in a harmonic potential is obtained in an exceedingly simple way and used for the calculation of S (t). In the limit of free particles coupled to a fractal heat bath, the results compare favorably with experiments acquired in human neuronal tissues. The effect of the trap is demonstrated by introducing a simple model for the generalized diffusion coefficient of the particle.
Invalidity of the spectral Fokker-Planck equation forCauchy noise driven Langevin equation
DEFF Research Database (Denmark)
Ditlevsen, Ove Dalager
2004-01-01
-called alpha-stable noise (or Levy noise) the Fokker-Planck equation no longer exists as a partial differential equation for the probability density because the property of finite variance is lost. In stead it has been attempted to formulate an equation for the characteristic function (the Fourier transform...
Stella, L.; Lorenz, C. D.; Kantorovich, L.
2014-04-01
The generalized Langevin equation (GLE) has been recently suggested to simulate the time evolution of classical solid and molecular systems when considering general nonequilibrium processes. In this approach, a part of the whole system (an open system), which interacts and exchanges energy with its dissipative environment, is studied. Because the GLE is derived by projecting out exactly the harmonic environment, the coupling to it is realistic, while the equations of motion are non-Markovian. Although the GLE formalism has already found promising applications, e.g., in nanotribology and as a powerful thermostat for equilibration in classical molecular dynamics simulations, efficient algorithms to solve the GLE for realistic memory kernels are highly nontrivial, especially if the memory kernels decay nonexponentially. This is due to the fact that one has to generate a colored noise and take account of the memory effects in a consistent manner. In this paper, we present a simple, yet efficient, algorithm for solving the GLE for practical memory kernels and we demonstrate its capability for the exactly solvable case of a harmonic oscillator coupled to a Debye bath.
AQUASOL: An efficient solver for the dipolar Poisson-Boltzmann-Langevin equation.
Koehl, Patrice; Delarue, Marc
2010-02-14
The Poisson-Boltzmann (PB) formalism is among the most popular approaches to modeling the solvation of molecules. It assumes a continuum model for water, leading to a dielectric permittivity that only depends on position in space. In contrast, the dipolar Poisson-Boltzmann-Langevin (DPBL) formalism represents the solvent as a collection of orientable dipoles with nonuniform concentration; this leads to a nonlinear permittivity function that depends both on the position and on the local electric field at that position. The differences in the assumptions underlying these two models lead to significant differences in the equations they generate. The PB equation is a second order, elliptic, nonlinear partial differential equation (PDE). Its response coefficients correspond to the dielectric permittivity and are therefore constant within each subdomain of the system considered (i.e., inside and outside of the molecules considered). While the DPBL equation is also a second order, elliptic, nonlinear PDE, its response coefficients are nonlinear functions of the electrostatic potential. Many solvers have been developed for the PB equation; to our knowledge, none of these can be directly applied to the DPBL equation. The methods they use may adapt to the difference; their implementations however are PBE specific. We adapted the PBE solver originally developed by Holst and Saied [J. Comput. Chem. 16, 337 (1995)] to the problem of solving the DPBL equation. This solver uses a truncated Newton method with a multigrid preconditioner. Numerical evidences suggest that it converges for the DPBL equation and that the convergence is superlinear. It is found however to be slow and greedy in memory requirement for problems commonly encountered in computational biology and computational chemistry. To circumvent these problems, we propose two variants, a quasi-Newton solver based on a simplified, inexact Jacobian and an iterative self-consistent solver that is based directly on the PBE
Fractional Diffusion Equations and Anomalous Diffusion
Evangelista, Luiz Roberto; Kaminski Lenzi, Ervin
2018-01-01
Preface; 1. Mathematical preliminaries; 2. A survey of the fractional calculus; 3. From normal to anomalous diffusion; 4. Fractional diffusion equations: elementary applications; 5. Fractional diffusion equations: surface effects; 6. Fractional nonlinear diffusion equation; 7. Anomalous diffusion: anisotropic case; 8. Fractional Schrödinger equations; 9. Anomalous diffusion and impedance spectroscopy; 10. The Poisson–Nernst–Planck anomalous (PNPA) models; References; Index.
International Nuclear Information System (INIS)
Wu, C.-H.; Lee, D.-S.
2005-01-01
We employ the Schwinger-Keldysh formalism to study the nonequilibrium dynamics of the mirror with perfect reflection moving in a quantum field. In the case where the mirror undergoes the small displacement, the coarse-grained effective action is obtained by integrating out the quantum field with the method of influence functional. The semiclassical Langevin equation is derived, and is found to involve two levels of backreaction effects on the dynamics of mirrors: radiation reaction induced by the motion of the mirror and backreaction dissipation arising from fluctuations in quantum field via a fluctuation-dissipation relation. Although the corresponding theorem of fluctuation and dissipation for the case with the small mirror's displacement is of model independence, the study from the first principles derivation shows that the theorem is also independent of the regulators introduced to deal with short-distance divergences from the quantum field. Thus, when the method of regularization is introduced to compute the dissipation and fluctuation effects, this theorem must be fulfilled as the results are obtained by taking the short-distance limit in the end of calculations. The backreaction effects from vacuum fluctuations on moving mirrors are found to be hardly detected while those effects from thermal fluctuations may be detectable
International Nuclear Information System (INIS)
Abdalla, E.; Carneiro, C.E.I.
1988-12-01
The O(3) model, the pure CP 1 model and the CP 1 model minimally coupled to fermions are numerically simulated. The equivalence between the O(3) and the bound state of the pure CP 1 model is investigated. It is shown that: the relations g O(3 ) = 2 g CP 1 and E O(3 )= 2E CP 1 + 2, for the coupling constants and energies hold beyond the classical level; the mass gap as a function of the coupling is the same for both models. The mass gap for the CP 1 minimally coupled to fermions is also calculated. The calculations are performed using different techniques. The proposal by Namiki and colaborators to enforce constraints on Langevin equations and Parisi's technique to calculate correlation functions via Langevin equations is tested. The results are compared with those obtained using the multi-hit Metropolis algorithm. (author) [pt
International Nuclear Information System (INIS)
Wu, Fuke; Tian, Tianhai; Rawlings, James B.; Yin, George
2016-01-01
The frequently used reduction technique is based on the chemical master equation for stochastic chemical kinetics with two-time scales, which yields the modified stochastic simulation algorithm (SSA). For the chemical reaction processes involving a large number of molecular species and reactions, the collection of slow reactions may still include a large number of molecular species and reactions. Consequently, the SSA is still computationally expensive. Because the chemical Langevin equations (CLEs) can effectively work for a large number of molecular species and reactions, this paper develops a reduction method based on the CLE by the stochastic averaging principle developed in the work of Khasminskii and Yin [SIAM J. Appl. Math. 56, 1766–1793 (1996); ibid. 56, 1794–1819 (1996)] to average out the fast-reacting variables. This reduction method leads to a limit averaging system, which is an approximation of the slow reactions. Because in the stochastic chemical kinetics, the CLE is seen as the approximation of the SSA, the limit averaging system can be treated as the approximation of the slow reactions. As an application, we examine the reduction of computation complexity for the gene regulatory networks with two-time scales driven by intrinsic noise. For linear and nonlinear protein production functions, the simulations show that the sample average (expectation) of the limit averaging system is close to that of the slow-reaction process based on the SSA. It demonstrates that the limit averaging system is an efficient approximation of the slow-reaction process in the sense of the weak convergence.
International Nuclear Information System (INIS)
Migdal, A.A.; Polikarpov, M.I.; Veselov, A.I.; Yurov, V.P.
1983-01-01
The Langevin equation for the lattice theory with arbitrary gauge group is derived. The four-dimensional twisted Eguchi-Kawai model is investigated numerically. The results for the plaquette energy agree with those of the known Monte Carlo calculations. The new result is the distribution of eigenvalues of the plaquette matrix. In the strong coupling phase this distribution is smooth, whereas in the weak coupling phase a gap is clearly seen
Chu, Weiqi; Li, Xiantao
2018-01-01
We present some estimates for the memory kernel function in the generalized Langevin equation, derived using the Mori-Zwanzig formalism from a one-dimensional lattice model, in which the particles interactions are through nearest and second nearest neighbors. The kernel function can be explicitly expressed in a matrix form. The analysis focuses on the decay properties, both spatially and temporally, revealing a power-law behavior in both cases. The dependence on the level of coarse-graining is also studied.
International Nuclear Information System (INIS)
Nadler, Boaz; Schuss, Zeev; Singer, Amit; Eisenberg, R S
2004-01-01
Ionic diffusion through and near small domains is of considerable importance in molecular biophysics in applications such as permeation through protein channels and diffusion near the charged active sites of macromolecules. The motion of the ions in these settings depends on the specific nanoscale geometry and charge distribution in and near the domain, so standard continuum type approaches have obvious limitations. The standard machinery of equilibrium statistical mechanics includes microscopic details, but is also not applicable, because these systems are usually not in equilibrium due to concentration gradients and to the presence of an external applied potential, which drive a non-vanishing stationary current through the system. We present a stochastic molecular model for the diffusive motion of interacting particles in an external field of force and a derivation of effective partial differential equations and their boundary conditions that describe the stationary non-equilibrium system. The interactions can include electrostatic, Lennard-Jones and other pairwise forces. The analysis yields a new type of Poisson-Nernst-Planck equations, that involves conditional and unconditional charge densities and potentials. The conditional charge densities are the non-equilibrium analogues of the well studied pair correlation functions of equilibrium statistical physics. Our proposed theory is an extension of equilibrium statistical mechanics of simple fluids to stationary non-equilibrium problems. The proposed system of equations differs from the standard Poisson-Nernst-Planck system in two important aspects. First, the force term depends on conditional densities and thus on the finite size of ions, and second, it contains the dielectric boundary force on a discrete ion near dielectric interfaces. Recently, various authors have shown that both of these terms are important for diffusion through confined geometries in the context of ion channels
The Extended Fractional Subequation Method for Nonlinear Fractional Differential Equations
Zhao, Jianping; Tang, Bo; Kumar, Sunil; Hou, Yanren
2012-01-01
An extended fractional subequation method is proposed for solving fractional differential equations by introducing a new general ansätz and Bäcklund transformation of the fractional Riccati equation with known solutions. Being concise and straightforward, this method is applied to the space-time fractional coupled Burgers’ equations and coupled MKdV equations. As a result, many exact solutions are obtained. It is shown that the considered method provides a very effective, convenient, and powe...
A fractional Dirac equation and its solution
International Nuclear Information System (INIS)
Muslih, Sami I; Agrawal, Om P; Baleanu, Dumitru
2010-01-01
This paper presents a fractional Dirac equation and its solution. The fractional Dirac equation may be obtained using a fractional variational principle and a fractional Klein-Gordon equation; both methods are considered here. We extend the variational formulations for fractional discrete systems to fractional field systems defined in terms of Caputo derivatives. By applying the variational principle to a fractional action S, we obtain the fractional Euler-Lagrange equations of motion. We present a Lagrangian and a Hamiltonian for the fractional Dirac equation of order α. We also use a fractional Klein-Gordon equation to obtain the fractional Dirac equation which is the same as that obtained using the fractional variational principle. Eigensolutions of this equation are presented which follow the same approach as that for the solution of the standard Dirac equation. We also provide expressions for the path integral quantization for the fractional Dirac field which, in the limit α → 1, approaches to the path integral for the regular Dirac field. It is hoped that the fractional Dirac equation and the path integral quantization of the fractional field will allow further development of fractional relativistic quantum mechanics.
Fractional gradient and its application to the fractional advection equation
D'Ovidio, M.; Garra, R.
2013-01-01
In this paper we provide a definition of fractional gradient operators, related to directional derivatives. We develop a fractional vector calculus, providing a probabilistic interpretation and mathematical tools to treat multidimensional fractional differential equations. A first application is discussed in relation to the d-dimensional fractional advection-dispersion equation. We also study the connection with multidimensional L\\'evy processes.
Fractional hydrodynamic equations for fractal media
International Nuclear Information System (INIS)
Tarasov, Vasily E.
2005-01-01
We use the fractional integrals in order to describe dynamical processes in the fractal medium. We consider the 'fractional' continuous medium model for the fractal media and derive the fractional generalization of the equations of balance of mass density, momentum density, and internal energy. The fractional generalization of Navier-Stokes and Euler equations are considered. We derive the equilibrium equation for fractal media. The sound waves in the continuous medium model for fractional media are considered
International Nuclear Information System (INIS)
Calif, Rudy
2012-01-01
Highlights: ► Probability Density Functions are proposed to fit the wind speed fluctuations distributions for three representative classes. ► Stochastic simulations are performed using a Langevin equation for each class. ► The properties of simulated and measured wind speed sequences are close. -- Abstract: Wind energy production is very sensitive to turbulent wind speed. Thus rapid variation of wind speed due to changes in the local meteorological conditions can lead to electrical power variations of the order of the nominal power output, in particular when wind power variations on very short time scales, range at few seconds to 1 h, are considered. In small grid as they exist on islands (Guadeloupean Archipelago: French West Indies) such fluctuations can cause instabilities in case of intermediate power shortages. The developed analysis in reveals three main classes of time series for the wind speed fluctuations. In this work, Probability Density Functions (PDFs) are proposed to fit the wind speed fluctuations distributions in each class. After, to simulate wind speed fluctuations sequences, we use a stochastic differential equation, the Langevin equation considering Gaussian turbulence PDF (class I), Gram–Charlier PDF (class II) and a mixture of gaussian PDF (class III). The statistical and dynamical properties of simulated wind sequences are close to those of measured wind sequences, for each class.
Fractional dynamic calculus and fractional dynamic equations on time scales
Georgiev, Svetlin G
2018-01-01
Pedagogically organized, this monograph introduces fractional calculus and fractional dynamic equations on time scales in relation to mathematical physics applications and problems. Beginning with the definitions of forward and backward jump operators, the book builds from Stefan Hilger’s basic theories on time scales and examines recent developments within the field of fractional calculus and fractional equations. Useful tools are provided for solving differential and integral equations as well as various problems involving special functions of mathematical physics and their extensions and generalizations in one and more variables. Much discussion is devoted to Riemann-Liouville fractional dynamic equations and Caputo fractional dynamic equations. Intended for use in the field and designed for students without an extensive mathematical background, this book is suitable for graduate courses and researchers looking for an introduction to fractional dynamic calculus and equations on time scales. .
Generalized Langevin quantization
International Nuclear Information System (INIS)
Defendi, A.; Roncadelli, M.
1994-01-01
The recently proposed Langevin formulation of quantum dynamics yields the quantum mechanical propagator at imaginary time as a noise average which involves the solutions of a Langevin equation in configuration space with a Gaussian white noise. This strategy does not require any knowledge about the ground-state quantum dynamics and has been successful in dealing with certain as yet unsolved problems. Here we sketch a generalization of this approach which is based on a similar Langevin equation, whose drift however contains an arbitrary function. As it turns out, this freedom leads to a great simplification in the treatment of several quantum mechanical systems as compared to the original Langevin formulation (this point is illustrated by taking the forced harmonic oscillator as an example). We also show that when the above-mentioned arbitrary function obeys the imaginary-time Hamilton-Jacobi equation, then the new formulation of quantum dynamics exhibits a manifest connection with classical mechanics (at imaginary time). (orig.)
Conformable Fractional Bessel Equation and Bessel Functions
Gökdoğan, Ahmet; Ünal, Emrah; Çelik, Ercan
2015-01-01
In this work, we study the fractional power series solutions around regular singular point x=0 of conformable fractional Bessel differential equation and fractional Bessel functions. Then, we compare fractional solutions with ordinary solutions. In addition, we present certain property of fractional Bessel functions.
Discrete fractional solutions of a Legendre equation
Yılmazer, Resat
2018-01-01
One of the most popular research interests of science and engineering is the fractional calculus theory in recent times. Discrete fractional calculus has also an important position in fractional calculus. In this work, we acquire new discrete fractional solutions of the homogeneous and non homogeneous Legendre differential equation by using discrete fractional nabla operator.
Fractional Stochastic Differential Equations Satisfying Fluctuation-Dissipation Theorem
Li, Lei; Liu, Jian-Guo; Lu, Jianfeng
2017-10-01
We propose in this work a fractional stochastic differential equation (FSDE) model consistent with the over-damped limit of the generalized Langevin equation model. As a result of the `fluctuation-dissipation theorem', the differential equations driven by fractional Brownian noise to model memory effects should be paired with Caputo derivatives, and this FSDE model should be understood in an integral form. We establish the existence of strong solutions for such equations and discuss the ergodicity and convergence to Gibbs measure. In the linear forcing regime, we show rigorously the algebraic convergence to Gibbs measure when the `fluctuation-dissipation theorem' is satisfied, and this verifies that satisfying `fluctuation-dissipation theorem' indeed leads to the correct physical behavior. We further discuss possible approaches to analyze the ergodicity and convergence to Gibbs measure in the nonlinear forcing regime, while leave the rigorous analysis for future works. The FSDE model proposed is suitable for systems in contact with heat bath with power-law kernel and subdiffusion behaviors.
Fractional Number Operator and Associated Fractional Diffusion Equations
Rguigui, Hafedh
2018-03-01
In this paper, we study the fractional number operator as an analog of the finite-dimensional fractional Laplacian. An important relation with the Ornstein-Uhlenbeck process is given. Using a semigroup approach, the solution of the Cauchy problem associated to the fractional number operator is presented. By means of the Mittag-Leffler function and the Laplace transform, we give the solution of the Caputo time fractional diffusion equation and Riemann-Liouville time fractional diffusion equation in infinite dimensions associated to the fractional number operator.
Chaos in discrete fractional difference equations
Indian Academy of Sciences (India)
2016-09-07
Sep 7, 2016 ... chaotic behaviour of fractional difference equations for the tent map, Gauss map and 2x(mod 1) map are studied ..... (4) No significant change is observed by changing .... (3) In fractional case, the rational initial condition.
Inelastic X-ray scattering on liquid benzene analyzed using a generalized Langevin equation
Yoshida, Koji; Fukuyama, Nami; Yamaguchi, Toshio; Hosokawa, Shinya; Uchiyama, Hiroshi; Tsutsui, Satoshi; Baron, Alfred Q. R.
2017-07-01
The dynamic structure factor, S(Q,ω), of liquid benzene was measured by meV-resolved inelastic X-ray scattering (IXS) and analyzed using a generalized Langevin model with a memory function including fast, μ-relaxation and slow, structural, α-relaxation. The model well reproduced the experimental S(Q,ω) of liquid benzene. The dispersion relation of the collective excitation energy yields the high-frequency sound velocity for liquid benzene as related to the α-relaxation. The ratio of the high-frequency to the adiabatic sound velocity is approximately 1.5, larger to that of carbon tetrachloride and smaller than those of methanol and water, reflecting the nature of intermolecular interactions.
On Fractional Order Hybrid Differential Equations
Directory of Open Access Journals (Sweden)
Mohamed A. E. Herzallah
2014-01-01
Full Text Available We develop the theory of fractional hybrid differential equations with linear and nonlinear perturbations involving the Caputo fractional derivative of order 0<α<1. Using some fixed point theorems we prove the existence of mild solutions for two types of hybrid equations. Examples are given to illustrate the obtained results.
Generalized fractional Schroedinger equation with space-time fractional derivatives
International Nuclear Information System (INIS)
Wang Shaowei; Xu Mingyu
2007-01-01
In this paper the generalized fractional Schroedinger equation with space and time fractional derivatives is constructed. The equation is solved for free particle and for a square potential well by the method of integral transforms, Fourier transform and Laplace transform, and the solution can be expressed in terms of Mittag-Leffler function. The Green function for free particle is also presented in this paper. Finally, we discuss the relationship between the cases of the generalized fractional Schroedinger equation and the ones in standard quantum
Symmetry properties of fractional diffusion equations
Energy Technology Data Exchange (ETDEWEB)
Gazizov, R K; Kasatkin, A A; Lukashchuk, S Yu [Ufa State Aviation Technical University, Karl Marx strausse 12, Ufa (Russian Federation)], E-mail: gazizov@mail.rb.ru, E-mail: alexei_kasatkin@mail.ru, E-mail: lsu@mail.rb.ru
2009-10-15
In this paper, nonlinear anomalous diffusion equations with time fractional derivatives (Riemann-Liouville and Caputo) of the order of 0-2 are considered. Lie point symmetries of these equations are investigated and compared. Examples of using the obtained symmetries for constructing exact solutions of the equations under consideration are presented.
Energy Technology Data Exchange (ETDEWEB)
Basharov, A. M., E-mail: basharov@gmail.com [National Research Centre ' Kurchatov Institute,' (Russian Federation)
2012-09-15
It is shown that the effective Hamiltonian representation, as it is formulated in author's papers, serves as a basis for distinguishing, in a broadband environment of an open quantum system, independent noise sources that determine, in terms of the stationary quantum Wiener and Poisson processes in the Markov approximation, the effective Hamiltonian and the equation for the evolution operator of the open system and its environment. General stochastic differential equations of generalized Langevin (non-Wiener) type for the evolution operator and the kinetic equation for the density matrix of an open system are obtained, which allow one to analyze the dynamics of a wide class of localized open systems in the Markov approximation. The main distinctive features of the dynamics of open quantum systems described in this way are the stabilization of excited states with respect to collective processes and an additional frequency shift of the spectrum of the open system. As an illustration of the general approach developed, the photon dynamics in a single-mode cavity without losses on the mirrors is considered, which contains identical intracavity atoms coupled to the external vacuum electromagnetic field. For some atomic densities, the photons of the cavity mode are 'locked' inside the cavity, thus exhibiting a new phenomenon of radiation trapping and non-Wiener dynamics.
International Nuclear Information System (INIS)
Basharov, A. M.
2012-01-01
It is shown that the effective Hamiltonian representation, as it is formulated in author’s papers, serves as a basis for distinguishing, in a broadband environment of an open quantum system, independent noise sources that determine, in terms of the stationary quantum Wiener and Poisson processes in the Markov approximation, the effective Hamiltonian and the equation for the evolution operator of the open system and its environment. General stochastic differential equations of generalized Langevin (non-Wiener) type for the evolution operator and the kinetic equation for the density matrix of an open system are obtained, which allow one to analyze the dynamics of a wide class of localized open systems in the Markov approximation. The main distinctive features of the dynamics of open quantum systems described in this way are the stabilization of excited states with respect to collective processes and an additional frequency shift of the spectrum of the open system. As an illustration of the general approach developed, the photon dynamics in a single-mode cavity without losses on the mirrors is considered, which contains identical intracavity atoms coupled to the external vacuum electromagnetic field. For some atomic densities, the photons of the cavity mode are “locked” inside the cavity, thus exhibiting a new phenomenon of radiation trapping and non-Wiener dynamics.
Frank, T. D.
2008-02-01
We discuss two central claims made in the study by Bassler et al. [K.E. Bassler, G.H. Gunaratne, J.L. McCauley, Physica A 369 (2006) 343]. Bassler et al. claimed that Green functions and Langevin equations cannot be defined for nonlinear diffusion equations. In addition, they claimed that nonlinear diffusion equations are linear partial differential equations disguised as nonlinear ones. We review bottom-up and top-down approaches that have been used in the literature to derive Green functions for nonlinear diffusion equations and, in doing so, show that the first claim needs to be revised. We show that the second claim as well needs to be revised. To this end, we point out similarities and differences between non-autonomous linear Fokker-Planck equations and autonomous nonlinear Fokker-Planck equations. In this context, we raise the question whether Bassler et al.’s approach to financial markets is physically plausible because it necessitates the introduction of external traders and causes. Such external entities can easily be eliminated when taking self-organization principles and concepts of nonextensive thermostatistics into account and modeling financial processes by means of nonlinear Fokker-Planck equations.
International Nuclear Information System (INIS)
Feng Qing-Hua
2014-01-01
In this paper, a new fractional projective Riccati equation method is proposed to establish exact solutions for fractional partial differential equations in the sense of modified Riemann—Liouville derivative. This method can be seen as the fractional version of the known projective Riccati equation method. For illustrating the validity of this method, we apply this method to solve the space-time fractional Whitham—Broer—Kaup (WBK) equations and the nonlinear fractional Sharma—Tasso—Olever (STO) equation, and as a result, some new exact solutions for them are obtained. (general)
Kawai, Shinnosuke; Komatsuzaki, Tamiki
2009-12-14
We present a novel theory which enables us to explore the mechanism of reaction selectivity and robust functions in complex systems persisting under thermal fluctuation. The theory constructs a nonlinear coordinate transformation so that the equation of motion for the new reaction coordinate is independent of the other nonreactive coordinates in the presence of thermal fluctuation. In this article we suppose that reacting systems subject to thermal noise are described by a multidimensional Langevin equation without a priori assumption for the form of potential. The reaction coordinate is composed not only of all the coordinates and velocities associated with the system (solute) but also of the random force exerted by the environment (solvent) with friction constants. The sign of the reaction coordinate at any instantaneous moment in the region of a saddle determines the fate of the reaction, i.e., whether the reaction will proceed through to the products or go back to the reactants. By assuming the statistical properties of the random force, one can know a priori a well-defined boundary of the reaction which separates the full position-velocity space in the saddle region into mainly reactive and mainly nonreactive regions even under thermal fluctuation. The analytical expression of the reaction coordinate provides the firm foundation on the mechanism of how and why reaction proceeds in thermal fluctuating environments.
Oscillation results for certain fractional difference equations
Directory of Open Access Journals (Sweden)
Zhiyun WANG
2017-08-01
Full Text Available Fractional calculus is a theory that studies the properties and application of arbitrary order differentiation and integration. It can describe the physical properties of some systems more accurately, and better adapt to changes in the system, playing an important role in many fields. For example, it can describe the process of tumor growth (growth stimulation and growth inhibition in biomedical science. The oscillation of solutions of two kinds of fractional difference equations is studied, mainly using the proof by contradiction, that is, assuming the equation has a nonstationary solution. For the first kind of equation, the function symbol is firstly determined, and by constructing the Riccati function, the difference is calculated. Then the condition of the function is used to satisfy the contradiction, that is, the assumption is false, which verifies the oscillation of the solution. For the second kind of equation with initial condition, the equivalent fractional sum form of the fractional difference equation are firstly proved. With considering 0<α≤1 and α>1, respectively, by using the properties of Stirling formula and factorial function, the contradictory is got through enhanced processing, namely the assuming is not established, and the sufficient condition for the bounded solutions of the fractional difference equation is obtained. The above results will optimize the relevant conclusions and enrich the relevant results. The results are applied to the specific equations, and the oscillation of the solutions of equations is proved.
Numerical study of fractional nonlinear Schrodinger equations
Klein, Christian; Sparber, Christof; Markowich, Peter A.
2014-01-01
Using a Fourier spectral method, we provide a detailed numerical investigation of dispersive Schrödinger-type equations involving a fractional Laplacian in an one-dimensional case. By an appropriate choice of the dispersive exponent, both mass
Subordination principle for fractional evolution equations
Bazhlekova, E.G.
2000-01-01
The abstract Cauchy problem for the fractional evolution equation Daa = Au, a > 0, (1) where A is a closed densely de??ned operator in a Banach space, is investigated. The subordination principle, presented earlier in [J. P r ??u s s, Evolutionary In- tegral Equations and Applications. Birkh??auser,
Fractional Bhatnagar-Gross-Krook kinetic equation
Goychuk, Igor
2017-11-01
The linear Boltzmann equation (LBE) approach is generalized to describe fractional superdiffusive transport of the Lévy walk type in external force fields. The time distribution between scattering events is assumed to have a finite mean value and infinite variance. It is completely characterized by the two scattering rates, one fractional and a normal one, which defines also the mean scattering rate. We formulate a general fractional LBE approach and exemplify it with a particularly simple case of the Bohm and Gross scattering integral leading to a fractional generalization of the Bhatnagar, Gross and Krook (BGK) kinetic equation. Here, at each scattering event the particle velocity is completely randomized and takes a value from equilibrium Maxwell distribution at a given fixed temperature. We show that the retardation effects are indispensable even in the limit of infinite mean scattering rate and argue that this novel fractional kinetic equation provides a viable alternative to the fractional Kramers-Fokker-Planck (KFP) equation by Barkai and Silbey and its generalization by Friedrich et al. based on the picture of divergent mean time between scattering events. The case of divergent mean time is also discussed at length and compared with the earlier results obtained within the fractional KFP. Also a phenomenological fractional BGK equation without retardation effects is proposed in the limit of infinite scattering rates. It cannot be, however, rigorously derived from a scattering model, being rather clever postulated. It this respect, this retardationless equation is similar to the fractional KFP by Barkai and Silbey. However, it corresponds to the opposite, much more physical limit and, therefore, also presents a viable alternative.
Some physical applications of fractional Schroedinger equation
International Nuclear Information System (INIS)
Guo Xiaoyi; Xu Mingyu
2006-01-01
The fractional Schroedinger equation is solved for a free particle and for an infinite square potential well. The fundamental solution of the Cauchy problem for a free particle, the energy levels and the normalized wave functions of a particle in a potential well are obtained. In the barrier penetration problem, the reflection coefficient and transmission coefficient of a particle from a rectangular potential wall is determined. In the quantum scattering problem, according to the fractional Schroedinger equation, the Green's function of the Lippmann-Schwinger integral equation is given
International Nuclear Information System (INIS)
Tang, Bo; He, Yinnian; Wei, Leilei; Zhang, Xindong
2012-01-01
In this Letter, a generalized fractional sub-equation method is proposed for solving fractional differential equations with variable coefficients. Being concise and straightforward, this method is applied to the space–time fractional Gardner equation with variable coefficients. As a result, many exact solutions are obtained including hyperbolic function solutions, trigonometric function solutions and rational solutions. It is shown that the considered method provides a very effective, convenient and powerful mathematical tool for solving many other fractional differential equations in mathematical physics. -- Highlights: ► Study of fractional differential equations with variable coefficients plays a role in applied physical sciences. ► It is shown that the proposed algorithm is effective for solving fractional differential equations with variable coefficients. ► The obtained solutions may give insight into many considerable physical processes.
Indian Academy of Sciences (India)
tions not only in physics but also in various other fields such as chemistry, biology and ... the required tools for the development of t~e special theory of relativity and would ... During the second world war Langevin became a vocal anti-.
Fractional diffusion equations and anomalous diffusion
Evangelista, Luiz Roberto
2018-01-01
Anomalous diffusion has been detected in a wide variety of scenarios, from fractal media, systems with memory, transport processes in porous media, to fluctuations of financial markets, tumour growth, and complex fluids. Providing a contemporary treatment of this process, this book examines the recent literature on anomalous diffusion and covers a rich class of problems in which surface effects are important, offering detailed mathematical tools of usual and fractional calculus for a wide audience of scientists and graduate students in physics, mathematics, chemistry and engineering. Including the basic mathematical tools needed to understand the rules for operating with the fractional derivatives and fractional differential equations, this self-contained text presents the possibility of using fractional diffusion equations with anomalous diffusion phenomena to propose powerful mathematical models for a large variety of fundamental and practical problems in a fast-growing field of research.
Numerical study of fractional nonlinear Schrodinger equations
Klein, Christian
2014-10-08
Using a Fourier spectral method, we provide a detailed numerical investigation of dispersive Schrödinger-type equations involving a fractional Laplacian in an one-dimensional case. By an appropriate choice of the dispersive exponent, both mass and energy sub- and supercritical regimes can be identified. This allows us to study the possibility of finite time blow-up versus global existence, the nature of the blow-up, the stability and instability of nonlinear ground states and the long-time dynamics of solutions. The latter is also studied in a semiclassical setting. Moreover, we numerically construct ground state solutions of the fractional nonlinear Schrödinger equation.
Stochastic Evolution Equations Driven by Fractional Noises
2016-11-28
paper is to establish the weak convergence, in the topology of the Skorohod space, of the ν-symmetric Riemann sums for functionals of the fractional...stochastic heat equation with fractional-colored noise: existence of the solution. ALEA Lat. Am. J. Probab. Math . Stat. 4 (2008), 57–87. [8] P. Carmona, Y...Hu: Strong disorder implies strong localization for directed polymers in a random environment. ALEA Lat. Am. J. Probab. Math . Stat. 2 (2006), 217
Bouzat, Sebastián
2016-01-01
One-dimensional models coupling a Langevin equation for the cargo position to stochastic stepping dynamics for the motors constitute a relevant framework for analyzing multiple-motor microtubule transport. In this work we explore the consistence of these models focusing on the effects of the thermal noise. We study how to define consistent stepping and detachment rates for the motors as functions of the local forces acting on them in such a way that the cargo velocity and run-time match previously specified functions of the external load, which are set on the base of experimental results. We show that due to the influence of the thermal fluctuations this is not a trivial problem, even for the single-motor case. As a solution, we propose a motor stepping dynamics which considers memory on the motor force. This model leads to better results for single-motor transport than the approaches previously considered in the literature. Moreover, it gives a much better prediction for the stall force of the two-motor case, highly compatible with the experimental findings. We also analyze the fast fluctuations of the cargo position and the influence of the viscosity, comparing the proposed model to the standard one, and we show how the differences on the single-motor dynamics propagate to the multiple motor situations. Finally, we find that the one-dimensional character of the models impede an appropriate description of the fast fluctuations of the cargo position at small loads. We show how this problem can be solved by considering two-dimensional models.
Fractional Diffusion Limit for Collisional Kinetic Equations
Mellet, Antoine
2010-08-20
This paper is devoted to diffusion limits of linear Boltzmann equations. When the equilibrium distribution function is a Maxwellian distribution, it is well known that for an appropriate time scale, the small mean free path limit gives rise to a diffusion equation. In this paper, we consider situations in which the equilibrium distribution function is a heavy-tailed distribution with infinite variance. We then show that for an appropriate time scale, the small mean free path limit gives rise to a fractional diffusion equation. © 2010 Springer-Verlag.
Energy Technology Data Exchange (ETDEWEB)
Eslamizadeh, H. [Persian Gulf University, Department of Physics, Bushehr (Iran, Islamic Republic of)
2014-12-01
The dynamics of fission of excited nuclei has been studied by solving four-dimensional Langevin equations with dissipation generated through the chaos-weighted wall and window friction formula. The projection of the total spin of the compound nucleus to the symmetry axis, K, was considered as the fourth dimension in Langevin dynamical calculations. The average pre-scission neutron multiplicities, mean kinetic energy of fission fragments and the variances of the mass and kinetic energy have been calculated in a wide range of fissile parameter for compound nuclei {sup 162}Yb, {sup 172}Yb, {sup 215}Fr, {sup 224}Th, {sup 248}Cf, {sup 260}Rf and results compared with the experimental data. Calculations were performed with a constant dissipation coefficient of K, {sub γK} (MeV zs){sup -1/2}, and with a non-constant dissipation coefficient. Comparison of the theoretical results for the average pre-scission neutron multiplicities, mean kinetic energy of fission fragments and the variances of the mass and kinetic energy with the experimental data showed that the results of four-dimensional Langevin equations with a non-constant dissipation coefficient are in better agreement with the experimental data. Furthermore, the difference between the results of two models for compound nuclei with low fissile parameter is low whereas, for heavy compound nuclei, is high. (orig.)
