Nonlinear Terms of MHD Equations for Homogeneous Magnetized Shear Flow
Dimitrov, Z D; Hristov, T S; Mishonov, T M
2011-01-01
We have derived the full set of MHD equations for incompressible shear flow of a magnetized fluid and considered their solution in the wave-vector space. The linearized equations give the famous amplification of slow magnetosonic waves and describe the magnetorotational instability. The nonlinear terms in our analysis are responsible for the creation of turbulence and self-sustained spectral density of the MHD (Alfven and pseudo-Alfven) waves. Perspectives for numerical simulations of weak turbulence and calculation of the effective viscosity of accretion disks are shortly discussed in k-space.
Nonlinear helical MHD instability
Energy Technology Data Exchange (ETDEWEB)
Zueva, N.M.; Solov' ev, L.S.
1977-07-01
An examination is made of the boundary problem on the development of MHD instability in a toroidal plasma. Two types of local helical instability are noted - Alfven and thermal, and the corresponding criteria of instability are cited. An evaluation is made of the maximum attainable kinetic energy, limited by the degree to which the law of conservation is fulfilled. An examination is made of a precise solution to a kinematic problem on the helical evolution of a cylindrical magnetic configuration at a given velocity distribution in a plasma. A numerical computation of the development of MHD instability in a plasma cylinder by a computerized solution of MHD equations is made where the process's helical symmetry is conserved. The development of instability is of a resonance nature. The instability involves the entire cross section of the plasma and leads to an inside-out reversal of the magnetic surfaces when there is a maximum unstable equilibrium configuration in the nonlinear stage. The examined instability in the tore is apparently stabilized by a magnetic hole when certain limitations are placed on the distribution of flows in the plasma. 29 references, 8 figures.
Proceedings of the workshop on nonlinear MHD and extended MHD
Energy Technology Data Exchange (ETDEWEB)
NONE
1998-12-01
Nonlinear MHD simulations have proven their value in interpreting experimental results over the years. As magnetic fusion experiments reach higher performance regimes, more sophisticated experimental diagnostics coupled with ever expanding computer capabilities have increased both the need for and the feasibility of nonlinear global simulations using models more realistic than regular ideal and resistive MHD. Such extended-MHD nonlinear simulations have already begun to produce useful results. These studies are expected to lead to ever more comprehensive simulation models in the future and to play a vital role in fully understanding fusion plasmas. Topics include the following: (1) current state of nonlinear MHD and extended-MHD simulations; (2) comparisons to experimental data; (3) discussions between experimentalists and theorists; (4) /equations for extended-MHD models, kinetic-based closures; and (5) paths toward more comprehensive simulation models, etc. Selected papers have been indexed separately for inclusion in the Energy Science and Technology Database.
Problems in nonlinear resistive MHD
Energy Technology Data Exchange (ETDEWEB)
Turnbull, A.D.; Strait, E.J.; La Haye, R.J.; Chu, M.S.; Miller, R.L. [General Atomics, San Diego, CA (United States)
1998-12-31
Two experimentally relevant problems can relatively easily be tackled by nonlinear MHD codes. Both problems require plasma rotation in addition to the nonlinear mode coupling and full geometry already incorporated into the codes, but no additional physics seems to be crucial. These problems discussed here are: (1) nonlinear coupling and interaction of multiple MHD modes near the B limit and (2) nonlinear coupling of the m/n = 1/1 sawtooth mode with higher n gongs and development of seed islands outside q = 1.
The Nonlinear Magnetosphere: Expressions in MHD and in Kinetic Models
Hesse, Michael; Birn, Joachim
2011-01-01
Like most plasma systems, the magnetosphere of the Earth is governed by nonlinear dynamic evolution equations. The impact of nonlinearities ranges from large scales, where overall dynamics features are exhibiting nonlinear behavior, to small scale, kinetic, processes, where nonlinear behavior governs, among others, energy conversion and dissipation. In this talk we present a select set of examples of such behavior, with a specific emphasis on how nonlinear effects manifest themselves in MHD and in kinetic models of magnetospheric plasma dynamics.
Westerhof, E.; de Blank, H. J.; Pratt, J.
2016-03-01
Two dimensional reduced MHD simulations of neoclassical tearing mode growth and suppression by ECCD are performed. The perturbation of the bootstrap current density and the EC drive current density perturbation are assumed to be functions of the perturbed flux surfaces. In the case of ECCD, this implies that the applied power is flux surface averaged to obtain the EC driven current density distribution. The results are consistent with predictions from the generalized Rutherford equation using common expressions for Δ \\text{bs}\\prime and Δ \\text{ECCD}\\prime . These expressions are commonly perceived to describe only the effect on the tearing mode growth of the helical component of the respective current perturbation acting through the modification of Ohm’s law. Our results show that they describe in addition the effect of the poloidally averaged current density perturbation which acts through modification of the tearing mode stability index. Except for modulated ECCD, the largest contribution to the mode growth comes from this poloidally averaged current density perturbation.
Linear and Nonlinear MHD Wave Processes in Plasmas. Final Report
Energy Technology Data Exchange (ETDEWEB)
Tataronis, J. A.
2004-06-01
This program treats theoretically low frequency linear and nonlinear wave processes in magnetized plasmas. A primary objective has been to evaluate the effectiveness of MHD waves to heat plasma and drive current in toroidal configurations. The research covers the following topics: (1) the existence and properties of the MHD continua in plasma equilibria without spatial symmetry; (2) low frequency nonresonant current drive and nonlinear Alfven wave effects; and (3) nonlinear electron acceleration by rf and random plasma waves. Results have contributed to the fundamental knowledge base of MHD activity in symmetric and asymmetric toroidal plasmas. Among the accomplishments of this research effort, the following are highlighted: Identification of the MHD continuum mode singularities in toroidal geometry. Derivation of a third order ordinary differential equation that governs nonlinear current drive in the singular layers of the Alfvkn continuum modes in axisymmetric toroidal geometry. Bounded solutions of this ODE implies a net average current parallel to the toroidal equilibrium magnetic field. Discovery of a new unstable continuum of the linearized MHD equation in axially periodic circular plasma cylinders with shear and incompressibility. This continuum, which we named “accumulation continuum” and which is related to ballooning modes, arises as discrete unstable eigenfrequency accumulate on the imaginary frequency axis in the limit of large mode numbers. Development of techniques to control nonlinear electron acceleration through the action of multiple coherent and random plasmas waves. Two important elements of this program aye student participation and student training in plasma theory.
Dynamics of nonlinear resonant slow MHD waves in twisted flux tubes
Directory of Open Access Journals (Sweden)
R. Erdélyi
2002-01-01
Full Text Available Nonlinear resonant magnetohydrodynamic (MHD waves are studied in weakly dissipative isotropic plasmas in cylindrical geometry. This geometry is suitable and is needed when one intends to study resonant MHD waves in magnetic flux tubes (e.g. for sunspots, coronal loops, solar plumes, solar wind, the magnetosphere, etc. The resonant behaviour of slow MHD waves is confined in a narrow dissipative layer. Using the method of simplified matched asymptotic expansions inside and outside of the narrow dissipative layer, we generalise the so-called connection formulae obtained in linear MHD for the Eulerian perturbation of the total pressure and for the normal component of the velocity. These connection formulae for resonant MHD waves across the dissipative layer play a similar role as the well-known Rankine-Hugoniot relations connecting solutions at both sides of MHD shock waves. The key results are the nonlinear connection formulae found in dissipative cylindrical MHD which are an important extension of their counterparts obtained in linear ideal MHD (Sakurai et al., 1991, linear dissipative MHD (Goossens et al., 1995; Erdélyi, 1997 and in nonlinear dissipative MHD derived in slab geometry (Ruderman et al., 1997. These generalised connection formulae enable us to connect solutions obtained at both sides of the dissipative layer without solving the MHD equations in the dissipative layer possibly saving a considerable amount of CPU-time when solving the full nonlinear resonant MHD problem.
Nonlinear MHD dynamo operating at equipartition
DEFF Research Database (Denmark)
Archontis, V.; Dorch, Bertil; Nordlund, Åke
2007-01-01
Context.We present results from non linear MHD dynamo experiments with a three-dimensional steady and smooth flow that drives fast dynamo action in the kinematic regime. In the saturation regime, the system yields strong magnetic fields, which undergo transitions between an energy-equipartition a......Context.We present results from non linear MHD dynamo experiments with a three-dimensional steady and smooth flow that drives fast dynamo action in the kinematic regime. In the saturation regime, the system yields strong magnetic fields, which undergo transitions between an energy......-equipartition and a turbulent state. The generation and evolution of such strong magnetic fields is relevant for the understanding of dynamo action that occurs in stars and other astrophysical objects. Aims.We study the mode of operation of this dynamo, in the linear and non-linear saturation regimes. We also consider...... the effect of varying the magnetic and fluid Reymolds number on the non-linear behaviour of the system. Methods.We perform three-dimensional non-linear MHD simulations and visualization using a high resolution numerical scheme. Results.We find that this dynamo has a high growth rate in the linear regime...
MAGNETOHYDRODYNAMIC EQUATIONS (MHD GENERATION CODE
Directory of Open Access Journals (Sweden)
Francisco Frutos Alfaro
2017-04-01
Full Text Available A program to generate codes in Fortran and C of the full magnetohydrodynamic equations is shown. The program uses the free computer algebra system software REDUCE. This software has a package called EXCALC, which is an exterior calculus program. The advantage of this program is that it can be modified to include another complex metric or spacetime. The output of this program is modified by means of a LINUX script which creates a new REDUCE program to manipulate the magnetohydrodynamic equations to obtain a code that can be used as a seed for a magnetohydrodynamic code for numerical applications. As an example, we present part of the output of our programs for Cartesian coordinates and how to do the discretization.
NONLINEAR MHD WAVES IN A PROMINENCE FOOT
Energy Technology Data Exchange (ETDEWEB)
Ofman, L. [Catholic University of America, Washington, DC 20064 (United States); Knizhnik, K.; Kucera, T. [NASA Goddard Space Flight Center, Code 671, Greenbelt, MD 20771 (United States); Schmieder, B. [LESIA, Observatoire de Paris, PSL Research University, CNRS, Sorbonne Universités, UPMC Univ. Paris 06, Univ. Paris-Diderot, Sorbonne Paris Cit, 5 place Jules Janssen, F-92195 Meudon (France)
2015-11-10
We study nonlinear waves in a prominence foot using a 2.5D MHD model motivated by recent high-resolution observations with Hinode/Solar Optical Telescope in Ca ii emission of a prominence on 2012 October 10 showing highly dynamic small-scale motions in the prominence material. Observations of Hα intensities and of Doppler shifts show similar propagating fluctuations. However, the optically thick nature of the emission lines inhibits a unique quantitative interpretation in terms of density. Nevertheless, we find evidence of nonlinear wave activity in the prominence foot by examining the relative magnitude of the fluctuation intensity (δI/I ∼ δn/n). The waves are evident as significant density fluctuations that vary with height and apparently travel upward from the chromosphere into the prominence material with quasi-periodic fluctuations with a typical period in the range of 5–11 minutes and wavelengths <2000 km. Recent Doppler shift observations show the transverse displacement of the propagating waves. The magnetic field was measured with the THEMIS instrument and was found to be 5–14 G. For the typical prominence density the corresponding fast magnetosonic speed is ∼20 km s{sup −1}, in qualitative agreement with the propagation speed of the detected waves. The 2.5D MHD numerical model is constrained with the typical parameters of the prominence waves seen in observations. Our numerical results reproduce the nonlinear fast magnetosonic waves and provide strong support for the presence of these waves in the prominence foot. We also explore gravitational MHD oscillations of the heavy prominence foot material supported by dipped magnetic field structure.
MHD Equations with Regularity in One Direction
Directory of Open Access Journals (Sweden)
Zujin Zhang
2014-01-01
Full Text Available We consider the 3D MHD equations and prove that if one directional derivative of the fluid velocity, say, ∂3u∈Lp0, T;LqR3, with 2/p + 3/q = γ ∈ [1,3/2, 3/γ ≤ q ≤ 1/(γ - 1, then the solution is in fact smooth. This improves previous results greatly.
3-D nonlinear evolution of MHD instabilities
Energy Technology Data Exchange (ETDEWEB)
Bateman, G.; Hicks, H. R.; Wooten, J. W.
1977-03-01
The nonlinear evolution of ideal MHD internal instabilities is investigated in straight cylindrical geometry by means of a 3-D initial-value computer code. These instabilities are characterized by pairs of velocity vortex cells rolling off each other and helically twisted down the plasma column. The cells persist until the poloidal velocity saturates at a few tenths of the Alfven velocity. The nonlinear phase is characterized by convection around these essentially fixed vortex cells. For example, the initially centrally peaked temperature profile is convected out and around to form an annulus of high temperature surrounding a small region of lower temperature. Weak, centrally localized instabilities do not alter the edge of the plasma. Strong, large-scale instabilities, resulting from a stronger longitudinal equilibrium current, drive the plasma against the wall. After three examples of instability are analyzed in detail, the numerical methods and their verification are discussed.
Directory of Open Access Journals (Sweden)
Wei Khim Ng
2009-02-01
Full Text Available We construct nonlinear extensions of Dirac's relativistic electron equation that preserve its other desirable properties such as locality, separability, conservation of probability and Poincaré invariance. We determine the constraints that the nonlinear term must obey and classify the resultant non-polynomial nonlinearities in a double expansion in the degree of nonlinearity and number of derivatives. We give explicit examples of such nonlinear equations, studying their discrete symmetries and other properties. Motivated by some previously suggested applications we then consider nonlinear terms that simultaneously violate Lorentz covariance and again study various explicit examples. We contrast our equations and construction procedure with others in the literature and also show that our equations are not gauge equivalent to the linear Dirac equation. Finally we outline various physical applications for these equations.
Nonlinear evolution of parallel propagating Alfven waves: Vlasov - MHD simulation
Nariyuki, Y; Kumashiro, T; Hada, T
2009-01-01
Nonlinear evolution of circularly polarized Alfv\\'en waves are discussed by using the recently developed Vlasov-MHD code, which is a generalized Landau-fluid model. The numerical results indicate that as far as the nonlinearity in the system is not so large, the Vlasov-MHD model can validly solve time evolution of the Alfv\\'enic turbulence both in the linear and nonlinear stages. The present Vlasov-MHD model is proper to discuss the solar coronal heating and solar wind acceleration by Alfve\\'n waves propagating from the photosphere.
Nonlinear MHD waves in a Prominence Foot
Ofman, Leon; Kucera, Therese; Schmieder, Brigitte
2015-01-01
We study nonlinear waves in a prominence foot using 2.5D MHD model motivated by recent high-resolution observations with Hinode/SOT in Ca~II emission of a prominence on October 10, 2012 showing highly dynamic small-scale motions in the prominence material. Observations of H$\\alpha$ intensities and of Doppler shifts show similar propagating fluctuations. However the optically thick nature of the emission lines inhibits unique quantitative interpretation in terms of density. Nevertheless, we find evidence of nonlinear wave activity in the prominence foot by examining the relative magnitude of the fluctuation intensity ($\\delta I/I\\sim \\delta n/n$). The waves are evident as significant density fluctuations that vary with height, and apparently travel upward from the chromosphere into the prominence material with quasi-periodic fluctuations with typical period in the range of 5-11 minutes, and wavelengths $\\sim <$2000 km. Recent Doppler shift observations show the transverse displacement of the propagating wav...
Wu Zhuo Qun; Li Hui Lai; Zhao Jun Ning
2001-01-01
Nonlinear diffusion equations, an important class of parabolic equations, come from a variety of diffusion phenomena which appear widely in nature. They are suggested as mathematical models of physical problems in many fields, such as filtration, phase transition, biochemistry and dynamics of biological groups. In many cases, the equations possess degeneracy or singularity. The appearance of degeneracy or singularity makes the study more involved and challenging. Many new ideas and methods have been developed to overcome the special difficulties caused by the degeneracy and singularity, which
Asymptotics for dissipative nonlinear equations
Hayashi, Nakao; Kaikina, Elena I; Shishmarev, Ilya A
2006-01-01
Many of problems of the natural sciences lead to nonlinear partial differential equations. However, only a few of them have succeeded in being solved explicitly. Therefore different methods of qualitative analysis such as the asymptotic methods play a very important role. This is the first book in the world literature giving a systematic development of a general asymptotic theory for nonlinear partial differential equations with dissipation. Many typical well-known equations are considered as examples, such as: nonlinear heat equation, KdVB equation, nonlinear damped wave equation, Landau-Ginzburg equation, Sobolev type equations, systems of equations of Boussinesq, Navier-Stokes and others.
MHD flow of a viscous fluid on a nonlinear porous shrinking sheet with homotopy analysis method
Institute of Scientific and Technical Information of China (English)
S. Nadeem; Anwar Hussain
2009-01-01
The present paper investigates the magnetohydrodynamic (MHD) flow of a viscous fluid towards a nonlinear porous shrinking sheet. The governing equations are simplified by similarity transformations. The reduced problem is then solved by the homotopy analysis method. The pertinent parameters appearing in the problem are discussed graphically and presented in tables. It is found that the shrinking solutions exist in the presence of MHD. It is also observed from the tables that the solutions for f"(0) with different values of parameters are convergent.
Energy Technology Data Exchange (ETDEWEB)
Golovin, Sergey V., E-mail: sergey@hydro.nsc.r [Lavrentyev Institute of Hydrodynamics SB RAS, 630090 Novosibirsk (Russian Federation); Department of Mechanics and Mathematics, Novosibirsk State University, 630090 Novosibirsk (Russian Federation)
2011-01-17
Equations of magnetohydrodynamics (MHD) in the natural curvilinear system of coordinates where trajectories and magnetic lines play a role of coordinate curves are reduced to the non-linear vector wave equation coupled with the incompressibility condition in the form of the generalized Cauchy integral. The symmetry group of obtained equation, equivalence transformation, and group classification with respect to the constitutive equation are calculated. New exact solutions with functional arbitrariness describing non-stationary incompressible flows with constant total pressure are given by explicit formulae. The corresponding magnetic surfaces have the shape of deformed nested cylinders, tori, or knotted tubes.
Global existence and uniqueness of nonlinear evolutionary fluid equations
Qin, Yuming; Wang, Taige
2015-01-01
This book presents recent results on nonlinear evolutionary fluid equations such as the compressible (radiative) magnetohydrodynamics (MHD) equations, compressible viscous micropolar fluid equations, the full non-Newtonian fluid equations and non-autonomous compressible Navier-Stokes equations. These types of partial differential equations arise in many fields of mathematics, but also in other branches of science such as physics and fluid dynamics. This book will be a valuable resource for graduate students and researchers interested in partial differential equations, and will also benefit practitioners in physics and engineering.
Solving Nonlinear Wave Equations by Elliptic Equation
Institute of Scientific and Technical Information of China (English)
FU Zun-Tao; LIU Shi-Da; LIU Shi-Kuo
2003-01-01
The elliptic equation is taken as a transformation and applied to solve nonlinear wave equations. It is shown that this method is more powerful to give more kinds of solutions, such as rational solutions, solitary wave solutions,periodic wave solutions and so on, so it can be taken as a generalized method.
Asymmetric and Moving-Frame Approaches to MHD Equations
Institute of Scientific and Technical Information of China (English)
Bin Tao CAO
2012-01-01
The magnetohydrodynamic (MHD) equations of incompressible viscous fluids with finite electrical conductivity describe the motion of viscous electrically conducting fluids in a magnetic field.In this paper,we find eight families of solutions of these equations by Xu's asymmetric and moving frame methods.A family of singular solutions may reflect basic characteristics of vortices.The other solutions are globally analytic with respect to the spacial variables.Our solutions may help engineers to develop more effective algorithms to find physical numeric solutions to practical models.
Introduction to nonlinear dispersive equations
Linares, Felipe
2015-01-01
This textbook introduces the well-posedness theory for initial-value problems of nonlinear, dispersive partial differential equations, with special focus on two key models, the Korteweg–de Vries equation and the nonlinear Schrödinger equation. A concise and self-contained treatment of background material (the Fourier transform, interpolation theory, Sobolev spaces, and the linear Schrödinger equation) prepares the reader to understand the main topics covered: the initial-value problem for the nonlinear Schrödinger equation and the generalized Korteweg–de Vries equation, properties of their solutions, and a survey of general classes of nonlinear dispersive equations of physical and mathematical significance. Each chapter ends with an expert account of recent developments and open problems, as well as exercises. The final chapter gives a detailed exposition of local well-posedness for the nonlinear Schrödinger equation, taking the reader to the forefront of recent research. The second edition of Introdu...