Higher Order and Fractional Diffusive Equations
Directory of Open Access Journals (Sweden)
D. Assante
2015-07-01
Full Text Available We discuss the solution of various generalized forms of the Heat Equation, by means of different tools ranging from the use of Hermite-Kampé de Fériet polynomials of higher and fractional order to operational techniques. We show that these methods are useful to obtain either numerical or analytical solutions.
Remarks for one-dimensional fractional equations
Directory of Open Access Journals (Sweden)
Massimiliano Ferrara
2014-01-01
Full Text Available In this paper we study a class of one-dimensional Dirichlet boundary value problems involving the Caputo fractional derivatives. The existence of infinitely many solutions for this equations is obtained by exploiting a recent abstract result. Concrete examples of applications are presented.
Multidimensional fractional Schrödinger equation
Rodrigues, M. M.; Vieira, N.
2012-11-01
This work is intended to investigate the multi-dimensional space-time fractional Schrödinger equation of the form (CDt0+αu)(t,x) = iħ/2m(C∇βu)(t,x), with ħ the Planck's constant divided by 2π, m is the mass and u(t,x) is a wave function of the particle. Here (CDt0+α,C∇β are operators of the Caputo fractional derivatives, where α ∈]0,1] and β ∈]1,2]. The wave function is obtained using Laplace and Fourier transforms methods and a symbolic operational form of solutions in terms of the Mittag-Leffler functions is exhibited. It is presented an expression for the wave function and for the quantum mechanical probability density. Using Banach fixed point theorem, the existence and uniqueness of solutions is studied for this kind of fractional differential equations.
On the solution of fractional evolution equations
International Nuclear Information System (INIS)
Kilbas, Anatoly A; Pierantozzi, Teresa; Trujillo, Juan J; Vazquez, Luis
2004-01-01
This paper is devoted to the solution of the bi-fractional differential equation ( C D α t u)(t, x) = λ( L D β x u)(t, x) (t>0, -∞ 0 and λ ≠ 0, with the initial conditions lim x→±∞ u(t,x) = 0 u(0+,x)=g(x). Here ( C D α t u)(t, x) is the partial derivative coinciding with the Caputo fractional derivative for 0 L D β x u)(t, x)) is the Liouville partial fractional derivative ( L D β t u)(t, x)) of order β > 0. The Laplace and Fourier transforms are applied to solve the above problem in closed form. The fundamental solution of these problems is established and its moments are calculated. The special case α = 1/2 and β = 1 is presented, and its application is given to obtain the Dirac-type decomposition for the ordinary diffusion equation
Particle Simulation of Fractional Diffusion Equations
Allouch, Samer
2017-07-12
This work explores different particle-based approaches to the simulation of one-dimensional fractional subdiffusion equations in unbounded domains. We rely on smooth particle approximations, and consider four methods for estimating the fractional diffusion term. The first method is based on direct differentiation of the particle representation, it follows the Riesz definition of the fractional derivative and results in a non-conservative scheme. The other three methods follow the particle strength exchange (PSE) methodology and are by construction conservative, in the sense that the total particle strength is time invariant. The first PSE algorithm is based on using direct differentiation to estimate the fractional diffusion flux, and exploiting the resulting estimates in an integral representation of the divergence operator. Meanwhile, the second one relies on the regularized Riesz representation of the fractional diffusion term to derive a suitable interaction formula acting directly on the particle representation of the diffusing field. A third PSE construction is considered that exploits the Green\\'s function of the fractional diffusion equation. The performance of all four approaches is assessed for the case of a one-dimensional diffusion equation with constant diffusivity. This enables us to take advantage of known analytical solutions, and consequently conduct a detailed analysis of the performance of the methods. This includes a quantitative study of the various sources of error, namely filtering, quadrature, domain truncation, and time integration, as well as a space and time self-convergence analysis. These analyses are conducted for different values of the order of the fractional derivatives, and computational experiences are used to gain insight that can be used for generalization of the present constructions.
Particle Simulation of Fractional Diffusion Equations
Allouch, Samer; Lucchesi, Marco; Maî tre, O. P. Le; Mustapha, K. A.; Knio, Omar
2017-01-01
This work explores different particle-based approaches to the simulation of one-dimensional fractional subdiffusion equations in unbounded domains. We rely on smooth particle approximations, and consider four methods for estimating the fractional diffusion term. The first method is based on direct differentiation of the particle representation, it follows the Riesz definition of the fractional derivative and results in a non-conservative scheme. The other three methods follow the particle strength exchange (PSE) methodology and are by construction conservative, in the sense that the total particle strength is time invariant. The first PSE algorithm is based on using direct differentiation to estimate the fractional diffusion flux, and exploiting the resulting estimates in an integral representation of the divergence operator. Meanwhile, the second one relies on the regularized Riesz representation of the fractional diffusion term to derive a suitable interaction formula acting directly on the particle representation of the diffusing field. A third PSE construction is considered that exploits the Green's function of the fractional diffusion equation. The performance of all four approaches is assessed for the case of a one-dimensional diffusion equation with constant diffusivity. This enables us to take advantage of known analytical solutions, and consequently conduct a detailed analysis of the performance of the methods. This includes a quantitative study of the various sources of error, namely filtering, quadrature, domain truncation, and time integration, as well as a space and time self-convergence analysis. These analyses are conducted for different values of the order of the fractional derivatives, and computational experiences are used to gain insight that can be used for generalization of the present constructions.
q-fractional calculus and equations
Annaby, Mahmoud H
2012-01-01
This nine-chapter monograph introduces a rigorous investigation of q-difference operators in standard and fractional settings. It starts with elementary calculus of q-differences and integration of Jackson’s type before turning to q-difference equations. The existence and uniqueness theorems are derived using successive approximations, leading to systems of equations with retarded arguments. Regular q-Sturm–Liouville theory is also introduced; Green’s function is constructed and the eigenfunction expansion theorem is given. The monograph also discusses some integral equations of Volterra and Abel type, as introductory material for the study of fractional q-calculi. Hence fractional q-calculi of the types Riemann–Liouville; Grünwald–Letnikov; Caputo; Erdélyi–Kober and Weyl are defined analytically. Fractional q-Leibniz rules with applications in q-series are also obtained with rigorous proofs of the formal results of Al-Salam-Verma, which remained unproved for decades. In working ...
Integrable coupling system of fractional soliton equation hierarchy
Energy Technology Data Exchange (ETDEWEB)
Yu Fajun, E-mail: yfajun@163.co [College of Maths and Systematic Science, Shenyang Normal University, Shenyang 110034 (China)
2009-10-05
In this Letter, we consider the derivatives and integrals of fractional order and present a class of the integrable coupling system of the fractional order soliton equations. The fractional order coupled Boussinesq and KdV equations are the special cases of this class. Furthermore, the fractional AKNS soliton equation hierarchy is obtained.
On the solution of fractional evolution equations
Energy Technology Data Exchange (ETDEWEB)
Kilbas, Anatoly A [Department of Mathematics and Mechanics, Belarusian State University, 220050 Minsk (Belarus); Pierantozzi, Teresa [Departamento de Matematica Aplicada, Facultad de Informatica, Universidad Complutense, E-28040 Madrid (Spain); Trujillo, Juan J [Departamento de Analisis Matematico, Universidad de la Laguna, 38271 La Laguna-Tenerife (Spain); Vazquez, Luis [Departamento de Matematica Aplicada, Facultad de Informatica, Universidad Complutense, E-28040 Madrid (Spain)
2004-03-05
This paper is devoted to the solution of the bi-fractional differential equation ({sup C}D{sup {alpha}}{sub t}u)(t, x) = {lambda}({sup L}D{sup {beta}}{sub x}u)(t, x) (t>0, -{infinity}
Fractional Schrodinger equations with new conditions
Directory of Open Access Journals (Sweden)
Abderrazek Benhassine
2018-01-01
Full Text Available In this article, we study the nonlinear fractional Schrodinger equation $$\\displaylines{ (-\\Delta^{\\alpha}u+ V(xu= f(x,u\\cr u\\in H^{\\alpha}(\\mathbb{R}^{n},\\mathbb{R}, }$$ where $(-\\Delta^{\\alpha}(\\alpha \\in (0, 1$ stands for the fractional Laplacian of order $\\alpha$, $x\\in \\mathbb{R}^{n}$, $V\\in C(\\mathbb{R}^{n},\\mathbb{R}$ may change sign and f is only locally defined near the origin with respect to u. Under some new assumptions on V and f, we show that the above system has infinitely many solutions near the origin. Some examples are also given to illustrate our main theoretical result.
Fractional diffusion equation for heterogeneous medium
International Nuclear Information System (INIS)
Polo L, M. A.; Espinosa M, E. G.; Espinosa P, G.; Del Valle G, E.
2011-11-01
The asymptotic diffusion approximation for the Boltzmann (transport) equation was developed in 1950 decade in order to describe the diffusion of a particle in an isotropic medium, considers that the particles have a diffusion infinite velocity. In this work is developed a new approximation where is considered that the particles have a finite velocity, with this model is possible to describe the behavior in an anomalous medium. According with these ideas the model was obtained from the Fick law, where is considered that the temporal term of the current vector is not negligible. As a result the diffusion equation of fractional order which describes the dispersion of particles in a highly heterogeneous or disturbed medium is obtained, i.e., in a general medium. (Author)
Semianalytic Solution of Space-Time Fractional Diffusion Equation
Directory of Open Access Journals (Sweden)
A. Elsaid
2016-01-01
Full Text Available We study the space-time fractional diffusion equation with spatial Riesz-Feller fractional derivative and Caputo fractional time derivative. The continuation of the solution of this fractional equation to the solution of the corresponding integer order equation is proved. The series solution of this problem is obtained via the optimal homotopy analysis method (OHAM. Numerical simulations are presented to validate the method and to show the effect of changing the fractional derivative parameters on the solution behavior.
Fractional Complex Transform and exp-Function Methods for Fractional Differential Equations
Directory of Open Access Journals (Sweden)
Ahmet Bekir
2013-01-01
Full Text Available The exp-function method is presented for finding the exact solutions of nonlinear fractional equations. New solutions are constructed in fractional complex transform to convert fractional differential equations into ordinary differential equations. The fractional derivatives are described in Jumarie's modified Riemann-Liouville sense. We apply the exp-function method to both the nonlinear time and space fractional differential equations. As a result, some new exact solutions for them are successfully established.
Directory of Open Access Journals (Sweden)
Wei Li
2014-01-01
Full Text Available Based on a general fractional Riccati equation and with Jumarie’s modified Riemann-Liouville derivative to an extended fractional Riccati expansion method for solving the time fractional Burgers equation and the space-time fractional Cahn-Hilliard equation, the exact solutions expressed by the hyperbolic functions and trigonometric functions are obtained. The obtained results show that the presented method is effective and appropriate for solving nonlinear fractional differential equations.
Directory of Open Access Journals (Sweden)
M. L. Kavvas
2017-10-01
Full Text Available Using fractional calculus, a dimensionally consistent governing equation of transient, saturated groundwater flow in fractional time in a multi-fractional confined aquifer is developed. First, a dimensionally consistent continuity equation for transient saturated groundwater flow in fractional time and in a multi-fractional, multidimensional confined aquifer is developed. For the equation of water flux within a multi-fractional multidimensional confined aquifer, a dimensionally consistent equation is also developed. The governing equation of transient saturated groundwater flow in a multi-fractional, multidimensional confined aquifer in fractional time is then obtained by combining the fractional continuity and water flux equations. To illustrate the capability of the proposed governing equation of groundwater flow in a confined aquifer, a numerical application of the fractional governing equation to a confined aquifer groundwater flow problem was also performed.
Langevin formulation of quantum dynamics
International Nuclear Information System (INIS)
Roncadelli, M.
1989-03-01
We first show that nonrelativistic quantum mechanics formulated at imaginary-(h/2 π) can formally be viewed as the Fokker-Planck description of a frictionless brownian motion, which occurs (in general) in an absorbing medium. We next offer a new formulation of quantum mechanics, which is basically the Langevin treatment of this brownian motion. Explicitly, we derive a noise-average representation for the transition probability W(X'',t''|X',t'), in terms of the solutions to a Langevin equation with a Gaussian white-noise. Upon analytic continuation back to real-(h/2 π),W(X'',t''|X',t') becomes the propagator of the original Schroedinger equation. Our approach allows for a straightforward application to quantum dynamical problems of the mathematical techniques of classical stochastic processes. Moreover, computer simulations of quantum mechanical systems can be carried out by using numerical programs based on the Langevin dynamics. (author). 19 refs, 1 tab
International Nuclear Information System (INIS)
Lu, Bin
2012-01-01
In this Letter, the fractional derivatives in the sense of modified Riemann–Liouville derivative and the Bäcklund transformation of fractional Riccati equation are employed for constructing the exact solutions of nonlinear fractional partial differential equations. The power of this manageable method is presented by applying it to several examples. This approach can also be applied to other nonlinear fractional differential equations. -- Highlights: ► Backlund transformation of fractional Riccati equation is presented. ► A new method for solving nonlinear fractional differential equations is proposed. ► Three important fractional differential equations are solved successfully. ► Some new exact solutions of the fractional differential equations are obtained.
Series expansion in fractional calculus and fractional differential equations
Li, Ming-Fan; Ren, Ji-Rong; Zhu, Tao
2009-01-01
Fractional calculus is the calculus of differentiation and integration of non-integer orders. In a recently paper (Annals of Physics 323 (2008) 2756-2778), the Fundamental Theorem of Fractional Calculus is highlighted. Based on this theorem, in this paper we introduce fractional series expansion method to fractional calculus. We define a kind of fractional Taylor series of an infinitely fractionally-differentiable function. Further, based on our definition we generalize hypergeometric functio...
M. L. Kavvas; T. Tu; A. Ercan; J. Polsinelli
2017-01-01
Using fractional calculus, a dimensionally consistent governing equation of transient, saturated groundwater flow in fractional time in a multi-fractional confined aquifer is developed. First, a dimensionally consistent continuity equation for transient saturated groundwater flow in fractional time and in a multi-fractional, multidimensional confined aquifer is developed. For the equation of water flux within a multi-fractional multidimensional confined aquifer, a dimensionally...
Exp-function method for solving fractional partial differential equations.
Zheng, Bin
2013-01-01
We extend the Exp-function method to fractional partial differential equations in the sense of modified Riemann-Liouville derivative based on nonlinear fractional complex transformation. For illustrating the validity of this method, we apply it to the space-time fractional Fokas equation and the nonlinear fractional Sharma-Tasso-Olver (STO) equation. As a result, some new exact solutions for them are successfully established.
Similarity Solutions for Multiterm Time-Fractional Diffusion Equation
Elsaid, A.; Abdel Latif, M. S.; Maneea, M.
2016-01-01
Similarity method is employed to solve multiterm time-fractional diffusion equation. The orders of the fractional derivatives belong to the interval (0,1] and are defined in the Caputo sense. We illustrate how the problem is reduced from a multiterm two-variable fractional partial differential equation to a multiterm ordinary fractional differential equation. Power series solution is obtained for the resulting ordinary problem and the convergence of the series solution is discussed. Based on ...
dimensional generalised time-fractional Hirota equation
Indian Academy of Sciences (India)
Youwei Zhang
2018-02-09
Feb 9, 2018 ... Fractional calculus has attracted much attention in ... cally proved that the fractional calculus theory is non- ... calculus and various definitions of fractional integration .... basic features of the tanh-expansion are outlined as.
On solutions of variable-order fractional differential equations
Directory of Open Access Journals (Sweden)
Ali Akgül
2017-01-01
solutions to fractional differential equations are compelling to get in real applications, due to the nonlocality and complexity of the fractional differential operators, especially for variable-order fractional differential equations. Therefore, it is significant to enhanced numerical methods for fractional differential equations. In this work, we consider variable-order fractional differential equations by reproducing kernel method. There has been much attention in the use of reproducing kernels for the solutions to many problems in the recent years. We give two examples to demonstrate how efficiently our theory can be implemented in practice.
The fundamental solutions for fractional evolution equations of parabolic type
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Mahmoud M. El-Borai
2004-01-01
Full Text Available The fundamental solutions for linear fractional evolution equations are obtained. The coefficients of these equations are a family of linear closed operators in the Banach space. Also, the continuous dependence of solutions on the initial conditions is studied. A mixed problem of general parabolic partial differential equations with fractional order is given as an application.
THE NEW SOLUTION OF TIME FRACTIONAL WAVE EQUATION WITH CONFORMABLE FRACTIONAL DERIVATIVE DEFINITION
Çenesiz, Yücel; Kurt, Ali
2015-01-01
– In this paper, we used new fractional derivative definition, the conformable fractional derivative, for solving two and three dimensional time fractional wave equation. This definition is simple and very effective in the solution procedures of the fractional differential equations that have complicated solutions with classical fractional derivative definitions like Caputo, Riemann-Liouville and etc. The results show that conformable fractional derivative definition is usable and convenient ...
Long-Term Dynamics of Autonomous Fractional Differential Equations
Liu, Tao; Xu, Wei; Xu, Yong; Han, Qun
This paper aims to investigate long-term dynamic behaviors of autonomous fractional differential equations with effective numerical method. The long-term dynamic behaviors predict where systems are heading after long-term evolution. We make some modification and transplant cell mapping methods to autonomous fractional differential equations. The mapping time duration of cell mapping is enlarged to deal with the long memory effect. Three illustrative examples, i.e. fractional Lotka-Volterra equation, fractional van der Pol oscillator and fractional Duffing equation, are studied with our revised generalized cell mapping method. We obtain long-term dynamics, such as attractors, basins of attraction, and saddles. Compared with some existing stability and numerical results, the validity of our method is verified. Furthermore, we find that the fractional order has its effect on the long-term dynamics of autonomous fractional differential equations.
Exact solutions to the time-fractional differential equations via local fractional derivatives
Guner, Ozkan; Bekir, Ahmet
2018-01-01
This article utilizes the local fractional derivative and the exp-function method to construct the exact solutions of nonlinear time-fractional differential equations (FDEs). For illustrating the validity of the method, it is applied to the time-fractional Camassa-Holm equation and the time-fractional-generalized fifth-order KdV equation. Moreover, the exact solutions are obtained for the equations which are formed by different parameter values related to the time-fractional-generalized fifth-order KdV equation. This method is an reliable and efficient mathematical tool for solving FDEs and it can be applied to other non-linear FDEs.
Variational problems with fractional derivatives: Euler-Lagrange equations
International Nuclear Information System (INIS)
Atanackovic, T M; Konjik, S; Pilipovic, S
2008-01-01
We generalize the fractional variational problem by allowing the possibility that the lower bound in the fractional derivative does not coincide with the lower bound of the integral that is minimized. Also, for the standard case when these two bounds coincide, we derive a new form of Euler-Lagrange equations. We use approximations for fractional derivatives in the Lagrangian and obtain the Euler-Lagrange equations which approximate the initial Euler-Lagrange equations in a weak sense
Fractional differential equation with the fuzzy initial condition
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Sadia Arshad
2011-02-01
Full Text Available In this paper we study the existence and uniqueness of the solution for a class of fractional differential equation with fuzzy initial value. The fractional derivatives are considered in the Riemann-Liouville sense.
A novel approach for solving fractional Fisher equation using
Indian Academy of Sciences (India)
Differential transform method; fractional Fisher equation. ... confirmed by applying this method on different forms of functional equations. Author Affiliations. MIRZAZADEH M1. Department of Engineering Sciences, Faculty of Technology and ...
Almost Periodic Solutions for Impulsive Fractional Stochastic Evolution Equations
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Toufik Guendouzi
2014-08-01
Full Text Available In this paper, we consider the existence of square-mean piecewise almost periodic solutions for impulsive fractional stochastic evolution equations involving Caputo fractional derivative. The main results are obtained by means of the theory of operators semi-group, fractional calculus, fixed point technique and stochastic analysis theory and methods adopted directly from deterministic fractional equations. Some known results are improved and generalized.
Analytical Approach to Space- and Time-Fractional Burgers Equations
International Nuclear Information System (INIS)
Yıldırım, Ahmet; Mohyud-Din, Syed Tauseef
2010-01-01
A scheme is developed to study numerical solution of the space- and time-fractional Burgers equations under initial conditions by the homotopy analysis method. The fractional derivatives are considered in the Caputo sense. The solutions are given in the form of series with easily computable terms. Numerical solutions are calculated for the fractional Burgers equation to show the nature of solution as the fractional derivative parameter is changed
Integral transform method for solving time fractional systems and fractional heat equation
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Arman Aghili
2014-01-01
Full Text Available In the present paper, time fractional partial differential equation is considered, where the fractional derivative is defined in the Caputo sense. Laplace transform method has been applied to obtain an exact solution. The authors solved certain homogeneous and nonhomogeneous time fractional heat equations using integral transform. Transform method is a powerful tool for solving fractional singular Integro - differential equations and PDEs. The result reveals that the transform method is very convenient and effective.
Langevin dynamics for ramified structures
Méndez, Vicenç; Iomin, Alexander; Horsthemke, Werner; Campos, Daniel
2017-06-01
We propose a generalized Langevin formalism to describe transport in combs and similar ramified structures. Our approach consists of a Langevin equation without drift for the motion along the backbone. The motion along the secondary branches may be described either by a Langevin equation or by other types of random processes. The mean square displacement (MSD) along the backbone characterizes the transport through the ramified structure. We derive a general analytical expression for this observable in terms of the probability distribution function of the motion along the secondary branches. We apply our result to various types of motion along the secondary branches of finite or infinite length, such as subdiffusion, superdiffusion, and Langevin dynamics with colored Gaussian noise and with non-Gaussian white noise. Monte Carlo simulations show excellent agreement with the analytical results. The MSD for the case of Gaussian noise is shown to be independent of the noise color. We conclude by generalizing our analytical expression for the MSD to the case where each secondary branch is n dimensional.
Directory of Open Access Journals (Sweden)
Sheng-Ping Yan
2014-01-01
Full Text Available We perform a comparison between the local fractional Adomian decomposition and local fractional function decomposition methods applied to the Laplace equation. The operators are taken in the local sense. The results illustrate the significant features of the two methods which are both very effective and straightforward for solving the differential equations with local fractional derivative.
Boundary value problem for Caputo-Hadamard fractional differential equations
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Yacine Arioua
2017-09-01
Full Text Available The aim of this work is to study the existence and uniqueness solutions for boundary value problem of nonlinear fractional differential equations with Caputo-Hadamard derivative in bounded domain. We used the standard and Krasnoselskii's fixed point theorems. Some new results of existence and uniqueness solutions for Caputo-Hadamard fractional equations are obtained.
Existence of solutions of abstract fractional impulsive semilinear evolution equations
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K. Balachandran
2010-01-01
Full Text Available In this paper we prove the existence of solutions of fractional impulsive semilinear evolution equations in Banach spaces. A nonlocal Cauchy problem is discussed for the evolution equations. The results are obtained using fractional calculus and fixed point theorems. An example is provided to illustrate the theory.
Analysis of Caputo Impulsive Fractional Order Differential Equations with Applications
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Lakshman Mahto
2013-01-01
Full Text Available We use Sadovskii's fixed point method to investigate the existence and uniqueness of solutions of Caputo impulsive fractional differential equations of order with one example of impulsive logistic model and few other examples as well. We also discuss Caputo impulsive fractional differential equations with finite delay. The results proven are new and compliment the existing one.
On some impulsive fractional differential equations in Banach spaces
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JinRong Wang
2010-01-01
Full Text Available This paper deals with some impulsive fractional differential equations in Banach spaces. Utilizing the Leray-Schauder fixed point theorem and the impulsive nonlinear singular version of the Gronwall inequality, the existence of \\(PC\\-mild solutions for some fractional differential equations with impulses are obtained under some easily checked conditions. At last, an example is given for demonstration.
On Some Fractional Stochastic Integrodifferential Equations in Hilbert Space
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Hamdy M. Ahmed
2009-01-01
Full Text Available We study a class of fractional stochastic integrodifferential equations considered in a real Hilbert space. The existence and uniqueness of the Mild solutions of the considered problem is also studied. We also give an application for stochastic integropartial differential equations of fractional order.
Boundary value problems for multi-term fractional differential equations
Daftardar-Gejji, Varsha; Bhalekar, Sachin
2008-09-01
Multi-term fractional diffusion-wave equation along with the homogeneous/non-homogeneous boundary conditions has been solved using the method of separation of variables. It is observed that, unlike in the one term case, solution of multi-term fractional diffusion-wave equation is not necessarily non-negative, and hence does not represent anomalous diffusion of any kind.
On the singular perturbations for fractional differential equation.
Atangana, Abdon
2014-01-01
The goal of this paper is to examine the possible extension of the singular perturbation differential equation to the concept of fractional order derivative. To achieve this, we presented a review of the concept of fractional calculus. We make use of the Laplace transform operator to derive exact solution of singular perturbation fractional linear differential equations. We make use of the methodology of three analytical methods to present exact and approximate solution of the singular perturbation fractional, nonlinear, nonhomogeneous differential equation. These methods are including the regular perturbation method, the new development of the variational iteration method, and the homotopy decomposition method.
An Efficient Series Solution for Nonlinear Multiterm Fractional Differential Equations
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Moh’d Khier Al-Srihin
2017-01-01
Full Text Available In this paper, we introduce an efficient series solution for a class of nonlinear multiterm fractional differential equations of Caputo type. The approach is a generalization to our recent work for single fractional differential equations. We extend the idea of the Taylor series expansion method to multiterm fractional differential equations, where we overcome the difficulty of computing iterated fractional derivatives, which are difficult to be computed in general. The terms of the series are obtained sequentially using a closed formula, where only integer derivatives have to be computed. Several examples are presented to illustrate the efficiency of the new approach and comparison with the Adomian decomposition method is performed.
On the Singular Perturbations for Fractional Differential Equation
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Abdon Atangana
2014-01-01
Full Text Available The goal of this paper is to examine the possible extension of the singular perturbation differential equation to the concept of fractional order derivative. To achieve this, we presented a review of the concept of fractional calculus. We make use of the Laplace transform operator to derive exact solution of singular perturbation fractional linear differential equations. We make use of the methodology of three analytical methods to present exact and approximate solution of the singular perturbation fractional, nonlinear, nonhomogeneous differential equation. These methods are including the regular perturbation method, the new development of the variational iteration method, and the homotopy decomposition method.
Periodicity and positivity of a class of fractional differential equations.
Ibrahim, Rabha W; Ahmad, M Z; Mohammed, M Jasim
2016-01-01
Fractional differential equations have been discussed in this study. We utilize the Riemann-Liouville fractional calculus to implement it within the generalization of the well known class of differential equations. The Rayleigh differential equation has been generalized of fractional second order. The existence of periodic and positive outcome is established in a new method. The solution is described in a fractional periodic Sobolev space. Positivity of outcomes is considered under certain requirements. We develop and extend some recent works. An example is constructed.
A novel fractional technique for the modified point kinetics equations
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Ahmed E. Aboanber
2016-10-01
Full Text Available A fractional model for the modified point kinetics equations is derived and analyzed. An analytical method is used to solve the fractional model for the modified point kinetics equations. This methodical technique is based on the representation of the neutron density as a power series of the relaxation time as a small parameter. The validity of the fractional model is tested for different cases of step, ramp and sinusoidal reactivity. The results show that the fractional model for the modified point kinetics equations is the best representation of neutron density for subcritical and supercritical reactors.
Similarity Solutions for Multiterm Time-Fractional Diffusion Equation
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A. Elsaid
2016-01-01
Full Text Available Similarity method is employed to solve multiterm time-fractional diffusion equation. The orders of the fractional derivatives belong to the interval (0,1] and are defined in the Caputo sense. We illustrate how the problem is reduced from a multiterm two-variable fractional partial differential equation to a multiterm ordinary fractional differential equation. Power series solution is obtained for the resulting ordinary problem and the convergence of the series solution is discussed. Based on the obtained results, we propose a definition for a multiterm error function with generalized coefficients.
Closed form solutions of two time fractional nonlinear wave equations
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M. Ali Akbar
2018-06-01
Full Text Available In this article, we investigate the exact traveling wave solutions of two nonlinear time fractional wave equations. The fractional derivatives are described in the sense of conformable fractional derivatives. In addition, the traveling wave solutions are accomplished in the form of hyperbolic, trigonometric, and rational functions involving free parameters. To investigate such types of solutions, we implement the new generalized (G′/G-expansion method. The extracted solutions are reliable, useful and suitable to comprehend the optimal control problems, chaotic vibrations, global and local bifurcations and resonances, furthermore, fission and fusion phenomena occur in solitons, the relativistic energy-momentum relation, scalar electrodynamics, quantum relativistic one-particle theory, electromagnetic interactions etc. The results reveal that the method is very fruitful and convenient for exploring nonlinear differential equations of fractional order treated in theoretical physics. Keywords: Traveling wave solution, Soliton, Generalized (G′/G-expansion method, Time fractional Duffing equation, Time fractional Riccati equation
Stability analysis of a class of fractional delay differential equations
Indian Academy of Sciences (India)
In this paper we analyse stability of nonlinear fractional order delay differential equations of the form D y ( t ) = a f ( y ( t − ) ) − by ( t ) , where D is a Caputo fractional derivative of order 0 < ≤ 1. We describe stability regions using critical curves. To explain the proposed theory, we discuss fractional order logistic ...
Numerical solution of distributed order fractional differential equations
Katsikadelis, John T.
2014-02-01
In this paper a method for the numerical solution of distributed order FDEs (fractional differential equations) of a general form is presented. The method applies to both linear and nonlinear equations. The Caputo type fractional derivative is employed. The distributed order FDE is approximated with a multi-term FDE, which is then solved by adjusting appropriately the numerical method developed for multi-term FDEs by Katsikadelis. Several example equations are solved and the response of mechanical systems described by such equations is studied. The convergence and the accuracy of the method for linear and nonlinear equations are demonstrated through well corroborated numerical results.
Exact solutions of time-fractional heat conduction equation by the fractional complex transform
Directory of Open Access Journals (Sweden)
Li Zheng-Biao
2012-01-01
Full Text Available The Fractional Complex Transform is extended to solve exactly time-fractional differential equations with the modified Riemann-Liouville derivative. How to incorporate suitable boundary/initial conditions is also discussed.
Analysis of Nonlinear Fractional Nabla Difference Equations
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Jagan Mohan Jonnalagadda
2015-01-01
Full Text Available In this paper, we establish sufficient conditions on global existence and uniqueness of solutions of nonlinear fractional nabla difference systems and investigate the dependence of solutions on initial conditions and parameters.
The modified simple equation method for solving some fractional ...
Indian Academy of Sciences (India)
... and processes in various areas of natural science. Thus, many effective and powerful methods have been established and improved. In this study, we establish exact solutions of the time fractional biological population model equation and nonlinearfractional Klein–Gordon equation by using the modified simple equation ...
Directory of Open Access Journals (Sweden)
Ai-Min Yang
2014-01-01
Full Text Available The local fractional Laplace variational iteration method was applied to solve the linear local fractional partial differential equations. The local fractional Laplace variational iteration method is coupled by the local fractional variational iteration method and Laplace transform. The nondifferentiable approximate solutions are obtained and their graphs are also shown.
Fractional Diffusion Limit for Collisional Kinetic Equations
Mellet, Antoine; Mischler, Sté phane; Mouhot, Clé ment
2010-01-01
This paper is devoted to diffusion limits of linear Boltzmann equations. When the equilibrium distribution function is a Maxwellian distribution, it is well known that for an appropriate time scale, the small mean free path limit gives rise to a
Linear fractional diffusion-wave equation for scientists and engineers
Povstenko, Yuriy
2015-01-01
This book systematically presents solutions to the linear time-fractional diffusion-wave equation. It introduces the integral transform technique and discusses the properties of the Mittag-Leffler, Wright, and Mainardi functions that appear in the solutions. The time-nonlocal dependence between the flux and the gradient of the transported quantity with the “long-tail” power kernel results in the time-fractional diffusion-wave equation with the Caputo fractional derivative. Time-nonlocal generalizations of classical Fourier’s, Fick’s and Darcy’s laws are considered and different kinds of boundary conditions for this equation are discussed (Dirichlet, Neumann, Robin, perfect contact). The book provides solutions to the fractional diffusion-wave equation with one, two and three space variables in Cartesian, cylindrical and spherical coordinates. The respective sections of the book can be used for university courses on fractional calculus, heat and mass transfer, transport processes in porous media and ...
Fractional neutron point kinetics equations for nuclear reactor dynamics
International Nuclear Information System (INIS)
Espinosa-Paredes, Gilberto; Polo-Labarrios, Marco-A.; Espinosa-Martinez, Erick-G.; Valle-Gallegos, Edmundo del
2011-01-01
The fractional point-neutron kinetics model for the dynamic behavior in a nuclear reactor is derived and analyzed in this paper. The fractional model retains the main dynamic characteristics of the neutron motion in which the relaxation time associated with a rapid variation in the neutron flux contains a fractional order, acting as exponent of the relaxation time, to obtain the best representation of a nuclear reactor dynamics. The physical interpretation of the fractional order is related with non-Fickian effects from the neutron diffusion equation point of view. The numerical approximation to the solution of the fractional neutron point kinetics model, which can be represented as a multi-term high-order linear fractional differential equation, is calculated by reducing the problem to a system of ordinary and fractional differential equations. The numerical stability of the fractional scheme is investigated in this work. Results for neutron dynamic behavior for both positive and negative reactivity and for different values of fractional order are shown and compared with the classic neutron point kinetic equations. Additionally, a related review with the neutron point kinetics equations is presented, which encompasses papers written in English about this research topic (as well as some books and technical reports) published since 1940 up to 2010.
Modified Legendre Wavelets Technique for Fractional Oscillation Equations
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Syed Tauseef Mohyud-Din
2015-10-01
Full Text Available Physical Phenomena’s located around us are primarily nonlinear in nature and their solutions are of highest significance for scientists and engineers. In order to have a better representation of these physical models, fractional calculus is used. Fractional order oscillation equations are included among these nonlinear phenomena’s. To tackle with the nonlinearity arising, in these phenomena’s we recommend a new method. In the proposed method, Picard’s iteration is used to convert the nonlinear fractional order oscillation equation into a fractional order recurrence relation and then Legendre wavelets method is applied on the converted problem. In order to check the efficiency and accuracy of the suggested modification, we have considered three problems namely: fractional order force-free Duffing–van der Pol oscillator, forced Duffing–van der Pol oscillator and higher order fractional Duffing equations. The obtained results are compared with the results obtained via other techniques.