Nonlinear evolution equations in QCD
Stasto, A. M.
2004-01-01
The following lectures are an introduction to the phenomena of partonic saturation and nonlinear evolution equations in Quantum Chromodynamics. After a short introduction to the linear evolution, the problems of unitarity bound and parton saturation are discussed. The nonlinear Balitsky-Kovchegov evolution equation in the high energy limit is introduced, and the progress towards the understanding of the properties of its solution is reviewed. We discuss the concepts of the saturation scale, g...
Equations of state for self-excited MHD generator studies
Energy Technology Data Exchange (ETDEWEB)
Rogers, F.J.; Ross, M.; Haggin, G.L.; Wong, L.K.
1980-02-26
We have constructed a state-of-the-art equation of state (EOS) for argon covering the temperature density range attainable by currently proposed self-excited MHD generators. The EOS for conditions in the flow channel was obtained primarily by a non-ideal plasma code (ACTEX) that is based on a many body activity expansion. For conditions in the driver chamber the EOS was primarily obtained from a fluid code (HDFP) that calculates the fluid properties from perturbation theory based on the insulator interatomic pair potential but including electronic excitations. The results are in agreement with several sets of experimental data in the 0.6 - 91 GPa pressure range.
Saaty, Thomas L
1981-01-01
Covers major types of classical equations: operator, functional, difference, integro-differential, and more. Suitable for graduate students as well as scientists, technologists, and mathematicians. "A welcome contribution." - Math Reviews. 1964 edition.
Directory of Open Access Journals (Sweden)
S.P. Anjali Devi
2010-01-01
Full Text Available Viscous and Joule dissipation effects are considered on MHD nonlinear flow and heat transfer past a stretching porous surface embedded in a porous medium under a transverse magnetic field. Analytical solutions of highly nonlinear momentum equation and confluent hypergeometric similarity solution of heat transfer equations in the case when the plate stretches with velocity varying linearly with distance are obtained. The effect of various parameters like suction parameter, Prandtl number, Magnetic parameter, and Eckert number entering into the velocity field, temperature distribution and skin friction co-efficient at the wall are discussed with the aid of graphs.
Stochastic nonlinear differential equations. I
Heilmann, O.J.; Kampen, N.G. van
1974-01-01
A solution method is developed for nonlinear differential equations having the following two properties. Their coefficients are stochastic through their dependence on a Markov process. The magnitude of the fluctuations, multiplied with their auto-correlation time, is a small quantity. Under these co
Vlaykov, Dimitar G; Schmidt, Wolfram; Schleicher, Dominik R G
2016-01-01
Compressible magnetohydrodynamic (MHD) turbulence is ubiquitous in astrophysical phenomena ranging from the intergalactic to the stellar scales. In studying them, numerical simulations are nearly inescapable, due to the large degree of nonlinearity involved. However the dynamical ranges of these phenomena are much larger than what is computationally accessible. In large eddy simulations (LES), the resulting limited resolution effects are addressed explicitly by introducing to the equations of motion additional terms associated with the unresolved, subgrid-scale (SGS) dynamics. This renders the system unclosed. We derive a set of nonlinear structural closures for the ideal MHD LES equations with particular emphasis on the effects of compressibility. The closures are based on a gradient expansion of the finite-resolution operator (W.K. Yeo CUP 1993, ed. Galperin & Orszag) and require no assumptions about the nature of the flow or magnetic field. Thus the scope of their applicability ranges from the sub- to ...
Standing waves for discrete nonlinear Schrodinger equations
Ming Jia
2016-01-01
The discrete nonlinear Schrodinger equation is a nonlinear lattice system that appears in many areas of physics such as nonlinear optics, biomolecular chains and Bose-Einstein condensates. By using critical point theory, we establish some new sufficient conditions on the existence results of standing waves for the discrete nonlinear Schrodinger equations. We give an appropriate example to illustrate the conclusion obtained.
Discrete nonlinear Schrodinger equations with arbitrarily high-order nonlinearities
DEFF Research Database (Denmark)
Khare, A.; Rasmussen, Kim Ø; Salerno, M.
2006-01-01
A class of discrete nonlinear Schrodinger equations with arbitrarily high-order nonlinearities is introduced. These equations are derived from the same Hamiltonian using different Poisson brackets and include as particular cases the saturable discrete nonlinear Schrodinger equation and the Ablowi......-Ladik equation. As a common property, these equations possess three kinds of exact analytical stationary solutions for which the Peierls-Nabarro barrier is zero. Several properties of these solutions, including stability, discrete breathers, and moving solutions, are investigated....
Energy Technology Data Exchange (ETDEWEB)
Mabood, F., E-mail: mabood1971@yahoo.com [School of Mathematical Sciences, Universiti Sains Malaysia, Penang 11800 (Malaysia); Khan, W.A., E-mail: wkhan_2000@yahoo.com [Department of Mechanical Engineering, University of Waterloo, Waterloo, ON, Canada N2L 3G1 (Canada); Ismail, A.I.M., E-mail: izani@cs.usm.my [School of Mathematical Sciences, Universiti Sains Malaysia, Penang 11800 (Malaysia)
2015-01-15
The MHD laminar boundary layer flow with heat and mass transfer of an electrically conducting water-based nanofluid over a nonlinear stretching sheet with viscous dissipation effect is investigated numerically. This is the extension of the previous study on flow and heat transfer of a nanofluid over nonlinear stretching sheet (Rana and Bhargava, Commun. Nonlinear Sci. Numer. Simul. 17 (2012) 212–226). The governing equations are reduced to nonlinear ordinary differential equations using suitable similarity transformation. The effects of the governing parameters on dimensionless quantities like velocity, temperature, nanoparticle concentration, friction factor, local Nusselt, and Sherwood numbers are explored. It is found that the dimensionless velocity decreases and temperature increases with magnetic parameter, and the thermal boundary layer thickness increases with Brownian motion and thermophoresis parameters. - Highlights: • MHD flow of nanofluid and heat transfer over a nonlinear stretching sheet has not been studied yet. • Numerical solutions are computed with Runge–Kutta Fehlberg fourth–fifth order method. • Previous published results can be obtained from present study. • Reduced Nusselt and Sherwood numbers decrease with magnetic parameter.
Li, Xujing; Zheng, Weiying
2016-10-01
A new parallel code based on discontinuous Galerkin (DG) method for hyperbolic conservation laws on three dimensional unstructured meshes is developed recently. This code can be used for simulations of MHD equations, which are very important in magnetic confined plasma research. The main challenges in MHD simulations in fusion include the complex geometry of the configurations, such as plasma in tokamaks, the possibly discontinuous solutions and large scale computing. Our new developed code is based on three dimensional unstructured meshes, i.e. tetrahedron. This makes the code flexible to arbitrary geometries. Second order polynomials are used on each element and HWENO type limiter are applied. The accuracy tests show that our scheme reaches the desired three order accuracy and the nonlinear shock test demonstrate that our code can capture the sharp shock transitions. Moreover, One of the advantages of DG compared with the classical finite element methods is that the matrices solved are localized on each element, making it easy for parallelization. Several simulations including the kink instabilities in toroidal geometry will be present here. Chinese National Magnetic Confinement Fusion Science Program 2015GB110003.
A spatial discretization of the MHD equations based on the finite volume - spectral method
Energy Technology Data Exchange (ETDEWEB)
Miyoshi, Takahiro [Japan Atomic Energy Research Inst., Naka, Ibaraki (Japan). Naka Fusion Research Establishment
2000-05-01
Based on the finite volume - spectral method, we present new discretization formulae for the spatial differential operators in the full system of the compressible MHD equations. In this approach, the cell-centered finite volume method is adopted in a bounded plane (poloidal plane), while the spectral method is applied to the differential with respect to the periodic direction perpendicular to the poloidal plane (toroidal direction). Here, an unstructured grid system composed of the arbitrary triangular elements is utilized for constructing the cell-centered finite volume method. In order to maintain the divergence free constraint of the magnetic field numerically, only the poloidal component of the rotation is defined at three edges of the triangular element. This poloidal component is evaluated under the assumption that the toroidal component of the operated vector times the radius, RA{sub {phi}}, is linearly distributed in the element. The present method will be applied to the nonlinear MHD dynamics in an realistic torus geometry without the numerical singularities. (author)
Quasi self-adjoint nonlinear wave equations
Energy Technology Data Exchange (ETDEWEB)
Ibragimov, N H [Department of Mathematics and Science, Blekinge Institute of Technology, SE-371 79 Karlskrona (Sweden); Torrisi, M; Tracina, R, E-mail: nib@bth.s, E-mail: torrisi@dmi.unict.i, E-mail: tracina@dmi.unict.i [Dipartimento di Matematica e Informatica, University of Catania (Italy)
2010-11-05
In this paper we generalize the classification of self-adjoint second-order linear partial differential equation to a family of nonlinear wave equations with two independent variables. We find a class of quasi self-adjoint nonlinear equations which includes the self-adjoint linear equations as a particular case. The property of a differential equation to be quasi self-adjoint is important, e.g. for constructing conservation laws associated with symmetries of the differential equation. (fast track communication)
Auxiliary equation method for solving nonlinear partial differential equations
Energy Technology Data Exchange (ETDEWEB)
Sirendaoreji,; Jiong, Sun
2003-03-31
By using the solutions of an auxiliary ordinary differential equation, a direct algebraic method is described to construct several kinds of exact travelling wave solutions for some nonlinear partial differential equations. By this method some physically important nonlinear equations are investigated and new exact travelling wave solutions are explicitly obtained with the aid of symbolic computation.
The myth about nonlinear differential equations
Radhakrishnan, C.
2002-01-01
Taking the example of Koretweg--de Vries equation, it is shown that soliton solutions need not always be the consequence of the trade-off between the nonlinear terms and the dispersive term in the nonlinear differential equation. Even the ordinary one dimensional linear partial differential equation can produce a soliton.
Standing waves for discrete nonlinear Schrodinger equations
Directory of Open Access Journals (Sweden)
Ming Jia
2016-07-01
Full Text Available The discrete nonlinear Schrodinger equation is a nonlinear lattice system that appears in many areas of physics such as nonlinear optics, biomolecular chains and Bose-Einstein condensates. By using critical point theory, we establish some new sufficient conditions on the existence results of standing waves for the discrete nonlinear Schrodinger equations. We give an appropriate example to illustrate the conclusion obtained.
Energy Technology Data Exchange (ETDEWEB)
Fierros Palacios, Angel [Instituto de Investigaciones Electricas, Temixco, Morelos (Mexico)
2001-02-01
In this work the complete set of differential field equations which describes the dynamic state of a continuos conducting media which flow in presence of a perturbed magnetic field is obtained. Then, the thermic equation of state, the wave equation and the conservation law of energy for the Alfven MHD waves are obtained. [Spanish] Es este trabajo se obtiene el conjunto completo de ecuaciones diferenciales de campo que describen el estado dinamico de un medio continuo conductor que se mueve en presencia de un campo magnetico externo perturbado. Asi, se obtiene la ecuacion termica de estado, la ecuacion de onda y la ley de la conservacion de la energia para las ondas de Alfven de la MHD.
Nonlinear Elliptic Differential Equations with Multivalued Nonlinearities
Indian Academy of Sciences (India)
Antonella Fiacca; Nikolaos Matzakos; Nikolaos S Papageorgiou; Raffaella Servadei
2001-11-01
In this paper we study nonlinear elliptic boundary value problems with monotone and nonmonotone multivalued nonlinearities. First we consider the case of monotone nonlinearities. In the first result we assume that the multivalued nonlinearity is defined on all $\\mathbb{R}$. Assuming the existence of an upper and of a lower solution, we prove the existence of a solution between them. Also for a special version of the problem, we prove the existence of extremal solutions in the order interval formed by the upper and lower solutions. Then we drop the requirement that the monotone nonlinearity is defined on all of $\\mathbb{R}$. This case is important because it covers variational inequalities. Using the theory of operators of monotone type we show that the problem has a solution. Finally in the last part we consider an eigenvalue problem with a nonmonotone multivalued nonlinearity. Using the critical point theory for nonsmooth locally Lipschitz functionals we prove the existence of at least two nontrivial solutions (multiplicity theorem).
Symmetrized solutions for nonlinear stochastic differential equations
Directory of Open Access Journals (Sweden)
G. Adomian
1981-01-01
Full Text Available Solutions of nonlinear stochastic differential equations in series form can be put into convenient symmetrized forms which are easily calculable. This paper investigates such forms for polynomial nonlinearities, i.e., equations of the form Ly+ym=x where x is a stochastic process and L is a linear stochastic operator.
Directory of Open Access Journals (Sweden)
E. M. E. Zayed
2014-01-01
Full Text Available We apply the generalized projective Riccati equations method to find the exact traveling wave solutions of some nonlinear evolution equations with any-order nonlinear terms, namely, the nonlinear Pochhammer-Chree equation, the nonlinear Burgers equation and the generalized, nonlinear Zakharov-Kuznetsov equation. This method presents wider applicability for handling many other nonlinear evolution equations in mathematical physics.
Generalized solutions of nonlinear partial differential equations
Rosinger, EE
1987-01-01
During the last few years, several fairly systematic nonlinear theories of generalized solutions of rather arbitrary nonlinear partial differential equations have emerged. The aim of this volume is to offer the reader a sufficiently detailed introduction to two of these recent nonlinear theories which have so far contributed most to the study of generalized solutions of nonlinear partial differential equations, bringing the reader to the level of ongoing research.The essence of the two nonlinear theories presented in this volume is the observation that much of the mathematics concernin
A CLASS OF TWO-STEP TVD MACCORMACK TYPE NUMERICAL SCHEME FOR MHD EQUATIONS
Institute of Scientific and Technical Information of China (English)
FENG Xueshang; WEI Fengsi; ZHONG Dingkun
2003-01-01
In this paper, a new numerical scheme of Total Variation Diminishing (TVD) MacCormack type for MagnetoHydroDynamic (MHD) equations is proposed by taking into account of the characteristics such as convergence, stability, resolution. This new scheme is established by solving the MHD equations with a TVD modified MacCormack scheme for the purpose of developing a scheme of quick convergence as well as of TVD property. To show the validation, simplicity and practicability of the scheme for modelling MHD problems, a self-similar Cauchy problem with the discontinuous initial data consisting of constant states, and the collision of two fast MHD shocks, and two-dimensional Orszag and Tang's MHD vortex problem are discussed with the numerical results conforming to the existing results obtained by the Roe type TVD, the high-order Godunov scheme,and Weighted Essentially Non-Oscillatory (WENO) scheme. The numerical tests show that this two-step TVD MacCormack numerical scheme for MHD system is of robust operation in the presence of very strong waves, thin shock fronts, thin contact and slip surface discontinuities.
A Numerical Approach to Solving the Hall MHD Equations Including Diamagnetic Drift (Preprint)
2008-02-19
Article 3. DATES COVERED (From - To) 4. TITLE AND SUBTITLE 5a. CONTRACT NUMBER FA9300-06-D-0002 0003 A Numerical Approach to Solving the Hall MHD...Loverich and U. Shumlak. Nonlinear full two-fluid study of m=0 sausage instabilities in an axisymmetric z pinch. Physics of Plasmas, (13), 2006. [19
Thermoviscous Model Equations in Nonlinear Acoustics
DEFF Research Database (Denmark)
Rasmussen, Anders Rønne
Four nonlinear acoustical wave equations that apply to both perfect gasses and arbitrary fluids with a quadratic equation of state are studied. Shock and rarefaction wave solutions to the equations are studied. In order to assess the accuracy of the wave equations, their solutions are compared...... to solutions of the basic equations from which the wave equations are derived. A straightforward weakly nonlinear equation is the most accurate for shock modeling. A higher order wave equation is the most accurate for modeling of smooth disturbances. Investigations of the linear stability properties...... of solutions to the wave equations, reveal that the solutions may become unstable. Such instabilities are not found in the basic equations. Interacting shocks and standing shocks are investigated....
Elliptic Equation and New Solutions to Nonlinear Wave Equations
Institute of Scientific and Technical Information of China (English)
FU Zun-Tao; LIU Shi-Kuo; LIU Shi-Da
2004-01-01
The new solutions to elliptic equation are shown, and then the elliptic equation is taken as a transformationand is applied to solve nonlinear wave equations. It is shown that more kinds of solutions are derived, such as periodicsolutions of rational form, solitary wave solutions of rational form, and so on.
Explicit Traveling Wave Solutions to Nonlinear Evolution Equations
Institute of Scientific and Technical Information of China (English)
Linghai ZHANG
2011-01-01
First of all,some technical tools are developed. Then the author studies explicit traveling wave solutions to nonlinear dispersive wave equations,nonlinear dissipative dispersive wave equations,nonlinear convection equations,nonlinear reaction diffusion equations and nonlinear hyperbolic equations,respectively.
EXACT SOLUTIONS TO NONLINEAR WAVE EQUATION
Institute of Scientific and Technical Information of China (English)
无
2011-01-01
In this paper,we use an invariant set to construct exact solutions to a nonlinear wave equation with a variable wave speed. Moreover,we obtain conditions under which the equation admits a nonclassical symmetry. Several different nonclassical symmetries for equations with different diffusion terms are presented.
ANALYTICAL SOLUTION OF NONLINEAR BAROTROPIC VORTICITY EQUATION
Institute of Scientific and Technical Information of China (English)
WANG Yue-peng; SHI Wei-hui
2008-01-01
The stability of nonlinear barotropic vorticity equation was proved. The necessary and sufficient conditions for the initial value problem to be well-posed were presented. Under the conditions of well-posedness, the corresponding analytical solution was also gained.
GLOBAL SOLUTIONS OF NONLINEAR SCHRODINGER EQUATIONS
Institute of Scientific and Technical Information of China (English)
Ye Yaojun
2005-01-01
In this paper we study the existence of global solutions to the Cauchy problem of nonlinear Schrodinger equation by establishing time weight function spaces and using the contraction mapping principle.
Some geometrical iteration methods for nonlinear equations
Institute of Scientific and Technical Information of China (English)
LU Xing-jiang; QIAN Chun
2008-01-01
This paper describes geometrical essentials of some iteration methods (e.g. Newton iteration,secant line method,etc.) for solving nonlinear equations and advances some geomet-rical methods of iteration that are flexible and efficient.
Homogenization of a nonlinear degenerate parabolic equation
Institute of Scientific and Technical Information of China (English)
无
2005-01-01
The homogenization of one kind of nonlinear parabolic equation is studied. The weak convergence and corrector results are obtained by combining carefully the compactness method and two-scale convergence method in the homogenization theory.
Nonlinear Electrostatic Wave Equations for Magnetized Plasmas
DEFF Research Database (Denmark)
Dysthe, K.B.; Mjølhus, E.; Pécseli, Hans
1984-01-01
The lowest order kinetic effects are included in the equations for nonlinear electrostatic electron waves in a magnetized plasma. The modifications of the authors' previous analysis based on a fluid model are discussed.......The lowest order kinetic effects are included in the equations for nonlinear electrostatic electron waves in a magnetized plasma. The modifications of the authors' previous analysis based on a fluid model are discussed....
Nonlinear elliptic equations of the second order
Han, Qing
2016-01-01
Nonlinear elliptic differential equations are a diverse subject with important applications to the physical and social sciences and engineering. They also arise naturally in geometry. In particular, much of the progress in the area in the twentieth century was driven by geometric applications, from the Bernstein problem to the existence of Kähler-Einstein metrics. This book, designed as a textbook, provides a detailed discussion of the Dirichlet problems for quasilinear and fully nonlinear elliptic differential equations of the second order with an emphasis on mean curvature equations and on Monge-Ampère equations. It gives a user-friendly introduction to the theory of nonlinear elliptic equations with special attention given to basic results and the most important techniques. Rather than presenting the topics in their full generality, the book aims at providing self-contained, clear, and "elementary" proofs for results in important special cases. This book will serve as a valuable resource for graduate stu...