International Nuclear Information System (INIS)
Pierantozzi, T.; Vazquez, L.
2005-01-01
Through fractional calculus and following the method used by Dirac to obtain his well-known equation from the Klein-Gordon equation, we analyze a possible interpolation between the Dirac and the diffusion equations in one space dimension. We study the transition between the hyperbolic and parabolic behaviors by means of the generalization of the D'Alembert formula for the classical wave equation and the invariance under space and time inversions of the interpolating fractional evolution equations Dirac like. Such invariance depends on the values of the fractional index and is related to the nonlocal property of the time fractional differential operator. For this system of fractional evolution equations, we also find an associated conserved quantity analogous to the Hamiltonian for the classical Dirac case
On the numerical solution of the neutron fractional diffusion equation
International Nuclear Information System (INIS)
Maleki Moghaddam, Nader; Afarideh, Hossein; Espinosa-Paredes, Gilberto
2014-01-01
Highlights: • The new version of neutron diffusion equation which established on the fractional derivatives is presented. • The Neutron Fractional Diffusion Equation (NFDE) is solved in the finite differences frame. • NFDE is solved using shifted Grünwald-Letnikov definition of fractional operators. • The results show that “K eff ” strongly depends on the order of fractional derivative. - Abstract: In order to core calculation in the nuclear reactors there is a new version of neutron diffusion equation which is established on the fractional partial derivatives, named Neutron Fractional Diffusion Equation (NFDE). In the NFDE model, neutron flux in each zone depends directly on the all previous zones (not only on the nearest neighbors). Under this circumstance, it can be said that the NFDE has the space history. We have developed a one-dimension code, NFDE-1D, which can simulate the reactor core using arbitrary exponent of differential operators. In this work a numerical solution of the NFDE is presented using shifted Grünwald-Letnikov definition of fractional derivative in finite differences frame. The model is validated with some numerical experiments where different orders of fractional derivative are considered (e.g. 0.999, 0.98, 0.96, and 0.94). The results show that the effective multiplication factor (K eff ) depends strongly on the order of fractional derivative
ALMOST AUTOMORPHIC MILD SOLUTIONS TO SOME FRACTIONAL DELAY DIFFERENTIAL EQUATIONS
Institute of Scientific and Technical Information of China (English)
无
2012-01-01
In this paper,a new and general existence and uniqueness theorem of almost automorphic mild solutions is obtained for some fractional delay differential equations,using sectorial operators and the Banach contraction principle.
Existence Results for a Family of Equations of Fractional Resolvent
International Nuclear Information System (INIS)
Ibrahim, R.W.; Qasem, S.A.; Zailan Siri
2015-01-01
This study deals with the presence and distinction of bounded m-solutions (type mild) for a family of generalized integral and differential equations of spot order with fractional resolvent and indefinite delay. (author)
A novel approach for solving fractional Fisher equation using ...
Indian Academy of Sciences (India)
Department of Engineering Sciences, Faculty of Technology and Engineering, East of ... The reliability, simplicity and cost-effectiveness of the method are confirmed by applying this ... Differential transform method; fractional Fisher equation.
Exact solutions of fractional mBBM equation and coupled system of fractional Boussinesq-Burgers
Javeed, Shumaila; Saif, Summaya; Waheed, Asif; Baleanu, Dumitru
2018-06-01
The new exact solutions of nonlinear fractional partial differential equations (FPDEs) are established by adopting first integral method (FIM). The Riemann-Liouville (R-L) derivative and the local conformable derivative definitions are used to deal with the fractional order derivatives. The proposed method is applied to get exact solutions for space-time fractional modified Benjamin-Bona-Mahony (mBBM) equation and coupled time-fractional Boussinesq-Burgers equation. The suggested technique is easily applicable and effectual which can be implemented successfully to obtain the solutions for different types of nonlinear FPDEs.
Solution of the Bagley Torvik equation by fractional DTM
Arora, Geeta; Pratiksha
2017-07-01
In this paper, fractional differential transform method(DTM) is implemented on the Bagley Torvik equation. This equation models the viscoelastic behavior of geological strata, metals, glasses etc. It explains the motion of a rigid plate immersed in a Newtonian fluid. DTM is a simple, reliable and efficient method that gives a series solution. Caputo fractional derivative is considered throughout this work. Two examples are given to demonstrate the validity and applicability of the method and comparison is made with the existing results.
Directory of Open Access Journals (Sweden)
Hossein Jafari
2016-04-01
Full Text Available In this paper, we consider the local fractional decomposition method, variational iteration method, and differential transform method for analytic treatment of linear and nonlinear local fractional differential equations, homogeneous or nonhomogeneous. The operators are taken in the local fractional sense. Some examples are given to demonstrate the simplicity and the efficiency of the presented methods.
Solving Fuzzy Fractional Differential Equations Using Zadeh's Extension Principle
Ahmad, M. Z.; Hasan, M. K.; Abbasbandy, S.
2013-01-01
We study a fuzzy fractional differential equation (FFDE) and present its solution using Zadeh's extension principle. The proposed study extends the case of fuzzy differential equations of integer order. We also propose a numerical method to approximate the solution of FFDEs. To solve nonlinear problems, the proposed numerical method is then incorporated into an unconstrained optimisation technique. Several numerical examples are provided. PMID:24082853
Fractional equations of kicked systems and discrete maps
International Nuclear Information System (INIS)
Tarasov, Vasily E; Zaslavsky, George M
2008-01-01
Starting from kicked equations of motion with derivatives of non-integer orders, we obtain 'fractional' discrete maps. These maps are generalizations of well-known universal, standard, dissipative, kicked damped rotator maps. The main property of the suggested fractional maps is a long-term memory. The memory effects in the fractional discrete maps mean that their present state evolution depends on all past states with special forms of weights. These forms are represented by combinations of power-law functions
Existence of a coupled system of fractional differential equations
International Nuclear Information System (INIS)
Ibrahim, Rabha W.; Siri, Zailan
2015-01-01
We manage the existence and uniqueness of a fractional coupled system containing Schrödinger equations. Such a system appears in quantum mechanics. We confirm that the fractional system under consideration admits a global solution in appropriate functional spaces. The solution is shown to be unique. The method is based on analytic technique of the fixed point theory. The fractional differential operator is considered from the virtue of the Riemann-Liouville differential operator
Existence of a coupled system of fractional differential equations
Energy Technology Data Exchange (ETDEWEB)
Ibrahim, Rabha W. [Multimedia unit, Department of Computer System and Technology Faculty of Computer Science & IT, University of Malaya, 50603 Kuala Lumpur (Malaysia); Siri, Zailan [Institute of Mathematical Sciences, University of Malaya, 50603 Kuala Lumpur (Malaysia)
2015-10-22
We manage the existence and uniqueness of a fractional coupled system containing Schrödinger equations. Such a system appears in quantum mechanics. We confirm that the fractional system under consideration admits a global solution in appropriate functional spaces. The solution is shown to be unique. The method is based on analytic technique of the fixed point theory. The fractional differential operator is considered from the virtue of the Riemann-Liouville differential operator.
A remark on fractional differential equation involving I-function
Mishra, Jyoti
2018-02-01
The present paper deals with the solution of the fractional differential equation using the Laplace transform operator and its corresponding properties in the fractional calculus; we derive an exact solution of a complex fractional differential equation involving a special function known as I-function. The analysis of the some fractional integral with two parameters is presented using the suggested Theorem 1. In addition, some very useful corollaries are established and their proofs presented in detail. Some obtained exact solutions are depicted to see the effect of each fractional order. Owing to the wider applicability of the I-function, we can conclude that, the obtained results in our work generalize numerous well-known results obtained by specializing the parameters.
Finite element method for time-space-fractional Schrodinger equation
Directory of Open Access Journals (Sweden)
Xiaogang Zhu
2017-07-01
Full Text Available In this article, we develop a fully discrete finite element method for the nonlinear Schrodinger equation (NLS with time- and space-fractional derivatives. The time-fractional derivative is described in Caputo's sense and the space-fractional derivative in Riesz's sense. Its stability is well derived; the convergent estimate is discussed by an orthogonal operator. We also extend the method to the two-dimensional time-space-fractional NLS and to avoid the iterative solvers at each time step, a linearized scheme is further conducted. Several numerical examples are implemented finally, which confirm the theoretical results as well as illustrate the accuracy of our methods.
Boundary value problemfor multidimensional fractional advection-dispersion equation
Directory of Open Access Journals (Sweden)
Khasambiev Mokhammad Vakhaevich
2015-05-01
Full Text Available In recent time there is a very great interest in the study of differential equations of fractional order, in which the unknown function is under the symbol of fractional derivative. It is due to the development of the theory of fractional integro-differential theory and application of it in different fields.The fractional integrals and derivatives of fractional integro-differential equations are widely used in modern investigations of theoretical physics, mechanics, and applied mathematics. The fractional calculus is a very powerful tool for describing physical systems, which have a memory and are non-local. Many processes in complex systems have nonlocality and long-time memory. Fractional integral operators and fractional differential operators allow describing some of these properties. The use of the fractional calculus will be helpful for obtaining the dynamical models, in which integro-differential operators describe power long-time memory by time and coordinates, and three-dimensional nonlocality for complex medium and processes.Differential equations of fractional order appear when we use fractal conception in physics of the condensed medium. The transfer, described by the operator with fractional derivatives at a long distance from the sources, leads to other behavior of relatively small concentrations as compared with classic diffusion. This fact redefines the existing ideas about safety, based on the ideas on exponential velocity of damping. Fractional calculus in the fractal theory and the systems with memory have the same importance as the classic analysis in mechanics of continuous medium.In recent years, the application of fractional derivatives for describing and studying the physical processes of stochastic transfer is very popular too. Many problems of filtration of liquids in fractal (high porous medium lead to the need to study boundary value problems for partial differential equations in fractional order.In this paper the
Positive Solutions for Coupled Nonlinear Fractional Differential Equations
Directory of Open Access Journals (Sweden)
Wenning Liu
2014-01-01
Full Text Available We consider the existence of positive solutions for a coupled system of nonlinear fractional differential equations with integral boundary values. Assume the nonlinear term is superlinear in one equation and sublinear in the other equation. By constructing two cones K1, K2 and computing the fixed point index in product cone K1×K2, we obtain that the system has a pair of positive solutions. It is remarkable that it is established on the Cartesian product of two cones, in which the feature of two equations can be opposite.
A procedure to construct exact solutions of nonlinear fractional differential equations.
Güner, Özkan; Cevikel, Adem C
2014-01-01
We use the fractional transformation to convert the nonlinear partial fractional differential equations with the nonlinear ordinary differential equations. The Exp-function method is extended to solve fractional partial differential equations in the sense of the modified Riemann-Liouville derivative. We apply the Exp-function method to the time fractional Sharma-Tasso-Olver equation, the space fractional Burgers equation, and the time fractional fmKdV equation. As a result, we obtain some new exact solutions.
The numerical solution of linear multi-term fractional differential equations: systems of equations
Edwards, John T.; Ford, Neville J.; Simpson, A. Charles
2002-11-01
In this paper, we show how the numerical approximation of the solution of a linear multi-term fractional differential equation can be calculated by reduction of the problem to a system of ordinary and fractional differential equations each of order at most unity. We begin by showing how our method applies to a simple class of problems and we give a convergence result. We solve the Bagley Torvik equation as an example. We show how the method can be applied to a general linear multi-term equation and give two further examples.
Closed form solutions of two time fractional nonlinear wave equations
Akbar, M. Ali; Ali, Norhashidah Hj. Mohd.; Roy, Ripan
2018-06-01
In this article, we investigate the exact traveling wave solutions of two nonlinear time fractional wave equations. The fractional derivatives are described in the sense of conformable fractional derivatives. In addition, the traveling wave solutions are accomplished in the form of hyperbolic, trigonometric, and rational functions involving free parameters. To investigate such types of solutions, we implement the new generalized (G‧ / G) -expansion method. The extracted solutions are reliable, useful and suitable to comprehend the optimal control problems, chaotic vibrations, global and local bifurcations and resonances, furthermore, fission and fusion phenomena occur in solitons, the relativistic energy-momentum relation, scalar electrodynamics, quantum relativistic one-particle theory, electromagnetic interactions etc. The results reveal that the method is very fruitful and convenient for exploring nonlinear differential equations of fractional order treated in theoretical physics.
Numerical simulation of fractional Cable equation of spiny neuronal dendrites
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N.H. Sweilam
2014-03-01
Full Text Available In this article, numerical study for the fractional Cable equation which is fundamental equations for modeling neuronal dynamics is introduced by using weighted average of finite difference methods. The stability analysis of the proposed methods is given by a recently proposed procedure similar to the standard John von Neumann stability analysis. A simple and an accurate stability criterion valid for different discretization schemes of the fractional derivative and arbitrary weight factor is introduced and checked numerically. Numerical results, figures, and comparisons have been presented to confirm the theoretical results and efficiency of the proposed method.
An efficient technique for higher order fractional differential equation.
Ali, Ayyaz; Iqbal, Muhammad Asad; Ul-Hassan, Qazi Mahmood; Ahmad, Jamshad; Mohyud-Din, Syed Tauseef
2016-01-01
In this study, we establish exact solutions of fractional Kawahara equation by using the idea of [Formula: see text]-expansion method. The results of different studies show that the method is very effective and can be used as an alternative for finding exact solutions of nonlinear evolution equations (NLEEs) in mathematical physics. The solitary wave solutions are expressed by the hyperbolic, trigonometric, exponential and rational functions. Graphical representations along with the numerical data reinforce the efficacy of the used procedure. The specified idea is very effective, expedient for fractional PDEs, and could be extended to other physical problems.
Solution of fractional differential equations by using differential transform method
International Nuclear Information System (INIS)
Arikoglu, Aytac; Ozkol, Ibrahim
2007-01-01
In this study, we implement a well known transformation technique, Differential Transform Method (DTM), to the area of fractional differential equations. Theorems that never existed before are introduced with their proofs. Also numerical examples are carried out for various types of problems, including the Bagley-Torvik, Ricatti and composite fractional oscillation equations for the application of the method. The results obtained are in good agreement with the existing ones in open literature and it is shown that the technique introduced here is robust, accurate and easy to apply
Solution of fractional differential equations by using differential transform method
Energy Technology Data Exchange (ETDEWEB)
Arikoglu, Aytac [Istanbul Technical University, Faculty of Aeronautics and Astronautics, Department of Aeronautical Engineering, Maslak, TR-34469 Istanbul (Turkey); Ozkol, Ibrahim [Istanbul Technical University, Faculty of Aeronautics and Astronautics, Department of Aeronautical Engineering, Maslak, TR-34469 Istanbul (Turkey)]. E-mail: ozkol@itu.edu.tr
2007-12-15
In this study, we implement a well known transformation technique, Differential Transform Method (DTM), to the area of fractional differential equations. Theorems that never existed before are introduced with their proofs. Also numerical examples are carried out for various types of problems, including the Bagley-Torvik, Ricatti and composite fractional oscillation equations for the application of the method. The results obtained are in good agreement with the existing ones in open literature and it is shown that the technique introduced here is robust, accurate and easy to apply.
Modified Legendre Wavelets Technique for Fractional Oscillation Equations
Mohyud-Din, Syed; Iqbal, Muhammad; Hassan, Saleh
2015-01-01
Physical Phenomena’s located around us are primarily nonlinear in nature and their solutions are of highest significance for scientists and engineers. In order to have a better representation of these physical models, fractional calculus is used. Fractional order oscillation equations are included among these nonlinear phenomena’s. To tackle with the nonlinearity arising, in these phenomena’s we recommend a new method. In the proposed method, Picard’s iteration is used to convert the nonlinea...
Reply to "Comment on 'Fractional quantum mechanics' and 'Fractional Schrödinger equation' ".
Laskin, Nick
2016-06-01
The fractional uncertainty relation is a mathematical formulation of Heisenberg's uncertainty principle in the framework of fractional quantum mechanics. Two mistaken statements presented in the Comment have been revealed. The origin of each mistaken statement has been clarified and corrected statements have been made. A map between standard quantum mechanics and fractional quantum mechanics has been presented to emphasize the features of fractional quantum mechanics and to avoid misinterpretations of the fractional uncertainty relation. It has been shown that the fractional probability current equation is correct in the area of its applicability. Further studies have to be done to find meaningful quantum physics problems with involvement of the fractional probability current density vector and the extra term emerging in the framework of fractional quantum mechanics.
On the solutions of fractional reaction-diffusion equations
Directory of Open Access Journals (Sweden)
Jagdev Singh
2013-05-01
Full Text Available In this paper, we obtain the solution of a fractional reaction-diffusion equation associated with the generalized Riemann-Liouville fractional derivative as the time derivative and Riesz-Feller fractional derivative as the space-derivative. The results are derived by the application of the Laplace and Fourier transforms in compact and elegant form in terms of Mittag-Leffler function and H-function. The results obtained here are of general nature and include the results investigated earlier by many authors.
The Klein–Gordon–Zakharov equations with the positive fractional
Indian Academy of Sciences (India)
... and the new special types of KGZ equations with the positive fractional power terms (gKGZE) are presented. ... exact solutions of four special types of the gKGZE are derived, which are the bell-type ... Pramana – Journal of Physics | News.
On the Fractional Nagumo Equation with Nonlinear Diffusion and Convection
Directory of Open Access Journals (Sweden)
Abdon Atangana
2014-01-01
Full Text Available We presented the Nagumo equation using the concept of fractional calculus. With the help of two analytical techniques including the homotopy decomposition method (HDM and the new development of variational iteration method (NDVIM, we derived an approximate solution. Both methods use a basic idea of integral transform and are very simple to be used.
On Comparison Theorems for Conformable Fractional Differential Equations
Directory of Open Access Journals (Sweden)
Mehmet Zeki Sarikaya
2016-10-01
Full Text Available In this paper the more general comparison theorems for conformable fractional differential equations is proposed and tested. Thus we prove some inequalities for conformable integrals by using the generalization of Sturm's separation and Sturm's comparison theorems. The results presented here would provide generalizations of those given in earlier works. The numerical example is also presented to verify the proposed theorem.
Asymptotic behavior of solutions of forced fractional differential equations
Directory of Open Access Journals (Sweden)
Said Grace
2016-09-01
where $y(t=\\left( a(tx^{\\prime }(t\\right ^{\\prime }$, $c_{0}=\\frac{y(c}{\\Gamma (1}=y(c$, and $c_{0}$ is a real constant. The technique used in obtaining their results will apply to related fractional differential equations with Caputo derivatives of any order. Examples illustrate the results obtained in this paper.
Hipergeometric solutions to some nonhomogeneous equations of fractional order
Olivares, Jorge; Martin, Pablo; Maass, Fernando
2017-12-01
In this paper a study is performed to the solution of the linear non homogeneous fractional order alpha differential equation equal to I 0(x), where I 0(x) is the modified Bessel function of order zero, the initial condition is f(0)=0 and 0 definition for the fractional derivatives is considered. Fractional derivatives have become important in physical and chemical phenomena as visco-elasticity and visco-plasticity, anomalous diffusion and electric circuits. In particular in this work the values of alpha=1/2, 1/4 and 3/4. are explicitly considered . In these cases Laplace transform is applied, and later the inverse Laplace transform leads to the solutions of the differential equation, which become hypergeometric functions.
Directory of Open Access Journals (Sweden)
Hamid A. Jalab
2014-01-01
Full Text Available The interest in using fractional mask operators based on fractional calculus operators has grown for image denoising. Denoising is one of the most fundamental image restoration problems in computer vision and image processing. This paper proposes an image denoising algorithm based on convex solution of fractional heat equation with regularized fractional power parameters. The performances of the proposed algorithms were evaluated by computing the PSNR, using different types of images. Experiments according to visual perception and the peak signal to noise ratio values show that the improvements in the denoising process are competent with the standard Gaussian filter and Wiener filter.
Fractional equivalent Lagrangian densities for a fractional higher-order equation
International Nuclear Information System (INIS)
Fujioka, J
2014-01-01
In this communication we show that the equivalent Lagrangian densities (ELDs) of a fractional higher-order nonlinear Schrödinger equation with stable soliton-like solutions can be related in a hitherto unknown way. This new relationship is described in terms of a new fractional operator that includes both left- and right-sided fractional derivatives. Using this operator it is possible to generate new ELDs that contain different fractional parts, in addition to the already known ELDs, which only differ by a sum of first-order partial derivatives of two arbitrary functions. (fast track communications)
A Solution to the Fundamental Linear Fractional Order Differential Equation
Hartley, Tom T.; Lorenzo, Carl F.
1998-01-01
This paper provides a solution to the fundamental linear fractional order differential equation, namely, (sub c)d(sup q, sub t) + ax(t) = bu(t). The impulse response solution is shown to be a series, named the F-function, which generalizes the normal exponential function. The F-function provides the basis for a qth order "fractional pole". Complex plane behavior is elucidated and a simple example, the inductor terminated semi- infinite lossy line, is used to demonstrate the theory.
Exact solutions of nonlinear differential equations using continued fractions
International Nuclear Information System (INIS)
Ditto, W.L.; Pickett, T.J.
1990-01-01
The continued-fraction conversion method (J. Math. Phys. (N.Y.), 29, 1761 (1988)) is used to generate a homologous family of exact solutions to the Lane-Emden equation φ(r) '' + 2φ(r)'/r + αφ(r) p = 0, for p=5. An exact solution is also obtained for a generalization of the Lane-Emden equation of the form -φ '' (r) -2φ(r)'/r + αφ(r) 2p+1 + λφ(r) 4p+1 = 0 for arbitrary α, γ and p. A condition is established for the generation of exact solutions from the method
Numerical Solution of Stochastic Nonlinear Fractional Differential Equations
El-Beltagy, Mohamed A.; Al-Juhani, Amnah
2015-01-01
Using Wiener-Hermite expansion (WHE) technique in the solution of the stochastic partial differential equations (SPDEs) has the advantage of converting the problem to a system of deterministic equations that can be solved efficiently using the standard deterministic numerical methods [1]. WHE is the only known expansion that handles the white/colored noise exactly. This work introduces a numerical estimation of the stochastic response of the Duffing oscillator with fractional or variable order damping and driven by white noise. The WHE technique is integrated with the Grunwald-Letnikov approximation in case of fractional order and with Coimbra approximation in case of variable-order damping. The numerical solver was tested with the analytic solution and with Monte-Carlo simulations. The developed mixed technique was shown to be efficient in simulating SPDEs.
Numerical Solution of Stochastic Nonlinear Fractional Differential Equations
El-Beltagy, Mohamed A.
2015-01-07
Using Wiener-Hermite expansion (WHE) technique in the solution of the stochastic partial differential equations (SPDEs) has the advantage of converting the problem to a system of deterministic equations that can be solved efficiently using the standard deterministic numerical methods [1]. WHE is the only known expansion that handles the white/colored noise exactly. This work introduces a numerical estimation of the stochastic response of the Duffing oscillator with fractional or variable order damping and driven by white noise. The WHE technique is integrated with the Grunwald-Letnikov approximation in case of fractional order and with Coimbra approximation in case of variable-order damping. The numerical solver was tested with the analytic solution and with Monte-Carlo simulations. The developed mixed technique was shown to be efficient in simulating SPDEs.
Ground state solutions for non-local fractional Schrodinger equations
Directory of Open Access Journals (Sweden)
Yang Pu
2015-08-01
Full Text Available In this article, we study a time-independent fractional Schrodinger equation with non-local (regional diffusion $$ (-\\Delta^{\\alpha}_{\\rho}u + V(xu = f(x,u \\quad \\text{in }\\mathbb{R}^{N}, $$ where $\\alpha \\in (0,1$, $N > 2\\alpha$. We establish the existence of a non-negative ground state solution by variational methods.
TRAVELING WAVE SOLUTIONS OF SOME FRACTIONAL DIFFERENTIAL EQUATIONS
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SERIFE MUGE EGE
2016-07-01
Full Text Available The modified Kudryashov method is powerful, efficient and can be used as an alternative to establish new solutions of different type of fractional differential equations applied in mathematical physics. In this article, we’ve constructed new traveling wave solutions including symmetrical Fibonacci function solutions, hyperbolic function solutions and rational solutions of the space-time fractional Cahn Hillihard equation D_t^α u − γD_x^α u − 6u(D_x^α u^2 − (3u^2 − 1D_x^α (D_x^α u + D_x^α(D_x^α(D_x^α(D_x^α u = 0 and the space-time fractional symmetric regularized long wave (SRLW equation D_t^α(D_t^α u + D_x^α(D_x^α u + uD_t^α(D_x^α u + D_x^α u D_t^α u + D_t^α(D_t^α(D_x^α(D_x^α u = 0 via modified Kudryashov method. In addition, some of the solutions are described in the figures with the help of Mathematica.
Robust fast controller design via nonlinear fractional differential equations.
Zhou, Xi; Wei, Yiheng; Liang, Shu; Wang, Yong
2017-07-01
A new method for linear system controller design is proposed whereby the closed-loop system achieves both robustness and fast response. The robustness performance considered here means the damping ratio of closed-loop system can keep its desired value under system parameter perturbation, while the fast response, represented by rise time of system output, can be improved by tuning the controller parameter. We exploit techniques from both the nonlinear systems control and the fractional order systems control to derive a novel nonlinear fractional order controller. For theoretical analysis of the closed-loop system performance, two comparison theorems are developed for a class of fractional differential equations. Moreover, the rise time of the closed-loop system can be estimated, which facilitates our controller design to satisfy the fast response performance and maintain the robustness. Finally, numerical examples are given to illustrate the effectiveness of our methods. Copyright © 2017 ISA. Published by Elsevier Ltd. All rights reserved.
Exact solutions of space-time fractional EW and modified EW equations
International Nuclear Information System (INIS)
Korkmaz, Alper
2017-01-01
The bright soliton solutions and singular solutions are constructed for the space-time fractional EW and the space-time fractional modified EW (MEW) equations. Both equations are reduced to ordinary differential equations by the use of fractional complex transform (FCT) and properties of modified Riemann–Liouville derivative. Then, various ansatz method are implemented to construct the solutions for both equations.
Xiao-Li Ding; Juan J. Nieto
2018-01-01
In this paper, we investigate analytical solutions of multi-time scale fractional stochastic differential equations driven by fractional Brownian motions. We firstly decompose homogeneous multi-time scale fractional stochastic differential equations driven by fractional Brownian motions into independent differential subequations, and give their analytical solutions. Then, we use the variation of constant parameters to obtain the solutions of nonhomogeneous multi-time scale fractional stochast...
Exact solutions of nonlinear fractional differential equations by (G′/G)-expansion method
International Nuclear Information System (INIS)
Bekir Ahmet; Güner Özkan
2013-01-01
In this paper, we use the fractional complex transform and the (G′/G)-expansion method to study the nonlinear fractional differential equations and find the exact solutions. The fractional complex transform is proposed to convert a partial fractional differential equation with Jumarie's modified Riemann—Liouville derivative into its ordinary differential equation. It is shown that the considered transform and method are very efficient and powerful in solving wide classes of nonlinear fractional order equations
International Nuclear Information System (INIS)
Guo, Shimin; Mei, Liquan; Li, Ying; Sun, Youfa
2012-01-01
By introducing a new general ansätz, the improved fractional sub-equation method is proposed to construct analytical solutions of nonlinear evolution equations involving Jumarie's modified Riemann–Liouville derivative. By means of this method, the space–time fractional Whitham–Broer–Kaup and generalized Hirota–Satsuma coupled KdV equations are successfully solved. The obtained results show that the proposed method is quite effective, promising and convenient for solving nonlinear fractional differential equations. -- Highlights: ► We propose a novel method for nonlinear fractional differential equations. ► Two important fractional differential equations in fluid mechanics are solved successfully. ► Some new exact solutions of the fractional differential equations are obtained. ► These solutions will advance the understanding of nonlinear physical phenomena.
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Coşkun Yakar
2010-01-01
Full Text Available The qualitative behavior of a perturbed fractional-order differential equation with Caputo's derivative that differs in initial position and initial time with respect to the unperturbed fractional-order differential equation with Caputo's derivative has been investigated. We compare the classical notion of stability to the notion of initial time difference stability for fractional-order differential equations in Caputo's sense. We present a comparison result which again gives the null solution a central role in the comparison fractional-order differential equation when establishing initial time difference stability of the perturbed fractional-order differential equation with respect to the unperturbed fractional-order differential equation.
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Xiao-Li Ding
2018-01-01
Full Text Available In this paper, we investigate analytical solutions of multi-time scale fractional stochastic differential equations driven by fractional Brownian motions. We firstly decompose homogeneous multi-time scale fractional stochastic differential equations driven by fractional Brownian motions into independent differential subequations, and give their analytical solutions. Then, we use the variation of constant parameters to obtain the solutions of nonhomogeneous multi-time scale fractional stochastic differential equations driven by fractional Brownian motions. Finally, we give three examples to demonstrate the applicability of our obtained results.
Fendzi-Donfack, Emmanuel; Nguenang, Jean Pierre; Nana, Laurent
2018-02-01
We use the fractional complex transform with the modified Riemann-Liouville derivative operator to establish the exact and generalized solutions of two fractional partial differential equations. We determine the solutions of fractional nonlinear electrical transmission lines (NETL) and the perturbed nonlinear Schroedinger (NLS) equation with the Kerr law nonlinearity term. The solutions are obtained for the parameters in the range (0<α≤1) of the derivative operator and we found the traditional solutions for the limiting case of α =1. We show that according to the modified Riemann-Liouville derivative, the solutions found can describe physical systems with memory effect, transient effects in electrical systems and nonlinear transmission lines, and other systems such as optical fiber.
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F. Ghomanjani
2016-10-01
Full Text Available In the present paper, we apply the Bezier curves method for solving fractional optimal control problems (OCPs and fractional Riccati differential equations. The main advantage of this method is that it can reduce the error of the approximate solutions. Hence, the solutions obtained using the Bezier curve method give good approximations. Some numerical examples are provided to confirm the accuracy of the proposed method. All of the numerical computations have been performed on a PC using several programs written in MAPLE 13.
On the solution of the Schroedinger equation through continued fractions
International Nuclear Information System (INIS)
Mignaco, J.A.
1979-05-01
The domain of interest for the applications of a method to solve the Schroedinger equation through continued fractions is studied. It is argued that the method applies almost equally well to quantum mechanical regimes (lower energy levels, low energy scattering) as well as to semiclassical ones simultaneously; this is illustrated by the example of the central power law potentials r sup(ν)(ν>o). The explanation of this behaviour is given in terms of the mathematical approximations involved and its relationship to physically interesting quantities. (Author) [pt
On Solution of a Fractional Diffusion Equation by Homotopy Transform Method
International Nuclear Information System (INIS)
Salah, A.; Hassan, S.S.A.
2012-01-01
The homotopy analysis transform method (HATM) is applied in this work in order to find the analytical solution of fractional diffusion equations (FDE). These equations are obtained from standard diffusion equations by replacing a second-order space derivative by a fractional derivative of order α and a first order time derivative by a fractional derivative. Furthermore, some examples are given. Numerical results show that the homotopy analysis transform method is easy to implement and accurate when applied to a fractional diffusion equations.
Oscillation of a class of fractional differential equations with damping term.
Qin, Huizeng; Zheng, Bin
2013-01-01
We investigate the oscillation of a class of fractional differential equations with damping term. Based on a certain variable transformation, the fractional differential equations are converted into another differential equations of integer order with respect to the new variable. Then, using Riccati transformation, inequality, and integration average technique, some new oscillatory criteria for the equations are established. As for applications, oscillation for two certain fractional differential equations with damping term is investigated by the use of the presented results.
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M.H.T. Alshbool
2017-01-01
Full Text Available An algorithm for approximating solutions to fractional differential equations (FDEs in a modified new Bernstein polynomial basis is introduced. Writing x→xα(0<α<1 in the operational matrices of Bernstein polynomials, the fractional Bernstein polynomials are obtained and then transformed into matrix form. Furthermore, using Caputo fractional derivative, the matrix form of the fractional derivative is constructed for the fractional Bernstein matrices. We convert each term of the problem to the matrix form by means of fractional Bernstein matrices. A basic matrix equation which corresponds to a system of fractional equations is utilized, and a new system of nonlinear algebraic equations is obtained. The method is given with some priori error estimate. By using the residual correction procedure, the absolute error can be estimated. Illustrative examples are included to demonstrate the validity and applicability of the presented technique.
A study of fractional Schrödinger equation composed of Jumarie ...
Indian Academy of Sciences (India)
In this paper we have derived the fractional-order Schrödinger equation composed of Jumarie fractional derivative. The solution of this fractional-order Schrödinger equation is obtained in terms of Mittag–Leffler function with complex arguments, and fractional trigonometric functions. A few important properties of the ...
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Dumitru Baleanu
2014-01-01
Full Text Available We perform a comparison between the fractional iteration and decomposition methods applied to the wave equation on Cantor set. The operators are taken in the local sense. The results illustrate the significant features of the two methods which are both very effective and straightforward for solving the differential equations with local fractional derivative.
Langevin- Science and vigilance
International Nuclear Information System (INIS)
Bensaude-Vincent, B.
1987-01-01
Paul Langevin personifies the figure of popular scientist, buried in the Pantheon, because he was in all great fights: to diffuse knowledge, for improvement and democratization of teaching, for justice and peace. Great theoretician of physics, comparable to Einstein whom he rejoined the approach, he was a fertile discoverer, author, in particular, of a proceeding to detect submarines. He fought politically, regardless of his career, in the ranges of pacifists and opponents of fascism. This book reveals, in his warm diversity, a badly known personality, in spite of the legend in his lifetime. (N.C.)
Fast-forward Langevin dynamics with momentum flips
Hijazi, Mahdi; Wilkins, David M.; Ceriotti, Michele
2018-05-01
Stochastic thermostats based on the Langevin equation, in which a system is coupled to an external heat bath, are popular methods for temperature control in molecular dynamics simulations due to their ergodicity and their ease of implementation. Traditionally, these thermostats suffer from sluggish behavior in the limit of high friction, unlike thermostats of the Nosé-Hoover family whose performance degrades more gently in the strong coupling regime. We propose a simple and easy-to-implement modification to the integration scheme of the Langevin algorithm that addresses the fundamental source of the overdamped behavior of high-friction Langevin dynamics: if the action of the thermostat causes the momentum of a particle to change direction, it is flipped back. This fast-forward Langevin equation preserves the momentum distribution and so guarantees the correct equilibrium sampling. It mimics the quadratic behavior of Nosé-Hoover thermostats and displays similarly good performance in the strong coupling limit. We test the efficiency of this scheme by applying it to a 1-dimensional harmonic oscillator, as well as to water and Lennard-Jones polymers. The sampling efficiency of the fast-forward Langevin equation thermostat, measured by the correlation time of relevant system variables, is at least as good as the traditional Langevin thermostat, and in the overdamped regime, the fast-forward thermostat performs much better, improving the efficiency by an order of magnitude at the highest frictions we considered.