Variational Integration for Ideal MHD with Built-in Advection Equations
Energy Technology Data Exchange (ETDEWEB)
Zhou, Yao [Princeton Plasma Physics Lab. (PPPL), Princeton, NJ (United States); Qin, Hong [Princeton Plasma Physics Lab. (PPPL), Princeton, NJ (United States); Burby, J. W. [Princeton Plasma Physics Lab. (PPPL), Princeton, NJ (United States); Bhattacharjee, A. [Princeton Plasma Physics Lab. (PPPL), Princeton, NJ (United States)
2014-08-05
Newcomb's Lagrangian for ideal MHD in Lagrangian labeling is discretized using discrete exterior calculus. Variational integrators for ideal MHD are derived thereafter. Besides being symplectic and momentum preserving, the schemes inherit built-in advection equations from Newcomb's formulation, and therefore avoid solving them and the accompanying error and dissipation. We implement the method in 2D and show that numerical reconnection does not take place when singular current sheets are present. We then apply it to studying the dynamics of the ideal coalescence instability with multiple islands. The relaxed equilibrium state with embedded current sheets is obtained numerically.
Nonlinear excitation of low-n harmonics in reduced MHD simulations of edge-localized modes
Krebs, Isabel; Lackner, Karl; Guenter, Sibylle
2013-01-01
Nonlinear simulations of the early ELMphase based on a typical type-I ELMy ASDEX Upgrade discharge have been carried out using the reduced MHD code JOREK. The analysis is focused on the evolution of the toroidal Fourier spectrum. It is found that during the nonlinear evolution, linearly subdominant low-n Fourier components, in particular the n = 1, grow to energies comparable with linearly dominant harmonics. A simple model is developed, based on the idea that energy is transferred among the toroidal harmonics via second order nonlinear interaction. The simple model reproduces and explains very well the early nonlinear evolution of the toroidal spectrum in the JOREK simulations. Furthermore, it is shown for the n = 1 harmonic, that its spatial structure changes significantly during the transition from linear to nonlinearly driven growth. The rigidly growing structure of the linearly barely unstable n = 1 reaches far into the plasma core. In contrast, the nonlinearly driven n = 1 has a rigidly growing structur...
The Nonlinear Analytical Envelope Equation in quadratic nonlinear crystals
Bache, Morten
2016-01-01
We here derive the so-called Nonlinear Analytical Envelope Equation (NAEE) inspired by the work of Conforti et al. [M. Conforti, A. Marini, T. X. Tran, D. Faccio, and F. Biancalana, "Interaction between optical fields and their conjugates in nonlinear media," Opt. Express 21, 31239-31252 (2013)], whose notation we follow. We present a complete model that includes $\\chi^{(2)}$ terms [M. Conforti, F. Baronio, and C. De Angelis, "Nonlinear envelope equation for broadband optical pulses in quadratic media," Phys. Rev. A 81, 053841 (2010)], $\\chi^{(3)}$ terms, and then extend the model to delayed Raman effects in the $\\chi^{(3)}$ term. We therefore get a complete model for ultrafast pulse propagation in quadratic nonlinear crystals similar to the Nonlinear Wave Equation in Frequency domain [H. Guo, X. Zeng, B. Zhou, and M. Bache, "Nonlinear wave equation in frequency domain: accurate modeling of ultrafast interaction in anisotropic nonlinear media," J. Opt. Soc. Am. B 30, 494-504 (2013)], but where the envelope is...
Nonlinear tearing mode study using the almost ideal magnetohydrodynamics (MHD) constraint
Energy Technology Data Exchange (ETDEWEB)
Ren, C.; Callen, J.D. [Univ. of Wisconsin, Madison, WI (United States); Jensen, T.H. [General Atomics, San Diego, CA (United States)
1998-12-31
The tearing mode is an important resistive magnetohydrodynamics (MHD) mode. It perturbs the initial equilibrium magnetic flux surfaces through magnetic field line reconnection to form new flux surfaces with magnetic islands. In the study of the tearing mode, usually the initial equilibria are one dimensional with two ignorable coordinates and the perturbed equilibria are two dimensional with one ignorable coordinate. The tearing mode can be linearly unstable and its growth saturates at a fine amplitude. The neoclassical tearing mode theory shows that the mode can be nonlinearly driven by the bootstrap current even when it is linearly stable to the classical tearing mode. It is important to study the nonlinear behavior of the tearing mode. As an intrinsically nonlinear approach, the use of the almost ideal MHD constraint is suited to study the nonlinear properties of the tearing mode. In this paper, as a validation of the method, the authors study two characteristics of the tearing mode using the almost ideal MHD constraint: (1) the linear stability condition for the initial one dimensional equilibrium; and (2) the final saturation level for the unstable case. In this work, they only consider the simplest case where no gradient of pressure or current density exists at the mode resonant surface.
Nonlinear Schrodinger equation with chaotic, random, and nonperiodic nonlinearity
Cardoso, W B; Avelar, A T; Bazeia, D; Hussein, M S
2009-01-01
In this paper we deal with a nonlinear Schr\\"{o}dinger equation with chaotic, random, and nonperiodic cubic nonlinearity. Our goal is to study the soliton evolution, with the strength of the nonlinearity perturbed in the space and time coordinates and to check its robustness under these conditions. Comparing with a real system, the perturbation can be related to, e.g., impurities in crystalline structures, or coupling to a thermal reservoir which, on the average, enhances the nonlinearity. We also discuss the relevance of such random perturbations to the dynamics of Bose-Einstein Condensates and their collective excitations and transport.
A stochastic approach to uncertainty in the equations of MHD kinematics
Energy Technology Data Exchange (ETDEWEB)
Phillips, Edward G., E-mail: egphillips@math.umd.edu [Applied Mathematics & Statistics, and Scientific Computation Program, University of Maryland, College Park, MD (United States); Elman, Howard C., E-mail: elman@cs.umd.edu [Department of Computer Science and Institute for Advanced Computer Studies, University of Maryland, College Park, MD (United States)
2015-03-01
The magnetohydrodynamic (MHD) kinematics model describes the electromagnetic behavior of an electrically conducting fluid when its hydrodynamic properties are assumed to be known. In particular, the MHD kinematics equations can be used to simulate the magnetic field induced by a given velocity field. While prescribing the velocity field leads to a simpler model than the fully coupled MHD system, this may introduce some epistemic uncertainty into the model. If the velocity of a physical system is not known with certainty, the magnetic field obtained from the model may not be reflective of the magnetic field seen in experiments. Additionally, uncertainty in physical parameters such as the magnetic resistivity may affect the reliability of predictions obtained from this model. By modeling the velocity and the resistivity as random variables in the MHD kinematics model, we seek to quantify the effects of uncertainty in these fields on the induced magnetic field. We develop stochastic expressions for these quantities and investigate their impact within a finite element discretization of the kinematics equations. We obtain mean and variance data through Monte Carlo simulation for several test problems. Toward this end, we develop and test an efficient block preconditioner for the linear systems arising from the discretized equations.
Extended Trial Equation Method for Nonlinear Partial Differential Equations
Gepreel, Khaled A.; Nofal, Taher A.
2015-04-01
The main objective of this paper is to use the extended trial equation method to construct a series of some new solutions for some nonlinear partial differential equations (PDEs) in mathematical physics. We will construct the solutions in many different functions such as hyperbolic function solutions, trigonometric function solutions, Jacobi elliptic function solutions, and rational functional solutions for the nonlinear PDEs when the balance number is a real number via the Zhiber-Shabat nonlinear differential equation. The balance number of this method is not constant as we shown in other methods, but it is changed by changing the trial equation derivative definition. This method allowed us to construct many new types of solutions. It is shown by using the Maple software package that all obtained solutions satisfy the original PDEs.
Wave equation with concentrated nonlinearities
Noja, Diego; Posilicano, Andrea
2004-01-01
In this paper we address the problem of wave dynamics in presence of concentrated nonlinearities. Given a vector field $V$ on an open subset of $\\CO^n$ and a discrete set $Y\\subset\\RE^3$ with $n$ elements, we define a nonlinear operator $\\Delta_{V,Y}$ on $L^2(\\RE^3)$ which coincides with the free Laplacian when restricted to regular functions vanishing at $Y$, and which reduces to the usual Laplacian with point interactions placed at $Y$ when $V$ is linear and is represented by an Hermitean m...
The Homoclinic Orbits in Nonlinear Schroedinger Equation
Institute of Scientific and Technical Information of China (English)
PengchengXU; BolingGUO; 等
1998-01-01
The persistence of Homoclinic orbits for perturbed nonlinear Schroedinger equation with five degree term under een periodic boundary conditions is considered.The exstences of the homoclinic orbits for the truncation equation is established by Melnikov's analysis and geometric singular perturbation theory.
Linearization of Systems of Nonlinear Diffusion Equations
Institute of Scientific and Technical Information of China (English)
KANG Jing; QU Chang-Zheng
2007-01-01
We investigate the linearization of systems of n-component nonlinear diffusion equations; such systems have physical applications in soil science, mathematical biology and invariant curve flows. Equivalence transformations of their auxiliary systems are used to identify the systems that can be linearized. We also provide several examples of systems with two-component equations, and show how to linearize them by nonlocal mappings.
The approximate solutions of nonlinear Boussinesq equation
Lu, Dianhen; Shen, Jie; Cheng, Yueling
2016-04-01
The homotopy analysis method (HAM) is introduced to solve the generalized Boussinesq equation. In this work, we establish the new analytical solution of the exponential function form. Applying the homotopy perturbation method to solve the variable coefficient Boussinesq equation. The results indicate that this method is efficient for the nonlinear models with variable coefficients.
SEMICLASSICAL LIMIT OF NONLINEAR SCHRODINGER EQUATION (Ⅱ)
Institute of Scientific and Technical Information of China (English)
张平
2002-01-01
In this paper, we use the Wigner measure approach to study the semiclassical limit of nonlinear Schrodinger equation in small time. We prove that: the limits of the quantum density: pε =: |ψε|2, and the quantum momentum: Jε =: εIm(ψεψε) satisfy the compressible Euler equations before the formation of singularities in the limit system.
Nonlinear second order elliptic equations involving measures
Marcus, Moshe
2013-01-01
This book presents a comprehensive study of boundary value problems for linear and semilinear second order elliptic equations with measure data,especially semilinear equations with absorption. The interactions between the diffusion operator and the absorption term give rise to a large class of nonlinear phenomena in the study of which singularities and boundary trace play a central role.
Nonlinear differentiation equation and analytic function spaces
Li, Hao; Li, Songxiao
2015-01-01
In this paper we consider the nonlinear complex differential equation $$(f^{(k)})^{n_{k}}+A_{k-1}(z)(f^{(k-1)})^{n_{k-1}}+\\cdot\\cdot\\cdot+A_{1}(z)(f')^{n_{1}}+A_{0}(z)f^{n_{0}}=0, $$where $ A_{j}(z)$, $ j=0, \\cdots, k-1 $, are analytic in the unit disk $ \\mathbb{D} $, $ n_{j}\\in R^{+} $ for all $ j=0, \\cdots, k $. We investigate this nonlinear differential equation from two aspects. On one hand, we provide some sufficient conditions on coefficients such that all solutions of this equation bel...
Advances in iterative methods for nonlinear equations
Busquier, Sonia
2016-01-01
This book focuses on the approximation of nonlinear equations using iterative methods. Nine contributions are presented on the construction and analysis of these methods, the coverage encompassing convergence, efficiency, robustness, dynamics, and applications. Many problems are stated in the form of nonlinear equations, using mathematical modeling. In particular, a wide range of problems in Applied Mathematics and in Engineering can be solved by finding the solutions to these equations. The book reveals the importance of studying convergence aspects in iterative methods and shows that selection of the most efficient and robust iterative method for a given problem is crucial to guaranteeing a good approximation. A number of sample criteria for selecting the optimal method are presented, including those regarding the order of convergence, the computational cost, and the stability, including the dynamics. This book will appeal to researchers whose field of interest is related to nonlinear problems and equations...
Exponential Attractor for a Nonlinear Boussinesq Equation
Institute of Scientific and Technical Information of China (English)
Ahmed Y. Abdallah
2006-01-01
This paper is devoted to prove the existence of an exponential attractor for the semiflow generated by a nonlinear Boussinesq equation. We formulate the Boussinesq equation as an abstract equation in the Hilbert space H20(0, 1) × L2(0, 1). The main step in this research is to show that there exists an absorbing set for the solution semiflow in the Hilbert space H03(0, 1) × H10(0, 1).
The Nonlinear Convection—Reaction—Diffusion Equation
Institute of Scientific and Technical Information of China (English)
ShiminTANG; MaochangCUI; 等
1996-01-01
A nonlinear convection-reaction-diffusion equation is used as a model equation of the El Nino events.In this model,the effects of convection,turbulent diffusion,linear feed-back and nolinear radiation on the anomaly of Sea Surface Temperature(SST) are considered.In the case of constant convection,this equation has exact kink-like travelling wave solutions,which can be used to explain the history of an El Nino event.
Orain, François; Bécoulet, M.; Morales, J.; Huijsmans, G. T. A.; Dif-Pradalier, G.; Hoelzl, M.; Garbet, X.; Pamela, S.; Nardon, E.; Passeron, C.; Latu, G.; Fil, A.; Cahyna, P.
2015-01-01
The dynamics of a multi-edge localized mode (ELM) cycle as well as the ELM mitigation by resonant magnetic perturbations (RMPs) are modeled in realistic tokamak X-point geometry with the non-linear reduced MHD code JOREK. The diamagnetic rotation is found to be a key parameter enabling us to reproduce the cyclical dynamics of the plasma relaxations and to model the near-symmetric ELM power deposition on the inner and outer divertor target plates consistently with experimental measurements. Moreover, the non-linear coupling of the RMPs with unstable modes are found to modify the edge magnetic topology and induce a continuous MHD activity in place of a large ELM crash, resulting in the mitigation of the ELMs. At larger diamagnetic rotation, a bifurcation from unmitigated ELMs—at low RMP current—towards fully suppressed ELMs—at large RMP current—is obtained.
Implementation of a 3-D nonlinear MHD calculation on the Intel hypercube
Energy Technology Data Exchange (ETDEWEB)
Drake, J.B.; Lawkins, W.F.; Carreras, B.A.; Hicks, H.R.
1987-08-01
As part of an exploratory study of the suitability of hypercube multiprocessors for scientific computations, the non-linear magnetohydrodynamics (MHD) code RSF was parallelized for use on an Intel iPSC hypercube. This report presents the numerical algorithm of RSF and the techniques used to obtain parallelism without sacrificing the numerical properties of the serial algorithm. Timing results are presented for a sample problem.
Directory of Open Access Journals (Sweden)
Boričić Zoran
2009-01-01
Full Text Available This paper concerns with unsteady two-dimensional temperature laminar magnetohydrodynamic (MHD boundary layer of incompressible fluid. It is assumed that induction of outer magnetic field is function of longitudinal coordinate with force lines perpendicular to the body surface on which boundary layer forms. Outer electric filed is neglected and magnetic Reynolds number is significantly lower then one i.e. considered problem is in inductionless approximation. Characteristic properties of fluid are constant because velocity of flow is much lower than speed of light and temperature difference is small enough (under 50ºC . Introduced assumptions simplify considered problem in sake of mathematical solving, but adopted physical model is interesting from practical point of view, because its relation with large number of technically significant MHD flows. Obtained partial differential equations can be solved with modern numerical methods for every particular problem. Conclusions based on these solutions are related only with specific temperature MHD boundary layer problem. In this paper, quite different approach is used. First new variables are introduced and then sets of similarity parameters which transform equations on the form which don't contain inside and in corresponding boundary conditions characteristics of particular problems and in that sense equations are considered as universal. Obtained universal equations in appropriate approximation can be solved numerically once for all. So-called universal solutions of equations can be used to carry out general conclusions about temperature MHD boundary layer and for calculation of arbitrary particular problems. To calculate any particular problem it is necessary also to solve corresponding momentum integral equation.
Zhai, Xiaoping; Yin, Zhaoyang
2017-02-01
The present paper is dedicated to the global well-posedness for the 3D inhomogeneous incompressible Navier-Stokes equations, in critical Besov spaces without smallness assumption on the variation of the density. We aim at extending the work by Abidi, Gui and Zhang (2012) [2], and (2013) [3] to a lower regularity index about the initial velocity. The key to that improvement is a new a priori estimate for an elliptic equation with nonconstant coefficients in Besov spaces which have the same degree as L2 in R3. Finally, we also generalize our well-posedness result to the inhomogeneous incompressible MHD equations.
Numerical methods for nonlinear partial differential equations
Bartels, Sören
2015-01-01
The description of many interesting phenomena in science and engineering leads to infinite-dimensional minimization or evolution problems that define nonlinear partial differential equations. While the development and analysis of numerical methods for linear partial differential equations is nearly complete, only few results are available in the case of nonlinear equations. This monograph devises numerical methods for nonlinear model problems arising in the mathematical description of phase transitions, large bending problems, image processing, and inelastic material behavior. For each of these problems the underlying mathematical model is discussed, the essential analytical properties are explained, and the proposed numerical method is rigorously analyzed. The practicality of the algorithms is illustrated by means of short implementations.
On implicit abstract neutral nonlinear differential equations
Energy Technology Data Exchange (ETDEWEB)
Hernández, Eduardo, E-mail: lalohm@ffclrp.usp.br [Universidade de São Paulo, Departamento de Computação e Matemática, Faculdade de Filosofia Ciências e Letras de Ribeirão Preto (Brazil); O’Regan, Donal, E-mail: donal.oregan@nuigalway.ie [National University of Ireland, School of Mathematics, Statistics and Applied Mathematics (Ireland)
2016-04-15
In this paper we continue our developments in Hernández and O’Regan (J Funct Anal 261:3457–3481, 2011) on the existence of solutions for abstract neutral differential equations. In particular we extend the results in Hernández and O’Regan (J Funct Anal 261:3457–3481, 2011) for the case of implicit nonlinear neutral equations and we focus on applications to partial “nonlinear” neutral differential equations. Some applications involving partial neutral differential equations are presented.
Directory of Open Access Journals (Sweden)
S.K. Parida
2015-12-01
Full Text Available This work considers the two-dimensional steady MHD boundary layer flow of heat and mass transfer over a flat plate with partial slip at the surface subjected to the convective heat flux. The particular attraction lies in searching the effects of variable viscosity and variable thermal diffusivity on the behavior of the flow. In addition, non-linear thermal radiation effects and thermophoresis are taken into account. The governing nonlinear partial differential equations for the flow, heat and mass transfer are transformed into a set of coupled nonlinear ordinary differential equations by using similarity variable, which are solved numerically by applying Runge–Kutta fourth–fifth order integration scheme in association with quasilinear shooting technique. The novel results for the dimensionless velocity, temperature, concentration and ambient Prandtl number within the boundary layer are displayed graphically for various parameters that characterize the flow. The local skin friction, Nusselt number and Sherwood number are shown graphically. The numerical results obtained for the particular case are fairly in good agreement with the result of Rahman [6].
TAYLOR EXPANSION METHOD FOR NONLINEAR EVOLUTION EQUATIONS
Institute of Scientific and Technical Information of China (English)
HE Yin-nian
2005-01-01
A new numerical method of integrating the nonlinear evolution equations, namely the Taylor expansion method, was presented. The standard Galerkin method can be viewed as the 0-th order Taylor expansion method; while the nonlinear Galerkin method can be viewed as the 1-st order modified Taylor expansion method. Moreover, the existence of the numerical solution and its convergence rate were proven. Finally, a concrete example,namely, the two-dimensional Navier-Stokes equations with a non slip boundary condition,was provided. The result is that the higher order Taylor expansion method is of the higher convergence rate under some assumptions about the regularity of the solution.
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.
Explicit solutions of nonlinear wave equation systems
Institute of Scientific and Technical Information of China (English)
Ahmet Bekir; Burcu Ayhan; M.Naci (O)zer
2013-01-01
We apply the (G'/G)-expansion method to solve two systems of nonlinear differential equations and construct traveling wave solutions expressed in terms of hyperbolic functions,trigonometric functions,and rational functions with arbitrary parameters.We highlight the power of the (G'/G)-expansion method in providing generalized solitary wave solutions of different physical structures.It is shown that the (G'/G)-expansion method is very effective and provides a powerful mathematical tool to solve nonlinear differential equation systems in mathematical physics.
Exact solutions for nonlinear foam drainage equation
Zayed, E. M. E.; Al-Nowehy, Abdul-Ghani
2016-09-01
In this paper, the modified simple equation method, the exp-function method, the soliton ansatz method, the Riccati equation expansion method and the ( G^' }/G) -expansion method are used to construct exact solutions with parameters of the nonlinear foam drainage equation. When these parameters are taken to be special values, the solitary wave solutions and the trigonometric function solutions are derived from the exact solutions. The obtained results confirm that the proposed methods are efficient techniques for analytic treatments of a wide variety of nonlinear partial differential equations in mathematical physics. We compare our results together with each other yielding from these integration tools. Also, our results have been compared with the well-known results of others.