Positive solutions of fractional differential equations with derivative terms
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Cuiping Cheng
2012-11-01
Full Text Available In this article, we are concerned with the existence of positive solutions for nonlinear fractional differential equation whose nonlinearity contains the first-order derivative, $$displaylines{ D_{0^+}^{alpha}u(t+f(t,u(t,u'(t=0,quad tin (0,1,; n-1
Stable Numerical Approach for Fractional Delay Differential Equations
Singh, Harendra; Pandey, Rajesh K.; Baleanu, D.
2017-12-01
In this paper, we present a new stable numerical approach based on the operational matrix of integration of Jacobi polynomials for solving fractional delay differential equations (FDDEs). The operational matrix approach converts the FDDE into a system of linear equations, and hence the numerical solution is obtained by solving the linear system. The error analysis of the proposed method is also established. Further, a comparative study of the approximate solutions is provided for the test examples of the FDDE by varying the values of the parameters in the Jacobi polynomials. As in special case, the Jacobi polynomials reduce to the well-known polynomials such as (1) Legendre polynomial, (2) Chebyshev polynomial of second kind, (3) Chebyshev polynomial of third and (4) Chebyshev polynomial of fourth kind respectively. Maximum absolute error and root mean square error are calculated for the illustrated examples and presented in form of tables for the comparison purpose. Numerical stability of the presented method with respect to all four kind of polynomials are discussed. Further, the obtained numerical results are compared with some known methods from the literature and it is observed that obtained results from the proposed method is better than these methods.
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Veyis Turut
2013-01-01
Full Text Available Two tecHniques were implemented, the Adomian decomposition method (ADM and multivariate Padé approximation (MPA, for solving nonlinear partial differential equations of fractional order. The fractional derivatives are described in Caputo sense. First, the fractional differential equation has been solved and converted to power series by Adomian decomposition method (ADM, then power series solution of fractional differential equation was put into multivariate Padé series. Finally, numerical results were compared and presented in tables and figures.
International Nuclear Information System (INIS)
Yao Ruo-Xia; Wang Wei; Chen Ting-Hua
2014-01-01
Motivated by the widely used ansätz method and starting from the modified Riemann—Liouville derivative together with a fractional complex transformation that can be utilized to transform nonlinear fractional partial differential equations to nonlinear ordinary differential equations, new types of exact traveling wave solutions to three important nonlinear space- and time-fractional partial differential equations are obtained simultaneously in terms of solutions of a Riccati equation. The results are new and first reported in this paper. (general)
Stationarity-conservation laws for fractional differential equations with variable coefficients
Energy Technology Data Exchange (ETDEWEB)
Klimek, Malgorzata [Institute of Mathematics and Computer Science, Technical University of Czestochowa, Czestochowa (Poland)
2002-08-09
In this paper, we study linear fractional differential equations with variable coefficients. It is shown that, by assuming some conditions for the coefficients, the stationarity-conservation laws can be derived. The area where these are valid is restricted by the asymptotic properties of solutions of the respective equation. Applications of the proposed procedure include the fractional Fokker-Planck equation in (1+1)- and (d+1)-dimensional space and the fractional Klein-Kramers equation. (author)
Stationarity-conservation laws for fractional differential equations with variable coefficients
International Nuclear Information System (INIS)
Klimek, Malgorzata
2002-01-01
In this paper, we study linear fractional differential equations with variable coefficients. It is shown that, by assuming some conditions for the coefficients, the stationarity-conservation laws can be derived. The area where these are valid is restricted by the asymptotic properties of solutions of the respective equation. Applications of the proposed procedure include the fractional Fokker-Planck equation in (1+1)- and (d+1)-dimensional space and the fractional Klein-Kramers equation. (author)
Directory of Open Access Journals (Sweden)
Bashir Ahmad
2013-02-01
Full Text Available In this article, we discuss the existence of solutions for a boundary-value problem of integro-differential equations of fractional order with nonlocal fractional boundary conditions by means of some standard tools of fixed point theory. Our problem describes a more general form of fractional stochastic dynamic model for financial asset. An illustrative example is also presented.
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Olaniyi Samuel Iyiola
2014-09-01
Full Text Available In this paper, we obtain analytical solutions of homogeneous time-fractional Gardner equation and non-homogeneous time-fractional models (including Buck-master equation using q-Homotopy Analysis Method (q-HAM. Our work displays the elegant nature of the application of q-HAM not only to solve homogeneous non-linear fractional differential equations but also to solve the non-homogeneous fractional differential equations. The presence of the auxiliary parameter h helps in an effective way to obtain better approximation comparable to exact solutions. The fraction-factor in this method gives it an edge over other existing analytical methods for non-linear differential equations. Comparisons are made upon the existence of exact solutions to these models. The analysis shows that our analytical solutions converge very rapidly to the exact solutions.
International Nuclear Information System (INIS)
Feng Qinghua
2013-01-01
In this paper, an extended Riccati sub-ODE method is proposed to establish new exact solutions for fractional differential-difference equations in the sense of modified Riemann—Liouville derivative. By a fractional complex transformation, a given fractional differential-difference equation can be turned into another differential-difference equation of integer order. The validity of the method is illustrated by applying it to solve the fractional Hybrid lattice equation and the fractional relativistic Toda lattice system. As a result, some new exact solutions including hyperbolic function solutions, trigonometric function solutions and rational solutions are established. (general)
Snook, Ian
2007-01-01
The Langevin and Generalised Langevin Approach To The Dynamics Of Atomic, Polymeric And Colloidal Systems is concerned with the description of aspects of the theory and use of so-called random processes to describe the properties of atomic, polymeric and colloidal systems in terms of the dynamics of the particles in the system. It provides derivations of the basic equations, the development of numerical schemes to solve them on computers and gives illustrations of application to typical systems.Extensive appendices are given to enable the reader to carry out computations to illustrate many of the points made in the main body of the book.* Starts from fundamental equations* Gives up-to-date illustration of the application of these techniques to typical systems of interest* Contains extensive appendices including derivations, equations to be used in practice and elementary computer codes
Ghosh, Uttam; Banerjee, Joydip; Sarkar, Susmita; Das, Shantanu
2018-06-01
Klein-Gordon equation is one of the basic steps towards relativistic quantum mechanics. In this paper, we have formulated fractional Klein-Gordon equation via Jumarie fractional derivative and found two types of solutions. Zero-mass solution satisfies photon criteria and non-zero mass satisfies general theory of relativity. Further, we have developed rest mass condition which leads us to the concept of hidden wave. Classical Klein-Gordon equation fails to explain a chargeless system as well as a single-particle system. Using the fractional Klein-Gordon equation, we can overcome the problem. The fractional Klein-Gordon equation also leads to the smoothness parameter which is the measurement of the bumpiness of space. Here, by using this smoothness parameter, we have defined and interpreted the various cases.
Comment on "Fractional quantum mechanics" and "Fractional Schrödinger equation".
Wei, Yuchuan
2016-06-01
In this Comment we point out some shortcomings in two papers [N. Laskin, Phys. Rev. E 62, 3135 (2000)10.1103/PhysRevE.62.3135; N. Laskin, Phys. Rev. E 66, 056108 (2002)10.1103/PhysRevE.66.056108]. We prove that the fractional uncertainty relation does not hold generally. The probability continuity equation in fractional quantum mechanics has a missing source term, which leads to particle teleportation, i.e., a particle can teleport from a place to another. Since the relativistic kinetic energy can be viewed as an approximate realization of the fractional kinetic energy, the particle teleportation should be an observable relativistic effect in quantum mechanics. With the help of this concept, superconductivity could be viewed as the teleportation of electrons from one side of a superconductor to another and superfluidity could be viewed as the teleportation of helium atoms from one end of a capillary tube to the other. We also point out how to teleport a particle to an arbitrary destination.
International Nuclear Information System (INIS)
Zabadal, J.; Vilhena, M.T.; Segatto, C.F.; Pazos, R.P.Ruben Panta.
2002-01-01
In this work we construct a closed-form solution for the multidimensional transport equation rewritten in integral form which is expressed in terms of a fractional derivative of the angular flux. We determine the unknown order of the fractional derivative comparing the kernel of the integral equation with the one of the Riemann-Liouville definition of fractional derivative. We report numerical simulations
Energy Technology Data Exchange (ETDEWEB)
Zabadal, J. E-mail: jorge.zabadal@ufrgs.br; Vilhena, M.T. E-mail: vilhena@mat.ufrgs.br; Segatto, C.F. E-mail: cynthia@mat.ufrgs.br; Pazos, R.P.Ruben Panta. E-mail: rpp@mat.pucrgs.br
2002-07-01
In this work we construct a closed-form solution for the multidimensional transport equation rewritten in integral form which is expressed in terms of a fractional derivative of the angular flux. We determine the unknown order of the fractional derivative comparing the kernel of the integral equation with the one of the Riemann-Liouville definition of fractional derivative. We report numerical simulations.
Application of the Lie Symmetry Analysis for second-order fractional differential equations
Directory of Open Access Journals (Sweden)
Mousa Ilie
2017-12-01
Full Text Available Obtaining analytical or numerical solution of fractional differential equations is one of the troublesome and challenging issue among mathematicians and engineers, specifically in recent years. The purpose of this paper Lie Symmetry method is developed to solve second-order fractional differential equations, based on conformable fractional derivative. Some numerical examples are presented to illustrate the proposed approach.
A non-differentiable solution for the local fractional telegraph equation
Directory of Open Access Journals (Sweden)
Li Jie
2017-01-01
Full Text Available In this paper, we consider the linear telegraph equations with local fractional derivative. The local fractional Laplace series expansion method is used to handle the local fractional telegraph equation. The analytical solution with the non-differentiable graphs is discussed in detail. The proposed method is efficient and accurate.
New Hamiltonian structure of the fractional C-KdV soliton equation hierarchy
International Nuclear Information System (INIS)
Yu Fajun; Zhang Hongqing
2008-01-01
A generalized Hamiltonian structure of the fractional soliton equation hierarchy is presented by using of differential forms and exterior derivatives of fractional orders. Example of the fractional Hamiltonian system of the C-KdV soliton equation hierarchy is constructed, which is a new Hamiltonian structure
Lie Group Classifications and Non-differentiable Solutions for Time-Fractional Burgers Equation
International Nuclear Information System (INIS)
Wu Guocheng
2011-01-01
Lie group method provides an efficient tool to solve nonlinear partial differential equations. This paper suggests Lie group method for fractional partial differential equations. A time-fractional Burgers equation is used as an example to illustrate the effectiveness of the Lie group method and some classes of exact solutions are obtained. (electromagnetism, optics, acoustics, heat transfer, classical mechanics, and fluid dynamics)
Solution of Moving Boundary Space-Time Fractional Burger’s Equation
Directory of Open Access Journals (Sweden)
E. A.-B. Abdel-Salam
2014-01-01
Full Text Available The fractional Riccati expansion method is used to solve fractional differential equations with variable coefficients. To illustrate the effectiveness of the method, the moving boundary space-time fractional Burger’s equation is studied. The obtained solutions include generalized trigonometric and hyperbolic function solutions. Among these solutions, some are found for the first time. The linear and periodic moving boundaries for the kink solution of the Burger’s equation are presented graphically and discussed.
Asymptotic behavior of solutions of linear multi-order fractional differential equation systems
Diethelm, Kai; Siegmund, Stefan; Tuan, H. T.
2017-01-01
In this paper, we investigate some aspects of the qualitative theory for multi-order fractional differential equation systems. First, we obtain a fundamental result on the existence and uniqueness for multi-order fractional differential equation systems. Next, a representation of solutions of homogeneous linear multi-order fractional differential equation systems in series form is provided. Finally, we give characteristics regarding the asymptotic behavior of solutions to some classes of line...
Spline Collocation Method for Nonlinear Multi-Term Fractional Differential Equation
Choe, Hui-Chol; Kang, Yong-Suk
2013-01-01
We study an approximation method to solve nonlinear multi-term fractional differential equations with initial conditions or boundary conditions. First, we transform the nonlinear multi-term fractional differential equations with initial conditions and boundary conditions to nonlinear fractional integral equations and consider the relations between them. We present a Spline Collocation Method and prove the existence, uniqueness and convergence of approximate solution as well as error estimatio...
Directory of Open Access Journals (Sweden)
Guo Zheng-Hong
2016-01-01
Full Text Available In this article, the Sumudu transform series expansion method is used to handle the local fractional Laplace equation arising in the steady fractal heat-transfer problem via local fractional calculus.
Exact solutions of a class of fractional Hamiltonian equations involving Caputo derivatives
Energy Technology Data Exchange (ETDEWEB)
Baleanu, Dumitru [Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Cankaya University, Ankara 06530 (Turkey); Trujillo, Juan J [Departamento de Analisis Matematico, University of La Laguna, 38271 La Laguna, Tenerife (Spain)], E-mail: dumitru@cankaya.edu.tr, E-mail: JTrujill@ullmat.es, E-mail: baleanu@venus.nipne.ro
2009-11-15
The fractional Hamiltonian equations corresponding to the Lagrangians of constrained systems within Caputo derivatives are investigated. The fractional phase space is obtained and the exact solutions of some constrained systems are obtained.
An Efficient Implicit FEM Scheme for Fractional-in-Space Reaction-Diffusion Equations
Burrage, Kevin; Hale, Nicholas; Kay, David
2012-01-01
Fractional differential equations are becoming increasingly used as a modelling tool for processes associated with anomalous diffusion or spatial heterogeneity. However, the presence of a fractional differential operator causes memory (time
Equations for calculating interfacial drag and shear from void fraction correlations
International Nuclear Information System (INIS)
Putney, J.M.
1988-12-01
Equations are developed for calculating interfacial drag and shear coefficients for dispersed vapour flows from void fraction correlations. The equations have a sound physical basis and lead to physically correct coefficients in all flow situations. (author)
Existence of high-energy solutions for supercritical fractional Schrodinger equations in R^N
Directory of Open Access Journals (Sweden)
Lu Gan
2016-12-01
Full Text Available In this article, we study supercritical fractional Schr\\"odinger equations. Applying the finite-dimensional reduction method and the penalization method, we obtain the high-energy solutions for this equation.
Energy Technology Data Exchange (ETDEWEB)
Uchaikin, V V; Sibatov, R T, E-mail: vuchaikin@gmail.com, E-mail: ren_sib@bk.ru [Ulyanovsk State University, 432000, 42 Leo Tolstoy str., Ulyanovsk (Russian Federation)
2011-04-08
The fractional Boltzmann equation for resonance radiation transport in plasma is proposed. We start with the standard Boltzmann equation; averaging over photon frequencies leads to the appearance of a fractional derivative. This fact is in accordance with the conception of latent variables leading to hereditary and non-local dynamics (in particular, fractional dynamics). The presence of a fractional material derivative in the equation is concordant with heavy tailed distribution of photon path lengths and with spatiotemporal coupling peculiar to the process. We discuss some methods of solving the obtained equation and demonstrate numerical results in some simple cases.
International Nuclear Information System (INIS)
Uchaikin, V V; Sibatov, R T
2011-01-01
The fractional Boltzmann equation for resonance radiation transport in plasma is proposed. We start with the standard Boltzmann equation; averaging over photon frequencies leads to the appearance of a fractional derivative. This fact is in accordance with the conception of latent variables leading to hereditary and non-local dynamics (in particular, fractional dynamics). The presence of a fractional material derivative in the equation is concordant with heavy tailed distribution of photon path lengths and with spatiotemporal coupling peculiar to the process. We discuss some methods of solving the obtained equation and demonstrate numerical results in some simple cases.
Non-perturbative analytical solutions of the space- and time-fractional Burgers equations
International Nuclear Information System (INIS)
Momani, Shaher
2006-01-01
Non-perturbative analytical solutions for the generalized Burgers equation with time- and space-fractional derivatives of order α and β, 0 < α, β ≤ 1, are derived using Adomian decomposition method. The fractional derivatives are considered in the Caputo sense. The solutions are given in the form of series with easily computable terms. Numerical solutions are calculated for the fractional Burgers equation to show the nature of solution as the fractional derivative parameter is changed
New Approaches for Solving Fokker Planck Equation on Cantor Sets within Local Fractional Operators
Directory of Open Access Journals (Sweden)
Hassan Kamil Jassim
2015-01-01
Full Text Available We discuss new approaches to handling Fokker Planck equation on Cantor sets within local fractional operators by using the local fractional Laplace decomposition and Laplace variational iteration methods based on the local fractional calculus. The new approaches maintain the efficiency and accuracy of the analytical methods for solving local fractional differential equations. Illustrative examples are given to show the accuracy and reliable results.
An Efficient Implicit FEM Scheme for Fractional-in-Space Reaction-Diffusion Equations
Burrage, Kevin
2012-01-01
Fractional differential equations are becoming increasingly used as a modelling tool for processes associated with anomalous diffusion or spatial heterogeneity. However, the presence of a fractional differential operator causes memory (time fractional) or nonlocality (space fractional) issues that impose a number of computational constraints. In this paper we develop efficient, scalable techniques for solving fractional-in-space reaction diffusion equations using the finite element method on both structured and unstructured grids via robust techniques for computing the fractional power of a matrix times a vector. Our approach is show-cased by solving the fractional Fisher and fractional Allen-Cahn reaction-diffusion equations in two and three spatial dimensions, and analyzing the speed of the traveling wave and size of the interface in terms of the fractional power of the underlying Laplacian operator. © 2012 Society for Industrial and Applied Mathematics.
Regarding on the exact solutions for the nonlinear fractional differential equations
Directory of Open Access Journals (Sweden)
Kaplan Melike
2016-01-01
Full Text Available In this work, we have considered the modified simple equation (MSE method for obtaining exact solutions of nonlinear fractional-order differential equations. The space-time fractional equal width (EW and the modified equal width (mEW equation are considered for illustrating the effectiveness of the algorithm. It has been observed that all exact solutions obtained in this paper verify the nonlinear ordinary differential equations which was obtained from nonlinear fractional-order differential equations under the terms of wave transformation relationship. The obtained results are shown graphically.
Directory of Open Access Journals (Sweden)
Resat Yilmazer
2016-02-01
Full Text Available In this work; we present a method for solving the second-order linear ordinary differential equation of hypergeometric type. The solutions of this equation are given by the confluent hypergeometric functions (CHFs. Unlike previous studies, we obtain some different new solutions of the equation without using the CHFs. Therefore, we obtain new discrete fractional solutions of the homogeneous and non-homogeneous confluent hypergeometric differential equation (CHE by using a discrete fractional Nabla calculus operator. Thus, we obtain four different new discrete complex fractional solutions for these equations.
Maxwell’s Equations on Cantor Sets: A Local Fractional Approach
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Yang Zhao
2013-01-01
Full Text Available Maxwell’s equations on Cantor sets are derived from the local fractional vector calculus. It is shown that Maxwell’s equations on Cantor sets in a fractal bounded domain give efficiency and accuracy for describing the fractal electric and magnetic fields. Local fractional differential forms of Maxwell’s equations on Cantor sets in the Cantorian and Cantor-type cylindrical coordinates are obtained. Maxwell's equations on Cantor set with local fractional operators are the first step towards a unified theory of Maxwell’s equations for the dynamics of cold dark matter.
Stability analysis of a class of fractional delay differential equations
Indian Academy of Sciences (India)
Abstract. In this paper we analyse stability of nonlinear fractional order delay differential equa- tions of the form Dα y(t) = af (y(t − τ )) − by(t), where Dα is a Caputo fractional derivative of order 0 < α ≤ 1. We describe stability regions using critical curves. To explain the proposed theory, we discuss fractional order logistic ...
International Nuclear Information System (INIS)
Zhang, Xiao; Wei, Chaozhen; Liu, Yingming; Luo, Maokang
2014-01-01
In this paper we use Dirac function to construct a fractional operator called fractional corresponding operator, which is the general form of momentum corresponding operator. Then we give a judging theorem for this operator and with this judging theorem we prove that R–L, G–L, Caputo, Riesz fractional derivative operator and fractional derivative operator based on generalized functions, which are the most popular ones, coincide with the fractional corresponding operator. As a typical application, we use the fractional corresponding operator to construct a new fractional quantization scheme and then derive a uniform fractional Schrödinger equation in form. Additionally, we find that the five forms of fractional Schrödinger equation belong to the particular cases. As another main result of this paper, we use fractional corresponding operator to generalize fractional quantization scheme by using Lévy path integral and use it to derive the corresponding general form of fractional Schrödinger equation, which consequently proves that these two quantization schemes are equivalent. Meanwhile, relations between the theory in fractional quantum mechanics and that in classic quantum mechanics are also discussed. As a physical example, we consider a particle in an infinite potential well. We give its wave functions and energy spectrums in two ways and find that both results are the same
Rusyaman, E.; Parmikanti, K.; Chaerani, D.; Asefan; Irianingsih, I.
2018-03-01
One of the application of fractional ordinary differential equation is related to the viscoelasticity, i.e., a correlation between the viscosity of fluids and the elasticity of solids. If the solution function develops into function with two or more variables, then its differential equation must be changed into fractional partial differential equation. As the preliminary study for two variables viscoelasticity problem, this paper discusses about convergence analysis of function sequence which is the solution of the homogenous fractional partial differential equation. The method used to solve the problem is Homotopy Analysis Method. The results show that if given two real number sequences (αn) and (βn) which converge to α and β respectively, then the solution function sequences of fractional partial differential equation with order (αn, βn) will also converge to the solution function of fractional partial differential equation with order (α, β).
Directory of Open Access Journals (Sweden)
A. A. Hemeda
2013-01-01
Full Text Available An extension of the so-called new iterative method (NIM has been used to handle linear and nonlinear fractional partial differential equations. The main property of the method lies in its flexibility and ability to solve nonlinear equations accurately and conveniently. Therefore, a general framework of the NIM is presented for analytical treatment of fractional partial differential equations in fluid mechanics. The fractional derivatives are described in the Caputo sense. Numerical illustrations that include the fractional wave equation, fractional Burgers equation, fractional KdV equation, fractional Klein-Gordon equation, and fractional Boussinesq-like equation are investigated to show the pertinent features of the technique. Comparison of the results obtained by the NIM with those obtained by both Adomian decomposition method (ADM and the variational iteration method (VIM reveals that the NIM is very effective and convenient. The basic idea described in this paper is expected to be further employed to solve other similar linear and nonlinear problems in fractional calculus.
Bright and dark soliton solutions for some nonlinear fractional differential equations
International Nuclear Information System (INIS)
Guner, Ozkan; Bekir, Ahmet
2016-01-01
In this work, we propose a new approach, namely ansatz method, for solving fractional differential equations based on a fractional complex transform and apply it to the nonlinear partial space–time fractional modified Benjamin–Bona–Mahoney (mBBM) equation, the time fractional mKdV equation and the nonlinear fractional Zoomeron equation which gives rise to some new exact solutions. The physical parameters in the soliton solutions: amplitude, inverse width, free parameters and velocity are obtained as functions of the dependent model coefficients. This method is suitable and more powerful for solving other kinds of nonlinear fractional PDEs arising in mathematical physics. Since the fractional derivatives are described in the modified Riemann–Liouville sense. (paper)
Jain, Sonal
2018-01-01
In this paper, we aim to use the alternative numerical scheme given by Gnitchogna and Atangana for solving partial differential equations with integer and non-integer differential operators. We applied this method to fractional diffusion model and fractional Buckmaster models with non-local fading memory. The method yields a powerful numerical algorithm for fractional order derivative to implement. Also we present in detail the stability analysis of the numerical method for solving the diffusion equation. This proof shows that this method is very stable and also converges very quickly to exact solution and finally some numerical simulation is presented.
Directory of Open Access Journals (Sweden)
Waleed M. Abd-Elhameed
2016-09-01
Full Text Available Herein, two numerical algorithms for solving some linear and nonlinear fractional-order differential equations are presented and analyzed. For this purpose, a novel operational matrix of fractional-order derivatives of Fibonacci polynomials was constructed and employed along with the application of the tau and collocation spectral methods. The convergence and error analysis of the suggested Fibonacci expansion were carefully investigated. Some numerical examples with comparisons are presented to ensure the efficiency, applicability and high accuracy of the proposed algorithms. Two accurate semi-analytic polynomial solutions for linear and nonlinear fractional differential equations are the result.
High Performance Computing for Solving Fractional Differential Equations with Applications
Zhang, Wei
2014-01-01
Fractional calculus is the generalization of integer-order calculus to rational order. This subject has at least three hundred years of history. However, it was traditionally regarded as a pure mathematical field and lacked real world applications for a very long time. In recent decades, fractional calculus has re-attracted the attention of scientists and engineers. For example, many researchers have found that fractional calculus is a useful tool for describing hereditary materials and p...
Directory of Open Access Journals (Sweden)
Chen Jie-Dong
2016-01-01
Full Text Available In this paper, we investigate the local fractional Laplace equation in the steady heat-conduction problem. The solutions involving the non-differentiable graph are obtained by using the characteristic equation method (CEM via local fractional derivative. The obtained results are given to present the accuracy of the technology to solve the steady heat-conduction in fractal media.
Directory of Open Access Journals (Sweden)
Ai-Min Yang
2014-01-01
Full Text Available We use the local fractional series expansion method to solve the Klein-Gordon equations on Cantor sets within the local fractional derivatives. The analytical solutions within the nondifferential terms are discussed. The obtained results show the simplicity and efficiency of the present technique with application to the problems of the liner differential equations on Cantor sets.
Local Fractional Series Expansion Method for Solving Wave and Diffusion Equations on Cantor Sets
Directory of Open Access Journals (Sweden)
Ai-Min Yang
2013-01-01
Full Text Available We proposed a local fractional series expansion method to solve the wave and diffusion equations on Cantor sets. Some examples are given to illustrate the efficiency and accuracy of the proposed method to obtain analytical solutions to differential equations within the local fractional derivatives.
Solving Fokker-Planck Equations on Cantor Sets Using Local Fractional Decomposition Method
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Shao-Hong Yan
2014-01-01
Full Text Available The local fractional decomposition method is applied to approximate the solutions for Fokker-Planck equations on Cantor sets with local fractional derivative. The obtained results give the present method that is very effective and simple for solving the differential equations on Cantor set.
The fractional coupled KdV equations: Exact solutions and white noise functional approach
International Nuclear Information System (INIS)
Ghany, Hossam A.; El Bab, A. S. Okb; Zabel, A. M.; Hyder, Abd-Allah
2013-01-01
Variable coefficients and Wick-type stochastic fractional coupled KdV equations are investigated. By using the modified fractional sub-equation method, Hermite transform, and white noise theory the exact travelling wave solutions and white noise functional solutions are obtained, including the generalized exponential, hyperbolic, and trigonometric types. (general)
Approximate solution of integro-differential equation of fractional (arbitrary order
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Asma A. Elbeleze
2016-01-01
Full Text Available In the present paper, we study the integro-differential equations which are combination of differential and Fredholm–Volterra equations that have the fractional order with constant coefficients by the homotopy perturbation and the variational iteration. The fractional derivatives are described in Caputo sense. Some illustrative examples are presented.
EXISTENCE AND UNIQUENESS OF SOLUTIONS TO A NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION
Institute of Scientific and Technical Information of China (English)
无
2011-01-01
The initial value problem of a nonlinear fractional differential equation is discussed in this paper. Using the nonlinear alternative of Leray-Schauder type and the contraction mapping principle,we obtain the existence and uniqueness of solutions to the fractional differential equation,which extend some results of the previous papers.
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Hossein Jafari
2016-04-01
Full Text Available The non-differentiable solution of the linear and non-linear partial differential equations on Cantor sets is implemented in this article. The reduced differential transform method is considered in the local fractional operator sense. The four illustrative examples are given to show the efficiency and accuracy features of the presented technique to solve local fractional partial differential equations.
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Erkinjon Karimov
2017-10-01
Full Text Available In this work we discuss higher order multi-term partial differential equation (PDE with the Caputo-Fabrizio fractional derivative in time. Using method of separation of variables, we reduce fractional order partial differential equation to the integer order. We represent explicit solution of formulated problem in particular case by Fourier series.
Existence of smooth solutions of multi-term Caputo-type fractional differential equations
Sin, Chung-Sik; Cheng, Shusen; Ri, Gang-Il; Kim, Mun-Chol
2017-01-01
This paper deals with the initial value problem for the multi-term fractional differential equation. The fractional derivative is defined in the Caputo sense. Firstly the initial value problem is transformed into a equivalent Volterra-type integral equation under appropriate assumptions. Then new existence results for smooth solutions are established by using the Schauder fixed point theorem.
Erkinjon Karimov; Sardor Pirnafasov
2017-01-01
In this work we discuss higher order multi-term partial differential equation (PDE) with the Caputo-Fabrizio fractional derivative in time. Using method of separation of variables, we reduce fractional order partial differential equation to the integer order. We represent explicit solution of formulated problem in particular case by Fourier series.
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Wu Guo-Cheng
2012-01-01
Full Text Available This note presents a Laplace transform approach in the determination of the Lagrange multiplier when the variational iteration method is applied to time fractional heat diffusion equation. The presented approach is more straightforward and allows some simplification in application of the variational iteration method to fractional differential equations, thus improving the convergence of the successive iterations.
Another higher order Langevin algorithm for QCD
International Nuclear Information System (INIS)
Kronfeld, A.S.
1986-01-01
This note provides an algorithm for integrating the Langevin equation which is second order. It introduces a term into the drift force which is a product of the Gaussian noise and a second derivative of the action. The specific application presented here is for nonabelian gauge theories interacting with fermions, e.g. QCD, for which it requires less memory than the Runge-Kutta algorithm of the same order. The memory and computational requirements of Euler, Runge-Kutta, and the present algorithm are compared. (orig.)
International Nuclear Information System (INIS)
Aslan İsmail
2014-01-01
The extended simplest equation method is used to solve exactly a new differential-difference equation of fractional-type, proposed by Narita [J. Math. Anal. Appl. 381 (2011) 963] quite recently, related to the discrete MKdV equation. It is shown that the model supports three types of exact solutions with arbitrary parameters: hyperbolic, trigonometric and rational, which have not been reported before. (general)
Lie symmetry analysis and conservation laws for the time fractional fourth-order evolution equation
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Wang Li
2017-06-01
Full Text Available In this paper, we study Lie symmetry analysis and conservation laws for the time fractional nonlinear fourth-order evolution equation. Using the method of Lie point symmetry, we provide the associated vector fields, and derive the similarity reductions of the equation, respectively. The method can be applied wisely and efficiently to get the reduced fractional ordinary differential equations based on the similarity reductions. Finally, by using the nonlinear self-adjointness method and Riemann-Liouville time-fractional derivative operator as well as Euler-Lagrange operator, the conservation laws of the equation are obtained.
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Ji Juan-Juan
2017-01-01
Full Text Available A table lookup method for solving nonlinear fractional partial differential equations (fPDEs is proposed in this paper. Looking up the corresponding tables, we can quickly obtain the exact analytical solutions of fPDEs by using this method. To illustrate the validity of the method, we apply it to construct the exact analytical solutions of four nonlinear fPDEs, namely, the time fractional simplified MCH equation, the space-time fractional combined KdV-mKdV equation, the (2+1-dimensional time fractional Zoomeron equation, and the space-time fractional ZKBBM equation. As a result, many new types of exact analytical solutions are obtained including triangular periodic solution, hyperbolic function solution, singular solution, multiple solitary wave solution, and Jacobi elliptic function solution.
The Klein–Gordon–Zakharov equations with the positive fractional ...
Indian Academy of Sciences (India)
2016-11-03
Nov 3, 2016 ... Gan et al [12,13] studied the instability of standing waves for KGZ equations. Linear stability analysis for periodic travelling waves of KGZ equa- tions are performed in refs [14,15]. In refs [16–19], finite difference schemes are proposed for the initial- boundary problem of the KGZ equations. The rest of the ...
New Iterative Method for Fractional Gas Dynamics and Coupled Burger’s Equations
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Mohamed S. Al-luhaibi
2015-01-01
Full Text Available This paper presents the approximate analytical solutions to solve the nonlinear gas dynamics and coupled Burger’s equations with fractional time derivative. By using initial values, the explicit solutions of the equations are solved by using a reliable algorithm. Numerical results show that the new iterative method is easy to implement and accurate when applied to time-fractional partial differential equations.
Ford, Neville J.; Connolly, Joseph A.
2009-07-01
We give a comparison of the efficiency of three alternative decomposition schemes for the approximate solution of multi-term fractional differential equations using the Caputo form of the fractional derivative. The schemes we compare are based on conversion of the original problem into a system of equations. We review alternative approaches and consider how the most appropriate numerical scheme may be chosen to solve a particular equation.
Estimates for a general fractional relaxation equation and application to an inverse source problem
Bazhlekova, Emilia
2018-01-01
A general fractional relaxation equation is considered with a convolutional derivative in time introduced by A. Kochubei (Integr. Equ. Oper. Theory 71 (2011), 583-600). This equation generalizes the single-term, multi-term and distributed-order fractional relaxation equations. The fundamental and the impulse-response solutions are studied in detail. Properties such as analyticity and subordination identities are established and employed in the proof of an upper and a lower bound. The obtained...
Investigation of the Dirac Equation by Using the Conformable Fractional Derivative
Mozaffari, F. S.; Hassanabadi, H.; Sobhani, H.; Chung, W. S.
2018-05-01
In this paper,the Dirac equation is constructed using the conformable fractional derivative so that in its limit for the fractional parameter, the normal version is recovered. Then, the Cornell potential is considered as the interaction of the system. In this case, the wave function and the energy eigenvalue equation are derived with the aim of the bi-confluent Heun functions. use of the conformable fractional derivative is proven to lead to a branching treatment for the energy of the system. Such a treatment is obvious for small values of the fractional parameter, and a united value as the fractional parameter approaches unity.
NUMERICAL METHODS FOR SOLVING THE MULTI-TERM TIME-FRACTIONAL WAVE-DIFFUSION EQUATION.
Liu, F; Meerschaert, M M; McGough, R J; Zhuang, P; Liu, Q
2013-03-01
In this paper, the multi-term time-fractional wave-diffusion equations are considered. The multi-term time fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0,1], [1,2), [0,2), [0,3), [2,3) and [2,4), respectively. Some computationally effective numerical methods are proposed for simulating the multi-term time-fractional wave-diffusion equations. The numerical results demonstrate the effectiveness of theoretical analysis. These methods and techniques can also be extended to other kinds of the multi-term fractional time-space models with fractional Laplacian.