Exact solutions for nonlinear foam drainage equation
Zayed, E. M. E.; Al-Nowehy, Abdul-Ghani
2017-02-01
In this paper, the modified simple equation method, the exp-function method, the soliton ansatz method, the Riccati equation expansion method and the ( G^' }/G)-expansion method are used to construct exact solutions with parameters of the nonlinear foam drainage equation. When these parameters are taken to be special values, the solitary wave solutions and the trigonometric function solutions are derived from the exact solutions. The obtained results confirm that the proposed methods are efficient techniques for analytic treatments of a wide variety of nonlinear partial differential equations in mathematical physics. We compare our results together with each other yielding from these integration tools. Also, our results have been compared with the well-known results of others.
Global attractivity in a nonlinear difference equation
Directory of Open Access Journals (Sweden)
Chuanxi Qian
2007-02-01
Full Text Available In this paper, we study the asymptotic behavior of positive solutions of the nonlinear difference equation $$ x_{n+1}=x_n f(x_{n-k}, $$ where $f:[0,inftyo(0, infty$ is a unimodal function, and $k$ is a nonnegative integer. Sufficient conditions for the positive equilibrium to be a global attractor of all positive solutions are established. Our results can be applied to to some difference equations derived from mathematical biology.
Energy Equality and Uniqueness of Weak Solutions to MHD Equations in L∞(O,T;Ln(Ω))
Institute of Scientific and Technical Information of China (English)
Yan YONG; Quan Sen JIU
2009-01-01
In this paper, we study the energy equality and the uniqueness of weak solutions to the MHD equations in the critical space L∞(O,T; Ln(Ω)). We prove that if the velocity u belongs to the critical space L∞(O,T; Ln(Ω)), the energy equality holds. On the basis of the energy equality, we further prove that the weak solution to the MHD equations is unique.
Energy Technology Data Exchange (ETDEWEB)
Haverkort, J.W. [Centrum Wiskunde & Informatica, P.O. Box 94079, 1090 GB Amsterdam (Netherlands); Dutch Institute for Fundamental Energy Research, P.O. Box 6336, 5600 HH Eindhoven (Netherlands); Blank, H.J. de [Dutch Institute for Fundamental Energy Research, P.O. Box 6336, 5600 HH Eindhoven (Netherlands); Huysmans, G.T.A. [ITER Organization, Route de Vinon sur Verdon, 13115 Saint Paul Lez Durance (France); Pratt, J. [Dutch Institute for Fundamental Energy Research, P.O. Box 6336, 5600 HH Eindhoven (Netherlands); Koren, B., E-mail: b.koren@tue.nl [Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven (Netherlands)
2016-07-01
Numerical simulations form an indispensable tool to understand the behavior of a hot plasma that is created inside a tokamak for providing nuclear fusion energy. Various aspects of tokamak plasmas have been successfully studied through the reduced magnetohydrodynamic (MHD) model. The need for more complete modeling through the full MHD equations is addressed here. Our computational method is presented along with measures against possible problems regarding pollution, stability, and regularity. The problem of ensuring continuity of solutions in the center of a polar grid is addressed in the context of a finite element discretization of the full MHD equations. A rigorous and generally applicable solution is proposed here. Useful analytical test cases are devised to verify the correct implementation of the momentum and induction equation, the hyperdiffusive terms, and the accuracy with which highly anisotropic diffusion can be simulated. A striking observation is that highly anisotropic diffusion can be treated with the same order of accuracy as isotropic diffusion, even on non-aligned grids, as long as these grids are generated with sufficient care. This property is shown to be associated with our use of a magnetic vector potential to describe the magnetic field. Several well-known instabilities are simulated to demonstrate the capabilities of the new method. The linear growth rate of an internal kink mode and a tearing mode are benchmarked against the results of a linear MHD code. The evolution of a tearing mode and the resulting magnetic islands are simulated well into the nonlinear regime. The results are compared with predictions from the reduced MHD model. Finally, a simulation of a ballooning mode illustrates the possibility to use our method as an ideal MHD method without the need to add any physical dissipation.
Approximating parameters in nonlinear reaction diffusion equations
Directory of Open Access Journals (Sweden)
Robert R. Ferdinand
2001-07-01
Full Text Available We present a model describing population dynamics in an environment. The model is a nonlinear, nonlocal, reaction diffusion equation with Neumann boundary conditions. An inverse method, involving minimization of a least-squares cost functional, is developed to identify unknown model parameters. Finally, numerical results are presented which display estimates of these parameters using computationally generated data.
Exact solutions for nonlinear partial fractional differential equations
Institute of Scientific and Technical Information of China (English)
Khaled A.Gepreel; Saleh Omran
2012-01-01
In this article,we use the fractional complex transformation to convert nonlinear partial fractional differential equations to nonlinear ordinary differential equations.We use the improved (G’/G)-expansion function method to calculate the exact solutions to the time-and space-fractional derivative foam drainage equation and the time-and space-fractional derivative nonlinear KdV equation.This method is efficient and powerful for solving wide classes of nonlinear evolution fractional order equations.
An Optimal Homotopy Asymptotic Approach Applied to Nonlinear MHD Jeffery-Hamel Flow
Directory of Open Access Journals (Sweden)
Vasile Marinca
2011-01-01
Full Text Available A simple and effective procedure is employed to propose a new analytic approximate solution for nonlinear MHD Jeffery-Hamel flow. This technique called the Optimal Homotopy Asymptotic Method (OHAM does not depend upon any small/large parameters and provides us with a convenient way to control the convergence of the solution. The examples given in this paper lead to the conclusion that the accuracy of the obtained results is growing along with increasing the number of constants in the auxiliary function, which are determined using a computer technique. The results obtained through the proposed method are in very good agreement with the numerical results.
Possible signatures of nonlinear MHD waves in the solar wind: UVCS observations and models
Ofman, L.; Romoli, M.; Davila, J. M.; Poletto, G.; Kohl, J.; Noci, G.
1997-01-01
Recent ultraviolet coronagraph spectrometer (UVCS) white light channel observations are discussed. These data indicated quasi-periodic variations in the polarized brightness in the polar coronal holes. The Fourier power spectrum analysis showed significant peaks at about six minutes and possible fluctuations on longer time scales. The observations are consistent with the predictions of the nonlinear solitary-like wave model. The purpose of a planned study on plume and inter-plume regions of coronal holes, motivated by the result of a 2.5 magnetohydrodynamic model (MHD), is explained.
Thurgood, J. O.; McLaughlin, J. A.
2012-09-01
Context. Coronal magnetic null points have been implicated as possible locations for localised heating events in 2D models. We investigate this possibility about fully 3D null points. Aims: We investigate the nature of the fast magnetoacoustic wave about a fully 3D magnetic null point, with a specific interest in its propagation, and we look for evidence of MHD mode coupling and/or conversion to the Alfvén mode. Methods: A special fieldline and flux-based coordinate system was constructed to permit the introduction of a pure fast magnetoacoustic wave in the vicinity of proper and improper 3D null points. We considered the ideal, β = 0, MHD equations, which are solved using the LARE3D numerical code. The constituent modes of the resulting wave were isolated and identified using the special coordinate system. Numerical results were supported by analytical work derived from perturbation theory and a linear implementation of the WKB method. Results: An initially pure fast wave is found to be permanently decoupled from the Alfvén mode both linearly and nonlinearly for both proper and improper 3D null points. The pure fast mode also generates and sustains a nonlinear disturbance aligned along the equilibrium magnetic field. The resulting pure fast magnetoacoustic pulse has transient behaviour, which is found to be governed by the (equilibrium) Alfvén-speed profile, and a refraction effect focuses all the wave energy towards the null point. Conclusions: Thus, the main results from previous 2D work do indeed carry over to the fully 3D magnetic null points and so we conclude that 3D null points are locations for preferential heating in the corona by 3D fast magnetoacoustic waves.
Nonlinear Acoustics -- Perturbation Theory and Webster's Equation
Jorge, Rogério
2013-01-01
Webster's horn equation (1919) offers a one-dimensional approximation for low-frequency sound waves along a rigid tube with a variable cross-sectional area. It can be thought as a wave equation with a source term that takes into account the nonlinear geometry of the tube. In this document we derive this equation using a simplified fluid model of an ideal gas. By a simple change of variables, we convert it to a Schr\\"odinger equation and use the well-known variational and perturbative methods to seek perturbative solutions. As an example, we apply these methods to the Gabriel's Horn geometry, deriving the first order corrections to the linear frequency. An algorithm to the harmonic modes in any order for a general horn geometry is derived.
Directory of Open Access Journals (Sweden)
Mohammed Almakki
2017-07-01
Full Text Available The entropy generation in unsteady three-dimensional axisymmetric magnetohydrodynamics (MHD nanofluid flow over a non-linearly stretching sheet is investigated. The flow is subject to thermal radiation and a chemical reaction. The conservation equations are solved using the spectral quasi-linearization method. The novelty of the work is in the study of entropy generation in three-dimensional axisymmetric MHD nanofluid and the choice of the spectral quasi-linearization method as the solution method. The effects of Brownian motion and thermophoresis are also taken into account. The nanofluid particle volume fraction on the boundary is passively controlled. The results show that as the Hartmann number increases, both the Nusselt number and the Sherwood number decrease, whereas the skin friction increases. It is further shown that an increase in the thermal radiation parameter corresponds to a decrease in the Nusselt number. Moreover, entropy generation increases with respect to some physical parameters.
The Lie algebra of infinitesimal symmetries of nonlinear diffusion equations
Kersten, Paul H.M.; Gragert, Peter K.H.
1983-01-01
By using developed software for solving overdetermined systems of partial differential equations, the authors establish the complete Lie algebra of infinitesimal symmetries of nonlinear diffusion equations.
Lattice Boltzmann model for nonlinear convection-diffusion equations.
Shi, Baochang; Guo, Zhaoli
2009-01-01
A lattice Boltzmann model for convection-diffusion equation with nonlinear convection and isotropic-diffusion terms is proposed through selecting equilibrium distribution function properly. The model can be applied to the common real and complex-valued nonlinear evolutionary equations, such as the nonlinear Schrödinger equation, complex Ginzburg-Landau equation, Burgers-Fisher equation, nonlinear heat conduction equation, and sine-Gordon equation, by using a real and complex-valued distribution function and relaxation time. Detailed simulations of these equations are performed, and it is found that the numerical results agree well with the analytical solutions and the numerical solutions reported in previous studies.
Exact Solutions for Nonlinear Differential Difference Equations in Mathematical Physics
Directory of Open Access Journals (Sweden)
Khaled A. Gepreel
2013-01-01
Full Text Available We modified the truncated expansion method to construct the exact solutions for some nonlinear differential difference equations in mathematical physics via the general lattice equation, the discrete nonlinear Schrodinger with a saturable nonlinearity, the quintic discrete nonlinear Schrodinger equation, and the relativistic Toda lattice system. Also, we put a rational solitary wave function method to find the rational solitary wave solutions for some nonlinear differential difference equations. The proposed methods are more effective and powerful to obtain the exact solutions for nonlinear difference differential equations.
Nonsmooth analysis of doubly nonlinear evolution equations
Mielke, Alexander; Savare', Giuseppe
2011-01-01
In this paper we analyze a broad class of abstract doubly nonlinear evolution equations in Banach spaces, driven by nonsmooth and nonconvex energies. We provide some general sufficient conditions, on the dissipation potential and the energy functional,for existence of solutions to the related Cauchy problem. We prove our main existence result by passing to the limit in a time-discretization scheme with variational techniques. Finally, we discuss an application to a material model in finite-strain elasticity.
Solving Nonlinear Euler Equations with Arbitrary Accuracy
Dyson, Rodger W.
2005-01-01
A computer program that efficiently solves the time-dependent, nonlinear Euler equations in two dimensions to an arbitrarily high order of accuracy has been developed. The program implements a modified form of a prior arbitrary- accuracy simulation algorithm that is a member of the class of algorithms known in the art as modified expansion solution approximation (MESA) schemes. Whereas millions of lines of code were needed to implement the prior MESA algorithm, it is possible to implement the present MESA algorithm by use of one or a few pages of Fortran code, the exact amount depending on the specific application. The ability to solve the Euler equations to arbitrarily high accuracy is especially beneficial in simulations of aeroacoustic effects in settings in which fully nonlinear behavior is expected - for example, at stagnation points of fan blades, where linearizing assumptions break down. At these locations, it is necessary to solve the full nonlinear Euler equations, and inasmuch as the acoustical energy is of the order of 4 to 5 orders of magnitude below that of the mean flow, it is necessary to achieve an overall fractional error of less than 10-6 in order to faithfully simulate entropy, vortical, and acoustical waves.
A nonlinear Schroedinger wave equation with linear quantum behavior
Energy Technology Data Exchange (ETDEWEB)
Richardson, Chris D.; Schlagheck, Peter; Martin, John; Vandewalle, Nicolas; Bastin, Thierry [Departement de Physique, University of Liege, 4000 Liege (Belgium)
2014-07-01
We show that a nonlinear Schroedinger wave equation can reproduce all the features of linear quantum mechanics. This nonlinear wave equation is obtained by exploring, in a uniform language, the transition from fully classical theory governed by a nonlinear classical wave equation to quantum theory. The classical wave equation includes a nonlinear classicality enforcing potential which when eliminated transforms the wave equation into the linear Schroedinger equation. We show that it is not necessary to completely cancel this nonlinearity to recover the linear behavior of quantum mechanics. Scaling the classicality enforcing potential is sufficient to have quantum-like features appear and is equivalent to scaling Planck's constant.
Energy Technology Data Exchange (ETDEWEB)
Furuya, Atsushi [Kyushu Univ., Interdisciplinary Graduate School of Engineering Sciences, Kasuga, Fukuoka (Japan); Yagi, Masatoshi; Itoh, Sanae-I. [Kyushu Univ., Research Institute for Applied Mechanics, Kasuga, Fukuoka (Japan)
2003-02-01
The linear neoclassical tearing mode is investigated using the four-field reduced neoclassical MHD equations, in which the fluctuating ion parallel flow and ion neoclassical viscosity are taken into account. The dependences of the neoclassical tearing mode on collisionality, diamagnetic drift and q profile are investigated. These results are compared with the results from the conventional three-field model. It is shown that the linear neoclassical tearing mode is stabilized by the ion neoclassical viscosity in the banana regime even if {delta}' > 0. (author)
Alfven Wave Tomography for Cold MHD Plasmas
Energy Technology Data Exchange (ETDEWEB)
I.Y. Dodin; N.J. Fisch
2001-09-07
Alfven waves propagation in slightly nonuniform cold plasmas is studied by means of ideal magnetohydrodynamics (MHD) nonlinear equations. The evolution of the MHD spectrum is shown to be governed by a matrix linear differential equation with constant coefficients determined by the spectrum of quasi-static plasma density perturbations. The Alfven waves are shown not to affect the plasma density inhomogeneities, as they scatter off of them. The application of the MHD spectrum evolution equation to the inverse scattering problem allows tomographic measurements of the plasma density profile by scanning the plasma volume with Alfven radiation.
Local Existence for the Non-Resistive MHD Equations in Nearly Optimal Sobolev Spaces
Fefferman, Charles L.; McCormick, David S.; Robinson, James C.; Rodrigo, Jose L.
2017-02-01
This paper establishes the local-in-time existence and uniqueness of solutions to the viscous, non-resistive magnetohydrodynamics (MHD) equations in {R^d}, where d = 2, 3, with initial data {B_0in H^s(R^d)} and {u_0in H^{s-1+ɛ}(R^d)} for {s > d/2} and any {0 < ɛ < 1}. The proof relies on maximal regularity estimates for the Stokes equation. The obstruction to taking {ɛ=0} is explained by the failure of solutions of the heat equation with initial data {u_0in H^{s-1}} to satisfy {uin L^1(0,T; H^{s+1})}; we provide an explicit example of this phenomenon.
Hoelzl, M.; Huijsmans, G. T. A.; Merkel, P.; Atanasiu, C.; Lackner, K.; Nardon, E.; Aleynikova, K.; Liu, F.; Strumberger, E.; McAdams, R.; Chapman, I.; Fil, A.
2014-11-01
The dynamics of large scale plasma instabilities can be strongly influenced by the mutual interaction with currents flowing in conducting vessel structures. Especially eddy currents caused by time-varying magnetic perturbations and halo currents flowing directly from the plasma into the walls are important. The relevance of a resistive wall model is directly evident for Resistive Wall Modes (RWMs) or Vertical Displacement Events (VDEs). However, also the linear and non-linear properties of most other large-scale instabilities may be influenced significantly by the interaction with currents in conducting structures near the plasma. The understanding of halo currents arising during disruptions and VDEs, which are a serious concern for ITER as they may lead to strong asymmetric forces on vessel structures, could also benefit strongly from these non-linear modeling capabilities. Modeling the plasma dynamics and its interaction with wall currents requires solving the magneto-hydrodynamic (MHD) equations in realistic toroidal X-point geometry consistently coupled with a model for the vacuum region and the resistive conducting structures. With this in mind, the non-linear finite element MHD code JOREK [1, 2] has been coupled [3] with the resistive wall code STARWALL [4], which allows us to include the effects of eddy currents in 3D conducting structures in non-linear MHD simulations. This article summarizes the capabilities of the coupled JOREK-STARWALL system and presents benchmark results as well as first applications to non-linear simulations of RWMs, VDEs, disruptions triggered by massive gas injection, and Quiescent H-Mode. As an outlook, the perspectives for extending the model to halo currents are described.
EL-Dabe, N. T.; Attia, H. A.; Essawy, M. A. I.; Ramadan, A. A.; Abdel-Hamid, A. H.
2016-11-01
The steady MHD axisymmetric flow of an incompressible viscous electrically conducting nanofluid impinging on a permeable plate is investigated with heat and mass transfer. An external uniform magnetic field as well as a uniform inflow, in the presence of either suction or injection, are applied normal to the plate. The effects of heat (generation/absorption) and chemical reaction have been accentuated. This study indicates the incorporated influence of both the thermophoresis phenomenon and the Brownian behavior. Numerical solutions for the governing non-linear momentum, energy and nanoparticle equations have been obtained. The rates of heat and mass transfer are presented and discussed.
Explicit Integration of Friedmann's Equation with Nonlinear Equations of State
Chen, Shouxin; Yang, Yisong
2015-01-01
This paper is a continuation of our earlier study on the integrability of the Friedmann equations in the light of the Chebyshev theorem. Our main focus will be on a series of important, yet not previously touched, problems when the equation of state for the perfect-fluid universe is nonlinear. These include the generalized Chaplygin gas, two-term energy density, trinomial Friedmann, Born--Infeld, and two-fluid models. We show that some of these may be integrated using Chebyshev's result while other are out of reach by the theorem but may be integrated explicitly by other methods. With the explicit integration, we are able to understand exactly the roles of the physical parameters in various models play in the cosmological evolution. For example, in the Chaplygin gas universe, it is seen that, as far as there is a tiny presence of nonlinear matter, linear matter makes contribution to the dark matter, which becomes significant near the phantom divide line. The Friedmann equations also arise in areas of physics ...
The tanh-coth method combined with the Riccati equation for solving non-linear equation
Energy Technology Data Exchange (ETDEWEB)
Bekir, Ahmet [Dumlupinar University, Art-Science Faculty, Department of Mathematics, Kuetahya (Turkey)], E-mail: abekir@dumlupinar.edu.tr
2009-05-15
In this work, we established abundant travelling wave solutions for some non-linear evolution equations. This method was used to construct solitons and traveling wave solutions of non-linear evolution equations. The tanh-coth method combined with Riccati equation presents a wider applicability for handling non-linear wave equations.