Hosseini, Kamyar; Mayeli, Peyman; Bekir, Ahmet; Guner, Ozkan
2018-01-01
In this article, a special type of fractional differential equations (FDEs) named the density-dependent conformable fractional diffusion-reaction (DDCFDR) equation is studied. Aforementioned equation has a significant role in the modelling of some phenomena arising in the applied science. The well-organized methods, including the \\exp (-φ (\\varepsilon )) -expansion and modified Kudryashov methods are exerted to generate the exact solutions of this equation such that some of the solutions are new and have been reported for the first time. Results illustrate that both methods have a great performance in handling the DDCFDR equation.
Differential-difference equations associated with the fractional Lax operators
Energy Technology Data Exchange (ETDEWEB)
Adler, V E [LD Landau Institute for Theoretical Physics, 1A Ak. Semenov, Chernogolovka 142432 (Russian Federation); Postnikov, V V, E-mail: adler@itp.ac.ru, E-mail: postnikofvv@mail.ru [Sochi Branch of Peoples' Friendship University of Russia, 32 Kuibyshev str., 354000 Sochi (Russian Federation)
2011-10-14
We study integrable hierarchies associated with spectral problems of the form P{psi} = {lambda}Q{psi}, where P and Q are difference operators. The corresponding nonlinear differential-difference equations can be viewed as inhomogeneous generalizations of the Bogoyavlensky-type lattices. While the latter turn into the Korteweg-de Vries equation under the continuous limit, the lattices under consideration provide discrete analogs of the Sawada-Kotera and Kaup-Kupershmidt equations. The r-matrix formulation and several of the simplest explicit solutions are presented. (paper)
The modified simple equation method for solving some fractional ...
Indian Academy of Sciences (India)
2016-06-21
Jun 21, 2016 ... and AHMET BEKIR. Art-Science Faculty, Department of Mathematics-Computer, Eskisehir Osmangazi University, Eskisehir, Turkey ... chemistry, biology, engineering and in numerous other applications. ... fractional calculus.
Wavelet Methods for Solving Fractional Order Differential Equations
A. K. Gupta; S. Saha Ray
2014-01-01
Fractional calculus is a field of applied mathematics which deals with derivatives and integrals of arbitrary orders. The fractional calculus has gained considerable importance during the past decades mainly due to its application in diverse fields of science and engineering such as viscoelasticity, diffusion of biological population, signal processing, electromagnetism, fluid mechanics, electrochemistry, and many more. In this paper, we review different wavelet methods for solving both linea...
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Taohua Liu
2017-01-01
Full Text Available Fractional advection-dispersion equations, as generalizations of classical integer-order advection-dispersion equations, are used to model the transport of passive tracers carried by fluid flow in a porous medium. In this paper, we develop an implicit finite difference method for fractional advection-dispersion equations with fractional derivative boundary conditions. First-order consistency, solvability, unconditional stability, and first-order convergence of the method are proven. Then, we present a fast iterative method for the implicit finite difference scheme, which only requires storage of O(K and computational cost of O(KlogK. Traditionally, the Gaussian elimination method requires storage of O(K2 and computational cost of O(K3. Finally, the accuracy and efficiency of the method are checked with a numerical example.
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Mostafa M.A. Khater
Full Text Available In this article and for the first time, we introduce and describe Khater method which is a new technique for solving nonlinear partial differential equations (PDEs.. We apply this method for each of the following models Bogoyavlenskii equation, couple Boiti-Leon-Pempinelli system and Time-fractional Cahn-Allen equation. Khater method is very powerful, Effective, felicitous and fabulous method to get exact and solitary wave solution of (PDEs.. Not only just like that but it considers too one of the general methods for solving that kind of equations since it involves some methods as we will see in our discuss of the results. We make a comparison between the results of this new method and another method. Keywords: Bogoyavlenskii equations system, Couple Boiti-Leon-Pempinelli equations system, Time-fractional Cahn-Allen equation, Khater method, Traveling wave solutions, Solitary wave solutions
Ding, Xiao-Li; Nieto, Juan J.
2017-11-01
In this paper, we consider the analytical solutions of coupling fractional partial differential equations (FPDEs) with Dirichlet boundary conditions on a finite domain. Firstly, the method of successive approximations is used to obtain the analytical solutions of coupling multi-term time fractional ordinary differential equations. Then, the technique of spectral representation of the fractional Laplacian operator is used to convert the coupling FPDEs to the coupling multi-term time fractional ordinary differential equations. By applying the obtained analytical solutions to the resulting multi-term time fractional ordinary differential equations, the desired analytical solutions of the coupling FPDEs are given. Our results are applied to derive the analytical solutions of some special cases to demonstrate their applicability.
Analysis of a time fractional wave-like equation with the homotopy analysis method
International Nuclear Information System (INIS)
Xu Hang; Cang Jie
2008-01-01
The time fractional wave-like differential equation with a variable coefficient is studied analytically. By using a simple transformation, the governing equation is reduced to two fractional ordinary differential equations. Then the homotopy analysis method is employed to derive the solutions of these equations. The accurate series solutions are obtained. Especially, when h f =h g =-1, these solutions are exactly the same as those results given by the Adomian decomposition method. The present work shows the validity and great potential of the homotopy analysis method for solving nonlinear fractional differential equations. The basic idea described in this Letter is expected to be further employed to solve other similar nonlinear problems in fractional calculus
Zúñiga-Aguilar, C. J.; Coronel-Escamilla, A.; Gómez-Aguilar, J. F.; Alvarado-Martínez, V. M.; Romero-Ugalde, H. M.
2018-02-01
In this paper, we approximate the solution of fractional differential equations with delay using a new approach based on artificial neural networks. We consider fractional differential equations of variable order with the Mittag-Leffler kernel in the Liouville-Caputo sense. With this new neural network approach, an approximate solution of the fractional delay differential equation is obtained. Synaptic weights are optimized using the Levenberg-Marquardt algorithm. The neural network effectiveness and applicability were validated by solving different types of fractional delay differential equations, linear systems with delay, nonlinear systems with delay and a system of differential equations, for instance, the Newton-Leipnik oscillator. The solution of the neural network was compared with the analytical solutions and the numerical simulations obtained through the Adams-Bashforth-Moulton method. To show the effectiveness of the proposed neural network, different performance indices were calculated.
Conservation laws for certain time fractional nonlinear systems of partial differential equations
Singla, Komal; Gupta, R. K.
2017-12-01
In this study, an extension of the concept of nonlinear self-adjointness and Noether operators is proposed for calculating conserved vectors of the time fractional nonlinear systems of partial differential equations. In our recent work (J Math Phys 2016; 57: 101504), by proposing the symmetry approach for time fractional systems, the Lie symmetries for some fractional nonlinear systems have been derived. In this paper, the obtained infinitesimal generators are used to find conservation laws for the corresponding fractional systems.
International Nuclear Information System (INIS)
Li Xicheng; Xu Mingyu; Wang Shaowei
2008-01-01
In this paper, we give similarity solutions of partial differential equations of fractional order with a moving boundary condition. The solutions are given in terms of a generalized Wright function. The time-fractional Caputo derivative and two types of space-fractional derivatives are considered. The scale-invariant variable and the form of the solution of the moving boundary are obtained by the Lie group analysis. A comparison between the solutions corresponding to two types of fractional derivative is also given
Exact solutions of fractional Schroedinger-like equation with a nonlocal term
International Nuclear Information System (INIS)
Jiang Xiaoyun; Xu Mingyu; Qi Haitao
2011-01-01
We study the time-space fractional Schroedinger equation with a nonlocal potential. By the method of Fourier transform and Laplace transform, the Green function, and hence the wave function, is expressed in terms of H-functions. Graphical analysis demonstrates that the influence of both the space-fractal parameter α and the nonlocal parameter ν on the fractional quantum system is strong. Indeed, the nonlocal potential may act similar to a fractional spatial derivative as well as fractional time derivative.
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Fukang Yin
2013-01-01
Full Text Available A numerical method is presented to obtain the approximate solutions of the fractional partial differential equations (FPDEs. The basic idea of this method is to achieve the approximate solutions in a generalized expansion form of two-dimensional fractional-order Legendre functions (2D-FLFs. The operational matrices of integration and derivative for 2D-FLFs are first derived. Then, by these matrices, a system of algebraic equations is obtained from FPDEs. Hence, by solving this system, the unknown 2D-FLFs coefficients can be computed. Three examples are discussed to demonstrate the validity and applicability of the proposed method.
Introduction to fractional and pseudo-differential equations with singular symbols
Umarov, Sabir
2015-01-01
The book systematically presents the theories of pseudo-differential operators with symbols singular in dual variables, fractional order derivatives, distributed and variable order fractional derivatives, random walk approximants, and applications of these theories to various initial and multi-point boundary value problems for pseudo-differential equations. Fractional Fokker-Planck-Kolmogorov equations associated with a large class of stochastic processes are presented. A complex version of the theory of pseudo-differential operators with meromorphic symbols based on the recently introduced complex Fourier transform is developed and applied for initial and boundary value problems for systems of complex differential and pseudo-differential equations.
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Özkan Güner
2014-01-01
Full Text Available We apply the functional variable method, exp-function method, and (G′/G-expansion method to establish the exact solutions of the nonlinear fractional partial differential equation (NLFPDE in the sense of the modified Riemann-Liouville derivative. As a result, some new exact solutions for them are obtained. The results show that these methods are very effective and powerful mathematical tools for solving nonlinear fractional equations arising in mathematical physics. As a result, these methods can also be applied to other nonlinear fractional differential equations.
Existence of solutions to boundary value problem of fractional differential equations with impulsive
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Weihua JIANG
2016-12-01
Full Text Available In order to solve the boundary value problem of fractional impulsive differential equations with countable impulses and integral boundary conditions on the half line, the existence of solutions to the boundary problem is specifically studied. By defining suitable Banach spaces, norms and operators, using the properties of fractional calculus and applying the contraction mapping principle and Krasnoselskii's fixed point theorem, the existence of solutions for the boundary value problem of fractional impulsive differential equations with countable impulses and integral boundary conditions on the half line is proved, and examples are given to illustrate the existence of solutions to this kind of equation boundary value problems.
Periodic Solutions, Eigenvalue Curves, and Degeneracy of the Fractional Mathieu Equation
International Nuclear Information System (INIS)
Parra-Hinojosa, A; Gutiérrez-Vega, J C
2016-01-01
We investigate the eigenvalue curves, the behavior of the periodic solutions, and the orthogonality properties of the Mathieu equation with an additional fractional derivative term using the method of harmonic balance. The addition of the fractional derivative term breaks the hermiticity of the equation in such a way that its eigenvalues need not be real nor its eigenfunctions orthogonal. We show that for a certain choice of parameters the eigenvalue curves reveal the appearance of degenerate eigenvalues. We offer a detailed discussion of the matrix representation of the differential operator corresponding to the fractional Mathieu equation, as well as some numerical examples of its periodic solutions. (paper)
The Oscillation of a Class of the Fractional-Order Delay Differential Equations
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Qianli Lu
2014-01-01
Full Text Available Several oscillation results are proposed including necessary and sufficient conditions for the oscillation of fractional-order delay differential equations with constant coefficients, the sufficient or necessary and sufficient conditions for the oscillation of fractional-order delay differential equations by analysis method, and the sufficient or necessary and sufficient conditions for the oscillation of delay partial differential equation with three different boundary conditions. For this, α-exponential function which is a kind of functions that play the same role of the classical exponential functions of fractional-order derivatives is used.
A Novel Method for Analytical Solutions of Fractional Partial Differential Equations
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Mehmet Ali Akinlar
2013-01-01
Full Text Available A new solution technique for analytical solutions of fractional partial differential equations (FPDEs is presented. The solutions are expressed as a finite sum of a vector type functional. By employing MAPLE software, it is shown that the solutions might be extended to an arbitrary degree which makes the present method not only different from the others in the literature but also quite efficient. The method is applied to special Bagley-Torvik and Diethelm fractional differential equations as well as a more general fractional differential equation.
Computational Challenge of Fractional Differential Equations and the Potential Solutions: A Survey
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Chunye Gong
2015-01-01
Full Text Available We present a survey of fractional differential equations and in particular of the computational cost for their numerical solutions from the view of computer science. The computational complexities of time fractional, space fractional, and space-time fractional equations are O(N2M, O(NM2, and O(NM(M + N compared with O(MN for the classical partial differential equations with finite difference methods, where M, N are the number of space grid points and time steps. The potential solutions for this challenge include, but are not limited to, parallel computing, memory access optimization (fractional precomputing operator, short memory principle, fast Fourier transform (FFT based solutions, alternating direction implicit method, multigrid method, and preconditioner technology. The relationships of these solutions for both space fractional derivative and time fractional derivative are discussed. The authors pointed out that the technologies of parallel computing should be regarded as a basic method to overcome this challenge, and some attention should be paid to the fractional killer applications, high performance iteration methods, high order schemes, and Monte Carlo methods. Since the computation of fractional equations with high dimension and variable order is even heavier, the researchers from the area of mathematics and computer science have opportunity to invent cornerstones in the area of fractional calculus.
Chen, Shanzhen; Jiang, Xiaoyun
2012-08-01
In this paper, analytical solutions to time-fractional partial differential equations in a multi-layer annulus are presented. The final solutions are obtained in terms of Mittag-Leffler function by using the finite integral transform technique and Laplace transform technique. In addition, the classical diffusion equation (α=1), the Helmholtz equation (α→0) and the wave equation (α=2) are discussed as special cases. Finally, an illustrative example problem for the three-layer semi-circular annular region is solved and numerical results are presented graphically for various kind of order of fractional derivative.
A fractional spline collocation-Galerkin method for the time-fractional diffusion equation
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Pezza L.
2018-03-01
Full Text Available The aim of this paper is to numerically solve a diffusion differential problem having time derivative of fractional order. To this end we propose a collocation-Galerkin method that uses the fractional splines as approximating functions. The main advantage is in that the derivatives of integer and fractional order of the fractional splines can be expressed in a closed form that involves just the generalized finite difference operator. This allows us to construct an accurate and efficient numerical method. Several numerical tests showing the effectiveness of the proposed method are presented.
International Nuclear Information System (INIS)
Saha Ray, S.; Patra, A.
2012-01-01
Highlights: ► In this paper fractional neutron point kinetic equation has been analyzed. ► The numerical solution for fractional neutron point kinetic equation is obtained. ► Explicit Finite Difference Method has been applied. ► Supercritical reactivity, critical reactivity and subcritical reactivity analyzed. ► Comparison between fractional and classical neutron density is presented. - Abstract: In the present article, a numerical procedure to efficiently calculate the solution for fractional point kinetics equation in nuclear reactor dynamics is investigated. The Explicit Finite Difference Method is applied to solve the fractional neutron point kinetic equation with the Grunwald–Letnikov (GL) definition (). Fractional Neutron Point Kinetic Model has been analyzed for the dynamic behavior of the neutron motion in which the relaxation time associated with a variation in the neutron flux involves a fractional order acting as exponent of the relaxation time, to obtain the best operation of a nuclear reactor dynamics. Results for neutron dynamic behavior for subcritical reactivity, supercritical reactivity and critical reactivity and also for different values of fractional order have been presented and compared with the classical neutron point kinetic (NPK) equation as well as the results obtained by the learned researchers .
Estimation of Supercapacitor Energy Storage Based on Fractional Differential Equations.
Kopka, Ryszard
2017-12-22
In this paper, new results on using only voltage measurements on supercapacitor terminals for estimation of accumulated energy are presented. For this purpose, a study based on application of fractional-order models of supercapacitor charging/discharging circuits is undertaken. Parameter estimates of the models are then used to assess the amount of the energy accumulated in supercapacitor. The obtained results are compared with energy determined experimentally by measuring voltage and current on supercapacitor terminals. All the tests are repeated for various input signal shapes and parameters. Very high consistency between estimated and experimental results fully confirm suitability of the proposed approach and thus applicability of the fractional calculus to modelling of supercapacitor energy storage.
Multiple Positive Solutions for Nonlinear Semipositone Fractional Differential Equations
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Wen-Xue Zhou
2012-01-01
Full Text Available We present some new multiplicity of positive solutions results for nonlinear semipositone fractional boundary value problem D0+αu(t=p(tf(t,u(t-q(t,0
Stochastic fractional differential equations: Modeling, method and analysis
International Nuclear Information System (INIS)
Pedjeu, Jean-C.; Ladde, Gangaram S.
2012-01-01
By introducing a concept of dynamic process operating under multi-time scales in sciences and engineering, a mathematical model described by a system of multi-time scale stochastic differential equations is formulated. The classical Picard–Lindelöf successive approximations scheme is applied to the model validation problem, namely, existence and uniqueness of solution process. Naturally, this leads to the problem of finding closed form solutions of both linear and nonlinear multi-time scale stochastic differential equations of Itô–Doob type. Finally, to illustrate the scope of ideas and presented results, multi-time scale stochastic models for ecological and epidemiological processes in population dynamic are outlined.
Analytical approach to linear fractional partial differential equations arising in fluid mechanics
International Nuclear Information System (INIS)
Momani, Shaher; Odibat, Zaid
2006-01-01
In this Letter, we implement relatively new analytical techniques, the variational iteration method and the Adomian decomposition method, for solving linear fractional partial differential equations arising in fluid mechanics. The fractional derivatives are described in the Caputo sense. The two methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for different types of fractional differential equations. In these methods, the solution takes the form of a convergent series with easily computable components. The corresponding solutions of the integer order equations are found to follow as special cases of those of fractional order equations. Some numerical examples are presented to illustrate the efficiency and reliability of the two methods
Directory of Open Access Journals (Sweden)
Guotao Wang
2012-01-01
Full Text Available We study nonlinear impulsive differential equations of fractional order with irregular boundary conditions. Some existence and uniqueness results are obtained by applying standard fixed-point theorems. For illustration of the results, some examples are discussed.
Direct and inverse source problems for a space fractional advection dispersion equation
Aldoghaither, Abeer; Laleg-Kirati, Taous-Meriem; Liu, Da Yan
2016-01-01
In this paper, direct and inverse problems for a space fractional advection dispersion equation on a finite domain are studied. The inverse problem consists in determining the source term from final observations. We first derive the analytic
Numerical solution for multi-term fractional (arbitrary) orders differential equations
El-Sayed, A. M. A.; El-Mesiry, A. E. M.; El-Saka, H. A. A.
2004-01-01
Our main concern here is to give a numerical scheme to solve a nonlinear multi-term fractional (arbitrary) orders differential equation. Some results concerning the existence and uniqueness have been also obtained.
Stable multi-domain spectral penalty methods for fractional partial differential equations
Xu, Qinwu; Hesthaven, Jan S.
2014-01-01
We propose stable multi-domain spectral penalty methods suitable for solving fractional partial differential equations with fractional derivatives of any order. First, a high order discretization is proposed to approximate fractional derivatives of any order on any given grids based on orthogonal polynomials. The approximation order is analyzed and verified through numerical examples. Based on the discrete fractional derivative, we introduce stable multi-domain spectral penalty methods for solving fractional advection and diffusion equations. The equations are discretized in each sub-domain separately and the global schemes are obtained by weakly imposed boundary and interface conditions through a penalty term. Stability of the schemes are analyzed and numerical examples based on both uniform and nonuniform grids are considered to highlight the flexibility and high accuracy of the proposed schemes.
On Generalized Fractional Kinetic Equations Involving Generalized Bessel Function of the First Kind
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Dinesh Kumar
2015-01-01
Full Text Available We develop a new and further generalized form of the fractional kinetic equation involving generalized Bessel function of the first kind. The manifold generality of the generalized Bessel function of the first kind is discussed in terms of the solution of the fractional kinetic equation in the paper. The results obtained here are quite general in nature and capable of yielding a very large number of known and (presumably new results.
On the Asymptotic Behavior of Positive Solutions of Certain Fractional Differential Equations
Said R. Grace
2015-01-01
This paper deals with the asymptotic behavior of positive solutions of certain forced fractional differential equations of the form DcαCyt=et+ft, xt, c>1, α∈0,1, where yt=atx′t′, c0=y(c)/Γ(1) =yc, and c0 is a real constant. From the obtained results, we derive a technique which can be applied to some related fractional differential equations.
A Novel Method for Analytical Solutions of Fractional Partial Differential Equations
Mehmet Ali Akinlar; Muhammet Kurulay
2013-01-01
A new solution technique for analytical solutions of fractional partial differential equations (FPDEs) is presented. The solutions are expressed as a finite sum of a vector type functional. By employing MAPLE software, it is shown that the solutions might be extended to an arbitrary degree which makes the present method not only different from the others in the literature but also quite efficient. The method is applied to special Bagley-Torvik and Diethelm fractional differential equations as...
Numerical solutions of multi-order fractional differential equations by Boubaker polynomials
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Bolandtalat A.
2016-01-01
Full Text Available In this paper, we have applied a numerical method based on Boubaker polynomials to obtain approximate numerical solutions of multi-order fractional differential equations. We obtain an operational matrix of fractional integration based on Boubaker polynomials. Using this operational matrix, the given problem is converted into a set of algebraic equations. Illustrative examples are are given to demonstrate the efficiency and simplicity of this technique.
Yaşar, Elif; Yıldırım, Yakup; Yaşar, Emrullah
2018-06-01
This paper devotes to conformable fractional space-time perturbed Gerdjikov-Ivanov (GI) equation which appears in nonlinear fiber optics and photonic crystal fibers (PCF). We consider the model with full nonlinearity in order to give a generalized flavor. The sine-Gordon equation approach is carried out to model equation for retrieving the dark, bright, dark-bright, singular and combined singular optical solitons. The constraint conditions are also reported for guaranteeing the existence of these solitons. We also present some graphical simulations of the solutions for better understanding the physical phenomena of the behind the considered model.
Generalized Langevin Equation Description of Stochastic ...
Indian Academy of Sciences (India)
dissipation theorems. The vertical ... luminosity is also obtained, and it is shown that it has non-standard values. The theoretical .... black hole (a), for A = 0.0015 and a = 0.03 (red curve), a = 0.1 (green curve) and a = 0.001. (black curve) ...
On solutions of nonlinear time-space fractional Swift–Hohenberg equation: A comparative study
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Najeeb Alam Khan
2014-03-01
Full Text Available In this paper, a comparison for the solutions of nonlinear Swift–Hohenberg equation with time-space fractional derivatives has been analyzed. The two most promising techniques, fractional variational iteration method (FVIM and the homotopy analysis method have been chosen for the comparison. The two different definitions of fractional calculus are considered to solve time-fractional derivative separately for the considered approaches. Also, the space fractional derivative is described in the Reisz sense. Analytical and numerical solutions for various combinations of the parameters are obtained. Numerical comparisons have been made for different values of parameters and depicted.
International Nuclear Information System (INIS)
Sandev, D. Trivche
2010-01-01
The fractional calculus basis, Mittag-Leffler functions, various relaxation-oscillation and diffusion-wave fractional order equation and systems of fractional order equations are considered in this thesis. To solve these fractional order equations analytical methods, such as the Laplace transform method and method of separation of variables are employed. Some applications of the fractional calculus are considered, particularly physical system with anomalous diffusive behavior. (Author)
The usage of Maxwell fractional equations for the investigation of the waveguide processes
International Nuclear Information System (INIS)
Maksyuta, M.V.; Slinchenko, Yu.A.; Grygoruk, V.I.
2016-01-01
By means of nabla operator written down with using both of some differential operators with integer orders and fractional differential Caputo operators, gradient, divergence and rotor operators are determined, it is checked up the fulfillment of vector relations in fractional vector analysis, fractional Green, Stocks and Ostrogradsky-Gauss formulas. For a specific expression of nabla operator (nabla components along x and y axes have a unit order and along z axis, correspondingly, a fractional value in the interval from zero till unit) Maxwell fractional equations are written down. Based on the following from them some fractional wave equations, dissipative and polarization processes at electromagnetic waves distribution both in rectangular (planar) and in cylindrical waveguide structures are analyzed.
Solving Nonlinear Fractional Differential Equation by Generalized Mittag-Leffler Function Method
Arafa, A. A. M.; Rida, S. Z.; Mohammadein, A. A.; Ali, H. M.
2013-06-01
In this paper, we use Mittag—Leffler function method for solving some nonlinear fractional differential equations. A new solution is constructed in power series. The fractional derivatives are described by Caputo's sense. To illustrate the reliability of the method, some examples are provided.
The Laplace series solution for local fractional Korteweg-de Vries equation
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Ye Shan-Shan
2016-01-01
Full Text Available In this paper, we consider a new application of the local fractional Laplace series expansion method to handle the local fractional Korteweg-de Vries equation. The obtained solution with non-differentiable type shows that the technology is accurate and efficient.
Non-probabilistic solutions of imprecisely defined fractional-order diffusion equations
International Nuclear Information System (INIS)
Chakraverty, S.; Tapaswini, Smita
2014-01-01
The fractional diffusion equation is one of the most important partial differential equations (PDEs) to model problems in mathematical physics. These PDEs are more practical when those are combined with uncertainties. Accordingly, this paper investigates the numerical solution of a non-probabilistic viz. fuzzy fractional-order diffusion equation subjected to various external forces. A fuzzy diffusion equation having fractional order 0 < α ≤ 1 with fuzzy initial condition is taken into consideration. Fuzziness appearing in the initial conditions is modelled through convex normalized triangular and Gaussian fuzzy numbers. A new computational technique is proposed based on double parametric form of fuzzy numbers to handle the fuzzy fractional diffusion equation. Using the single parametric form of fuzzy numbers, the original fuzzy diffusion equation is converted first into an interval-based fuzzy differential equation. Next, this equation is transformed into crisp form by using the proposed double parametric form of fuzzy numbers. Finally, the same is solved by Adomian decomposition method (ADM) symbolically to obtain the uncertain bounds of the solution. Computed results are depicted in terms of plots. Results obtained by the proposed method are compared with the existing results in special cases. (general)
A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations
Diethelm, Kai; Ford, Neville J.; Freed, Alan D.; Gray, Hugh R. (Technical Monitor)
2002-01-01
We discuss an Adams-type predictor-corrector method for the numerical solution of fractional differential equations. The method may be used both for linear and for nonlinear problems, and it may be extended to multi-term equations (involving more than one differential operator) too.
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Lihong Zhang
2017-11-01
Full Text Available In this article, a family of nonlinear diffusion equations involving multi-term Caputo-Fabrizio time fractional derivative is investigated. Some maximum principles are obtained. We also demonstrate the application of the obtained results by deriving some estimation for solution to reaction-diffusion equations.
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Hasan Bulut
2013-01-01
Full Text Available We introduce the rudiments of fractional calculus and the consequent applications of the Sumudu transform on fractional derivatives. Once this connection is firmly established in the general setting, we turn to the application of the Sumudu transform method (STM to some interesting nonhomogeneous fractional ordinary differential equations (FODEs. Finally, we use the solutions to form two-dimensional (2D graphs, by using the symbolic algebra package Mathematica Program 7.
Moving-boundary problems for the time-fractional diffusion equation
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Sabrina D. Roscani
2017-02-01
Full Text Available We consider a one-dimensional moving-boundary problem for the time-fractional diffusion equation. The time-fractional derivative of order $\\alpha\\in (0,1$ is taken in the sense of Caputo. We study the asymptotic behaivor, as t tends to infinity, of a general solution by using a fractional weak maximum principle. Also, we give some particular exact solutions in terms of Wright functions.
Homotopy decomposition method for solving one-dimensional time-fractional diffusion equation
Abuasad, Salah; Hashim, Ishak
2018-04-01
In this paper, we present the homotopy decomposition method with a modified definition of beta fractional derivative for the first time to find exact solution of one-dimensional time-fractional diffusion equation. In this method, the solution takes the form of a convergent series with easily computable terms. The exact solution obtained by the proposed method is compared with the exact solution obtained by using fractional variational homotopy perturbation iteration method via a modified Riemann-Liouville derivative.
A Fully Discrete Galerkin Method for a Nonlinear Space-Fractional Diffusion Equation
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Yunying Zheng
2011-01-01
Full Text Available The spatial transport process in fractal media is generally anomalous. The space-fractional advection-diffusion equation can be used to characterize such a process. In this paper, a fully discrete scheme is given for a type of nonlinear space-fractional anomalous advection-diffusion equation. In the spatial direction, we use the finite element method, and in the temporal direction, we use the modified Crank-Nicolson approximation. Here the fractional derivative indicates the Caputo derivative. The error estimate for the fully discrete scheme is derived. And the numerical examples are also included which are in line with the theoretical analysis.
Residual power series method for fractional Sharma-Tasso-Olever equation
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Amit Kumar
2016-02-01
Full Text Available In this paper, we introduce a modified analytical approximate technique to obtain solution of time fractional Sharma-Tasso-Olever equation. First, we present an alternative framework of the Residual power series method (RPSM which can be used simply and effectively to handle nonlinear fractional differential equations arising in several physical phenomena. This method is basically based on the generalized Taylor series formula and residual error function. A good result is found between our solution and the given solution. It is shown that the proposed method is reliable, efficient and easy to implement on all kinds of fractional nonlinear problems arising in science and technology.
NUMERICAL METHODS FOR SOLVING THE MULTI-TERM TIME-FRACTIONAL WAVE-DIFFUSION EQUATION
Liu, F.; Meerschaert, M.M.; McGough, R.J.; Zhuang, P.; Liu, Q.
2013-01-01
In this paper, the multi-term time-fractional wave-diffusion equations are considered. The multi-term time fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0,1], [1,2), [0,2), [0,3), [2,3) and [2,4), respectively. Some computationally effective numerical methods are proposed for simulating the multi-term time-fractional wave-diffusion equations. The numerical results demonstrate the effectiveness of theoretical analysis. These methods and technique...
Analytical Solution for Fractional Derivative Gas-Flow Equation in Porous Media
El-Amin, Mohamed; Radwan, Ahmed G.; Sun, Shuyu
2017-01-01
In this paper, we introduce an analytical solution of the fractional derivative gas transport equation using the power-series technique. We present a new universal transform, namely, generalized Boltzmann change of variable which depends on the fractional order, time and space. This universal transform is employed to transfer the partial differential equation into an ordinary differential equation. Moreover, the convergence of the solution has been investigated and found that solutions are unconditionally converged. Results are introduced and discussed for the universal variable and other physical parameters such as porosity and permeability of the reservoir; time and space.
Stability of Equilibrium Points of Fractional Difference Equations with Stochastic Perturbations
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Shaikhet Leonid
2008-01-01
Full Text Available It is supposed that the fractional difference equation , has an equilibrium point and is exposed to additive stochastic perturbations type of that are directly proportional to the deviation of the system state from the equilibrium point . It is shown that known results in the theory of stability of stochastic difference equations that were obtained via V. Kolmanovskii and L. Shaikhet general method of Lyapunov functionals construction can be successfully used for getting of sufficient conditions for stability in probability of equilibrium points of the considered stochastic fractional difference equation. Numerous graphical illustrations of stability regions and trajectories of solutions are plotted.
Analytical Solution for Fractional Derivative Gas-Flow Equation in Porous Media
El-Amin, Mohamed
2017-07-06
In this paper, we introduce an analytical solution of the fractional derivative gas transport equation using the power-series technique. We present a new universal transform, namely, generalized Boltzmann change of variable which depends on the fractional order, time and space. This universal transform is employed to transfer the partial differential equation into an ordinary differential equation. Moreover, the convergence of the solution has been investigated and found that solutions are unconditionally converged. Results are introduced and discussed for the universal variable and other physical parameters such as porosity and permeability of the reservoir; time and space.
Agarwal, P.; El-Sayed, A. A.
2018-06-01
In this paper, a new numerical technique for solving the fractional order diffusion equation is introduced. This technique basically depends on the Non-Standard finite difference method (NSFD) and Chebyshev collocation method, where the fractional derivatives are described in terms of the Caputo sense. The Chebyshev collocation method with the (NSFD) method is used to convert the problem into a system of algebraic equations. These equations solved numerically using Newton's iteration method. The applicability, reliability, and efficiency of the presented technique are demonstrated through some given numerical examples.
Solving Abel’s Type Integral Equation with Mikusinski’s Operator of Fractional Order
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Ming Li
2013-01-01
Full Text Available This paper gives a novel explanation of the integral equation of Abel’s type from the point of view of Mikusinski’s operational calculus. The concept of the inverse of Mikusinski’s operator of fractional order is introduced for constructing a representation of the solution to the integral equation of Abel’s type. The proof of the existence of the inverse of the fractional Mikusinski operator is presented, providing an alternative method of treating the integral equation of Abel’s type.
New analytical exact solutions of time fractional KdV-KZK equation by Kudryashov methods
S Saha, Ray
2016-04-01
In this paper, new exact solutions of the time fractional KdV-Khokhlov-Zabolotskaya-Kuznetsov (KdV-KZK) equation are obtained by the classical Kudryashov method and modified Kudryashov method respectively. For this purpose, the modified Riemann-Liouville derivative is used to convert the nonlinear time fractional KdV-KZK equation into the nonlinear ordinary differential equation. In the present analysis, the classical Kudryashov method and modified Kudryashov method are both used successively to compute the analytical solutions of the time fractional KdV-KZK equation. As a result, new exact solutions involving the symmetrical Fibonacci function, hyperbolic function and exponential function are obtained for the first time. The methods under consideration are reliable and efficient, and can be used as an alternative to establish new exact solutions of different types of fractional differential equations arising from mathematical physics. The obtained results are exhibited graphically in order to demonstrate the efficiencies and applicabilities of these proposed methods of solving the nonlinear time fractional KdV-KZK equation.