ACCELERATION METHODS OF NONLINEAR ITERATION FOR NONLINEAR PARABOLIC EQUATIONS
Institute of Scientific and Technical Information of China (English)
Guang-wei Yuan; Xu-deng Hang
2006-01-01
This paper discusses the accelerating iterative methods for solving the implicit scheme of nonlinear parabolic equations. Two new nonlinear iterative methods named by the implicit-explicit quasi-Newton (IEQN) method and the derivative free implicit-explicit quasi-Newton (DFIEQN) method are introduced, in which the resulting linear equations from the linearization can preserve the parabolic characteristics of the original partial differential equations. It is proved that the iterative sequence of the iteration method can converge to the solution of the implicit scheme quadratically. Moreover, compared with the Jacobian Free Newton-Krylov (JFNK) method, the DFIEQN method has some advantages, e.g., its implementation is easy, and it gives a linear algebraic system with an explicit coefficient matrix, so that the linear (inner) iteration is not restricted to the Krylov method. Computational results by the IEQN, DFIEQN, JFNK and Picard iteration meth-ods are presented in confirmation of the theory and comparison of the performance of these methods.
General Symmetry Approach to Solve Variable-Coefficient Nonlinear Equations
Institute of Scientific and Technical Information of China (English)
RUAN HangYu; CHEN YiXin; LOU SenYue
2001-01-01
After considering the variable coefficient of a nonlinear equation as a new dependent variable, some special types of variable-coefficient equation can be solved from the corresponding constant-coefficient equations by using the general classical Lie approach. Taking the nonlinear Schrodinger equation as a concrete example, the method is recommended in detail.``
Local 4/5-law and energy dissipation anomaly in turbulence of incompressible MHD Equations
Guo, Shanshan; Tan, Zhong
2016-12-01
In this paper, we establish the longitudinal and transverse local energy balance equation of distributional solutions of the incompressible three-dimensional MHD equations. In particular, we find that the functions D_L^ɛ (u,B) and D_T^ɛ (u,B) appeared in the energy balance, all converging to the defect distribution (in the sense of distributions) D(u,B) which has been defined in Gao et al. (Acta Math Sci 33:865-871, 2013). Furthermore, we give a simpler form of defect distribution term, which is similar to the relation in turbulence theory, called the "4 / 3-law." As a corollary, we give the analogous "4 / 5-law" holds in the local sense.
DYNAMIC BIFURCATION OF NONLINEAR EVOLUTION EQUATIONS
Institute of Scientific and Technical Information of China (English)
MA TIAN; WANG SHOUHONG
2005-01-01
The authors introduce a notion of dynamic bifurcation for nonlinear evolution equations, which can be called attractor bifurcation. It is proved that as the control parameter crosses certain critical value, the system bifurcates from a trivial steady state solution to an attractor with dimension between m and m + 1, where m + 1 is the number of eigenvalues crossing the imaginary axis. The attractor bifurcation theory presented in this article generalizes the existing steady state bifurcations and the Hopf bifurcations. It provides a unified point of view on dynamic bifurcation and can be applied to many problems in physics and mechanics.
REITERATED HOMOGENIZATION OF DEGENERATE NONLINEAR ELLIPTIC EQUATIONS
Institute of Scientific and Technical Information of China (English)
无
2002-01-01
The authors study homogenization of some nonlinear partial differential equations of the form -div (a (hx,h2x,Duh)) = f,where a is periodic in the first two arguments and monotone in the third.In particular the case where a satisfies degenerated structure conditions is studied.It is proved that uh converges weakly in Wo1.1 (Ω) to the unique solution of a limit problem as h →∞.Moreover,explicit expressions for the limit problem are obtained.
Khan, Sabeel M.; Hammad, M.; Sunny, D. A.
2017-08-01
In this article, the influence of thermal relaxation time and chemical reaction is studied on the MHD upper-convected viscoelastic fluid with internal structure using the Cattaneo-Christov heat flux equation for the first time in the literature. The flow-governing equations are formulated and are converted into their respective ordinary differential equations (ODEs) with the application of similarity functions. The resulting system of coupled nonlinear ODEs is solved along with the prescribed conditions at boundary using a finite-difference code in MATLAB. Influence of chemical reaction, thermal relaxation time and internal material parameter on the macroscopic and micropolar velocities as well as on the temperature and concentration profiles is examined along with other physical parameters ( e.g., magnetic parameter, Eckert number, Prandtl number and fluid relaxation time). The accuracy of the obtained numerical solution is shown by comparing the physical parameters of interest with particular cases of existing results in the literature.
Coupled Nonlinear Schr\\"{o}dinger equation and Toda equation (the Root of Integrability)
Hisakado, Masato
1997-01-01
We consider the relation between the discrete coupled nonlinear Schr\\"{o}dinger equation and Toda equation. Introducing complex times we can show the intergability of the discrete coupled nonlinear Schr\\"{o}dinger equation. In the same way we can show the integrability in coupled case of dark and bright equations. Using this method we obtain several integrable equations.
Kanki, Takashi; Nagata, Masayoshi; Kagei, Yasuhiro
2011-10-01
The dynamics of structures of magnetic field, current density, and plasma flow generated during multi-pulsed coaxial helicity injection in spherical torus is investigated by 3-D nonlinear MHD simulations. During the driven phase, the flux and current amplifications occur due to the merging and magnetic reconnection between the preexisting plasma in the confinement region and the ejected plasma from the gun region involving the n = 1 helical kink distortion of the central open flux column (COFC). Interestingly, the diamagnetic poloidal flow which tends toward the gun region is then observed due to the steep pressure gradients of the COFC generated by ohmic heating through an injection current winding around the inboard field lines, resulting in the formation of the strong poloidal flow shear at the interface between the COFC and the core region. This result is consistent with the flow shear observed in the HIST. During the decay phase, the configuration approaches the axisymmetric MHD equilibrium state without flow because of the dissipation of magnetic fluctuation energy to increase the closed flux surfaces, suggesting the generation of ordered magnetic field structure. The parallel current density λ concentrated in the COFC then diffuses to the core region so as to reduce the gradient in λ, relaxing in the direction of the Taylor state.
Exact Solution of a Generalized Nonlinear Schrodinger Equation Dimer
DEFF Research Database (Denmark)
Christiansen, Peter Leth; Maniadis, P.; Tsironis, G.P.
1998-01-01
We present exact solutions for a nonlinear dimer system defined throught a discrete nonlinear Schrodinger equation that contains also an integrable Ablowitz-Ladik term. The solutions are obtained throught a transformation that maps the dimer into a double Sine-Gordon like ordinary nonlinear...... differential equation....
Nonlinear and Nonideal MHD. Final annual progress report, January 1, 2003 through December 31, 2003
Energy Technology Data Exchange (ETDEWEB)
Callen, James D
2003-04-30
This is the third and final annual progress report on the current 3-year ''Nonlinear and Nonideal MHD'' DoE grand DE-FG02-86ER53218 for the six months since the November 2002 progress report. During this grant year the funding level was $309k. The participating personnel and their approximate degree of funded involvement in this research project this grant year has been as follows: Professor J. D. Callen (PI, 1.8 months during academic year, 2.2 summer months); Professor C.C. Hegna (Co-PI: 2.3 months during academic year, 1.5 summer months); postdoc Dr. S. Gupta (100%); and graduate students A.L. Garcia-Perciante (50% RA) and X. Liu (50% RA).
Nonlinear Alfvén wave propagating in ideal MHD plasmas
Zheng, Jugao; Chen, Yinhua; Yu, Mingyang
2016-01-01
The behavior of nonlinear Alfvén waves propagating in ideal MHD plasmas is investigated numerically. It is found that in a one-dimensional weakly nonlinear system an Alfvén wave train can excite two longitudinal disturbances, namely an acoustic wave and a ponderomotively driven disturbance, which behave differently for β \\gt 1 and β \\lt 1, where β is the ratio of plasma-to-magnetic pressures. In a strongly nonlinear system, the Alfvén wave train is modulated and can steepen to form shocks, leading to significant dissipation due to appearance of current sheets at magnetic-pressure minima. For periodic boundary condition, we find that the Alfvén wave transfers its energy to the plasma and heats it during the shock formation. In two-dimensional systems, fast magneto-acoustic wave generation due to Alfvén wave phase mixing is considered. It is found that the process depends on the amplitude and frequency of the Alfvén waves, as well as their speed gradients and the pressure of the background plasma.
Soliton states of Maxwell’s equations and nonlinear Schrodinger equation
Institute of Scientific and Technical Information of China (English)
陈翼强
1997-01-01
Similarities and fundamental differences between Maxwell’s equations and nonlinear Schrodinger equation in predicting a soliton evolution in a uniform nonlinear anisotropic medium are analyzed.It is found that in some cases,the soliton solutions to the nonlinear Schrodinger equation cannot be recovered from Maxwell’s equations while in others the soliton solutions to Maxwell’s equations are lost from the nonlinear Schrodinger equation through approximation,although there are cases where the soliton solutions to the two sets of the equations demonstrate only quantitative difference.The origin of the differences is also discussed.
A new application of Riccati equation to some nonlinear evolution equations
Energy Technology Data Exchange (ETDEWEB)
Geng Tao [School of Science, PO Box 122, Beijing University of Posts and Telecommunications, Beijing 100876 (China)], E-mail: taogeng@yahoo.com.cn; Shan Wenrui [School of Science, PO Box 122, Beijing University of Posts and Telecommunications, Beijing 100876 (China)
2008-03-03
By means of symbolic computation, a new application of Riccati equation is presented to obtain novel exact solutions of some nonlinear evolution equations, such as nonlinear Klein-Gordon equation, generalized Pochhammer-Chree equation and nonlinear Schroedinger equation. Comparing with the existing tanh methods and the proposed modifications, we obtain the exact solutions in the form as a non-integer power polynomial of tanh (or tan) functions by using this method, and the availability of symbolic computation is demonstrated.
Institute of Scientific and Technical Information of China (English)
无
2006-01-01
A trial equation method to nonlinear evolution equation with rank inhomogeneous is given. As applications, the exact traveling wave solutions to some higher-order nonlinear equations such as generalized Boussinesq equation,generalized Pochhammer-Chree equation, KdV-Burgers equation, and KS equation and so on, are obtained. Among these, some results are new. The proposed method is based on the idea of reduction of the order of ODE. Some mathematical details of the proposed method are discussed.
Hyperbolic function method for solving nonlinear differential-different equations
Institute of Scientific and Technical Information of China (English)
Zhu Jia-Min
2005-01-01
An algorithm is devised to obtained exact travelling wave solutions of differential-different equations by means of hyperbolic function. For illustration, we apply the method to solve the discrete nonlinear (2+1)-dimensional Toda lattice equation and the discretized nonlinear mKdV lattice equation, and successfully constructed some explicit and exact travelling wave solutions.
Extension of Variable Separable Solutions for Nonlinear Evolution Equations
Institute of Scientific and Technical Information of China (English)
JIA Hua-Bing; ZHANG Shun-Li; XU Wei; ZHU Xiao-Ning; WANG Yong-Mao; LOU Sen-Yue
2008-01-01
We give the generalized definitions of variable separable solutions to nonlinear evolution equations, and characterize the relation between the functional separable solution and the derivative-dependent functional separablecation, we classify the generalized nonlinear diffusion equations that admit special functional separable solutions and obtain some exact solutions to the resulting equations.
Almost Periodic Viscosity Solutions of Nonlinear Parabolic Equations
Directory of Open Access Journals (Sweden)
Zhang Shilin
2009-01-01
Full Text Available We generalize the comparison result 2007 on Hamilton-Jacobi equations to nonlinear parabolic equations, then by using Perron's method to study the existence and uniqueness of time almost periodic viscosity solutions of nonlinear parabolic equations under usual hypotheses.
Positive periodic solutions for third-order nonlinear differential equations
Directory of Open Access Journals (Sweden)
Jingli Ren
2011-05-01
Full Text Available For several classes of third-order constant coefficient linear differential equations we obtain existence and uniqueness of periodic solutions utilizing explicit Green's functions. We discuss an iteration method for constant coefficient nonlinear differential equations and provide new conditions for the existence of periodic positive solutions for third-order time-varying nonlinear and neutral differential equations.
Exact periodic wave solutions for some nonlinear partial differential equations
Energy Technology Data Exchange (ETDEWEB)
El-Wakil, S.A. [Theoretical Physics Research Group, Department of Physics, Faculty of Science, Mansoura University, Mansoura 35516 (Egypt); Elgarayhi, A. [Theoretical Physics Research Group, Department of Physics, Faculty of Science, Mansoura University, Mansoura 35516 (Egypt)]. E-mail: elgarayhi@yahoo.com; Elhanbaly, A. [Theoretical Physics Research Group, Department of Physics, Faculty of Science, Mansoura University, Mansoura 35516 (Egypt)
2006-08-15
The periodic wave solutions for some nonlinear partial differential equations, including generalized Klein-Gordon equation, Kadomtsev-Petviashvili (KP) equation and Boussinesq equations, are obtained by using the solutions of Jacobi elliptic equation. Under limit conditions, exact solitary wave solutions, shock wave solutions and triangular periodic wave solutions have been recovered.
Optimal decay rates of classical solutions for the full compressible MHD equations
Gao, Jincheng; Tao, Qiang; Yao, Zheng-an
2016-04-01
In this paper, we are concerned with optimal decay rates for higher-order spatial derivatives of classical solutions to the full compressible MHD equations in three-dimensional whole space. If the initial perturbation is small in {H^3}-norm and bounded in {L^q(qin [1, 6/5 ))}-norm, we apply the Fourier splitting method by Schonbek (Arch Ration Mech Anal 88:209-222, 1985) to establish optimal decay rates for the second-order spatial derivatives of solutions and the third-order spatial derivatives of magnetic field in {L^2}-norm. These results improve the work of Pu and Guo (Z Angew Math Phys 64:519-538, 2013).
A discontinuous Galerkin method for solving the fluid and MHD equations in astrophysical simulations
Mocz, Philip; Sijacki, Debora; Hernquist, Lars
2013-01-01
A discontinuous Galerkin (DG) method suitable for large-scale astrophysical simulations on Cartesian meshes as well as arbitrary static and moving Voronoi meshes is presented. Most major astrophysical fluid dynamics codes use a finite volume (FV) approach. We demonstrate that the DG technique offers distinct advantages over FV formulations on both static and moving meshes. The DG method is also easily generalized to higher than second-order accuracy without requiring the use of extended stencils to estimate derivatives (thereby making the scheme highly parallelizable). We implement the technique in the AREPO code for solving the fluid and the magnetohydrodynamic (MHD) equations. By examining various test problems, we show that our new formulation provides improved accuracy over FV approaches of the same order, and reduces post-shock oscillations and artificial diffusion of angular momentum. In addition, the DG method makes it possible to represent magnetic fields in a locally divergence-free way, improving th...
Exact solitary wave solutions of nonlinear wave equations
Institute of Scientific and Technical Information of China (English)
无
2001-01-01
The hyperbolic function method for nonlinear wave equations ispresented. In support of a computer algebra system, many exact solitary wave solutions of a class of nonlinear wave equations are obtained via the method. The method is based on the fact that the solitary wave solutions are essentially of a localized nature. Writing the solitary wave solutions of a nonlinear wave equation as the polynomials of hyperbolic functions, the nonlinear wave equation can be changed into a nonlinear system of algebraic equations. The system can be solved via Wu Elimination or Grbner base method. The exact solitary wave solutions of the nonlinear wave equation are obtained including many new exact solitary wave solutions.
Hoelzl, M; Merkel, P; Atanasiu, C; Lackner, K; Nardon, E; Aleynikova, K; Liu, F; Strumberger, E; McAdams, R; Chapman, I; Fil, A
2014-01-01
The dynamics of large scale plasma instabilities can strongly be influenced by the mutual interaction with currents flowing in conducting vessel structures. Especially eddy currents caused by time-varying magnetic perturbations and halo currents flowing directly from the plasma into the walls are important. The relevance of a resistive wall model is directly evident for Resistive Wall Modes (RWMs) or Vertical Displacement Events (VDEs). However, also the linear and non-linear properties of most other large-scale instabilities may be influenced significantly by the interaction with currents in conducting structures near the plasma. The understanding of halo currents arising during disruptions and VDEs, which are a serious concern for ITER as they may lead to strong asymmetric forces on vessel structures, could also benefit strongly from these non-linear modeling capabilities. Modeling the plasma dynamics and its interaction with wall currents requires solving the magneto-hydrodynamic (MHD) equations in realist...
A remark on the Beale-Kato-Majda criterion for the 3D MHD equations with zero magnetic diffusivity
Gala, Sadek; Ragusa, Maria Alessandra
2016-06-01
In this work, we show that a smooth solution of the 3D MHD equations with zero magnetic diffusivity in the whole space ℝ3 breaks down if and only if a certain norm of the magnetic field blows up at the same time.
Some new solutions of nonlinear evolution equations with variable coefficients
Virdi, Jasvinder Singh
2016-05-01
We construct the traveling wave solutions of nonlinear evolution equations (NLEEs) with variable coefficients arising in physics. Some interesting nonlinear evolution equations are investigated by traveling wave solutions which are expressed by the hyperbolic functions, the trigonometric functions and rational functions. The applied method will be used in further works to establish more entirely new solutions for other kinds of such nonlinear evolution equations with variable coefficients arising in physics.
Logarithmic singularities of solutions to nonlinear partial differential equations
Tahara, Hidetoshi
2007-01-01
We construct a family of singular solutions to some nonlinear partial differential equations which have resonances in the sense of a paper due to T. Kobayashi. The leading term of a solution in our family contains a logarithm, possibly multiplied by a monomial. As an application, we study nonlinear wave equations with quadratic nonlinearities. The proof is by the reduction to a Fuchsian equation with singular coefficients.
New travelling wave solutions for nonlinear stochastic evolution equations
Indian Academy of Sciences (India)
Hyunsoo Kim; Rathinasamy Sakthivel
2013-06-01
The nonlinear stochastic evolution equations have a wide range of applications in physics, chemistry, biology, economics and finance from various points of view. In this paper, the (′/)-expansion method is implemented for obtaining new travelling wave solutions of the nonlinear (2 + 1)-dimensional stochastic Broer–Kaup equation and stochastic coupled Korteweg–de Vries (KdV) equation. The study highlights the significant features of the method employed and its capability of handling nonlinear stochastic problems.
Nonclassical Symmetries for Nonlinear Partial Differential Equations via Compatibility
Institute of Scientific and Technical Information of China (English)
Mostafa F. El-Sabbagh; Ahmad T. Ali
2011-01-01
The determining equations for the nonclassical symmetry reductions of nonlinear partial differential equations with arbitrary order can be obtained by requiring the compatibility between the original equations and the invariant surface conditions. The （2＋1）-dimensional shallow water wave equation, Boussinesq equation, and the dispersive wave equations in shallow water serve as examples i11ustrating how compatibility leads quickly and easily to the determining equations for their nonclassical symmetries.
Steen-Ermakov-Pinney equation and integrable nonlinear deformation of one-dimensional Dirac equation
Prykarpatskyy, Yarema
2017-01-01
The paper deals with nonlinear one-dimensional Dirac equation. We describe its invariants set by means of the deformed linear Dirac equation, using the fact that two ordinary differential equations are equivalent if their sets of invariants coincide.
Homoclinic orbits of second-order nonlinear difference equations
Directory of Open Access Journals (Sweden)
Haiping Shi
2015-06-01
Full Text Available We establish existence criteria for homoclinic orbits of second-order nonlinear difference equations by using the critical point theory in combination with periodic approximations.
Bifurcation and stability for a nonlinear parabolic partial differential equation
Chafee, N.
1973-01-01
Theorems are developed to support bifurcation and stability of nonlinear parabolic partial differential equations in the solution of the asymptotic behavior of functions with certain specified properties.
On the exact controllability of a nonlinear stochastic heat equation
Directory of Open Access Journals (Sweden)
Bui An Ton
2006-01-01
Full Text Available The exact controllability of a nonlinear stochastic heat equation with null Dirichlet boundary conditions, nonzero initial and target values, and an interior control is established.
SOLVABILITY FOR NONLINEAR ELLIPTIC EQUATION WITH BOUNDARY PERTURBATION
Institute of Scientific and Technical Information of China (English)
无
2007-01-01
The solvability of nonlinear elliptic equation with boundary perturbation is considered. The perturbed solution of original problem is obtained and the uniformly valid expansion of solution is proved.
OSCILLATION OF NONLINEAR IMPULSIVE PARABOLIC DIFFERENTIAL EQUATIONS WITH SEVERAL DELAYS
Institute of Scientific and Technical Information of China (English)
CuiChenpei; ZouMin; LiuAnping; XiaoLi
2005-01-01
In this paper, oscillatory properties for solutions of certain nonlinear impulsive parabolic equations with several delays are investigated and a series of new sufficient conditions for oscillations of the equation are established.
ALMOST PERIODIC SOLUTIONS TO SOME NONLINEAR DELAY DIFFERENTIAL EQUATION
Institute of Scientific and Technical Information of China (English)
无
2009-01-01
The existence of an almost periodic solutions to a nonlinear delay diffierential equation is considered in this paper. A set of sufficient conditions for the existence and uniqueness of almost periodic solutions to some delay diffierential equations is obtained.
Generalized Nonlinear Proca Equation and its Free-Particle Solutions
Nobre, F D
2016-01-01
We introduce a non-linear extension of Proca's field theory for massive vector (spin $1$) bosons. The associated relativistic nonlinear wave equation is related to recently advanced nonlinear extensions of the Schroedinger, Dirac, and Klein-Gordon equations inspired on the non-extensive generalized thermostatistics. This is a theoretical framework that has been applied in recent years to several problems in nuclear and particle physics, gravitational physics, and quantum field theory. The nonlinear Proca equation investigated here has a power-law nonlinearity characterized by a real parameter $q$ (formally corresponding to the Tsallis entropic parameter) in such a way that the standard linear Proca wave equation is recovered in the limit $q \\rightarrow 1$. We derive the nonlinear Proca equation from a Lagrangian that, besides the usual vectorial field $\\Psi^{\\mu}(\\vec{x},t)$, involves an additional field $\\Phi^{\\mu}(\\vec{x},t)$. We obtain exact time dependent soliton-like solutions for these fields having the...
Directory of Open Access Journals (Sweden)
Elsayed Mohamed Elsayed ZAYED
2014-07-01
Full Text Available In this article, many new exact solutions of the (2+1-dimensional nonlinear Boussinesq-Kadomtsev-Petviashvili equation and the (1+1-dimensional nonlinear heat conduction equation are constructed using the Riccati equation mapping method. By means of this method, many new exact solutions are successfully obtained. This method can be applied to many other nonlinear evolution equations in mathematical physics.doi:10.14456/WJST.2014.14
A nonlinear wave equation with a nonlinear integral equation involving the boundary value
Directory of Open Access Journals (Sweden)
Thanh Long Nguyen
2004-09-01
Full Text Available We consider the initial-boundary value problem for the nonlinear wave equation $$displaylines{ u_{tt}-u_{xx}+f(u,u_{t}=0,quad xin Omega =(0,1,; 0
Fuzzy Modeling for Uncertainty Nonlinear Systems with Fuzzy Equations
Directory of Open Access Journals (Sweden)
Raheleh Jafari
2017-01-01
Full Text Available The uncertain nonlinear systems can be modeled with fuzzy equations by incorporating the fuzzy set theory. In this paper, the fuzzy equations are applied as the models for the uncertain nonlinear systems. The nonlinear modeling process is to find the coefficients of the fuzzy equations. We use the neural networks to approximate the coefficients of the fuzzy equations. The approximation theory for crisp models is extended into the fuzzy equation model. The upper bounds of the modeling errors are estimated. Numerical experiments along with comparisons demonstrate the excellent behavior of the proposed method.
Power Series Solution for Solving Nonlinear Burgers-Type Equations
Directory of Open Access Journals (Sweden)
E. López-Sandoval
2015-01-01
Full Text Available Power series solution method has been traditionally used to solve ordinary and partial linear differential equations. However, despite their usefulness the application of this method has been limited to this particular kind of equations. In this work we use the method of power series to solve nonlinear partial differential equations. The method is applied to solve three versions of nonlinear time-dependent Burgers-type differential equations in order to demonstrate its scope and applicability.
A NONLINEAR LAVRENTIEV-BITSADZE MIXED TYPE EQUATION
Institute of Scientific and Technical Information of China (English)
Chen Shuxing
2011-01-01
In this paper the Tricomi problem for a nonlinear mixed type equation is studied.The coefficients of the mixed type equation are discontinuous on the line,where the equation changes its type.The existence of solution to this problem is proved.The method developed in this paper can be applied to study more difficult problems for nonlinear mixed type equations arising in gas dynamics.
Power Series Solution for Solving Nonlinear Burgers-Type Equations
López-Sandoval, E.; Mello, A.; Godina-Nava, J. J.; Samana, A. R.
2015-01-01
Power series solution method has been traditionally used to solve ordinary and partial linear differential equations. However, despite their usefulness the application of this method has been limited to this particular kind of equations. In this work we use the method of power series to solve nonlinear partial differential equations. The method is applied to solve three versions of nonlinear time-dependent Burgers-type differential equations in order to demonstrate its scope and applicability.
Solution and Positive Solution to Nonlinear Cantilever Beam Equations
Institute of Scientific and Technical Information of China (English)
无
2008-01-01
Using the decomposition technique of equation and the fixed point theorem, the existence of solution and positive solution is studied for a nonlinear cantilever beam equation. The equation describes the deformation of the elastic beam with a fixed end and a free end. The main results show that the equation has at least one solution or positive solution, provided that the "height" of nonlinear term is appropriate on a bounded set.
Hanasoge, S M; Gizon, L
2010-01-01
Perfectly matched layers are a very efficient and accurate way to absorb waves in media. We present a stable convolutional unsplit perfectly matched formulation designed for the linearized stratified Euler equations. However, the technique as applied to the Magneto-hydrodynamic (MHD) equations requires the use of a sponge, which, despite placing the perfectly matched status in question, is still highly efficient at absorbing outgoing waves. We study solutions of the equations in the backdrop of models of linearized wave propagation in the Sun. We test the numerical stability of the schemes by integrating the equations over a large number of wave periods.
Exact solutions for some nonlinear partial differential equations
Energy Technology Data Exchange (ETDEWEB)
Peng, Yan-Ze
2003-08-11
Exact solutions to some nonlinear partial differential equations, including (2+1)-dimensional breaking soliton equation, sine-Gordon equation and double sine-Gordon equation, are studied by means of the mapping method proposed by the author recently. Many new results are presented. A simple review of the method is finally given.
Simple equation method for nonlinear partial differential equations and its applications
Directory of Open Access Journals (Sweden)
Taher A. Nofal
2016-04-01
Full Text Available In this article, we focus on the exact solution of the some nonlinear partial differential equations (NLPDEs such as, Kodomtsev–Petviashvili (KP equation, the (2 + 1-dimensional breaking soliton equation and the modified generalized Vakhnenko equation by using the simple equation method. In the simple equation method the trial condition is the Bernoulli equation or the Riccati equation. It has been shown that the method provides a powerful mathematical tool for solving nonlinear wave equations in mathematical physics and engineering problems.
ANALYTICAL SOLUTIONS FOR SOME NONLINEAR EVOLUTION EQUATIONS
Institute of Scientific and Technical Information of China (English)
胡建兰; 张汉林
2003-01-01
The following partial differential equations are studied: generaliz ed fifth-orderKdV equation, water wave equation, Kupershmidt equation, couples KdV equation. Theanalytical solutions to these problems via using various ansaiz es by introducing a second-order ordinary differential equation are found out.
DIFFERENCE METHODS FOR A NON-LINEAR ELLIPTIC SYSTEM OF PARTIAL DIFFERENTIAL EQUATIONS,
DIFFERENCE EQUATIONS, ITERATIONS), (*ITERATIONS, DIFFERENCE EQUATIONS), (* PARTIAL DIFFERENTIAL EQUATIONS , BOUNDARY VALUE PROBLEMS), EQUATIONS, FUNCTIONS(MATHEMATICS), SEQUENCES(MATHEMATICS), NONLINEAR DIFFERENTIAL EQUATIONS
Exact Travelling Wave Solutions to a Coupled Nonlinear Evolution Equation[
Institute of Scientific and Technical Information of China (English)
HUANGDing-Jiang; ZHANGHong-Qing
2004-01-01
By using an improved hyperbola function method, several types of exact travelling wave solutions to a coupled nonlinear evolution equation are obtained, which include kink-shaped soliton solutions, bell-shaped soliton solutions, envelop solitary wave solutions, and new solitary waves. The method can be applied to other nonlinear evolution equations in mathematical physics.
Exact Travelling Wave Solutions to a Coupled Nonlinear Evolution Equation
Institute of Scientific and Technical Information of China (English)
HUANG Ding-Jiang; ZHANG Hong-Qing
2004-01-01
By using an improved hyperbola function method, several types of exact travelling wave solutions to a coupled nonlinear evolution equation are obtained, which include kink-shaped soliton solutions, bell-shaped soliton solutions, envelop solitary wave solutions, and new solitary waves. The method can be applied to other nonlinear evolution equations in mathematical physics.
A variational approach to nonlinear evolution equations in optics
Indian Academy of Sciences (India)
D Anderson; M Lisak; A Berntson
2001-11-01
A tutorial review is presented of the use of direct variational methods based on RayleighRitz optimization for ﬁnding approximate solutions to various nonlinear evolution equations. The practical application of the approach is demonstrated by some illustrative examples in connection with the nonlinear Schrödinger equation.
Collapse in a forced three-dimensional nonlinear Schrodinger equation
DEFF Research Database (Denmark)
Lushnikov, P.M.; Saffman, M.
2000-01-01
We derive sufficient conditions for the occurrence of collapse in a forced three-dimensional nonlinear Schrodinger equation without dissipation. Numerical studies continue the results to the case of finite dissipation.......We derive sufficient conditions for the occurrence of collapse in a forced three-dimensional nonlinear Schrodinger equation without dissipation. Numerical studies continue the results to the case of finite dissipation....
Prolongation Structure of Semi-discrete Nonlinear Evolution Equations
Institute of Scientific and Technical Information of China (English)
无
2007-01-01
Based on noncommutative differential calculus, we present a theory of prolongation structure for semi-discrete nonlinear evolution equations. As an illustrative example, a semi-discrete model of the nonlinear Schr(o)dinger equation is discussed in terms of this theory and the corresponding Lax pairs are also given.
Exact solutions for some nonlinear systems of partial differential equations
Energy Technology Data Exchange (ETDEWEB)
Darwish, A.A. [Department of Mathematics, Faculty of Science, Helwan University (Egypt)], E-mail: profdarwish@yahoo.com; Ramady, A. [Department of Mathematics, Faculty of Science, Beni-Suef University (Egypt)], E-mail: aramady@yahoo.com
2009-04-30
A direct and unified algebraic method for constructing multiple travelling wave solutions of nonlinear systems of partial differential equations (PDEs) is used and implemented in a computer algebraic system. New solutions for some nonlinear partial differential equations (NLPDEs) are obtained. Graphs of the solutions are displayed.
Alternative Forms of Enhanced Boussinesq Equations with Improved Nonlinearity
Directory of Open Access Journals (Sweden)
Kezhao Fang
2013-01-01
Full Text Available We propose alternative forms of the Boussinesq equations which extend the equations of Madsen and Schäffer by introducing extra nonlinear terms during enhancement. Theoretical analysis shows that nonlinear characteristics are considerably improved. A numerical implementation of one-dimensional equations is described. Three tests involving strongly nonlinear evolution, namely, regular waves propagating over an elevated bar feature in a tank with an otherwise constant depth, wave group transformation over constant water depth, and nonlinear shoaling of unsteady waves over a sloping beach, are simulated by the model. The model is found to be effective.
DEFF Research Database (Denmark)
Guo, Hairun; Zeng, Xianglong; Zhou, Binbin
2013-01-01
We interpret the purely spectral forward Maxwell equation with up to third-order induced polarizations for pulse propagation and interactions in quadratic nonlinear crystals. The interpreted equation, also named the nonlinear wave equation in the frequency domain, includes quadratic and cubic...
Marchenko Equation for the Derivative Nonlinear Schr(o)dinger Equation
Institute of Scientific and Technical Information of China (English)
HUANG Nian-Ning
2007-01-01
A simple derivation of the Marchenko equation is given for the derivative nonlinear Schr(o)dinger equation.The kernel of the Marchenko equation is demanded to satisfy the conditions given by the compatibility equations.the soliton solutions to the Marchenko equation are verified.The derivation is not concerned with the revisions of Kaup and Newell.
Estimation of saturation and coherence effects in the KGBJS equation - a non-linear CCFM equation
Deak, Michal
2012-01-01
We solve the modified non-linear extension of the CCFM equation - KGBJS equation - numerically for certain initial conditions and compare the resulting gluon Green functions with those obtained from solving the original CCFM equation and the BFKL and BK equations for the same initial conditions. We improve the low transversal momentum behaviour of the KGBJS equation by a small modification.
Exploring the Nonlinear Cloud and Rain Equation
Koren, Ilan; Feingold, Graham
2016-01-01
Marine stratocumulus cloud decks are regarded as the reflectors of the climate system, returning back to space a significant part of the income solar radiation, thus cooling the atmosphere. Such clouds can exist in two stable modes, open and closed cells, for a wide range of environmental conditions. This emergent behavior of the system, and its sensitivity to aerosol and environmental properties, is captured by a set of nonlinear equations. Here, using linear stability analysis, we express the transition from steady to a limit-cycle state analytically, showing how it depends on the model parameters. We show that the control of the droplet concentration (N) the environmental carrying-capacity (H0) and the cloud recovery parameter (tau) can be linked by a single nondimensional parameter mu=N/(alfa*tau*H0), suggesting that for deeper clouds the transition from open (oscillating) to closed (stable fixed point) cells will occur for higher droplet concentration (i.e. higher aerosol loading). The analytical calcula...
Lienard Equation and Exact Solutions for Some Soliton-Producing Nonlinear Equations
Institute of Scientific and Technical Information of China (English)
ZHANG Wei-Guo; CHANG Qian-Shun; ZHANG Qi-Ren
2004-01-01
In this paper, we first consider exact solutions for Lienard equation with nonlinear terms of any order. Then,explicit exact bell and kink profile solitary-wave solutions for many nonlinear evolution equations are obtained by means of results of the Lienard equation and proper deductions, which transform original partial differential equations into the Lienard one. These nonlinear equations include compound KdV, compound KdV-Burgers, generalized Boussinesq,generalized KP and Ginzburg-Landau equation. Some new solitary-wave solutions are found.
Smoothing and Decay Estimates for Nonlinear Diffusion Equations Equations of Porous Medium Type
Vázquez, Juan Luis
2006-01-01
This text is concerned with the quantitative aspects of the theory of nonlinear diffusion equations; equations which can be seen as nonlinear variations of the classical heat equation. They appear as mathematical models in different branches of Physics, Chemistry, Biology, and Engineering, and are also relevant in differential geometry and relativistic physics. Much of the modern theory of such equations is based on estimates and functional analysis.Concentrating on a class of equations with nonlinearities of power type that lead to degenerate or singular parabolicity ("equations of porou
Asymptotic behavior of solutions to nonlinear parabolic equation with nonlinear boundary conditions
Directory of Open Access Journals (Sweden)
Diabate Nabongo
2008-01-01
Full Text Available We show that solutions of a nonlinear parabolic equation of second order with nonlinear boundary conditions approach zero as t approaches infinity. Also, under additional assumptions, the solutions behave as a function determined here.
Techniques in Linear and Nonlinear Partial Differential Equations
1991-10-21
nonlinear partial differential equations , elliptic 15. NUMBER OF PAGES hyperbolic and parabolic. Variational methods. Vibration problems. Ordinary Five...NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS FINAL TECHNICAL REPORT PROFESSOR LOUIS NIRENBERG OCTOBER 21, 1991 NT)S CRA&I D FIC ,- U.S. ARMY RESEARCH OFFICE...Analysis and partial differential equations . ed. C. Sadowsky. Marcel Dekker (1990) 567-619. [7] Lin, Fanghua, Asymptotic behavior of area-minimizing
Linear and nonlinear degenerate abstract differential equations with small parameter
Shakhmurov, Veli B.
2016-01-01
The boundary value problems for linear and nonlinear regular degenerate abstract differential equations are studied. The equations have the principal variable coefficients and a small parameter. The linear problem is considered on a parameter-dependent domain (i.e., on a moving domain). The maximal regularity properties of linear problems and the optimal regularity of the nonlinear problem are obtained. In application, the well-posedness of the Cauchy problem for degenerate parabolic equation...
Travelling wave solutions for ( + 1)-dimensional nonlinear evolution equations
Indian Academy of Sciences (India)
Jonu Lee; Rathinasamy Sakthivel
2010-10-01
In this paper, we implement the exp-function method to obtain the exact travelling wave solutions of ( + 1)-dimensional nonlinear evolution equations. Four models, the ( + 1)-dimensional generalized Boussinesq equation, ( + 1)-dimensional sine-cosine-Gordon equation, ( + 1)-double sinh-Gordon equation and ( + 1)-sinh-cosinh-Gordon equation, are used as vehicles to conduct the analysis. New travelling wave solutions are derived.
International Conference on Differential Equations and Nonlinear Mechanics
2001-01-01
The International Conference on Differential Equations and Nonlinear Mechanics was hosted by the University of Central Florida in Orlando from March 17-19, 1999. One of the conference days was dedicated to Professor V. Lakshmikantham in th honor of his 75 birthday. 50 well established professionals (in differential equations, nonlinear analysis, numerical analysis, and nonlinear mechanics) attended the conference from 13 countries. Twelve of the attendees delivered hour long invited talks and remaining thirty-eight presented invited forty-five minute talks. In each of these talks, the focus was on the recent developments in differential equations and nonlinear mechanics and their applications. This book consists of 29 papers based on the invited lectures, and I believe that it provides a good selection of advanced topics of current interest in differential equations and nonlinear mechanics. I am indebted to the Department of Mathematics, College of Arts and Sciences, Department of Mechanical, Materials and Ae...
The effect of nonlinearity on unstable zones of Mathieu equation
Indian Academy of Sciences (India)
M GH SARYAZDI
2017-03-01
Mathieu equation is a well-known ordinary differential equation in which the excitation term appears as the non-constant coefficient. The mathematical modelling of many dynamic systems leads to Mathieu equation. The determination of the locus of unstable zone is important for the control of dynamic systems. In this paper, the stable and unstable regions of Mathieu equation are determined for three cases of linear and nonlinear equations using the homotopy perturbation method. The effect of nonlinearity is examined in the unstable zone. The results show that the transition curves of linear Mathieu equation depend on the frequency of the excitation term. However, for nonlinear equations, the curves depend also on initial conditions. In addition, increasing the amplitude of response leads to an increase in the unstable zone.
An FCT finite element scheme for ideal MHD equations in 1D and 2D
Basting, Melanie; Kuzmin, Dmitri
2017-06-01
This paper presents an implicit finite element (FE) scheme for solving the equations of ideal magnetohydrodynamics in 1D and 2D. The continuous Galerkin approximation is constrained using a flux-corrected transport (FCT) algorithm. The underlying low-order scheme is constructed using a Rusanov-type artificial viscosity operator based on scalar dissipation proportional to the fast wave speed. The accuracy of the low-order solution can be improved using a shock detector which makes it possible to prelimit the added viscosity in a monotonicity-preserving iterative manner. At the FCT correction step, the changes of conserved quantities are limited in a way which guarantees positivity preservation for the density and thermal pressure. Divergence-free magnetic fields are extracted using projections of the FCT predictor into staggered finite element spaces forming exact sequences. In the 2D case, the magnetic field is projected into the space of Raviart-Thomas finite elements. Numerical studies for standard test problems are performed to verify the ability of the proposed algorithms to enforce relevant constraints in applications to ideal MHD flows.
Si, Xin; Ye, Xia
2016-10-01
This paper concerns an initial-boundary value problem of the inhomogeneous incompressible MHD equations in a smooth bounded domain. The viscosity and resistivity coefficients are density-dependent. The global well-posedness of strong solutions is established, provided the initial norms of velocity and magnetic field are suitably small in some sense, or the lower bound of the transport coefficients are large enough. More importantly, there is not any smallness condition on the density and its gradient.
REDUCTION OF NONLINEAR PARTIAL DIFFERENTIAL EQUATION AND EXACT SOLUTIONS
Institute of Scientific and Technical Information of China (English)
YeCaier; PanZuliang
2003-01-01
Nonlinear partial differetial equation(NLPDE)is converted into ordinary differential equation(ODE)via a new ansatz.Using undetermined function method,the ODE obtained above is replaced by a set of algebraic equations which are solved out with the aid of Mathematica.The exact solutions and solitary solutions of NLPDE are obtained.
EXACT SOLITARY WAVE SOLUTIONS OF THETWO NONLINEAR EVOLUTION EQUATIONS
Institute of Scientific and Technical Information of China (English)
ZhuYanjuan; ZhangChunhua
2005-01-01
The solitary wave solutions of the combined KdV-mKdV-Burgers equation and the Kolmogorov-Petrovskii-Piskunov equation are obtained by means of the direct algebra method, which can be generalized to deal with high dimensional nonlinear evolution equations.