New analytical exact solutions of time fractional KdV–KZK equation by Kudryashov methods
International Nuclear Information System (INIS)
Saha Ray, S
2016-01-01
In this paper, new exact solutions of the time fractional KdV–Khokhlov–Zabolotskaya–Kuznetsov (KdV–KZK) equation are obtained by the classical Kudryashov method and modified Kudryashov method respectively. For this purpose, the modified Riemann–Liouville derivative is used to convert the nonlinear time fractional KdV–KZK equation into the nonlinear ordinary differential equation. In the present analysis, the classical Kudryashov method and modified Kudryashov method are both used successively to compute the analytical solutions of the time fractional KdV–KZK equation. As a result, new exact solutions involving the symmetrical Fibonacci function, hyperbolic function and exponential function are obtained for the first time. The methods under consideration are reliable and efficient, and can be used as an alternative to establish new exact solutions of different types of fractional differential equations arising from mathematical physics. The obtained results are exhibited graphically in order to demonstrate the efficiencies and applicabilities of these proposed methods of solving the nonlinear time fractional KdV–KZK equation. (paper)
Khater method for nonlinear Sharma Tasso-Olever (STO) equation of fractional order
Bibi, Sadaf; Mohyud-Din, Syed Tauseef; Khan, Umar; Ahmed, Naveed
In this work, we have implemented a direct method, known as Khater method to establish exact solutions of nonlinear partial differential equations of fractional order. Number of solutions provided by this method is greater than other traditional methods. Exact solutions of nonlinear fractional order Sharma Tasso-Olever (STO) equation are expressed in terms of kink, travelling wave, periodic and solitary wave solutions. Modified Riemann-Liouville derivative and Fractional complex transform have been used for compatibility with fractional order sense. Solutions have been graphically simulated for understanding the physical aspects and importance of the method. A comparative discussion between our established results and the results obtained by existing ones is also presented. Our results clearly reveal that the proposed method is an effective, powerful and straightforward technique to work out new solutions of various types of differential equations of non-integer order in the fields of applied sciences and engineering.
Cuahutenango-Barro, B.; Taneco-Hernández, M. A.; Gómez-Aguilar, J. F.
2017-12-01
Analytical solutions of the wave equation with bi-fractional-order and frictional memory kernel of Mittag-Leffler type are obtained via Caputo-Fabrizio fractional derivative in the Liouville-Caputo sense. Through the method of separation of variables and Laplace transform method we derive closed-form solutions and establish fundamental solutions. Special cases with homogeneous Dirichlet boundary conditions and nonhomogeneous initial conditions, as well as for the external force are considered. Numerical simulations of the special solutions were done and novel behaviors are obtained.
Xing, Yanyuan; Yan, Yubin
2018-03-01
Gao et al. [11] (2014) introduced a numerical scheme to approximate the Caputo fractional derivative with the convergence rate O (k 3 - α), 0 equation is sufficiently smooth, Lv and Xu [20] (2016) proved by using energy method that the corresponding numerical method for solving time fractional partial differential equation has the convergence rate O (k 3 - α), 0 equation has low regularity and in this case the numerical method fails to have the convergence rate O (k 3 - α), 0 quadratic interpolation polynomials. Based on this scheme, we introduce a time discretization scheme to approximate the time fractional partial differential equation and show by using Laplace transform methods that the time discretization scheme has the convergence rate O (k 3 - α), 0 0 for smooth and nonsmooth data in both homogeneous and inhomogeneous cases. Numerical examples are given to show that the theoretical results are consistent with the numerical results.
Langevin approach to synchronization of hyperchaotic time-delay dynamics
Energy Technology Data Exchange (ETDEWEB)
Budini, Adrian A [Consejo Nacional de Investigaciones CientIficas y Tecnicas, Centro Atomico Bariloche, Av. E Bustillo Km 9.5, (8400) Bariloche (Argentina); Consortium of the Americas for Interdisciplinary Science and Department of Physics and Astronomy, University of New Mexico, Albuquerque, NM 87131 (United States)
2008-11-07
In this paper, we characterize the synchronization phenomenon of hyperchaotic scalar nonlinear delay dynamics in a fully-developed chaos regime. Our results rely on the observation that, in that regime, the stationary statistical properties of a class of hyperchaotic attractors can be reproduced with a linear Langevin equation, defined by replacing the nonlinear delay force by a delta-correlated noise. Therefore, the synchronization phenomenon can be analytically characterized by a set of coupled Langevin equations. We apply this formalism to study anticipated synchronization dynamics subject to external noise fluctuations as well as for characterizing the effects of parameter mismatch in a hyperchaotic communication scheme. The same procedure is applied to second-order differential delay equations associated with synchronization in electro-optical devices. In all cases, the departure with respect to perfect synchronization is measured through a similarity function. Numerical simulations in discrete maps associated with the hyperchaotic dynamics support the formalism.
High-order fractional partial differential equation transform for molecular surface construction.
Hu, Langhua; Chen, Duan; Wei, Guo-Wei
2013-01-01
Fractional derivative or fractional calculus plays a significant role in theoretical modeling of scientific and engineering problems. However, only relatively low order fractional derivatives are used at present. In general, it is not obvious what role a high fractional derivative can play and how to make use of arbitrarily high-order fractional derivatives. This work introduces arbitrarily high-order fractional partial differential equations (PDEs) to describe fractional hyperdiffusions. The fractional PDEs are constructed via fractional variational principle. A fast fractional Fourier transform (FFFT) is proposed to numerically integrate the high-order fractional PDEs so as to avoid stringent stability constraints in solving high-order evolution PDEs. The proposed high-order fractional PDEs are applied to the surface generation of proteins. We first validate the proposed method with a variety of test examples in two and three-dimensional settings. The impact of high-order fractional derivatives to surface analysis is examined. We also construct fractional PDE transform based on arbitrarily high-order fractional PDEs. We demonstrate that the use of arbitrarily high-order derivatives gives rise to time-frequency localization, the control of the spectral distribution, and the regulation of the spatial resolution in the fractional PDE transform. Consequently, the fractional PDE transform enables the mode decomposition of images, signals, and surfaces. The effect of the propagation time on the quality of resulting molecular surfaces is also studied. Computational efficiency of the present surface generation method is compared with the MSMS approach in Cartesian representation. We further validate the present method by examining some benchmark indicators of macromolecular surfaces, i.e., surface area, surface enclosed volume, surface electrostatic potential and solvation free energy. Extensive numerical experiments and comparison with an established surface model
Baleanu, Dumitru; Inc, Mustafa; Yusuf, Abdullahi; Aliyu, Aliyu Isa
2018-06-01
In this work, we investigate the Lie symmetry analysis, exact solutions and conservation laws (Cls) to the time fractional Caudrey-Dodd-Gibbon-Sawada-Kotera (CDGDK) equation with Riemann-Liouville (RL) derivative. The time fractional CDGDK is reduced to nonlinear ordinary differential equation (ODE) of fractional order. New exact traveling wave solutions for the time fractional CDGDK are obtained by fractional sub-equation method. In the reduced equation, the derivative is in Erdelyi-Kober (EK) sense. Ibragimov's nonlocal conservation method is applied to construct Cls for time fractional CDGDK.
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Ahmad Bashir
2010-01-01
Full Text Available We study an initial value problem for a coupled Caputo type nonlinear fractional differential system of higher order. As a first problem, the nonhomogeneous terms in the coupled fractional differential system depend on the fractional derivatives of lower orders only. Then the nonhomogeneous terms in the fractional differential system are allowed to depend on the unknown functions together with the fractional derivative of lower orders. Our method of analysis is based on the reduction of the given system to an equivalent system of integral equations. Applying the nonlinear alternative of Leray-Schauder, we prove the existence of solutions of the fractional differential system. The uniqueness of solutions of the fractional differential system is established by using the Banach contraction principle. An illustrative example is also presented.
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Ahmet Bekir
2014-09-01
Full Text Available In this paper, the fractional partial differential equations are defined by modified Riemann–Liouville fractional derivative. With the help of fractional derivative and traveling wave transformation, these equations can be converted into the nonlinear nonfractional ordinary differential equations. Then G′G-expansion method is applied to obtain exact solutions of the space-time fractional Burgers equation, the space-time fractional KdV-Burgers equation and the space-time fractional coupled Burgers’ equations. As a result, many exact solutions are obtained including hyperbolic function solutions, trigonometric function solutions and rational solutions. These results reveal that the proposed method is very effective and simple in performing a solution to the fractional partial differential equation.
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Mourad Kerboua
2014-12-01
Full Text Available We introduce a new notion called fractional stochastic nonlocal condition, and then we study approximate controllability of class of fractional stochastic nonlinear differential equations of Sobolev type in Hilbert spaces. We use Hölder's inequality, fixed point technique, fractional calculus, stochastic analysis and methods adopted directly from deterministic control problems for the main results. A new set of sufficient conditions is formulated and proved for the fractional stochastic control system to be approximately controllable. An example is given to illustrate the abstract results.
Fractional diffusion equation with distributed-order material derivative. Stochastic foundations
International Nuclear Information System (INIS)
Magdziarz, M; Teuerle, M
2017-01-01
In this paper, we present the stochastic foundations of fractional dynamics driven by the fractional material derivative of distributed-order type. Before stating our main result, we present the stochastic scenario which underlies the dynamics given by the fractional material derivative. Then we introduce the Lévy walk process of distributed-order type to establish our main result, which is the scaling limit of the considered process. It appears that the probability density function of the scaling limit process fulfills, in a weak sense, the fractional diffusion equation with the material derivative of distributed-order type. (paper)
Lorenzo, C F; Hartley, T T; Malti, R
2013-05-13
A new and simplified method for the solution of linear constant coefficient fractional differential equations of any commensurate order is presented. The solutions are based on the R-function and on specialized Laplace transform pairs derived from the principal fractional meta-trigonometric functions. The new method simplifies the solution of such fractional differential equations and presents the solutions in the form of real functions as opposed to fractional complex exponential functions, and thus is directly applicable to real-world physics.
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Emrullah Yaşar
Full Text Available In this paper Lie symmetry analysis of the seventh-order time fractional Sawada–Kotera–Ito (FSKI equation with Riemann–Liouville derivative is performed. Using the Lie point symmetries of FSKI equation, it is shown that it can be transformed into a nonlinear ordinary differential equation of fractional order with a new dependent variable. In the reduced equation the derivative is in Erdelyi–Kober sense. Furthermore, adapting the Ibragimov’s nonlocal conservation method to time fractional partial differential equations, we obtain conservation laws of the underlying equation. In addition, we construct some exact travelling wave solutions for the FSKI equation using the sub-equation method. Keywords: Fractional Sawada–Kotera–Ito equation, Lie symmetry, Riemann–Liouville fractional derivative, Conservation laws, Exact solutions
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J. Prakash
2016-03-01
Full Text Available In this paper, a numerical algorithm based on a modified He-Laplace method (MHLM is proposed to solve space and time nonlinear fractional differential-difference equations (NFDDEs arising in physical phenomena such as wave phenomena in fluids, coupled nonlinear optical waveguides and nanotechnology fields. The modified He-Laplace method is a combined form of the fractional homotopy perturbation method and Laplace transforms method. The nonlinear terms can be easily decomposed by the use of He’s polynomials. This algorithm has been tested against time-fractional differential-difference equations such as the modified Lotka Volterra and discrete (modified KdV equations. The proposed scheme grants the solution in the form of a rapidly convergent series. Three examples have been employed to illustrate the preciseness and effectiveness of the proposed method. The achieved results expose that the MHLM is very accurate, efficient, simple and can be applied to other nonlinear FDDEs.
A multi-domain spectral method for time-fractional differential equations
Chen, Feng; Xu, Qinwu; Hesthaven, Jan S.
2015-07-01
This paper proposes an approach for high-order time integration within a multi-domain setting for time-fractional differential equations. Since the kernel is singular or nearly singular, two main difficulties arise after the domain decomposition: how to properly account for the history/memory part and how to perform the integration accurately. To address these issues, we propose a novel hybrid approach for the numerical integration based on the combination of three-term-recurrence relations of Jacobi polynomials and high-order Gauss quadrature. The different approximations used in the hybrid approach are justified theoretically and through numerical examples. Based on this, we propose a new multi-domain spectral method for high-order accurate time integrations and study its stability properties by identifying the method as a generalized linear method. Numerical experiments confirm hp-convergence for both time-fractional differential equations and time-fractional partial differential equations.
Ahmed, Naveed; Bibi, Sadaf; Khan, Umar; Mohyud-Din, Syed Tauseef
2018-02-01
We have modified the traditional exponential rational function method (ERFM) and have used it to find the exact solutions of two different fractional partial differential equations, one is the time fractional Boussinesq equation and the other is the (2+1)-dimensional time fractional Zoomeron equation. In both the cases it is observed that the modified scheme provides more types of solutions than the traditional one. Moreover, a comparison of the recent solutions is made with some already existing solutions. We can confidently conclude that the modified scheme works better and provides more types of solutions with almost similar computational cost. Our generalized solutions include periodic, soliton-like, singular soliton and kink solutions. A graphical simulation of all types of solutions is provided and the correctness of the solution is verified by direct substitution. The extended version of the solutions is expected to provide more flexibility to scientists working in the relevant field to test their simulation data.
Some properties of the fractional Ornstein-Uhlenbeck process
International Nuclear Information System (INIS)
Yan Litan; Lu Yunsheng; Xu Zhiqiang
2008-01-01
We consider the fractional analogue of the Ornstein-Uhlenbeck process, i.e. the solution of the Langevin equation driven by a fractional Brownian motion in place of the usual Brownian motion. We establish some properties of these processes. We show that the process is local nondeterminism. For a two-dimensional process we show that its renormalized self-intersection local time exists in L 2 if and only if 0< H<3/4
Parareal algorithms with local time-integrators for time fractional differential equations
Wu, Shu-Lin; Zhou, Tao
2018-04-01
It is challenge work to design parareal algorithms for time-fractional differential equations due to the historical effect of the fractional operator. A direct extension of the classical parareal method to such equations will lead to unbalance computational time in each process. In this work, we present an efficient parareal iteration scheme to overcome this issue, by adopting two recently developed local time-integrators for time fractional operators. In both approaches, one introduces auxiliary variables to localized the fractional operator. To this end, we propose a new strategy to perform the coarse grid correction so that the auxiliary variables and the solution variable are corrected separately in a mixed pattern. It is shown that the proposed parareal algorithm admits robust rate of convergence. Numerical examples are presented to support our conclusions.
Arqub, Omar Abu; El-Ajou, Ahmad; Momani, Shaher
2015-07-01
Building fractional mathematical models for specific phenomena and developing numerical or analytical solutions for these fractional mathematical models are crucial issues in mathematics, physics, and engineering. In this work, a new analytical technique for constructing and predicting solitary pattern solutions of time-fractional dispersive partial differential equations is proposed based on the generalized Taylor series formula and residual error function. The new approach provides solutions in the form of a rapidly convergent series with easily computable components using symbolic computation software. For method evaluation and validation, the proposed technique was applied to three different models and compared with some of the well-known methods. The resultant simulations clearly demonstrate the superiority and potentiality of the proposed technique in terms of the quality performance and accuracy of substructure preservation in the construct, as well as the prediction of solitary pattern solutions for time-fractional dispersive partial differential equations.
General solution of the Bagley-Torvik equation with fractional-order derivative
Wang, Z. H.; Wang, X.
2010-05-01
This paper investigates the general solution of the Bagley-Torvik equation with 1/2-order derivative or 3/2-order derivative. This fractional-order differential equation is changed into a sequential fractional-order differential equation (SFDE) with constant coefficients. Then the general solution of the SFDE is expressed as the linear combination of fundamental solutions that are in terms of α-exponential functions, a kind of functions that play the same role of the classical exponential function. Because the number of fundamental solutions of the SFDE is greater than 2, the general solution of the SFDE depends on more than two free (independent) constants. This paper shows that the general solution of the Bagley-Torvik equation involves actually two free constants only, and it can be determined fully by the initial displacement and initial velocity.
Energy Technology Data Exchange (ETDEWEB)
Saha Ray, S., E-mail: santanusaharay@yahoo.com; Patra, A.
2014-10-15
Highlights: • A stationary transport equation has been solved using the technique of Haar wavelet collocation method. • This paper intends to provide the great utility of Haar wavelets to nuclear science problem. • In the present paper, two-dimensional Haar wavelets are applied. • The proposed method is mathematically very simple, easy and fast. - Abstract: In this paper the numerical solution for the fractional order stationary neutron transport equation is presented using Haar wavelet Collocation Method (HWCM). Haar wavelet collocation method is efficient and powerful in solving wide class of linear and nonlinear differential equations. This paper intends to provide an application of Haar wavelets to nuclear science problems. This paper describes the application of Haar wavelets for the numerical solution of fractional order stationary neutron transport equation in homogeneous medium with isotropic scattering. The proposed method is mathematically very simple, easy and fast. To demonstrate about the efficiency and applicability of the method, two test problems are discussed.
A New Numerical Technique for Solving Systems Of Nonlinear Fractional Partial Differential Equations
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Mountassir Hamdi Cherif
2017-11-01
Full Text Available In this paper, we apply an efficient method called the Aboodh decomposition method to solve systems of nonlinear fractional partial differential equations. This method is a combined form of Aboodh transform with Adomian decomposition method. The theoretical analysis of this investigated for systems of nonlinear fractional partial differential equations is calculated in the explicit form of a power series with easily computable terms. Some examples are given to shows that this method is very efficient and accurate. This method can be applied to solve others nonlinear systems problems.
Multiple positive solutions to a coupled systems of nonlinear fractional differential equations.
Shah, Kamal; Khan, Rahmat Ali
2016-01-01
In this article, we study existence, uniqueness and nonexistence of positive solution to a highly nonlinear coupled system of fractional order differential equations. Necessary and sufficient conditions for the existence and uniqueness of positive solution are developed by using Perov's fixed point theorem for the considered problem. Further, we also established sufficient conditions for existence of multiplicity results for positive solutions. Also, we developed some conditions under which the considered coupled system of fractional order differential equations has no positive solution. Appropriate examples are also provided which demonstrate our results.
Khader, M M
2013-10-01
In this paper, an efficient numerical method for solving the fractional delay differential equations (FDDEs) is considered. The fractional derivative is described in the Caputo sense. The proposed method is based on the derived approximate formula of the Laguerre polynomials. The properties of Laguerre polynomials are utilized to reduce FDDEs to a linear or nonlinear system of algebraic equations. Special attention is given to study the error and the convergence analysis of the proposed method. Several numerical examples are provided to confirm that the proposed method is in excellent agreement with the exact solution.
On a system of differential equations with fractional derivatives arising in rod theory
International Nuclear Information System (INIS)
Atanackovic, Teodor M; Stankovic, Bogoljub
2004-01-01
We study a system of equations with fractional derivatives, that arises in the analysis of the lateral motion of an elastic column fixed at one end and loaded by a concentrated follower force at the other end. We assume that the column is positioned on a viscoelastic foundation described by a constitutive equation of fractional derivative type. The stability boundary is determined. It is shown that as in the case of an elastic (Winkler) type of foundation the stability boundary remains the same as for the column without a foundation! Thus, with the solution analysed here, the column exhibits the so-called Hermann-Smith paradox
International Nuclear Information System (INIS)
Basak, K C; Ray, P C; Bera, R K
2009-01-01
The aim of the present analysis is to apply the Adomian decomposition method and He's variational method for the approximate analytical solution of a nonlinear ordinary fractional differential equation. The solutions obtained by the above two methods have been numerically evaluated and presented in the form of tables and also compared with the exact solution. It was found that the results obtained by the above two methods are in excellent agreement with the exact solution. Finally, a surface plot of the approximate solutions of the fractional differential equation by the above two methods is drawn for 0≤t≤2 and 1<α≤2.
Analytical Solutions of Fractional Differential Equations Using the Convenient Adomian Series
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Xiang-Chao Shi
2014-01-01
Full Text Available Due to the memory trait of the fractional calculus, numerical or analytical solution of higher order becomes very difficult even impossible to obtain in real engineering problems. Recently, a new and convenient way was suggested to calculate the Adomian series and the higher order approximation was realized. In this paper, the Adomian decomposition method is applied to nonlinear fractional differential equation and the error analysis is given which shows the convenience.
Periodic Solutions and S-Asymptotically Periodic Solutions to Fractional Evolution Equations
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Jia Mu
2017-01-01
Full Text Available This paper deals with the existence and uniqueness of periodic solutions, S-asymptotically periodic solutions, and other types of bounded solutions for some fractional evolution equations with the Weyl-Liouville fractional derivative defined for periodic functions. Applying Fourier transform we give reasonable definitions of mild solutions. Then we accurately estimate the spectral radius of resolvent operator and obtain some existence and uniqueness results.
A Simple Map Between Fokker-Planck Equation and its Fractional form
International Nuclear Information System (INIS)
Zahran, M.A.; El-Shewy, E.K.
2008-01-01
A simple map between Fokker-Planck Equation (FPE) and its fractional form (FFPE), which recently formulates to describe sub diffusive processes, has been suggested. This connection based on a relation between k-orders for moments of ordinary time domain of FPE and the moments associated with fractional time domain of FFPE . Two classes of special interest of FFPE has been considered to outline this map
Ulam stability for fractional differential equations in the sense of Caputo operator
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Rabha W. Ibrahim
2012-12-01
Full Text Available In this paper, we consider the Hyers-Ulam stability for the following fractional differential equations, in the sense ofcomplex Caputo fractional derivative defined, in the unit disk: cDßzf(z=G(f(z, cDázf(z,zf‘(z;z 0<á<1<ß<2 . Furthermore,a generalization of the admissible functions in complex Banach spaces is imposed and applications are illustrated.
An analytic algorithm for the space-time fractional reaction-diffusion equation
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M. G. Brikaa
2015-11-01
Full Text Available In this paper, we solve the space-time fractional reaction-diffusion equation by the fractional homotopy analysis method. Solutions of different examples of the reaction term will be computed and investigated. The approximation solutions of the studied models will be put in the form of convergent series to be easily computed and simulated. Comparison with the approximation solution of the classical case of the studied modeled with their approximation errors will also be studied.
International Nuclear Information System (INIS)
Raslan, K. R.; Ali, Khalid K.; EL-Danaf, Talaat S.
2017-01-01
In the present paper, we established a traveling wave solution by using modified Kudryashov method for the space-time fractional nonlinear partial differential equations. The method is used to obtain the exact solutions for different types of the space-time fractional nonlinear partial differential equations such as, the space-time fractional coupled equal width wave equation (CEWE) and the space-time fractional coupled modified equal width wave equation (CMEW), which are the important soliton equations. Both equations are reduced to ordinary differential equations by the use of fractional complex transform and properties of modified Riemann–Liouville derivative. We plot the exact solutions for these equations at different time levels. (paper)
2018-03-14
UNIVERSITY OF TECHNOLOGY Final Report 03/14/2018 DISTRIBUTION A: Distribution approved for public release. AF Office Of Scientific Research (AFOSR...optimal control problems involving fractional-order differential equations Wang, Song Curtin University of Technology Kent Street, Bentley WA6102...Article history : Received 3 October 2016 Accepted 26 March 2017 Available online 29 April 2017 Keywords: Hamilton–Jacobi–Bellman equation Financial
Langevin theory of anomalous Brownian motion made simple
International Nuclear Information System (INIS)
Tothova, Jana; Vasziova, Gabriela; Lisy, VladimIr; Glod, Lukas
2011-01-01
During the century from the publication of the work by Einstein (1905 Ann. Phys. 17 549) Brownian motion has become an important paradigm in many fields of modern science. An essential impulse for the development of Brownian motion theory was given by the work of Langevin (1908 C. R. Acad. Sci., Paris 146 530), in which he proposed an 'infinitely more simple' description of Brownian motion than that by Einstein. The original Langevin approach has however strong limitations, which were rigorously stated after the creation of the hydrodynamic theory of Brownian motion (1945). Hydrodynamic Brownian motion is a special case of 'anomalous Brownian motion', now intensively studied both theoretically and in experiments. We show how some general properties of anomalous Brownian motion can be easily derived using an effective method that allows one to convert the stochastic generalized Langevin equation into a deterministic Volterra-type integro-differential equation for the mean square displacement of the particle. Within the Gibbs statistics, the method is applicable to linear equations of motion with any kind of memory during the evolution of the system. We apply it to memoryless Brownian motion in a harmonic potential well and to Brownian motion in fluids, taking into account the effects of hydrodynamic memory. Exploring the mathematical analogy between Brownian motion and electric circuits, which are at nanoscales also described by the generalized Langevin equation, we calculate the fluctuations of charge and current in RLC circuits that are in contact with the thermal bath. Due to the simplicity of our approach it could be incorporated into graduate courses of statistical physics. Once the method is established, it allows bringing to the attention of students and effectively solving a number of attractive problems related to Brownian motion.
Langevin diffusions on the torus
DEFF Research Database (Denmark)
García-Portugués, Eduardo; Sørensen, Michael; Mardia, Kanti V.
2018-01-01
We introduce stochastic models for continuous-time evolution of angles and develop their estimation. We focus on studying Langevin diffusions with stationary distributions equal to well-known distributions from directional statistics, since such diffusions can be regarded as toroidal analogues......) a likelihood based on the stationary distribution; (ii) toroidal adaptations of the Euler and Shoji–Ozaki pseudo-likelihoods; (iii) a likelihood based on a specific approximation to the transition density of the wrapped normal process. A simulation study compares, in dimensions one and two, the approximate...
From hard thermal loops to Langevin dynamics
International Nuclear Information System (INIS)
Boedeker, Dietrich
1999-01-01
In hot non-Abelian gauge theories, processes characterized by the momentum scale g 2 T (such as electroweak baryon number violation in the very early universe) are non-perturbative. An effective theory for the soft (vertical bar p vertical bar ∼ g 2 T) field modes is obtained by integrating out momenta larger than than g 2 T. Starting from the hard thermal loop effective theory, which is the result of integrating out the scale T, it is shown how to integrate out the scale gT in an expansion in the gauge coupling g. At leading order in g, one obtains Vlasov-Boltzmann equations for the soft field modes, which contain a Gaussian noise and a collision term. The 2-point function of the noise and the collision term are explicitly calculated in a leading logarithmic approximation. In this approximation the Boltzmann equation is solved. The resulting effective theory for the soft field modes is described by a Langevin equation. It determines the parametric form of the hot baryon number violation rate as Γ = κg 10 log(1/g)gT 4 , and it allows for a calculation for κ on the lattice
International Nuclear Information System (INIS)
Yang, Xiao-Jun; Srivastava, H.M.; He, Ji-Huan; Baleanu, Dumitru
2013-01-01
In this Letter, we propose to use the Cantor-type cylindrical-coordinate method in order to investigate a family of local fractional differential operators on Cantor sets. Some testing examples are given to illustrate the capability of the proposed method for the heat-conduction equation on a Cantor set and the damped wave equation in fractal strings. It is seen to be a powerful tool to convert differential equations on Cantor sets from Cantorian-coordinate systems to Cantor-type cylindrical-coordinate systems.
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Muhammad Asif Zahoor Raja
2011-01-01
Full Text Available A stochastic technique has been developed for the solution of fractional order system represented by Bagley-Torvik equation. The mathematical model of the equation was developed with the help of feed-forward artificial neural networks. The training of the networks was made with evolutionary computational intelligence based on genetic algorithm hybrid with pattern search technique. Designed scheme was successfully applied to different forms of the equation. Results are compared with standard approximate analytic, stochastic numerical solvers and exact solutions.
Mustapha, K.
2017-06-03
Anomalous diffusion is a phenomenon that cannot be modeled accurately by second-order diffusion equations, but is better described by fractional diffusion models. The nonlocal nature of the fractional diffusion operators makes substantially more difficult the mathematical analysis of these models and the establishment of suitable numerical schemes. This paper proposes and analyzes the first finite difference method for solving {\\\\em variable-coefficient} fractional differential equations, with two-sided fractional derivatives, in one-dimensional space. The proposed scheme combines first-order forward and backward Euler methods for approximating the left-sided fractional derivative when the right-sided fractional derivative is approximated by two consecutive applications of the first-order backward Euler method. Our finite difference scheme reduces to the standard second-order central difference scheme in the absence of fractional derivatives. The existence and uniqueness of the solution for the proposed scheme are proved, and truncation errors of order $h$ are demonstrated, where $h$ denotes the maximum space step size. The numerical tests illustrate the global $O(h)$ accuracy of our scheme, except for nonsmooth cases which, as expected, have deteriorated convergence rates.
López-Sánchez, Erick J.; Romero, Juan M.; Yépez-Martínez, Huitzilin
2017-09-01
Different experimental studies have reported anomalous diffusion in brain tissues and notably this anomalous diffusion is expressed through fractional derivatives. Axons are important to understand neurodegenerative diseases such as multiple sclerosis, Alzheimer's disease, and Parkinson's disease. Indeed, abnormal accumulation of proteins and organelles in axons is a hallmark of these diseases. The diffusion in the axons can become anomalous as a result of this abnormality. In this case the voltage propagation in axons is affected. Another hallmark of different neurodegenerative diseases is given by discrete swellings along the axon. In order to model the voltage propagation in axons with anomalous diffusion and swellings, in this paper we propose a fractional cable equation for a general geometry. This generalized equation depends on fractional parameters and geometric quantities such as the curvature and torsion of the cable. For a cable with a constant radius we show that the voltage decreases when the fractional effect increases. In cables with swellings we find that when the fractional effect or the swelling radius increases, the voltage decreases. Similar behavior is obtained when the number of swellings and the fractional effect increase. Moreover, we find that when the radius swelling (or the number of swellings) and the fractional effect increase at the same time, the voltage dramatically decreases.
Mustapha, K.; Furati, K.; Knio, Omar; Maitre, O. Le
2017-01-01
Anomalous diffusion is a phenomenon that cannot be modeled accurately by second-order diffusion equations, but is better described by fractional diffusion models. The nonlocal nature of the fractional diffusion operators makes substantially more difficult the mathematical analysis of these models and the establishment of suitable numerical schemes. This paper proposes and analyzes the first finite difference method for solving {\\em variable-coefficient} fractional differential equations, with two-sided fractional derivatives, in one-dimensional space. The proposed scheme combines first-order forward and backward Euler methods for approximating the left-sided fractional derivative when the right-sided fractional derivative is approximated by two consecutive applications of the first-order backward Euler method. Our finite difference scheme reduces to the standard second-order central difference scheme in the absence of fractional derivatives. The existence and uniqueness of the solution for the proposed scheme are proved, and truncation errors of order $h$ are demonstrated, where $h$ denotes the maximum space step size. The numerical tests illustrate the global $O(h)$ accuracy of our scheme, except for nonsmooth cases which, as expected, have deteriorated convergence rates.
Aguila-Camacho, Norelys; Duarte-Mermoud, Manuel A
2016-01-01
This paper presents the analysis of three classes of fractional differential equations appearing in the field of fractional adaptive systems, for the case when the fractional order is in the interval α ∈(0,1] and the Caputo definition for fractional derivatives is used. The boundedness of the solutions is proved for all three cases, and the convergence to zero of the mean value of one of the variables is also proved. Applications of the obtained results to fractional adaptive schemes in the context of identification and control problems are presented at the end of the paper, including numerical simulations which support the analytical results. Copyright © 2015 ISA. Published by Elsevier Ltd. All rights reserved.
Nonlinear $q$-fractional differential equations with nonlocal and sub-strip type boundary conditions
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Bashir Ahmad
2014-06-01
Full Text Available This paper is concerned with new boundary value problems of nonlinear $q$-fractional differential equations with nonlocal and sub-strip type boundary conditions. Our results are new in the present setting and rely on the contraction mapping principle and a fixed point theorem due to O'Regan. Some illustrative examples are also presented.
Existence of Solution of Neutral Fractional Impulsive Diﬀerential Equations with Inﬁnite Delay
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K.Hilal
2017-04-01
Full Text Available In this work we deﬁne the solution of the fractional neutral impulsive diﬀerential equation with inﬁnite delay. The results of existence are obtained by using the Banach contraction and Schafer ﬁxed point. Some proprieties of probability density functions and semi-group theory are also used.
Ashyralyev, Allaberen; Cakir, Zafer
2016-08-01
In this work, we investigate initial-boundary value problems for fractional parabolic equations with the Neumann boundary condition. Stability estimates for the solution of this problem are established. Difference schemes for approximate solution of initial-boundary value problem are constructed. Furthermore, we give theorem on coercive stability estimates for the solution of the difference schemes.
On the solution set for a class of sequential fractional differential equations
International Nuclear Information System (INIS)
Baleanu, Dumitru; Mustafa, Octavian G; Agarwal, Ravi P
2010-01-01
We establish here that under some simple restrictions on the functional coefficient a(t) the solution set of the fractional differential equation ( 0 D α t x)' + a(t)x = 0 splits between eventually small and eventually large solutions as t → +∞, where 0 D α t designates the Riemann-Liouville derivative of the order α in (0, 1).
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A. Anguraj
2014-02-01
Full Text Available We study in this paper,the existence of solutions for fractional integro differential equations with impulsive and integral conditions by using fixed point method. We establish the Sufficient conditions and unique solution for given problem. An Example is also explained to the main results.
New Exact Solutions of Time Fractional Gardner Equation by Using New Version of F -Expansion Method
International Nuclear Information System (INIS)
Pandir, Yusuf; Duzgun, Hasan Huseyin
2017-01-01
In this article, we consider analytical solutions of the time fractional derivative Gardner equation by using the new version of F-expansion method. With this proposed method multiple Jacobi elliptic functions are situated in the solution function. As a result, various exact analytical solutions consisting of single and combined Jacobi elliptic functions solutions are obtained. (paper)
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Azizollah Babakhani
2010-01-01
Full Text Available We investigate the existence and uniqueness of positive solution for system of nonlinear fractional differential equations in two dimensions with delay. Our analysis relies on a nonlinear alternative of Leray-Schauder type and Krasnoselskii's fixed point theorem in a cone.
On the energy-critical fractional Sch\\"odinger equation in the radial case
Guo, Zihua; Sire, Yannick; Wang, Yuzhao; Zhao, Lifeng
2013-01-01
We consider the Cauchy problem for the energy-critical nonlinear Schr\\"odinger equation with fractional Laplacian (fNLS) in the radial case. We obtain global well-posedness and scattering in the energy space in the defocusing case, and in the focusing case with energy below the ground state.
van Breukelen, B.M.
2007-01-01
The Rayleigh equation relates the change in isotope ratio of an element in a substrate to the extent of substrate consumption via a single kinetic isotopic fractionation factor (α). Substrate consumption is, however, commonly distributed over several metabolic pathways each potentially having a
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V. Vijayakumar
2014-09-01
Full Text Available In this article, we study the existence of mild solutions for nonlocal Cauchy problem for fractional neutral evolution equations with infinite delay. The results are obtained by using the Banach contraction principle. Finally, an application is given to illustrate the theory.
Space-Time Fractional Diffusion-Advection Equation with Caputo Derivative
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José Francisco Gómez Aguilar
2014-01-01
Full Text Available An alternative construction for the space-time fractional diffusion-advection equation for the sedimentation phenomena is presented. The order of the derivative is considered as 0<β, γ≤1 for the space and time domain, respectively. The fractional derivative of Caputo type is considered. In the spatial case we obtain the fractional solution for the underdamped, undamped, and overdamped case. In the temporal case we show that the concentration has amplitude which exhibits an algebraic decay at asymptotically large times and also shows numerical simulations where both derivatives are taken in simultaneous form. In order that the equation preserves the physical units of the system two auxiliary parameters σx and σt are introduced characterizing the existence of fractional space and time components, respectively. A physical relation between these parameters is reported and the solutions in space-time are given in terms of the Mittag-Leffler function depending on the parameters β and γ. The generalization of the fractional diffusion-advection equation in space-time exhibits anomalous behavior.