A Family of Exact Solutions for the Nonlinear Schrodinger Equation
Institute of Scientific and Technical Information of China (English)
无
2001-01-01
In this paper, the nonlinear Schrodinger (NLS) equation was analytically solved. Firstly, the stationary solutions of NLSequation were explicitly given by the elliptic functions. Then a family of exact solutions of NLS equation were obtained from these sta-tionary solutions by a method for finding new exact solutions from the stationary solutions of integrable evolution equations.
Boundary controllability for a nonlinear beam equation
Directory of Open Access Journals (Sweden)
Xiao-Min Cao
2015-09-01
Full Text Available This article concerns a nonlinear system modeling the bending vibrations of a nonlinear beam of length $L>0$. First, we derive the existence of long time solutions near an equilibrium. Then we prove that the nonlinear beam is locally exact controllable around the equilibrium in $H^4(0,L$ and with control functions in $H^2(0,T$. The approach we used are open mapping theorem, local controllability established by linearization, and the induction.
Lectures on nonlinear evolution equations initial value problems
Racke, Reinhard
2015-01-01
This book mainly serves as an elementary, self-contained introduction to several important aspects of the theory of global solutions to initial value problems for nonlinear evolution equations. The book employs the classical method of continuation of local solutions with the help of a priori estimates obtained for small data. The existence and uniqueness of small, smooth solutions that are defined for all values of the time parameter are investigated. Moreover, the asymptotic behavior of the solutions is described as time tends to infinity. The methods for nonlinear wave equations are discussed in detail. Other examples include the equations of elasticity, heat equations, the equations of thermoelasticity, Schrödinger equations, Klein-Gordon equations, Maxwell equations and plate equations. To emphasize the importance of studying the conditions under which small data problems offer global solutions, some blow-up results are briefly described. Moreover, the prospects for corresponding initial-boundary value p...
Energy Technology Data Exchange (ETDEWEB)
Mancas, Stefan C. [Department of Mathematics, Embry–Riddle Aeronautical University, Daytona Beach, FL 32114-3900 (United States); Rosu, Haret C., E-mail: hcr@ipicyt.edu.mx [IPICYT, Instituto Potosino de Investigacion Cientifica y Tecnologica, Apdo Postal 3-74 Tangamanga, 78231 San Luis Potosí, SLP (Mexico)
2013-09-02
We emphasize two connections, one well known and another less known, between the dissipative nonlinear second order differential equations and the Abel equations which in their first-kind form have only cubic and quadratic terms. Then, employing an old integrability criterion due to Chiellini, we introduce the corresponding integrable dissipative equations. For illustration, we present the cases of some integrable dissipative Fisher, nonlinear pendulum, and Burgers–Huxley type equations which are obtained in this way and can be of interest in applications. We also show how to obtain Abel solutions directly from the factorization of second order nonlinear equations.
Abdel-Wahed, Mohamed; Akl, Mohamed
2016-09-01
Analysis of the MHD Nanofluid boundary layer flow over a rotating disk with a constant velocity in the presence of hall current and non-linear thermal radiation has been covered in this work. The variation of viscosity and thermal conductivity of the fluid due to temperature and nanoparticles concentration and size is considered. The problem described by a system of P.D.E that converted to a system of ordinary differential equations by the similarity transformation technique, the obtained system solved analytically using Optimal Homotopy Asymptotic Method (OHAM) with association of mathematica program. The velocity profiles and temperature profiles of the boundary layer over the disk are plotted and investigated in details. Moreover, the surface shear stress, rate of heat transfer explained in details.
Analytic solutions of a class of nonlinear partial differential equations
Institute of Scientific and Technical Information of China (English)
ZHANG Hong-qing; DING Qi
2008-01-01
An approach is presented for computing the adjoint operator vector of a class of nonlinear (that is,partial-nonlinear) operator matrices by using the properties of conjugate operators to generalize a previous method proposed by the author.A unified theory is then given to solve a class of nonlinear (partial-nonlinear and including all linear)and non-homogeneous differential equations with a mathematical mechanization method.In other words,a transformation is constructed by homogenization and triangulation,which reduces the original system to a simpler diagonal system.Applications are given to solve some elasticity equations.
Solutions to nonlinear Schrodinger equations for special initial data
Directory of Open Access Journals (Sweden)
Takeshi Wada
2015-11-01
Full Text Available This article concerns the solvability of the nonlinear Schrodinger equation with gauge invariant power nonlinear term in one space dimension. The well-posedness of this equation is known only for $H^s$ with $s\\ge 0$. Under some assumptions on the nonlinearity, this paper shows that this equation is uniquely solvable for special but typical initial data, namely the linear combinations of $\\delta(x$ and p.v. (1/x, which belong to $H^{-1/2-0}$. The proof in this article allows $L^2$-perturbations on the initial data.
Indian Academy of Sciences (India)
Yusuf Gurefe; Abdullah Sonmezoglu; Emine Misirli
2011-12-01
In this paper some exact solutions including soliton solutions for the KdV equation with dual power law nonlinearity and the (, ) equation with generalized evolution are obtained using the trial equation method. Also a more general trial equation method is proposed.
SPHERICAL NONLINEAR PULSES FOR THE SOLUTIONS OF NONLINEAR WAVE EQUATIONS Ⅱ, NONLINEAR CAUSTIC
Institute of Scientific and Technical Information of China (English)
无
2007-01-01
This article discusses spherical pulse like solutions of the system of semilinear wave equations with the pulses focusing at a point and emerging outgoing in three space variables. In small initial data case, it shows that the nonlinearities have a strong effect at the focal point. Scattering operator is introduced to describe the caustic crossing. With the aid of the L∞ norms, it analyzes the relative errors in approximate solutions.
Kinetic equation for nonlinear resonant wave-particle interaction
Artemyev, A. V.; Neishtadt, A. I.; Vasiliev, A. A.; Mourenas, D.
2016-09-01
We investigate the nonlinear resonant wave-particle interactions including the effects of particle (phase) trapping, detrapping, and scattering by high-amplitude coherent waves. After deriving the relationship between probability of trapping and velocity of particle drift induced by nonlinear scattering (phase bunching), we substitute this relation and other characteristic equations of wave-particle interaction into a kinetic equation for the particle distribution function. The final equation has the form of a Fokker-Planck equation with peculiar advection and collision terms. This equation fully describes the evolution of particle momentum distribution due to particle diffusion, nonlinear drift, and fast transport in phase-space via trapping. Solutions of the obtained kinetic equation are compared with results of test particle simulations.
Linearized oscillation theory for a nonlinear delay impulsive equation
Berezansky, Leonid; Braverman, Elena
2003-12-01
For a scalar nonlinear impulsive delay differential equationwith rk(t)≥0,hk(t)≤t, limj-->∞ τj=∞, such an auxiliary linear impulsive delay differential equationis constructed that oscillation (nonoscillation) of the nonlinear equation can be deduced from the corresponding properties of the linear equation. Coefficients rk(t) and delays are not assumed to be continuous. Explicit oscillation and nonoscillation conditions are established for some nonlinear impulsive models of population dynamics, such as the impulsive logistic equation and the impulsive generalized Lasota-Wazewska equation which describes the survival of red blood cells. It is noted that unlike nonimpulsive delay logistic equations a solution of a delay impulsive logistic equation may become negative.
Painlevé analysis for nonlinear partial differential equations
Musette, M
1998-01-01
The Painlevé analysis introduced by Weiss, Tabor and Carnevale (WTC) in 1983 for nonlinear partial differential equations (PDE's) is an extension of the method initiated by Painlevé and Gambier at the beginning of this century for the classification of algebraic nonlinear differential equations (ODE's) without movable critical points. In these lectures we explain the WTC method in its invariant version introduced by Conte in 1989 and its application to solitonic equations in order to find algorithmically their associated so-called ``integrable'' equations but they are generically no more valid for equations modelising physical phenomema. Belonging to this second class, some equations called ``partially integrable'' sometimes keep remnants of integrability. In that case, the singularity analysis may also be useful for building closed form analytic solutions, which necessarily % Conte agree with the singularity structure of the equations. We display the privileged role played by the Riccati equation and syste...
New Exact Solutions for New Model Nonlinear Partial Differential Equation
Directory of Open Access Journals (Sweden)
A. Maher
2013-01-01
Full Text Available In this paper we propose a new form of Padé-II equation, namely, a combined Padé-II and modified Padé-II equation. The mapping method is a promising method to solve nonlinear evaluation equations. Therefore, we apply it, to solve the combined Padé-II and modified Padé-II equation. Exact travelling wave solutions are obtained and expressed in terms of hyperbolic functions, trigonometric functions, rational functions, and elliptic functions.
Singularity analysis of a new discrete nonlinear Schrodinger equation
Sakovich, Sergei
2001-01-01
We apply the Painleve test for integrability to a new discrete (differential-difference) nonlinear Schrodinger equation introduced by Leon and Manna. Since the singular expansions of solutions of this equation turn out to contain nondominant logarithmic terms, we conclude that the studied equation is nonintegrable. This result supports the observation of Levi and Yamilov that the Leon-Manna equation does not admit high-order generalized symmetries. As a byproduct of the singularity analysis c...
DEFF Research Database (Denmark)
Bang, O.; Juul Rasmussen, J.; Christiansen, P.L.
1994-01-01
Discretizing the continuous nonlinear Schrodinger equation with arbitrary power nonlinearity influences the time evolution of its ground state solitary solution. In the subcritical case, for grid resolutions above a certain transition value, depending on the degree of nonlinearity, the solution w...
Institute of Scientific and Technical Information of China (English)
HANG Chao; HUANG Guo-Xiang
2006-01-01
We investigate the nonlinear localized structures of optical pulses propagating in a one-dimensional photonic crystal with a quadratic nonlinearity. Using a method of multiple scales we show that the nonlinear evolution of a wave packet, formed by the superposition of short-wavelength excitations, and long-wavelength mean fields, generated by the self-interaction of the wave packet, are governed by a set of coupled high-dimensional nonlinear envelope equations, which can be reduced to Davey-Stewartson equations and thus support dromionlike high-dimensional nonlinear excitations in the system.
Bifurcation methods of dynamical systems for handling nonlinear wave equations
Indian Academy of Sciences (India)
Dahe Feng; Jibin Li
2007-05-01
By using the bifurcation theory and methods of dynamical systems to construct the exact travelling wave solutions for nonlinear wave equations, some new soliton solutions, kink (anti-kink) solutions and periodic solutions with double period are obtained.
Comparative study of homotopy continuation methods for nonlinear algebraic equations
Nor, Hafizudin Mohamad; Ismail, Ahmad Izani Md.; Majid, Ahmad Abd.
2014-07-01
We compare some recent homotopy continuation methods to see which method has greater applicability and greater accuracy. We test the methods on systems of nonlinear algebraic equations. The results obtained indicate the superior accuracy of Newton Homotopy Continuation Method (NHCM).
Lipschitz regularity results for nonlinear strictly elliptic equations and applications
Ley, Olivier; Nguyen, Vinh Duc
2017-10-01
Most of Lipschitz regularity results for nonlinear strictly elliptic equations are obtained for a suitable growth power of the nonlinearity with respect to the gradient variable (subquadratic for instance). For equations with superquadratic growth power in gradient, one usually uses weak Bernstein-type arguments which require regularity and/or convex-type assumptions on the gradient nonlinearity. In this article, we obtain new Lipschitz regularity results for a large class of nonlinear strictly elliptic equations with possibly arbitrary growth power of the Hamiltonian with respect to the gradient variable using some ideas coming from Ishii-Lions' method. We use these bounds to solve an ergodic problem and to study the regularity and the large time behavior of the solution of the evolution equation.
Hamiltonian Formalism of the Derivative Nonlinear Schrodinger Equation
Institute of Scientific and Technical Information of China (English)
CAI Hao; LIU Feng-Min; HUANG Nian-Ning
2003-01-01
A particular form of poisson bracket is introduced for the derivative nonlinear Schrodinger (DNLS) equation.And its Hamiltonian formalism is developed by a linear combination method. Action-angle variables are found.
Nonlinear damped Schrodinger equation in two space dimensions
Directory of Open Access Journals (Sweden)
Tarek Saanouni
2015-04-01
Full Text Available In this article, we study the initial value problem for a semi-linear damped Schrodinger equation with exponential growth nonlinearity in two space dimensions. We show global well-posedness and exponential decay.
IDENTIFICATION OF PARAMETERS IN PARABOLIC EQUATIONS WITH NONLINEARITY
Institute of Scientific and Technical Information of China (English)
无
2010-01-01
In this paper, we consider the identification of parameters in parabolic equations with nonlinearity. Some approximation processes for the identification problem are given. Our results improve and generalize the previous results.
Differentiability at lateral boundary for fully nonlinear parabolic equations
Ma, Feiyao; Moreira, Diego R.; Wang, Lihe
2017-09-01
For fully nonlinear uniformly parabolic equations, the first derivatives regularity of viscosity solutions at lateral boundary is studied under new Dini type conditions for the boundary, which is called Reifenberg Dini conditions and is weaker than usual Dini conditions.
Analysis of Nonlinear Fractional Nabla Difference Equations
Directory of Open Access Journals (Sweden)
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 theorem on existence of singular solutions to nonlinear equations
Directory of Open Access Journals (Sweden)
Prusinska А.
2005-01-01
Full Text Available The aim of this paper is to present some applications of pregularity theory to investigations of nonlinear multivalued mappings. The main result addresses to the problem of existence of solutions to nonlinear equations in the degenerate case when the linear part is singular at the considered initial point. We formulate conditions for existence of solutions of equation F(x = 0 when first p - 1 derivatives of F are singular.
MULTISCALE HOMOGENIZATION OF NONLINEAR HYPERBOLIC EQUATIONS WITH SEVERAL TIME SCALES
Institute of Scientific and Technical Information of China (English)
Jean Louis Woukeng; David Dongo
2011-01-01
We study the multiscale homogenization of a nonlinear hyperbolic equation in a periodic setting. We obtain an accurate homogenization result. We also show that as the nonlinear term depends on the microscopic time variable, the global homogenized problem thus obtained is a system consisting of two hyperbolic equations. It is also shown that in spite of the presence of several time scales, the global homogenized problem is not a reiterated one.
Modified Homotopy Analysis Method for Nonlinear Fractional Partial Differential Equations
Directory of Open Access Journals (Sweden)
D. Ziane
2017-05-01
Full Text Available In this paper, a combined form of natural transform with homotopy analysis method is proposed to solve nonlinear fractional partial differential equations. This method is called the fractional homotopy analysis natural transform method (FHANTM. The FHANTM can easily be applied to many problems and is capable of reducing the size of computational work. The fractional derivative is described in the Caputo sense. The results show that the FHANTM is an appropriate method for solving nonlinear fractional partial differentia equation.
Ehrenfest theorem, Galilean invariance and nonlinear Schr"odinger equations
Kälbermann, G
2003-01-01
Galilean invariant Schr"odinger equations possessing nonlinear terms coupling the amplitude and the phase of the wave function can violate the Ehrenfest theorem. An example of this kind is provided. The example leads to the proof of the theorem: A Galilean invariant Schr"odinger equation derived from a lagrangian density obeys the Ehrenfest theorem. The theorem holds for any linear or nonlinear lagrangian.
NONLINEAR BOUNDARY STABILIZATION OF WAVE EQUATIONS WITH VARIABLE C OEFFICIENTS
Institute of Scientific and Technical Information of China (English)
冯绍继; 冯德兴
2003-01-01
The wave equation with variable coefficients with a nonlinear dissipative boundary feedbackis studied. By the Riemannian geometry method and the multiplier technique, it is shown thatthe closed loop system decays exponentially or asymptotically, and hence the relation betweenthe decay rate of the system energy and the nonlinearity behavior of the feedback function isestablished.
Applications of Elliptic Equation to Nonlinear Coupled Systems
Institute of Scientific and Technical Information of China (English)
FUZun-Tao; LIUShi-Da; LIUShi-Kuo
2003-01-01
The elliptic equation is taken as a transformation and applied to solve nonlinear coupled systems. It is shown that this method is more powerful to give more kinds of solutions, such as rational solutions, solitary wave solutions, periodic wave solutions and so on, so this method can be taken as a unified method in solving nonlinear coupled systems.
Applications of Elliptic Equation to Nonlinear Coupled Systems
Institute of Scientific and Technical Information of China (English)
FU Zun-Tao; LIU Shi-Da; LIU Shi-Kuo
2003-01-01
The elliptic equation is taken as a transformation and applied to solve nonlinear coupled systems. Itis shown that this method is more powerful to give more kinds of solutions, such as rational solutions, solitary wavesolutions, periodic wave solutions and so on, so this method can be taken as a unified method in solving nonlinear coupled systems.
Approximate solution of a nonlinear partial differential equation
Vajta, M.
2007-01-01
Nonlinear partial differential equations (PDE) are notorious to solve. In only a limited number of cases can we find an analytic solution. In most cases, we can only apply some numerical scheme to simulate the process described by a nonlinear PDE. Therefore, approximate solutions are important for t
A NEW SMOOTHING EQUATIONS APPROACH TO THE NONLINEAR COMPLEMENTARITY PROBLEMS
Institute of Scientific and Technical Information of China (English)
Chang-feng Ma; Pu-yan Nie; Guo-ping Liang
2003-01-01
The nonlinear complementarity problem can be reformulated as a nonsmooth equation. In this paper we propose a new smoothing Newton algorithm for the solution of the nonlinear complementarity problem by constructing a new smoothing approximation function. Global and local superlinear convergence results of the algorithm are obtained under suitable conditions. Numerical experiments confirm the good theoretical properties of the algorithm.
Nonlinear Kramers equation associated with nonextensive statistical mechanics.
Mendes, G A; Ribeiro, M S; Mendes, R S; Lenzi, E K; Nobre, F D
2015-05-01
Stationary and time-dependent solutions of a nonlinear Kramers equation, as well as its associated nonlinear Fokker-Planck equations, are investigated within the context of Tsallis nonextensive statistical mechanics. Since no general analytical time-dependent solutions are found for such a nonlinear Kramers equation, an ansatz is considered and the corresponding asymptotic behavior is studied and compared with those known for the standard linear Kramers equation. The H-theorem is analyzed for this equation and its connection with Tsallis entropy is investigated. An application is discussed, namely the motion of Hydra cells in two-dimensional cellular aggregates, for which previous measurements have verified q-Gaussian distributions for velocity components and superdiffusion. The present analysis is in quantitative agreement with these experimental results.
Bilal, S.; Khalil-ur-Rehman; Malik, M. Y.; Hussain, Arif; Khan, Mair
Present work is communicated to identify characteristics of magnetohydrodynamic (MHD) three dimensional boundary layer flow of Williamson fluid confined by a bidirectional stretched surface. Conductivity of working fluid is assumed to be temperature dependent. Generative/absorptive heat transfer is also taken into account. Mathematical model is formulated in the form of partial expressions and then transmuted into ordinary differential equations with the help of newfangled set of similarity transformations. The resulting non-linear differential system of equations is solved numerically with the aid of Runge-Kutta algorithm supported by shooting method. Flow features are exemplified quantitatively through graphs. Scintillating results for friction factor and convective heat transfer are computed and scrutinized tabularly. Furthermore, the accuracy of present results is tested with existing literature and we found an excellent agreement. It is inferred that velocity along x-direction mounts whereas along y-direction depreciates for incrementing values of stretching ratio parameter. Moreover, it is also elucidated that non-linearity index tends to decrement the velocity and thermal distributions of fluid flow.