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Sun Huan
2016-01-01
Full Text Available In this paper, we use the Laplace transform series expansion method to find the analytical solution for the local fractional heat-transfer equation defined on Cantor sets via local fractional calculus.
Langlands, T A M; Henry, B I; Wearne, S L
2009-12-01
We introduce fractional Nernst-Planck equations and derive fractional cable equations as macroscopic models for electrodiffusion of ions in nerve cells when molecular diffusion is anomalous subdiffusion due to binding, crowding or trapping. The anomalous subdiffusion is modelled by replacing diffusion constants with time dependent operators parameterized by fractional order exponents. Solutions are obtained as functions of the scaling parameters for infinite cables and semi-infinite cables with instantaneous current injections. Voltage attenuation along dendrites in response to alpha function synaptic inputs is computed. Action potential firing rates are also derived based on simple integrate and fire versions of the models. Our results show that electrotonic properties and firing rates of nerve cells are altered by anomalous subdiffusion in these models. We have suggested electrophysiological experiments to calibrate and validate the models.
Approximate solution of space and time fractional higher order phase field equation
Shamseldeen, S.
2018-03-01
This paper is concerned with a class of space and time fractional partial differential equation (STFDE) with Riesz derivative in space and Caputo in time. The proposed STFDE is considered as a generalization of a sixth-order partial phase field equation. We describe the application of the optimal homotopy analysis method (OHAM) to obtain an approximate solution for the suggested fractional initial value problem. An averaged-squared residual error function is defined and used to determine the optimal convergence control parameter. Two numerical examples are studied, considering periodic and non-periodic initial conditions, to justify the efficiency and the accuracy of the adopted iterative approach. The dependence of the solution on the order of the fractional derivative in space and time and model parameters is investigated.
Fractional Order Stochastic Differential Equation with Application in European Option Pricing
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Qing Li
2014-01-01
Full Text Available Memory effect is an important phenomenon in financial systems, and a number of research works have been carried out to study the long memory in the financial markets. In recent years, fractional order ordinary differential equation is used as an effective instrument for describing the memory effect in complex systems. In this paper, we establish a fractional order stochastic differential equation (FSDE model to describe the effect of trend memory in financial pricing. We, then, derive a European option pricing formula based on the FSDE model and prove the existence of the trend memory (i.e., the mean value function in the option pricing formula when the Hurst index is between 0.5 and 1. In addition, we make a comparison analysis between our proposed model, the classic Black-Scholes model, and the stochastic model with fractional Brownian motion. Numerical results suggest that our model leads to more accurate and lower standard deviation in the empirical study.
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Mohamed Abdalla Darwish
2014-01-01
Full Text Available We study a generalized fractional quadratic functional-integral equation of Erdélyi-Kober type in the Banach space BC(ℝ+. We show that this equation has at least one asymptotically stable solution.
RDTM solution of Caputo time fractional-order hyperbolic telegraph equation
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Vineet K. Srivastava
2013-03-01
Full Text Available In this study, a mathematical model has been developed for the second order hyperbolic one-dimensional time fractional Telegraph equation (TFTE. The fractional derivative has been described in the Caputo sense. The governing equations have been solved by a recent reliable semi-analytic method known as the reduced differential transformation method (RDTM. The method is a powerful mathematical technique for solving wide range of problems. Using RDTM method, it is possible to find exact solution as well as closed approximate solution of any ordinary or partial differential equation. Three numerical examples of TFTE have been provided in order to check the effectiveness, accuracy and convergence of the method. The computed results are also depicted graphically.
Fourier spectral methods for fractional-in-space reaction-diffusion equations
Bueno-Orovio, Alfonso
2014-04-01
© 2014, Springer Science+Business Media Dordrecht. Fractional differential equations are becoming increasingly used as a powerful modelling approach for understanding the many aspects of nonlocality and spatial heterogeneity. However, the numerical approximation of these models is demanding and imposes a number of computational constraints. In this paper, we introduce Fourier spectral methods as an attractive and easy-to-code alternative for the integration of fractional-in-space reaction-diffusion equations described by the fractional Laplacian in bounded rectangular domains of ℝ. The main advantages of the proposed schemes is that they yield a fully diagonal representation of the fractional operator, with increased accuracy and efficiency when compared to low-order counterparts, and a completely straightforward extension to two and three spatial dimensions. Our approach is illustrated by solving several problems of practical interest, including the fractional Allen–Cahn, FitzHugh–Nagumo and Gray–Scott models, together with an analysis of the properties of these systems in terms of the fractional power of the underlying Laplacian operator.
International Nuclear Information System (INIS)
Jumarie, Guy
2004-01-01
There are presently two different models of fractional Brownian motions available in the literature: the Riemann-Liouville fractional derivative of white noise on the one hand, and the complex-valued Brownian motion of order n defined by using a random walk in the complex plane, on the other hand. The paper provides a comparison between these two approaches, and in addition, takes this opportunity to contribute some complements. These two models are more or less equivalent on the theoretical standpoint for fractional order between 0 and 1/2, but their practical significances are quite different. Otherwise, for order larger than 1/2, the fractional derivative model has no counterpart in the complex plane. These differences are illustrated by an example drawn from mathematical finance. Taylor expansion of fractional order provides the expression of fractional difference in terms of finite difference, and this allows us to improve the derivation of Fokker-Planck equation and Kramers-Moyal expansion, and to get more insight in their relation with stochastic differential equations of fractional order. In the case of multi-fractal systems, the Fokker-Planck equation can be solved by using path integrals, and the fractional dynamic equations of the state moments of the stochastic system can be easily obtained. By combining fractional derivative and complex white noise of order n, one obtains a family of complex-valued fractional Brownian motions which exhibits long-range dependence. The conclusion outlines suggestions for further research, mainly regarding Lorentz transformation of fractional noises
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Yanning Wang
2016-01-01
Full Text Available Using conformable fractional calculus on time scales, we first introduce fractional Sobolev spaces on time scales, characterize them, and define weak conformable fractional derivatives. Second, we prove the equivalence of some norms in the introduced spaces and derive their completeness, reflexivity, uniform convexity, and compactness of some imbeddings, which can be regarded as a novelty item. Then, as an application, we present a recent approach via variational methods and critical point theory to obtain the existence of solutions for a p-Laplacian conformable fractional differential equation boundary value problem on time scale T: Tα(Tαup-2Tα(u(t=∇F(σ(t,u(σ(t, Δ-a.e. t∈a,bTκ2, u(a-u(b=0, Tα(u(a-Tα(u(b=0, where Tα(u(t denotes the conformable fractional derivative of u of order α at t, σ is the forward jump operator, a,b∈T, 01, and F:[0,T]T×RN→R. By establishing a proper variational setting, we obtain three existence results. Finally, we present two examples to illustrate the feasibility and effectiveness of the existence results.
Langevin dynamics for vector variables driven by multiplicative white noise: A functional formalism
Moreno, Miguel Vera; Arenas, Zochil González; Barci, Daniel G.
2015-04-01
We discuss general multidimensional stochastic processes driven by a system of Langevin equations with multiplicative white noise. In particular, we address the problem of how time reversal diffusion processes are affected by the variety of conventions available to deal with stochastic integrals. We present a functional formalism to build up the generating functional of correlation functions without any type of discretization of the Langevin equations at any intermediate step. The generating functional is characterized by a functional integration over two sets of commuting variables, as well as Grassmann variables. In this representation, time reversal transformation became a linear transformation in the extended variables, simplifying in this way the complexity introduced by the mixture of prescriptions and the associated calculus rules. The stochastic calculus is codified in our formalism in the structure of the Grassmann algebra. We study some examples such as higher order derivative Langevin equations and the functional representation of the micromagnetic stochastic Landau-Lifshitz-Gilbert equation.
Liu, Da-Yan; Tian, Yang; Boutat, Driss; Laleg-Kirati, Taous-Meriem
2015-01-01
This paper aims at designing a digital fractional order differentiator for a class of signals satisfying a linear differential equation to estimate fractional derivatives with an arbitrary order in noisy case, where the input can be unknown or known with noises. Firstly, an integer order differentiator for the input is constructed using a truncated Jacobi orthogonal series expansion. Then, a new algebraic formula for the Riemann-Liouville derivative is derived, which is enlightened by the algebraic parametric method. Secondly, a digital fractional order differentiator is proposed using a numerical integration method in discrete noisy case. Then, the noise error contribution is analyzed, where an error bound useful for the selection of the design parameter is provided. Finally, numerical examples illustrate the accuracy and the robustness of the proposed fractional order differentiator.
Liu, Da-Yan
2015-04-30
This paper aims at designing a digital fractional order differentiator for a class of signals satisfying a linear differential equation to estimate fractional derivatives with an arbitrary order in noisy case, where the input can be unknown or known with noises. Firstly, an integer order differentiator for the input is constructed using a truncated Jacobi orthogonal series expansion. Then, a new algebraic formula for the Riemann-Liouville derivative is derived, which is enlightened by the algebraic parametric method. Secondly, a digital fractional order differentiator is proposed using a numerical integration method in discrete noisy case. Then, the noise error contribution is analyzed, where an error bound useful for the selection of the design parameter is provided. Finally, numerical examples illustrate the accuracy and the robustness of the proposed fractional order differentiator.
Application of the fractional neutron point kinetic equation: Start-up of a nuclear reactor
International Nuclear Information System (INIS)
Polo-Labarrios, M.-A.; Espinosa-Paredes, G.
2012-01-01
Highlights: ► Neutron density behavior at reactor start up with fractional neutron point kinetics. ► There is a relaxation time associated with a rapid variation in the neutron flux. ► Physical interpretation of the fractional order is related with non-Fickian effects. ► Effect of the anomalous diffusion coefficient and the relaxation time is analyzed. ► Neutron density is related with speed and duration of the control rods lifting. - Abstract: In this paper we present the behavior of the variation of neutron density when the nuclear reactor power is increased using the fractional neutron point kinetic (FNPK) equation with a single-group of delayed neutron precursor. It is considered that there is a relaxation time associated with a rapid variation in the neutron flux and its physical interpretation of the fractional order is related with non-Fickian effects from the neutron diffusion equation point of view. We analyzed the case of increase the nuclear reactor power when reactor is cold start-up which is a process of inserting reactivity by lifting control rods discontinuously. The results show that for short time scales of the start-up the neutronic density behavior with FNPK shows sub-diffusive effects whose absorption are government by control rods velocity. For large times scale, the results shows that the classical equation of the neutron point kinetics over predicted the neutron density regarding to FNPK.
Langevin description of fission fragment charge distribution from excited nuclei
Karpov, A V
2002-01-01
A stochastic approach to fission dynamics based on a set of three-dimensional Langevin equations was applied to calculate fission-fragment charge distribution of compound nucleus sup 2 sup 3 sup 6 U. The following collective coordinates have been chosen - elongation coordinate, neck-thickness coordinate, and charge-asymmetry coordinate. The friction coefficient of charge mode has been calculated in the framework of one-body and two-body dissipation mechanisms. Analysis of the results has shown that Langevin approach is appropriate for investigation of isobaric distribution. Moreover, the dependences of the variance of the charge distribution on excitation energy and on the two-body viscosity coefficient has been studied
Generalized hydrodynamic correlations and fractional memory functions
Rodríguez, Rosalio F.; Fujioka, Jorge
2015-12-01
A fractional generalized hydrodynamic (GH) model of the longitudinal velocity fluctuations correlation, and its associated memory function, for a complex fluid is analyzed. The adiabatic elimination of fast variables introduces memory effects in the transport equations, and the dynamic of the fluctuations is described by a generalized Langevin equation with long-range noise correlations. These features motivate the introduction of Caputo time fractional derivatives and allows us to calculate analytic expressions for the fractional longitudinal velocity correlation function and its associated memory function. Our analysis eliminates a spurious constant term in the non-fractional memory function found in the non-fractional description. It also produces a significantly slower power-law decay of the memory function in the GH regime that reduces to the well-known exponential decay in the non-fractional Navier-Stokes limit.
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Zhigang Hu
2014-01-01
Full Text Available In this paper, we apply the method of the Nehari manifold to study the fractional differential equation (d/dt((1/2 0Dt-β(u′(t+(1/2 tDT-β(u′(t= f(t,u(t, a.e. t∈[0,T], and u0=uT=0, where 0Dt-β, tDT-β are the left and right Riemann-Liouville fractional integrals of order 0≤β<1, respectively. We prove the existence of a ground state solution of the boundary value problem.
The Galerkin finite element method for a multi-term time-fractional diffusion equation
Jin, Bangti
2015-01-01
© 2014 The Authors. We consider the initial/boundary value problem for a diffusion equation involving multiple time-fractional derivatives on a bounded convex polyhedral domain. We analyze a space semidiscrete scheme based on the standard Galerkin finite element method using continuous piecewise linear functions. Nearly optimal error estimates for both cases of initial data and inhomogeneous term are derived, which cover both smooth and nonsmooth data. Further we develop a fully discrete scheme based on a finite difference discretization of the time-fractional derivatives, and discuss its stability and error estimate. Extensive numerical experiments for one- and two-dimensional problems confirm the theoretical convergence rates.
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Aqlan Mohammed H.
2016-01-01
Full Text Available We develop the existence theory for sequential fractional differential equations involving Liouville-Caputo fractional derivative equipped with anti-periodic type (non-separated and nonlocal integral boundary conditions. Several existence criteria depending on the nonlinearity involved in the problems are presented by means of a variety of tools of the fixed point theory. The applicability of the results is shown with the aid of examples. Our results are not only new in the given configuration but also yield some new special cases for specific choices of parameters involved in the problems.
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S. Das
2013-12-01
Full Text Available In this article, optimal homotopy-analysis method is used to obtain approximate analytic solution of the time-fractional diffusion equation with a given initial condition. The fractional derivatives are considered in the Caputo sense. Unlike usual Homotopy analysis method, this method contains at the most three convergence control parameters which describe the faster convergence of the solution. Effects of parameters on the convergence of the approximate series solution by minimizing the averaged residual error with the proper choices of parameters are calculated numerically and presented through graphs and tables for different particular cases.
Infinitely many solutions for fractional Schr\\"odinger equations in R^N
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Caisheng Chen
2016-03-01
Full Text Available Using variational methods we prove the existence of infinitely many solutions to the fractional Schrodinger equation $$ (-\\Delta^su+V(xu=f(x,u, \\quad x\\in\\mathbb{R}^N, $$ where $N\\ge 2, s\\in (0,1$. $(-\\Delta^s$ stands for the fractional Laplacian. The potential function satisfies $V(x\\geq V_0>0$. The nonlinearity f(x,u is superlinear, has subcritical growth in u, and may or may not satisfy the (AR condition.
Optimal homotopy asymptotic method for solving fractional relaxation-oscillation equation
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Mohammad Hamarsheh
2015-11-01
Full Text Available In this paper, an approximate analytical solution of linear fractional relaxation-oscillation equations in which the fractional derivatives are given in the Caputo sense, is obtained by the optimal homotopy asymptotic method (OHAM. The studied OHAM is based on minimizing the residual error. The results given by OHAM are compared with the exact solutions and the solutions obtained by generalized Taylor matrix method. The reliability and efficiency of the proposed approach are demonstrated in three examples with the aid of the symbolic algebra program Maple.
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Gang-Ling Hou
2018-04-01
Full Text Available This article concerns the fractional Schrodinger type equations $$ (-\\Delta^\\alpha u+V(xu =f(x,u \\quad\\text{in } \\mathbb{R}^N, $$ where $N\\geq 2$, $\\alpha\\in(0,1$, $(-\\Delta^\\alpha$ stands for the fractional Laplacian, $V$ is a positive continuous potential, $f\\in C(\\mathbb{R}^N\\times\\mathbb{R},\\mathbb{R}$. We establish criteria that guarantee the existence of infinitely many solutions by using the genus properties in critical point theory.
Enriched reproducing kernel particle method for fractional advection-diffusion equation
Ying, Yuping; Lian, Yanping; Tang, Shaoqiang; Liu, Wing Kam
2018-06-01
The reproducing kernel particle method (RKPM) has been efficiently applied to problems with large deformations, high gradients and high modal density. In this paper, it is extended to solve a nonlocal problem modeled by a fractional advection-diffusion equation (FADE), which exhibits a boundary layer with low regularity. We formulate this method on a moving least-square approach. Via the enrichment of fractional-order power functions to the traditional integer-order basis for RKPM, leading terms of the solution to the FADE can be exactly reproduced, which guarantees a good approximation to the boundary layer. Numerical tests are performed to verify the proposed approach.
A Numerical Algorithm for Solving a Four-Point Nonlinear Fractional Integro-Differential Equations
Gao, Er; Song, Songhe; Zhang, Xinjian
2012-01-01
We provide a new algorithm for a four-point nonlocal boundary value problem of nonlinear integro-differential equations of fractional order q∈(1,2] based on reproducing kernel space method. According to our work, the analytical solution of the equations is represented in the reproducing kernel space which we construct and so the n-term approximation. At the same time, the n-term approximation is proved to converge to the analytical solution. An illustrative example is also presented, which sh...
The determination of an unknown source for a space fractional advection dispersion equation
Aldoghaither, Abeer
2014-09-01
In this paper, we are interested in the estimation of the source term for a space fractional advection dispersion equation using concentration and flux measurements at final time. An example of application is the identification of contamination source in groundwater transport. We propose to use the socalled modulating functions method which has been introduced for parameters estimation. This method allows to transfer the estimation problem into solving a system of algebraic equations. Numerical examples are given to illustrate the effectiveness and the robustness of the proposed method. Finally, a comparison between a Tikhonov-based optimization method and the modulating functions approach is presented.
The determination of an unknown boundary condition in a fractional diffusion equation
Rundell, William
2013-07-01
In this article we consider an inverse boundary problem, in which the unknown boundary function ∂u/∂v = f(u) is to be determined from overposed data in a time-fractional diffusion equation. Based upon the free space fundamental solution, we derive a representation for the solution f as a nonlinear Volterra integral equation of second kind with a weakly singular kernel. Uniqueness and reconstructibility by iteration is an immediate result of a priori assumption on f and applying the fixed point theorem. Numerical examples are presented to illustrate the validity and effectiveness of the proposed method. © 2013 Copyright Taylor and Francis Group, LLC.
A Numerical Algorithm for Solving a Four-Point Nonlinear Fractional Integro-Differential Equations
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Er Gao
2012-01-01
Full Text Available We provide a new algorithm for a four-point nonlocal boundary value problem of nonlinear integro-differential equations of fractional order q∈(1,2] based on reproducing kernel space method. According to our work, the analytical solution of the equations is represented in the reproducing kernel space which we construct and so the n-term approximation. At the same time, the n-term approximation is proved to converge to the analytical solution. An illustrative example is also presented, which shows that the new algorithm is efficient and accurate.
Nonmaxwell relaxation in disordered media: Physical mechanisms and fractional relaxation equations
International Nuclear Information System (INIS)
Arkhincheev, V.E.
2004-12-01
The problem of charge relaxation in disordered systems has been solved. It is shown, that due to the inhomogeneity of the medium the charge relaxation has a non-Maxwell character. The two physical mechanisms of a such behavior have been founded. The first one is connected with the 'fractality' of conducting ways. The second mechanism of nonexponential non-Maxwell behavior is connected with the frequency dispersion of effective conductivity of heterogeneous medium, initially consisting of conducting phases without dispersion. The new generalized relaxation equations in the form of fractional temporal integro-differential equations are deduced. (author)
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Gao Guo-Ping
2016-01-01
Full Text Available In this article, we investigate the local fractional 3-D compressible Navier-Stokes equation via local fractional derivative. We use the Cantor-type cylindrical co-ordinate method to transfer 3-D compressible Navier-Stokes equation from the Cantorian co-ordinate system to the Cantor-type cylindrical co-ordinate system.
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Marina Popolizio
2018-01-01
Full Text Available Multiterm fractional differential equations (MTFDEs nowadays represent a widely used tool to model many important processes, particularly for multirate systems. Their numerical solution is then a compelling subject that deserves great attention, not least because of the difficulties to apply general purpose methods for fractional differential equations (FDEs to this case. In this paper, we first transform the MTFDEs into equivalent systems of FDEs, as done by Diethelm and Ford; in this way, the solution can be expressed in terms of Mittag–Leffler (ML functions evaluated at matrix arguments. We then propose to compute it by resorting to the matrix approach proposed by Garrappa and Popolizio. Several numerical tests are presented that clearly show that this matrix approach is very accurate and fast, also in comparison with other numerical methods.
A New Monotone Iteration Principle in the Theory of Nonlinear Fractional Differential Equations
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Bapurao C. Dhage
2015-08-01
Full Text Available In this paper the author proves the algorithms for the existence as well as approximations of the solutions for the initial value problems of nonlinear fractional diﬀerential equations using the operator theoretic techniques in a partially ordered metric space. The main results rely on the Dhage iteration principle embodied in the recent hybrid ﬁxed point theorems of Dhage (2014 in a partially ordered normed linear space and the existence and approximations of the solutions of the considered nonlinear fractional diﬀerential equations are obtained under weak mixed partial continuity and partial Lipschitz conditions. Our hypotheses and existence and approximation results are also well illustrated by some numerical examples.
Analysis of the cable equation with non-local and non-singular kernel fractional derivative
Karaagac, Berat
2018-02-01
Recently a new concept of differentiation was introduced in the literature where the kernel was converted from non-local singular to non-local and non-singular. One of the great advantages of this new kernel is its ability to portray fading memory and also well defined memory of the system under investigation. In this paper the cable equation which is used to develop mathematical models of signal decay in submarine or underwater telegraphic cables will be analysed using the Atangana-Baleanu fractional derivative due to the ability of the new fractional derivative to describe non-local fading memory. The existence and uniqueness of the more generalized model is presented in detail via the fixed point theorem. A new numerical scheme is used to solve the new equation. In addition, stability, convergence and numerical simulations are presented.
Space-time spectral collocation algorithm for solving time-fractional Tricomi-type equations
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Abdelkawy M.A.
2016-01-01
Full Text Available We introduce a new numerical algorithm for solving one-dimensional time-fractional Tricomi-type equations (T-FTTEs. We used the shifted Jacobi polynomials as basis functions and the derivatives of fractional is evaluated by the Caputo definition. The shifted Jacobi Gauss-Lobatt algorithm is used for the spatial discretization, while the shifted Jacobi Gauss-Radau algorithmis applied for temporal approximation. Substituting these approximations in the problem leads to a system of algebraic equations that greatly simplifies the problem. The proposed algorithm is successfully extended to solve the two-dimensional T-FTTEs. Extensive numerical tests illustrate the capability and high accuracy of the proposed methodologies.
Abstract fractional integro-differential equations involving nonlocal initial conditions in α-norm
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Wang Rong-Nian
2011-01-01
Full Text Available Abstract In the present paper, we deal with the Cauchy problems of abstract fractional integro-differential equations involving nonlocal initial conditions in α-norm, where the operator A in the linear part is the generator of a compact analytic semigroup. New criterions, ensuring the existence of mild solutions, are established. The results are obtained by using the theory of operator families associated with the function of Wright type and the semigroup generated by A, Krasnoselkii's fixed point theorem and Schauder's fixed point theorem. An application to a fractional partial integro-differential equation with nonlocal initial condition is also considered. Mathematics subject classification (2000 26A33, 34G10, 34G20
Talib, Imran; Belgacem, Fethi Bin Muhammad; Asif, Naseer Ahmad; Khalil, Hammad
2017-01-01
In this research article, we derive and analyze an efficient spectral method based on the operational matrices of three dimensional orthogonal Jacobi polynomials to solve numerically the mixed partial derivatives type multi-terms high dimensions generalized class of fractional order partial differential equations. We transform the considered fractional order problem to an easily solvable algebraic equations with the aid of the operational matrices. Being easily solvable, the associated algebraic system leads to finding the solution of the problem. Some test problems are considered to confirm the accuracy and validity of the proposed numerical method. The convergence of the method is ensured by comparing our Matlab software simulations based obtained results with the exact solutions in the literature, yielding negligible errors. Moreover, comparative results discussed in the literature are extended and improved in this study.
Solutions to Time-Fractional Diffusion-Wave Equation in Cylindrical Coordinates
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Povstenko YZ
2011-01-01
Full Text Available Nonaxisymmetric solutions to time-fractional diffusion-wave equation with a source term in cylindrical coordinates are obtained for an infinite medium. The solutions are found using the Laplace transform with respect to time , the Hankel transform with respect to the radial coordinate , the finite Fourier transform with respect to the angular coordinate , and the exponential Fourier transform with respect to the spatial coordinate . Numerical results are illustrated graphically.
Bai, Yunru; Baleanu, Dumitru; Wu, Guo-Cheng
2018-06-01
We investigate a class of generalized differential optimization problems driven by the Caputo derivative. Existence of weak Carathe ´odory solution is proved by using Weierstrass existence theorem, fixed point theorem and Filippov implicit function lemma etc. Then a numerical approximation algorithm is introduced, and a convergence theorem is established. Finally, a nonlinear programming problem constrained by the fractional differential equation is illustrated and the results verify the validity of the algorithm.
Direct and inverse source problems for a space fractional advection dispersion equation
Aldoghaither, Abeer
2016-05-15
In this paper, direct and inverse problems for a space fractional advection dispersion equation on a finite domain are studied. The inverse problem consists in determining the source term from final observations. We first derive the analytic solution to the direct problem which we use to prove the uniqueness and the unstability of the inverse source problem using final measurements. Finally, we illustrate the results with a numerical example.
A numerical solution for a class of time fractional diffusion equations with delay
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Pimenov Vladimir G.
2017-09-01
Full Text Available This paper describes a numerical scheme for a class of fractional diffusion equations with fixed time delay. The study focuses on the uniqueness, convergence and stability of the resulting numerical solution by means of the discrete energy method. The derivation of a linearized difference scheme with convergence order O(τ2−α+ h4 in L∞-norm is the main purpose of this study. Numerical experiments are carried out to support the obtained theoretical results.
FEM for time-fractional diffusion equations, novel optimal error analyses
Mustapha, Kassem
2016-01-01
A semidiscrete Galerkin finite element method applied to time-fractional diffusion equations with time-space dependent diffusivity on bounded convex spatial domains will be studied. The main focus is on achieving optimal error results with respect to both the convergence order of the approximate solution and the regularity of the initial data. By using novel energy arguments, for each fixed time $t$, optimal error bounds in the spatial $L^2$- and $H^1$-norms are derived for both cases: smooth...
The Galerkin Finite Element Method for A Multi-term Time-Fractional Diffusion equation
Jin, Bangti; Lazarov, Raytcho; Liu, Yikan; Zhou, Zhi
2014-01-01
We consider the initial/boundary value problem for a diffusion equation involving multiple time-fractional derivatives on a bounded convex polyhedral domain. We analyze a space semidiscrete scheme based on the standard Galerkin finite element method using continuous piecewise linear functions. Nearly optimal error estimates for both cases of initial data and inhomogeneous term are derived, which cover both smooth and nonsmooth data. Further we develop a fully discrete scheme based on a finite...
GHOLAMI, SAEID; BABOLIAN, ESMAIL; JAVIDI, MOHAMMAD
2016-01-01
This paper presents a new numerical approach to solve single and multiterm time fractional diffusion equations. In this work, the space dimension is discretized to the Gauss$-$Lobatto points. We use the normalized Grunwald approximation for the time dimension and a pseudospectral successive integration matrix for the space dimension. This approach shows that with fewer numbers of points, we can approximate the solution with more accuracy. Some examples with numerical results in tables and fig...
New Insights into the Fractional Order Diffusion Equation Using Entropy and Kurtosis.
Ingo, Carson; Magin, Richard L; Parrish, Todd B
2014-11-01
Fractional order derivative operators offer a concise description to model multi-scale, heterogeneous and non-local systems. Specifically, in magnetic resonance imaging, there has been recent work to apply fractional order derivatives to model the non-Gaussian diffusion signal, which is ubiquitous in the movement of water protons within biological tissue. To provide a new perspective for establishing the utility of fractional order models, we apply entropy for the case of anomalous diffusion governed by a fractional order diffusion equation generalized in space and in time. This fractional order representation, in the form of the Mittag-Leffler function, gives an entropy minimum for the integer case of Gaussian diffusion and greater values of spectral entropy for non-integer values of the space and time derivatives. Furthermore, we consider kurtosis, defined as the normalized fourth moment, as another probabilistic description of the fractional time derivative. Finally, we demonstrate the implementation of anomalous diffusion, entropy and kurtosis measurements in diffusion weighted magnetic resonance imaging in the brain of a chronic ischemic stroke patient.
New Insights into the Fractional Order Diffusion Equation Using Entropy and Kurtosis
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Carson Ingo
2014-11-01
Full Text Available Fractional order derivative operators offer a concise description to model multi-scale, heterogeneous and non-local systems. Specifically, in magnetic resonance imaging, there has been recent work to apply fractional order derivatives to model the non-Gaussian diffusion signal, which is ubiquitous in the movement of water protons within biological tissue. To provide a new perspective for establishing the utility of fractional order models, we apply entropy for the case of anomalous diffusion governed by a fractional order diffusion equation generalized in space and in time. This fractional order representation, in the form of the Mittag–Leffler function, gives an entropy minimum for the integer case of Gaussian diffusion and greater values of spectral entropy for non-integer values of the space and time derivatives. Furthermore, we consider kurtosis, defined as the normalized fourth moment, as another probabilistic description of the fractional time derivative. Finally, we demonstrate the implementation of anomalous diffusion, entropy and kurtosis measurements in diffusion weighted magnetic resonance imaging in the brain of a chronic ischemic stroke patient.
International Nuclear Information System (INIS)
Sperotto, Fabiola Aiub; Segatto, Cynthia Feijo; Zabadal, Jorge
2002-01-01
In this work, we determine the dominant eigenvalue of the one-dimensional neutron transport equation in a slab constructing an integral form for the neutron transport equation which is the expressed in terms of fractional derivative of the angular flux. Equating the fractional derivative of the angular flux to the integrate equation, we determine the unknown order of the fractional derivative comparing the kernel of the integral equation with the one of Riemann-Liouville definition of fractional derivative. Once known the angular flux the dominant eigenvalue is calculated solving a transcendental equation resulting from the application of the boundary conditions. We report the methodology applied, for comparison with available results in literature. (author)
International Nuclear Information System (INIS)
Li, Gongsheng; Zhang, Dali; Jia, Xianzheng; Yamamoto, Masahiro
2013-01-01
This paper deals with an inverse problem of simultaneously identifying the space-dependent diffusion coefficient and the fractional order in the 1D time-fractional diffusion equation with smooth initial functions by using boundary measurements. The uniqueness results for the inverse problem are proved on the basis of the inverse eigenvalue problem, and the Lipschitz continuity of the solution operator is established. A modified optimal perturbation algorithm with a regularization parameter chosen by a sigmoid-type function is put forward for the discretization of the minimization problem. Numerical inversions are performed for the diffusion coefficient taking on different functional forms and the additional data having random noise. Several factors which have important influences on the realization of the algorithm are discussed, including the approximate space of the diffusion coefficient, the regularization parameter and the initial iteration. The inversion solutions are good approximations to the exact solutions with stability and adaptivity demonstrating that the optimal perturbation algorithm with the sigmoid-type regularization parameter is efficient for the simultaneous inversion. (paper)
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Soheil Salahshour
2015-02-01
Full Text Available In this paper, we apply the concept of Caputo’s H-differentiability, constructed based on the generalized Hukuhara difference, to solve the fuzzy fractional differential equation (FFDE with uncertainty. This is in contrast to conventional solutions that either require a quantity of fractional derivatives of unknown solution at the initial point (Riemann–Liouville or a solution with increasing length of their support (Hukuhara difference. Then, in order to solve the FFDE analytically, we introduce the fuzzy Laplace transform of the Caputo H-derivative. To the best of our knowledge, there is limited research devoted to the analytical methods to solve the FFDE under the fuzzy Caputo fractional differentiability. An analytical solution is presented to confirm the capability of the proposed method.
Pirnapasov, Sardor; Karimov, Erkinjon
2017-01-01
In the present work we discuss higher order multi-term partial differential equation (PDE) with the Caputo-Fabrizio fractional derivative in time. We investigate a boundary value problem for fractional heat equation involving higher order Caputo-Fabrizio derivatives in time-variable. Using method of separation of variables and integration by parts, we reduce fractional order PDE to the integer order. We represent explicit solution of formulated problem in particular case by Fourier series.
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Ya-Juan Hao
2013-01-01
Full Text Available The main object of this paper is to investigate the Helmholtz and diffusion equations on the Cantor sets involving local fractional derivative operators. The Cantor-type cylindrical-coordinate method is applied to handle the corresponding local fractional differential equations. Two illustrative examples for the Helmholtz and diffusion equations on the Cantor sets are shown by making use of the Cantorian and Cantor-type cylindrical coordinates.
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M. Bishehniasar
2017-01-01
Full Text Available The demand of many scientific areas for the usage of fractional partial differential equations (FPDEs to explain their real-world systems has been broadly identified. The solutions may portray dynamical behaviors of various particles such as chemicals and cells. The desire of obtaining approximate solutions to treat these equations aims to overcome the mathematical complexity of modeling the relevant phenomena in nature. This research proposes a promising approximate-analytical scheme that is an accurate technique for solving a variety of noninteger partial differential equations (PDEs. The proposed strategy is based on approximating the derivative of fractional-order and reducing the problem to the corresponding partial differential equation (PDE. Afterwards, the approximating PDE is solved by using a separation-variables technique. The method can be simply applied to nonhomogeneous problems and is proficient to diminish the span of computational cost as well as achieving an approximate-analytical solution that is in excellent concurrence with the exact solution of the original problem. In addition and to demonstrate the efficiency of the method, it compares with two finite difference methods including a nonstandard finite difference (NSFD method and standard finite difference (SFD technique, which are popular in the literature for solving engineering problems.
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Rahmatullah
2018-03-01
Full Text Available We have computed new exact traveling wave solutions, including complex solutions of fractional order Boussinesq-Like equations, occurring in physical sciences and engineering, by applying Exp-function method. The method is blended with fractional complex transformation and modified Riemann-Liouville fractional order operator. Our obtained solutions are verified by substituting back into their corresponding equations. To the best of our knowledge, no other technique has been reported to cope with the said fractional order nonlinear problems combined with variety of exact solutions. Graphically, fractional order solution curves are shown to be strongly related to each other and most importantly, tend to fixate on their integer order solution curve. Our solutions comprise high frequencies and very small amplitude of the wave responses. Keywords: Exp-function method, New exact traveling wave solutions, Modified Riemann-Liouville derivative, Fractional complex transformation, Fractional order Boussinesq-like equations, Symbolic computation
Energy Technology Data Exchange (ETDEWEB)
Angstmann, C.N.; Donnelly, I.C. [School of Mathematics and Statistics, UNSW Australia, Sydney NSW 2052 (Australia); Henry, B.I., E-mail: B.Henry@unsw.edu.au [School of Mathematics and Statistics, UNSW Australia, Sydney NSW 2052 (Australia); Jacobs, B.A. [School of Computer Science and Applied Mathematics, University of the Witwatersrand, Johannesburg, Private Bag 3, Wits 2050 (South Africa); DST–NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS) (South Africa); Langlands, T.A.M. [Department of Mathematics and Computing, University of Southern Queensland, Toowoomba QLD 4350 (Australia); Nichols, J.A. [School of Mathematics and Statistics, UNSW Australia, Sydney NSW 2052 (Australia)
2016-02-15
We have introduced a new explicit numerical method, based on a discrete stochastic process, for solving a class of fractional partial differential equations that model reaction subdiffusion. The scheme is derived from the master equations for the evolution of the probability density of a sum of discrete time random walks. We show that the diffusion limit of the master equations recovers the fractional partial differential equation of interest. This limiting procedure guarantees the consistency of the numerical scheme. The positivity of the solution and stability results are simply obtained, provided that the underlying process is well posed. We also show that the method can be applied to standard reaction–diffusion equations. This work highlights the broader applicability of using discrete stochastic processes to provide numerical schemes for partial differential equations, including fractional partial differential equations.
Comparison of Langevin and Markov channel noise models for neuronal signal generation.
Sengupta, B; Laughlin, S B; Niven, J E
2010-01-01
The stochastic opening and closing of voltage-gated ion channels produce noise in neurons. The effect of this noise on the neuronal performance has been modeled using either an approximate or Langevin model based on stochastic differential equations or an exact model based on a Markov process model of channel gating. Yet whether the Langevin model accurately reproduces the channel noise produced by the Markov model remains unclear. Here we present a comparison between Langevin and Markov models of channel noise in neurons using single compartment Hodgkin-Huxley models containing either Na+ and K+, or only K+ voltage-gated ion channels. The performance of the Langevin and Markov models was quantified over a range of stimulus statistics, membrane areas, and channel numbers. We find that in comparison to the Markov model, the Langevin model underestimates the noise contributed by voltage-gated ion channels, overestimating information rates for both spiking and nonspiking membranes. Even with increasing numbers of channels, the difference between the two models persists. This suggests that the Langevin model may not be suitable for accurately simulating channel noise in neurons, even in simulations with large numbers of ion channels.
M. L. Kavvas; A. Ercan; J. Polsinelli
2017-01-01
In this study dimensionally consistent governing equations of continuity and motion for transient soil water flow and soil water flux in fractional time and in fractional multiple space dimensions in anisotropic media are developed. Due to the anisotropy in the hydraulic conductivities of natural soils, the soil medium within which the soil water flow occurs is essentially anisotropic. Accordingly, in this study the fractional dimensions in two horizontal and one vertical di...
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Inci Cilingir Sungu
2015-01-01
Full Text Available A new application of the hybrid generalized differential transform and finite difference method is proposed by solving time fractional nonlinear reaction-diffusion equations. This method is a combination of the multi-time-stepping temporal generalized differential transform and the spatial finite difference methods. The procedure first converts the time-evolutionary equations into Poisson equations which are then solved using the central difference method. The temporal differential transform method as used in the paper takes care of stability and the finite difference method on the resulting equation results in a system of diagonally dominant linear algebraic equations. The Gauss-Seidel iterative procedure then used to solve the linear system thus has assured convergence. To have optimized convergence rate, numerical experiments were done by using a combination of factors involving multi-time-stepping, spatial step size, and degree of the polynomial fit in time. It is shown that the hybrid technique is reliable, accurate, and easy to apply.
Fractional Fokker-Planck equation and oscillatory behavior of cumulant moments
International Nuclear Information System (INIS)
Suzuki, N.; Biyajima, M.
2002-01-01
The Fokker-Planck equation is considered, which is connected to the birth and death process with immigration by the Poisson transform. The fractional derivative in time variable is introduced into the Fokker-Planck equation in order to investigate an origin of oscillatory behavior of cumulant moments. From its solution (the probability density function), the generating function (GF) for the corresponding probability distribution is derived. We consider the case when the GF reduces to that of the negative binomial distribution (NBD), if the fractional derivative is replaced to the ordinary one. The H j moment derived from the GF of the NBD decreases monotonically as the rank j increases. However, the H j moment derived in our approach oscillates, which is contrasted with the case of the NBD. Calculated H j moments are compared with those of charged multiplicities observed in pp-bar, e + e - , and e + p collisions. A phenomenological meaning of introducing the fractional derivative in time variable is discussed
Gong, Chunye; Bao, Weimin; Tang, Guojian; Jiang, Yuewen; Liu, Jie
2014-01-01
It is very time consuming to solve fractional differential equations. The computational complexity of two-dimensional fractional differential equation (2D-TFDE) with iterative implicit finite difference method is O(M(x)M(y)N(2)). In this paper, we present a parallel algorithm for 2D-TFDE and give an in-depth discussion about this algorithm. A task distribution model and data layout with virtual boundary are designed for this parallel algorithm. The experimental results show that the parallel algorithm compares well with the exact solution. The parallel algorithm on single Intel Xeon X5540 CPU runs 3.16-4.17 times faster than the serial algorithm on single CPU core. The parallel efficiency of 81 processes is up to 88.24% compared with 9 processes on a distributed memory cluster system. We do think that the parallel computing technology will become a very basic method for the computational intensive fractional applications in the near future.
Chen, Hao; Lv, Wen; Zhang, Tongtong
2018-05-01
We study preconditioned iterative methods for the linear system arising in the numerical discretization of a two-dimensional space-fractional diffusion equation. Our approach is based on a formulation of the discrete problem that is shown to be the sum of two Kronecker products. By making use of an alternating Kronecker product splitting iteration technique we establish a class of fixed-point iteration methods. Theoretical analysis shows that the new method converges to the unique solution of the linear system. Moreover, the optimal choice of the involved iteration parameters and the corresponding asymptotic convergence rate are computed exactly when the eigenvalues of the system matrix are all real. The basic iteration is accelerated by a Krylov subspace method like GMRES. The corresponding preconditioner is in a form of a Kronecker product structure and requires at each iteration the solution of a set of discrete one-dimensional fractional diffusion equations. We use structure preserving approximations to the discrete one-dimensional fractional diffusion operators in the action of the preconditioning matrix. Numerical examples are presented to illustrate the effectiveness of this approach.
Brouwers, J.J.H.
2010-01-01
We derive the Langevin equation describing the stochastic process of fluid particle motion in wall-induced turbulence (turbulent flow in pipes, channels, and boundary layers including the atmospheric surface layer). The analysis is based on the asymptotic behavior at a large Reynolds number. We use
Diffusion with space memory modelled with distributed order space fractional differential equations
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M. Caputo
2003-06-01
Full Text Available Distributed order fractional differential equations (Caputo, 1995, 2001; Bagley and Torvik, 2000a,b were fi rst used in the time domain; they are here considered in the space domain and introduced in the constitutive equation of diffusion. The solution of the classic problems are obtained, with closed form formulae. In general, the Green functions act as low pass fi lters in the frequency domain. The major difference with the case when a single space fractional derivative is present in the constitutive equations of diffusion (Caputo and Plastino, 2002 is that the solutions found here are potentially more fl exible to represent more complex media (Caputo, 2001a. The difference between the space memory medium and that with the time memory is that the former is more fl exible to represent local phenomena while the latter is more fl exible to represent variations in space. Concerning the boundary value problem, the difference with the solution of the classic diffusion medium, in the case when a constant boundary pressure is assigned and in the medium the pressure is initially nil, is that one also needs to assign the fi rst order space derivative at the boundary.
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Jinfeng Wang
2014-01-01
Full Text Available We discuss and analyze an H1-Galerkin mixed finite element (H1-GMFE method to look for the numerical solution of time fractional telegraph equation. We introduce an auxiliary variable to reduce the original equation into lower-order coupled equations and then formulate an H1-GMFE scheme with two important variables. We discretize the Caputo time fractional derivatives using the finite difference methods and approximate the spatial direction by applying the H1-GMFE method. Based on the discussion on the theoretical error analysis in L2-norm for the scalar unknown and its gradient in one dimensional case, we obtain the optimal order of convergence in space-time direction. Further, we also derive the optimal error results for the scalar unknown in H1-norm. Moreover, we derive and analyze the stability of H1-GMFE scheme and give the results of a priori error estimates in two- or three-dimensional cases. In order to verify our theoretical analysis, we give some results of numerical calculation by using the Matlab procedure.
Exact Solutions of the Time Fractional BBM-Burger Equation by Novel (G′/G-Expansion Method
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Muhammad Shakeel
2014-01-01
Full Text Available The fractional derivatives are used in the sense modified Riemann-Liouville to obtain exact solutions for BBM-Burger equation of fractional order. This equation can be converted into an ordinary differential equation by using a persistent fractional complex transform and, as a result, hyperbolic function solutions, trigonometric function solutions, and rational solutions are attained. The performance of the method is reliable, useful, and gives newer general exact solutions with more free parameters than the existing methods. Numerical results coupled with the graphical representation completely reveal the trustworthiness of the method.
Singh, Brajesh K; Srivastava, Vineet K
2015-04-01
The main goal of this paper is to present a new approximate series solution of the multi-dimensional (heat-like) diffusion equation with time-fractional derivative in Caputo form using a semi-analytical approach: fractional-order reduced differential transform method (FRDTM). The efficiency of FRDTM is confirmed by considering four test problems of the multi-dimensional time fractional-order diffusion equation. FRDTM is a very efficient, effective and powerful mathematical tool which provides exact or very close approximate solutions for a wide range of real-world problems arising in engineering and natural sciences, modelled in terms of differential equations.
Langevin dynamics of A+A reactions in one dimension
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Sancho, J M; Romero, A H; Lacasta, A M; Lindenberg, Katja
2007-01-01
We propose a set of Langevin equations of motion together with a reaction rule for the study of binary reactions. Our scheme is designed to address this problem for arbitrary friction γ and temperature T. It easily accommodates the inclusion of a substrate potential, and it lends itself to straightforward numerical integration. We test this approach on diffusion-limited (γ → ∞) as well as ballistic (γ = 0) A+A → P reactions for which there are extensive exact and approximate theoretical results as well as extensive Monte Carlo results. We reproduce the known results using our integration scheme, and also present new results for the ballistic reactions
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Dali Zhang
2012-01-01
Full Text Available This paper deals with an inverse problem for identifying multiparameters in 1D space fractional advection dispersion equation (FADE on a finite domain with final observations. The parameters to be identified are the fractional order, the diffusion coefficient, and the average velocity in the FADE. The forward problem is solved by a finite difference scheme, and then an optimal perturbation regularization algorithm is introduced to determine the three parameters simultaneously. Numerical inversions are performed both with the accurate data and noisy data, and several factors having influences on realization of the algorithm are discussed. The inversion solutions are in good approximations to the exact solutions demonstrating the efficiency of the proposed algorithm.
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Boergers, C.; Peskin, C.S.
1987-01-01
In the Lagrangian fractional step method introduced in this paper, the fluid velocity and pressure are defined on a collection of N fluid markers. At each time step, these markers are used to generate a Voronoi diagram, and this diagram is used to construct finite-difference operators corresponding to the divergence, gradient, and Laplacian. The splitting of the Navier--Stokes equations leads to discrete Helmholtz and Poisson problems, which we solve using a two-grid method. The nonlinear convection terms are modeled simply by the displacement of the fluid markers. We have implemented this method on a periodic domain in the plane. We describe an efficient algorithm for the numerical construction of periodic Voronoi diagrams, and we report on numerical results which indicate the the fractional step method is convergent of first order. The overall work per time step is proportional to N log N. copyright 1987 Academic Press, Inc
Existence of standing waves for Schrodinger equations involving the fractional Laplacian
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Everaldo S. de Medeiros
2017-03-01
Full Text Available We study a class of fractional Schrodinger equations of the form $$ \\varepsilon^{2\\alpha}(-\\Delta^\\alpha u+ V(xu = f(x,u \\quad\\text{in } \\mathbb{R}^N, $$ where $\\varepsilon$ is a positive parameter, $0 < \\alpha < 1$, $2\\alpha < N$, $(-\\Delta^\\alpha$ is the fractional Laplacian, $V:\\mathbb{R}^{N}\\to \\mathbb{R}$ is a potential which may be bounded or unbounded and the nonlinearity $f:\\mathbb{R}^{N}\\times \\mathbb{R}\\to \\mathbb{R}$ is superlinear and behaves like $|u|^{p-2}u$ at infinity for some $2
Error Analysis of a Finite Element Method for the Space-Fractional Parabolic Equation
Jin, Bangti; Lazarov, Raytcho; Pasciak, Joseph; Zhou, Zhi
2014-01-01
© 2014 Society for Industrial and Applied Mathematics We consider an initial boundary value problem for a one-dimensional fractional-order parabolic equation with a space fractional derivative of Riemann-Liouville type and order α ∈ (1, 2). We study a spatial semidiscrete scheme using the standard Galerkin finite element method with piecewise linear finite elements, as well as fully discrete schemes based on the backward Euler method and the Crank-Nicolson method. Error estimates in the L2(D)- and Hα/2 (D)-norm are derived for the semidiscrete scheme and in the L2(D)-norm for the fully discrete schemes. These estimates cover both smooth and nonsmooth initial data and are expressed directly in terms of the smoothness of the initial data. Extensive numerical results are presented to illustrate the theoretical results.
Existence of solution for a general fractional advection-dispersion equation
Torres Ledesma, César E.
2018-05-01
In this work, we consider the existence of solution to the following fractional advection-dispersion equation -d/dt ( p {_{-∞}}It^{β }(u'(t)) + q {t}I_{∞}^{β }(u'(t))) + b(t)u = f(t, u(t)),t\\in R where β \\in (0,1) , _{-∞}It^{β } and tI_{∞}^{β } denote left and right Liouville-Weyl fractional integrals of order β respectively, 0continuous functions. Due to the general assumption on the constant p and q, the problem (0.1) does not have a variational structure. Despite that, here we study it performing variational methods, combining with an iterative technique, and give an existence criteria of solution for the problem (0.1) under suitable assumptions.
Analytical modeling for fractional multi-dimensional diffusion equations by using Laplace transform
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Devendra Kumar
2015-01-01
Full Text Available In this paper, we propose a simple numerical algorithm for solving multi-dimensional diffusion equations of fractional order which describes density dynamics in a material undergoing diffusion by using homotopy analysis transform method. The fractional derivative is described in the Caputo sense. This homotopy analysis transform method is an innovative adjustment in Laplace transform method and makes the calculation much simpler. The technique is not limited to the small parameter, such as in the classical perturbation method. The scheme gives an analytical solution in the form of a convergent series with easily computable components, requiring no linearization or small perturbation. The numerical solutions obtained by the proposed method indicate that the approach is easy to implement and computationally very attractive.
Zhang, Jinggui
2018-06-01
In this paper, we investigate the dynamical behaviors of the modulation instability (MI) of copropagating optical beams in fractional coupled nonlinear Schrödinger equations (NLSE) with the aim of revealing some novel properties different from those in the conventional coupled NLSE. By applying the standard linear stability method, we first derive an expression for the gain resulting from the instability induced by cross-phase modulation (CPM) in the presence of the Lévy indexes related to fractional effects. It is found that the modulation instability of copropagating optical beams still occurs even in the fractional NLSE with self-defocusing nonlinearity. Then, the analysis of our results further reveals that such Lévy indexes increase the fastest growth frequency and the bandwidth of conventional instability not only for the self-focusing case but also for the self-defocusing case, but do not influence the corresponding maximum gain. Numerical simulations are performed to confirm theoretical predictions. These findings suggest that the novel fractional physical settings may open up new possibilities for the manipulation of MI and nonlinear waves.
Stability properties of solitary waves for fractional KdV and BBM equations
Angulo Pava, Jaime
2018-03-01
This paper sheds new light on the stability properties of solitary wave solutions associated with Korteweg-de Vries-type models when the dispersion is very low. Using a compact, analytic approach and asymptotic perturbation theory, we establish sufficient conditions for the existence of exponentially growing solutions to the linearized problem and so a criterium of spectral instability of solitary waves is obtained for both models. Moreover, the nonlinear stability and spectral instability of the ground state solutions for both models is obtained for some specific regimen of parameters. Via a Lyapunov strategy and a variational analysis, we obtain the stability of the blow-up of solitary waves for the critical fractional KdV equation. The arguments presented in this investigation show promise for use in the study of the instability of traveling wave solutions of other nonlinear evolution equations.
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Fakhrodin Mohammadi
2017-10-01
Full Text Available Stochastic fractional differential equations (SFDEs have been used for modeling many physical problems in the fields of turbulance, heterogeneous, flows and matrials, viscoelasticity and electromagnetic theory. In this paper, an efficient wavelet Galerkin method based on the second kind Chebyshev wavelets are proposed for approximate solution of SFDEs. In this approach, operational matrices of the second kind Chebyshev wavelets are used for reducing SFDEs to a linear system of algebraic equations that can be solved easily. Convergence and error analysis of the proposed method is considered. Some numerical examples are performed to confirm the applicability and efficiency of the proposed method.
Asymptotic integration of some nonlinear differential equations with fractional time derivative
International Nuclear Information System (INIS)
Baleanu, Dumitru; Agarwal, Ravi P; Mustafa, Octavian G; Cosulschi, Mirel
2011-01-01
We establish that, under some simple integral conditions regarding the nonlinearity, the (1 + α)-order fractional differential equation 0 D α t (x') + f(t, x) = 0, t > 0, has a solution x element of C([0,+∞),R) intersection C 1 ((0,+∞),R), with lim t→0 [t 1-α x'(t)] element of R, which can be expanded asymptotically as a + bt α + O(t α-1 ) when t → +∞ for given real numbers a, b. Our arguments are based on fixed point theory. Here, 0 D α t designates the Riemann-Liouville derivative of order α in (0, 1).
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Chuanzhi Bai
2010-06-01
Full Text Available This paper deals with the existence of positive solutions for a boundary value problem involving a nonlinear functional differential equation of fractional order $\\alpha$ given by $ D^{\\alpha} u(t + f(t, u_t = 0$, $t \\in (0, 1$, $2 < \\alpha \\le 3$, $ u^{\\prime}(0 = 0$, $u^{\\prime}(1 = b u^{\\prime}(\\eta$, $u_0 = \\phi$. Our results are based on the nonlinear alternative of Leray-Schauder type and Krasnosel'skii fixed point theorem.
A Note on a Semilinear Fractional Differential Equation of Neutral Type with Infinite Delay
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Gisle M. Mophou
2010-01-01
Full Text Available We deal in this paper with the mild solution for the semilinear fractional differential equation of neutral type with infinite delay: Dαx(t+Ax(t=f(t,xt, t∈[0,T], x(t=ϕ(t, t∈]−∞,0], with T>0 and 0<α<1. We prove the existence (and uniqueness of solutions, assuming that −A is a linear closed operator which generates an analytic semigroup (T(tt≥0 on a Banach space 𝕏 by means of the Banach's fixed point theorem. This generalizes some recent results.
Preconditioned iterative methods for space-time fractional advection-diffusion equations
Zhao, Zhi; Jin, Xiao-Qing; Lin, Matthew M.
2016-08-01
In this paper, we propose practical numerical methods for solving a class of initial-boundary value problems of space-time fractional advection-diffusion equations. First, we propose an implicit method based on two-sided Grünwald formulae and discuss its stability and consistency. Then, we develop the preconditioned generalized minimal residual (preconditioned GMRES) method and preconditioned conjugate gradient normal residual (preconditioned CGNR) method with easily constructed preconditioners. Importantly, because resulting systems are Toeplitz-like, fast Fourier transform can be applied to significantly reduce the computational cost. We perform numerical experiments to demonstrate the efficiency of our preconditioners, even in cases with variable coefficients.
Energy Technology Data Exchange (ETDEWEB)
Carella, Alfredo Raul
2012-09-15
Quantifying species transport rates is a main concern in chemical and petrochemical industries. In particular, the design and operation of many large-scale industrial chemical processes is as much dependent on diffusion as it is on reaction rates. However, the existing diffusion models sometimes fail to predict experimentally observed behaviors and their accuracy is usually insufficient for process optimization purposes. Fractional diffusion models offer multiple possibilities for generalizing Flick's law in a consistent manner in order to account for history dependence and nonlocal effects. These models have not been extensively applied to the study of real systems, mainly due to their computational cost and mathematical complexity. A least squares spectral formulation was developed for solving fractional differential equations. The proposed method was proven particularly well-suited for dealing with the numerical difficulties inherent to fractional differential operators. The practical implementation was explained in detail in order to enhance reproducibility, and directions were specified for extending it to multiple dimensions and arbitrarily shaped domains. A numerical framework based on the least-squares spectral element method was developed for studying and comparing anomalous diffusion models in pellets. This simulation tool is capable of solving arbitrary integro-differential equations and can be effortlessly adapted to various problems in any number of dimensions. Simulations of the flow around a cylindrical particle were achieved by extending the functionality of the developed framework. A test case was analyzed by coupling the boundary condition yielded by the fluid model with two families of anomalous diffusion models: hyperbolic diffusion and fractional diffusion. Qualitative guidelines for determining the suitability of diffusion models can be formulated by complementing experimental data with the results obtained from this approach.(Author)
Lin, Yezhi; Liu, Yinping; Li, Zhibin
2013-01-01
The Adomian decomposition method (ADM) is one of the most effective methods to construct analytic approximate solutions for nonlinear differential equations. In this paper, based on the new definition of the Adomian polynomials, Rach (2008) [22], the Adomian decomposition method and the Padé approximants technique, a new algorithm is proposed to construct analytic approximate solutions for nonlinear fractional differential equations with initial or boundary conditions. Furthermore, a MAPLE software package is developed to implement this new algorithm, which is user-friendly and efficient. One only needs to input the system equation, initial or boundary conditions and several necessary parameters, then our package will automatically deliver the analytic approximate solutions within a few seconds. Several different types of examples are given to illustrate the scope and demonstrate the validity of our package, especially for non-smooth initial value problems. Our package provides a helpful and easy-to-use tool in science and engineering simulations. Program summaryProgram title: ADMP Catalogue identifier: AENE_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AENE_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 12011 No. of bytes in distributed program, including test data, etc.: 575551 Distribution format: tar.gz Programming language: MAPLE R15. Computer: PCs. Operating system: Windows XP/7. RAM: 2 Gbytes Classification: 4.3. Nature of problem: Constructing analytic approximate solutions of nonlinear fractional differential equations with initial or boundary conditions. Non-smooth initial value problems can be solved by this program. Solution method: Based on the new definition of the Adomian polynomials [1], the Adomian decomposition method and the Pad
Ahmed M. A. El-Sayed; Ebtisam O. Bin-Taher
2011-01-01
In this article, we prove the existence of positive nondecreasing solutions for a multi-term fractional-order functional differential equations. We consider Cauchy boundary problems with: nonlocal conditions, two-point boundary conditions, integral conditions, and deviated arguments.
Inc, Mustafa; Yusuf, Abdullahi; Aliyu, Aliyu Isa; Baleanu, Dumitru
2018-04-01
This paper studies the symmetry analysis, explicit solutions, convergence analysis, and conservation laws (Cls) for two different space-time fractional nonlinear evolution equations with Riemann-Liouville (RL) derivative. The governing equations are reduced to nonlinear ordinary differential equation (ODE) of fractional order using their Lie point symmetries. In the reduced equations, the derivative is in Erdelyi-Kober (EK) sense, power series technique is applied to derive an explicit solutions for the reduced fractional ODEs. The convergence of the obtained power series solutions is also presented. Moreover, the new conservation theorem and the generalization of the Noether operators are developed to construct the nonlocal Cls for the equations . Some interesting figures for the obtained explicit solutions are presented.
Multi-wave solutions of the space–time fractional Burgers and Sharma–Tasso–Olver equations
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Emad A.-B. Abdel-Salam
2016-03-01
Full Text Available Based on the improved generalized exp-function method, the space–time fractional Burgers and Sharma–Tasso–Olver equations were studied. The single-wave, double-wave, three-wave and four-wave solution discussed. With the best of our knowledge, some of the results are obtained for the first time. The improved generalized exp-function method can be applied to other fractional differential equations.
Multi-wave solutions of the space–time fractional Burgers and Sharma–Tasso–Olver equations
Emad A.-B. Abdel-Salam; Gamal F. Hassan
2016-01-01
Based on the improved generalized exp-function method, the space–time fractional Burgers and Sharma–Tasso–Olver equations were studied. The single-wave, double-wave, three-wave and four-wave solution discussed. With the best of our knowledge, some of the results are obtained for the first time. The improved generalized exp-function method can be applied to other fractional differential equations.
Liu, Yikan
2015-01-01
In this paper, we establish a strong maximum principle for fractional diffusion equations with multiple Caputo derivatives in time, and investigate a related inverse problem of practical importance. Exploiting the solution properties and the involved multinomial Mittag-Leffler functions, we improve the weak maximum principle for the multi-term time-fractional diffusion equation to a stronger one, which is parallel to that for its single-term counterpart as expected. As a direct application, w...
Mehdinejadiani, Behrouz
2017-08-01
This study represents the first attempt to estimate the solute transport parameters of the spatial fractional advection-dispersion equation using Bees Algorithm. The numerical studies as well as the experimental studies were performed to certify the integrity of Bees Algorithm. The experimental ones were conducted in a sandbox for homogeneous and heterogeneous soils. A detailed comparative study was carried out between the results obtained from Bees Algorithm and those from Genetic Algorithm and LSQNONLIN routines in FracFit toolbox. The results indicated that, in general, the Bees Algorithm much more accurately appraised the sFADE parameters in comparison with Genetic Algorithm and LSQNONLIN, especially in the heterogeneous soil and for α values near to 1 in the numerical study. Also, the results obtained from Bees Algorithm were more reliable than those from Genetic Algorithm. The Bees Algorithm showed the relative similar performances for all cases, while the Genetic Algorithm and the LSQNONLIN yielded different performances for various cases. The performance of LSQNONLIN strongly depends on the initial guess values so that, compared to the Genetic Algorithm, it can more accurately estimate the sFADE parameters by taking into consideration the suitable initial guess values. To sum up, the Bees Algorithm was found to be very simple, robust and accurate approach to estimate the transport parameters of the spatial fractional advection-dispersion equation. Copyright © 2017 Elsevier B.V. All rights reserved.
Liang, Yingjie; Chen, Wen; Magin, Richard L.
2016-07-01
Analytical solutions to the fractional diffusion equation are often obtained by using Laplace and Fourier transforms, which conveniently encode the order of the time and the space derivatives (α and β) as non-integer powers of the conjugate transform variables (s, and k) for the spectral and the spatial frequencies, respectively. This study presents a new solution to the fractional diffusion equation obtained using the Laplace transform and expressed as a Fox's H-function. This result clearly illustrates the kinetics of the underlying stochastic process in terms of the Laplace spectral frequency and entropy. The spectral entropy is numerically calculated by using the direct integration method and the adaptive Gauss-Kronrod quadrature algorithm. Here, the properties of spectral entropy are investigated for the cases of sub-diffusion and super-diffusion. We find that the overall spectral entropy decreases with the increasing α and β, and that the normal or Gaussian case with α = 1 and β = 2, has the lowest spectral entropy (i.e., less information is needed to describe the state of a Gaussian process). In addition, as the neighborhood over which the entropy is calculated increases, the spectral entropy decreases, which implies a spatial averaging or coarse graining of the material properties. Consequently, the spectral entropy is shown to provide a new way to characterize the temporal correlation of anomalous diffusion. Future studies should be designed to examine changes of spectral entropy in physical, chemical and biological systems undergoing phase changes, chemical reactions and tissue regeneration.
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Lingju Kong
2013-04-01
Full Text Available We study the existence of multiple solutions to the boundary value problem $$displaylines{ frac{d}{dt}Big(frac12{}_0D_t^{-eta}(u'(t+frac12{}_tD_T^{-eta}(u'(t Big+lambda abla F(t,u(t=0,quad tin [0,T],cr u(0=u(T=0, }$$ where $T>0$, $lambda>0$ is a parameter, $0leqeta<1$, ${}_0D_t^{-eta}$ and ${}_tD_T^{-eta}$ are, respectively, the left and right Riemann-Liouville fractional integrals of order $eta$, $F: [0,T]imesmathbb{R}^Nomathbb{R}$ is a given function. Our interest in the above system arises from studying the steady fractional advection dispersion equation. By applying variational methods, we obtain sufficient conditions under which the above equation has at least three solutions. Our results are new even for the special case when $eta=0$. Examples are provided to illustrate the applicability of our results.
Mehdinejadiani, Behrouz
2017-08-01
This study represents the first attempt to estimate the solute transport parameters of the spatial fractional advection-dispersion equation using Bees Algorithm. The numerical studies as well as the experimental studies were performed to certify the integrity of Bees Algorithm. The experimental ones were conducted in a sandbox for homogeneous and heterogeneous soils. A detailed comparative study was carried out between the results obtained from Bees Algorithm and those from Genetic Algorithm and LSQNONLIN routines in FracFit toolbox. The results indicated that, in general, the Bees Algorithm much more accurately appraised the sFADE parameters in comparison with Genetic Algorithm and LSQNONLIN, especially in the heterogeneous soil and for α values near to 1 in the numerical study. Also, the results obtained from Bees Algorithm were more reliable than those from Genetic Algorithm. The Bees Algorithm showed the relative similar performances for all cases, while the Genetic Algorithm and the LSQNONLIN yielded different performances for various cases. The performance of LSQNONLIN strongly depends on the initial guess values so that, compared to the Genetic Algorithm, it can more accurately estimate the sFADE parameters by taking into consideration the suitable initial guess values. To sum up, the Bees Algorithm was found to be very simple, robust and accurate approach to estimate the transport parameters of the spatial fractional advection-dispersion equation.
Nonlinear von Neumann equations for quantum dissipative systems
International Nuclear Information System (INIS)
Messer, J.; Baumgartner, B.
1978-01-01
For pure states nonlinear Schroedinger equations, the so-called Schroedinger-Langevin equations are well-known to model quantum dissipative systems of the Langevin type. For mixtures it is shown that these wave equations do not extend to master equations, but to corresponding nonlinear von Neumann equations. Solutions for the damped harmonic oscillator are discussed. (Auth.)
Nonlinear von Neumann equations for quantum dissipative systems
International Nuclear Information System (INIS)
Messer, J.; Baumgartner, B.
For pure states nonlinear Schroedinger equations, the so-called Schroedinger-Langevin equations are well-known to model quantum dissipative systems of the Langevin type. For mixtures it is shown that these wave equations do not extend to master equations, but to corresponding nonlinear von Neumann equations. Solutions for the damped harmonic oscillator are discussed. (Author)
Li, Zhiyuan; Yamamoto, Masahiro
2014-01-01
This article proves the uniqueness for two kinds of inverse problems of identifying fractional orders in diffusion equations with multiple time-fractional derivatives by pointwise observation. By means of eigenfunction expansion and Laplace transform, we reduce the uniqueness for our inverse problems to the uniqueness of expansions of some special function and complete the proof.
Owolabi, Kolade M.
2017-03-01
In this paper, some nonlinear space-fractional order reaction-diffusion equations (SFORDE) on a finite but large spatial domain x ∈ [0, L], x = x(x , y , z) and t ∈ [0, T] are considered. Also in this work, the standard reaction-diffusion system with boundary conditions is generalized by replacing the second-order spatial derivatives with Riemann-Liouville space-fractional derivatives of order α, for 0 Fourier spectral method is introduced as a better alternative to existing low order schemes for the integration of fractional in space reaction-diffusion problems in conjunction with an adaptive exponential time differencing method, and solve a range of one-, two- and three-components SFORDE numerically to obtain patterns in one- and two-dimensions with a straight forward extension to three spatial dimensions in a sub-diffusive (0 reaction-diffusion case. With application to models in biology and physics, different spatiotemporal dynamics are observed and displayed.
Edelman, Mark
2015-07-01
In this paper, we consider a simple general form of a deterministic system with power-law memory whose state can be described by one variable and evolution by a generating function. A new value of the system's variable is a total (a convolution) of the generating functions of all previous values of the variable with weights, which are powers of the time passed. In discrete cases, these systems can be described by difference equations in which a fractional difference on the left hand side is equal to a total (also a convolution) of the generating functions of all previous values of the system's variable with the fractional Eulerian number weights on the right hand side. In the continuous limit, the considered systems can be described by the Grünvald-Letnikov fractional differential equations, which are equivalent to the Volterra integral equations of the second kind. New properties of the fractional Eulerian numbers and possible applications of the results are discussed.
International Nuclear Information System (INIS)
Schramm, Marcelo; Bodmann, Bardo E.J.; Vilhena, Marco T.M.B.; Petersen, Claudio Z.; Alvim, Antonio C.M.
2013-01-01
Following the quest to find analytical solutions, we extend the methodology applied successfully to timely fractional neutron point kinetics (FNPK) equations by adding the effects of temperature. The FNPK equations with temperature feedback correspond to a nonlinear system and “stiff” type for the neutron density and the concentration of delayed neutron precursors. These variables determine the behavior of a nuclear reactor power with time and are influenced by the position of control rods, for example. The solutions of kinetics equations provide time information about the dynamics in a nuclear reactor in operation and are useful, for example, to understand the power fluctuations with time that occur during startup or shutdown of the reactor, due to adjustments of the control rods. The inclusion of temperature feedback in the model introduces an estimate of the transient behavior of the power and other variables, which are strongly coupled. Normally, a single value of reactivity is used across the energy spectrum. Especially in case of power change, the neutron energy spectrum changes as well as physical parameters such as the average cross sections. However, even knowing the importance of temperature effects on the control of the reactor power, the character of the set of nonlinear equations governing this system makes it difficult to obtain a purely analytical solution. Studies have been published in this sense, using numerical approaches. Here the idea is to consider temperature effects to make the model more realistic and thus solve it in a semi-analytical way. Therefore, the main objective of this paper is to obtain an analytical representation of fractional neutron point kinetics equations with temperature feedback, without having to resort to approximations inherent in numerical methods. To this end, we will use the decomposition method, which has been successfully used by the authors to solve neutron point kinetics problems. The results obtained will