Analytical exact solution of the non-linear Schroedinger equation
Energy Technology Data Exchange (ETDEWEB)
Martins, Alisson Xavier; Rocha Filho, Tarcisio Marciano da [Universidade de Brasilia (UnB), DF (Brazil). Inst. de Fisica. Grupo de Fisica e Matematica
2011-07-01
Full text: In this work we present how to classify and obtain analytical solutions of the Schroedinger equation with a generic non-linearity in 1+1 dimensions. Our approach is based on the determination of Lie symmetry transformation mapping solutions into solutions, and non-classical symmetry transformations, mapping a given solution into itself. From these symmetries it is then possible to reduce the equation to a system of ordinary differential equations which can then be solved using standard methods. The generic non-linearity is handled by considering it as an additional unknown in the determining equations for the symmetry transformations. This results in an over-determined system of non-linear partial differential equations. Its solution can then be determined in some cases by reducing it to the so called involutive (triangular) form, and then solved. This reduction is very tedious and can only performed using a computer algebra system. Once the determining system is solved, we obtain the explicit form for the non-linearity admitting a Lie or non-classical symmetry. The analytical solutions are then derived by solving the reduced ordinary differential equations. The non-linear determining system for the non-classical symmetry transformations and Lie symmetry generators are obtaining using the computer algebra package SADE (symmetry analysis of differential equations), developed at our group. (author)
Nonlinear ordinary differential equations analytical approximation and numerical methods
Hermann, Martin
2016-01-01
The book discusses the solutions to nonlinear ordinary differential equations (ODEs) using analytical and numerical approximation methods. Recently, analytical approximation methods have been largely used in solving linear and nonlinear lower-order ODEs. It also discusses using these methods to solve some strong nonlinear ODEs. There are two chapters devoted to solving nonlinear ODEs using numerical methods, as in practice high-dimensional systems of nonlinear ODEs that cannot be solved by analytical approximate methods are common. Moreover, it studies analytical and numerical techniques for the treatment of parameter-depending ODEs. The book explains various methods for solving nonlinear-oscillator and structural-system problems, including the energy balance method, harmonic balance method, amplitude frequency formulation, variational iteration method, homotopy perturbation method, iteration perturbation method, homotopy analysis method, simple and multiple shooting method, and the nonlinear stabilized march...
Nonlinear Parabolic Equations with Singularities in Colombeau Vector Spaces
Institute of Scientific and Technical Information of China (English)
Mirjana STOJANOVI(C)
2006-01-01
We consider nonlinear parabolic equations with nonlinear non-Lipschitz's term and singular initial data like Dirac measure, its derivatives and powers. We prove existence-uniqueness theorems in Colombeau vector space gC1,w2,2([O,T),Rn),n ≤ 3. Due to high singularity in a case of parabolic equation with nonlinear conservative term we employ the regularized derivative for the conservative term, in order to obtain the global existence-uniqueness result in Colombeau vector space gC1,L2([O,T),Rn),n ≤ 3.
The nonlinear Schroedinger equation on a disordered chain
Energy Technology Data Exchange (ETDEWEB)
Scharf, R.; Bishop, A.R.
1990-01-01
The integrable lattice nonlinear Schroedinger equation is a unique model with which to investigate the effects of disorder on a discrete integrable dynamics, and its interplay with nonlinearity. We first review some features of the lattice nonlinear Schroedinger equation in the absence of disorder and introduce a 1- and 2-soliton collective variable approximation. Then we describe the effect of different types of disorder: attractive and repulsive isolated impurities, spatially periodic potentials, random potentials, and time dependent (kicked) long wavelength perturbations. 18 refs., 15 figs.
GHM method for obtaining rationalsolutions of nonlinear differential equations.
Vazquez-Leal, Hector; Sarmiento-Reyes, Arturo
2015-01-01
In this paper, we propose the application of the general homotopy method (GHM) to obtain rational solutions of nonlinear differential equations. It delivers a high precision representation of the nonlinear differential equation using a few linear algebraic terms. In order to assess the benefits of this proposal, three nonlinear problems are solved and compared against other semi-analytic methods or numerical methods. The obtained results show that GHM is a powerful tool, capable to generate highly accurate rational solutions. AMS subject classification 34L30.
Derivation of an Applied Nonlinear Schroedinger Equation.
Energy Technology Data Exchange (ETDEWEB)
Pitts, Todd Alan; Laine, Mark Richard; Schwarz, Jens; Rambo, Patrick K.; Karelitz, David B.
2015-01-01
We derive from first principles a mathematical physics model useful for understanding nonlinear optical propagation (including filamentation). All assumptions necessary for the development are clearly explained. We include the Kerr effect, Raman scattering, and ionization (as well as linear and nonlinear shock, diffraction and dispersion). We explain the phenomenological sub-models and each assumption required to arrive at a complete and consistent theoretical description. The development includes the relationship between shock and ionization and demonstrates why inclusion of Drude model impedance effects alters the nature of the shock operator. Unclassified Unlimited Release
Derivation of an applied nonlinear Schroedinger equation
Energy Technology Data Exchange (ETDEWEB)
Pitts, Todd Alan [Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States); Laine, Mark Richard [Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States); Schwarz, Jens [Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States); Rambo, Patrick K. [Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States); Karelitz, David B. [Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States)
2015-01-01
We derive from first principles a mathematical physics model useful for understanding nonlinear optical propagation (including filamentation). All assumptions necessary for the development are clearly explained. We include the Kerr effect, Raman scattering, and ionization (as well as linear and nonlinear shock, diffraction and dispersion). We explain the phenomenological sub-models and each assumption required to arrive at a complete and consistent theoretical description. The development includes the relationship between shock and ionization and demonstrates why inclusion of Drude model impedance effects alters the nature of the shock operator. Unclassified Unlimited Release
Reduction of the equation for lower hybrid waves in a plasma to a nonlinear Schroedinger equation
Karney, C. F. F.
1977-01-01
Equations describing the nonlinear propagation of waves in an anisotropic plasma are rarely exactly soluble. However it is often possible to make approximations that reduce the exact equations into a simpler equation. The use of MACSYMA to make such approximations, and so reduce the equation describing lower hybrid waves into the nonlinear Schrodinger equation which is soluble by the inverse scattering method is demonstrated. MACSYMA is used at several stages in the calculation only because there is a natural division between calculations that are easiest done by hand, and those that are easiest done by machine.
Institute of Scientific and Technical Information of China (English)
WANG Mei-Jiao; WANG Qi
2006-01-01
In this work, by means of a new more general ansatz and the symbolic computation system Maple, we extend the Riccati equation rational expansion method [Chaos, Solitons & Fractals 25 (2005) 1019] to uniformly construct a series of stochastic nontravelling wave solutions for nonlinear stochastic evolution equation. To illustrate the effectiveness of our method, we take the stochastic mKdV equation as an example, and successfully construct some new and more general solutions including a series of rational formal nontraveling wave and coefficient functions' soliton-like solutions and trigonometric-like function solutions. The method can also be applied to solve other nonlinear stochastic evolution equation or equations.
NEW EXACT TRAVELLING WAVE SOLUTIONS TO THREE NONLINEAR EVOLUTION EQUATIONS
Institute of Scientific and Technical Information of China (English)
Sirendaoreji
2004-01-01
Based on the computerized symbolic computation, some new exact travelling wave solutions to three nonlinear evolution equations are explicitly obtained by replacing the tanhξ in tanh-function method with the solutions of a new auxiliary ordinary differential equation.
Exact solutions for the cubic-quintic nonlinear Schroedinger equation
Energy Technology Data Exchange (ETDEWEB)
Zhu Jiamin [Department of Physics, Zhejiang Lishui University, Lishui 323000 (China)]. E-mail: zjm64@163.com; Ma Zhengyi [Department of Physics, Zhejiang Lishui University, Lishui 323000 (China); Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072 (China)
2007-08-15
In this paper, the cubic-quintic nonlinear Schroedinger equation is solved through the extended elliptic sub-equation method. As a consequence, many types of exact travelling wave solutions are obtained which including bell and kink profile solitary wave solutions, triangular periodic wave solutions and singular solutions.
Nonlinear partial differential equations: Integrability, geometry and related topics
Krasil'shchik, Joseph; Rubtsov, Volodya
2017-03-01
Geometry and Differential Equations became inextricably entwined during the last one hundred fifty years after S. Lie and F. Klein's fundamental insights. The two subjects go hand in hand and they mutually enrich each other, especially after the "Soliton Revolution" and the glorious streak of Symplectic and Poisson Geometry methods in the context of Integrability and Solvability problems for Non-linear Differential Equations.
An analysis of the nonlinear equation = (, ) + (, )$u^2_$ + ℎ(, ) + (, )$
Indian Academy of Sciences (India)
R M Edelstein; K S Govinder
2011-01-01
We use the method of preliminary group classiﬁcation to analyse a particular form of the nonlinear diffusion equation in which the inhomogeneity is quadratic in . The method yields an optimal system of one-dimensional subalgebras. As a result we obtain those explicit forms of the unknown functions , , ℎ and for which the equation admits additional point symmetries.
Multiple solutions to some singular nonlinear Schrodinger equations
Directory of Open Access Journals (Sweden)
Monica Lazzo
2001-01-01
Full Text Available We consider the equation $$ - h^2 Delta u + V_varepsilon(x u = |u|^{p-2} u $$ which arises in the study of standing waves of a nonlinear Schrodinger equation. We allow the potential $V_varepsilon$ to be unbounded below and prove existence and multiplicity results for positive solutions.
Several Dynamical Properties for a Nonlinear Shallow Water Equation
Directory of Open Access Journals (Sweden)
Ls Yong
2014-01-01
Full Text Available A nonlinear third order dispersive shallow water equation including the Degasperis-Procesi model is investigated. The existence of weak solutions for the equation is proved in the space L1(R∩BV (R under certain assumptions. The Oleinik type estimate and L2N(R (N is a natural number estimate for the solution are obtained.
Nonlinear eigenvalue approach to differential Riccati equations for contraction analysis
Kawano, Yu; Ohtsuka, Toshiyuki
2017-01-01
In this paper, we extend the eigenvalue method of the algebraic Riccati equation to the differential Riccati equation (DRE) in contraction analysis. One of the main results is showing that solutions to the DRE can be expressed as functions of nonlinear eigenvectors of the differential Hamiltonian ma
Structure of Dirac matrices and invariants for nonlinear Dirac equations
2004-01-01
We present invariants for nonlinear Dirac equations in space-time ${\\mathbb R}^{n+1}$, by which we prove that a special choice of the Cauchy data yields free solutions. Our argument works for Klein-Gordon-Dirac equations with Yukawa coupling as well. Related problems on the structure of Dirac matrices are studied.
LIMITED MEMORY BFGS METHOD FOR NONLINEAR MONOTONE EQUATIONS
Institute of Scientific and Technical Information of China (English)
Weijun Zhou; Donghui Li
2007-01-01
In this paper, we propose an algorithm for solving nonlinear monotone equations by combining the limited memory BFGS method (L-BFGS) with a projection method. We show that the method is globally convergent if the equation involves a Lipschitz continuous monotone function. We also present some preliminary numerical results.
STABILITY OF NONLINEAR NEUTRAL DIFFERENTIAL EQUATION VIA FIXED POINT
Institute of Scientific and Technical Information of China (English)
无
2012-01-01
In this paper,a nonlinear neutral differential equation is considered.By a fixed point theory,we give some conditions to ensure that the zero solution to the equation is asymptotically stable.Some existing results are improved and generalized.
Entropy and convexity for nonlinear partial differential equations.
Ball, John M; Chen, Gui-Qiang G
2013-12-28
Partial differential equations are ubiquitous in almost all applications of mathematics, where they provide a natural mathematical description of many phenomena involving change in physical, chemical, biological and social processes. The concept of entropy originated in thermodynamics and statistical physics during the nineteenth century to describe the heat exchanges that occur in the thermal processes in a thermodynamic system, while the original notion of convexity is for sets and functions in mathematics. Since then, entropy and convexity have become two of the most important concepts in mathematics. In particular, nonlinear methods via entropy and convexity have been playing an increasingly important role in the analysis of nonlinear partial differential equations in recent decades. This opening article of the Theme Issue is intended to provide an introduction to entropy, convexity and related nonlinear methods for the analysis of nonlinear partial differential equations. We also provide a brief discussion about the content and contributions of the papers that make up this Theme Issue.
The numerical dynamic for highly nonlinear partial differential equations
Lafon, A.; Yee, H. C.
1992-01-01
Problems associated with the numerical computation of highly nonlinear equations in computational fluid dynamics are set forth and analyzed in terms of the potential ranges of spurious behaviors. A reaction-convection equation with a nonlinear source term is employed to evaluate the effects related to spatial and temporal discretizations. The discretization of the source term is described according to several methods, and the various techniques are shown to have a significant effect on the stability of the spurious solutions. Traditional linearized stability analyses cannot provide the level of confidence required for accurate fluid dynamics computations, and the incorporation of nonlinear analysis is proposed. Nonlinear analysis based on nonlinear dynamical systems complements the conventional linear approach and is valuable in the analysis of hypersonic aerodynamics and combustion phenomena.
Relations between nonlinear Riccati equations and other equations in fundamental physics
Schuch, Dieter
2014-10-01
Many phenomena in the observable macroscopic world obey nonlinear evolution equations while the microscopic world is governed by quantum mechanics, a fundamental theory that is supposedly linear. In order to combine these two worlds in a common formalism, at least one of them must sacrifice one of its dogmas. Linearizing nonlinear dynamics would destroy the fundamental property of this theory, however, it can be shown that quantum mechanics can be reformulated in terms of nonlinear Riccati equations. In a first step, it will be shown that the information about the dynamics of quantum systems with analytical solutions can not only be obtainable from the time-dependent Schrödinger equation but equally-well from a complex Riccati equation. Comparison with supersymmetric quantum mechanics shows that even additional information can be obtained from the nonlinear formulation. Furthermore, the time-independent Schrödinger equation can also be rewritten as a complex Riccati equation for any potential. Extension of the Riccati formulation to include irreversible dissipative effects is straightforward. Via (real and complex) Riccati equations, other fields of physics can also be treated within the same formalism, e.g., statistical thermodynamics, nonlinear dynamical systems like those obeying a logistic equation as well as wave equations in classical optics, Bose- Einstein condensates and cosmological models. Finally, the link to abstract "quantizations" such as the Pythagorean triples and Riccati equations connected with trigonometric and hyperbolic functions will be shown.
Indian Academy of Sciences (India)
Ranjit Kumar
2012-09-01
Travelling and solitary wave solutions of certain coupled nonlinear diffusion-reaction equations have been constructed using the auxiliary equation method. These equations arise in a variety of contexts not only in biological, chemical and physical sciences but also in ecological and social sciences.
The Duffing Equation Nonlinear Oscillators and their Behaviour
Kovacic, Ivana
2011-01-01
The Duffing Equation: Nonlinear Oscillators and their Behaviour brings together the results of a wealth of disseminated research literature on the Duffing equation, a key engineering model with a vast number of applications in science and engineering, summarizing the findings of this research. Each chapter is written by an expert contributor in the field of nonlinear dynamics and addresses a different form of the equation, relating it to various oscillatory problems and clearly linking the problem with the mathematics that describe it. The editors and the contributors explain the mathematical
NEW ALTERNATING DIRECTION FINITE ELEMENT SCHEME FOR NONLINEAR PARABOLIC EQUATION
Institute of Scientific and Technical Information of China (English)
崔霞
2002-01-01
A new alternating direction (AD) finite element (FE) scheme for 3-dimensional nonlinear parabolic equation and parabolic integro-differential equation is studied. By using AD,the 3-dimensional problem is reduced to a family of single space variable problems, calculation work is simplified; by using FE, high accuracy is kept; by using various techniques for priori estimate for differential equations such as inductive hypothesis reasoning, the difficulty arising from the nonlinearity is treated. For both FE and ADFE schemes, the convergence properties are rigorously demonstrated, the optimal H1- and L2-norm space estimates and the O((△t)2) estimate for time variable are obtained.
Exact solutions of certain nonlinear chemotaxis diffusion reaction equations
Indian Academy of Sciences (India)
MISHRA AJAY; KAUSHAL R S; PRASAD AWADHESH
2016-05-01
Using the auxiliary equation method, we obtain exact solutions of certain nonlinear chemotaxis diffusion reaction equations in the presence of a stimulant. In particular, we account for the nonlinearities arising not only from the density-dependent source terms contributed by the particles and the stimulant but also from the coupling term of the stimulant. In addition to this, the diffusion of the stimulant and the effect of long-range interactions are also accounted for in theconstructed coupled differential equations. The results obtained here could be useful in the studies of several biological systems and processes, e.g., in bacterial infection, chemotherapy, etc.
Generalized nonlinear Proca equation and its free-particle solutions
Energy Technology Data Exchange (ETDEWEB)
Nobre, F.D. [Centro Brasileiro de Pesquisas Fisicas and National Institute of Science and Technology for Complex Systems, Rio de Janeiro, RJ (Brazil); Plastino, A.R. [Universidad Nacional Buenos Aires-Noreoeste, CeBio y Secretaria de Investigacion, Junin (Argentina)
2016-06-15
We introduce a nonlinear extension of Proca's field theory for massive vector (spin 1) bosons. The associated relativistic nonlinear wave equation is related to recently advanced nonlinear extensions of the Schroedinger, Dirac, and Klein-Gordon equations inspired on the non-extensive generalized thermostatistics. This is a theoretical framework that has been applied in recent years to several problems in nuclear and particle physics, gravitational physics, and quantum field theory. The nonlinear Proca equation investigated here has a power-law nonlinearity characterized by a real parameter q (formally corresponding to the Tsallis entropic parameter) in such a way that the standard linear Proca wave equation is recovered in the limit q → 1. We derive the nonlinear Proca equation from a Lagrangian, which, besides the usual vectorial field Ψ{sup μ}(vector x,t), involves an additional field Φ{sup μ}(vector x,t). We obtain exact time-dependent soliton-like solutions for these fields having the form of a q-plane wave, and we show that both field equations lead to the relativistic energy-momentum relation E{sup 2} = p{sup 2}c{sup 2} + m{sup 2}c{sup 4} for all values of q. This suggests that the present nonlinear theory constitutes a new field theoretical representation of particle dynamics. In the limit of massless particles the present q-generalized Proca theory reduces to Maxwell electromagnetism, and the q-plane waves yield localized, transverse solutions of Maxwell equations. Physical consequences and possible applications are discussed. (orig.)
Optimal Variational Asymptotic Method for Nonlinear Fractional Partial Differential Equations.
Baranwal, Vipul K; Pandey, Ram K; Singh, Om P
2014-01-01
We propose optimal variational asymptotic method to solve time fractional nonlinear partial differential equations. In the proposed method, an arbitrary number of auxiliary parameters γ 0, γ 1, γ 2,… and auxiliary functions H 0(x), H 1(x), H 2(x),… are introduced in the correction functional of the standard variational iteration method. The optimal values of these parameters are obtained by minimizing the square residual error. To test the method, we apply it to solve two important classes of nonlinear partial differential equations: (1) the fractional advection-diffusion equation with nonlinear source term and (2) the fractional Swift-Hohenberg equation. Only few iterations are required to achieve fairly accurate solutions of both the first and second problems.
Stochastic nonlinear differential equation generating 1/f noise.
Kaulakys, B; Ruseckas, J
2004-08-01
Starting from the simple point process model of 1/f noise, we derive a stochastic nonlinear differential equation for the signal exhibiting 1/f noise, in any desirably wide range of frequency. A stochastic differential equation (the general Langevin equation with a multiplicative noise) that gives 1/f noise is derived. The solution of the equation exhibits the power-law distribution. The process with 1/f noise is demonstrated by the numerical solution of the derived equation with the appropriate restriction of the diffusion of the signal in some finite interval.
Exact travelling wave solutions of nonlinear partial differential equations
Energy Technology Data Exchange (ETDEWEB)
Soliman, A.A. [Department of Mathematics, Faculty of Education (AL-Arish) Suez Canal University, AL-Arish 45111 (Egypt)]. E-mail: asoliman_99@yahoo.com; Abdou, M.A. [Theoretical Research Group, Department of Physics, Faculty of Science, Mansoura University, Mansoura 35516 (Egypt)]. E-mail: m_abdou_eg@yahoo.com
2007-04-15
An extended Fan-sub equation method is developed for searching exact travelling wave solutions of nonlinear partial differential equations. The key idea of this method is to take full advantage of the general elliptic equation, involving five parameters, which has more new solutions and whose degeneracies can lead to special sub equation involving three parameters. As an illustration of the extended Fan method, more new solutions are obtained for three models namely, generalized KdV, Drinfeld-Sokolov system and RLW equation.
The Swift-Hohenberg equation with a nonlocal nonlinearity
2013-01-01
It is well known that aspects of the formation of localised states in a one-dimensional Swift--Hohenberg equation can be described by Ginzburg--Landau-type envelope equations. This paper extends these multiple scales analyses to cases where an additional nonlinear integral term, in the form of a convolution, is present. The presence of a kernel function introduces a new lengthscale into the problem, and this results in additional complexity in both the derivation of envelope equations and in ...
Adomian solution of a nonlinear quadratic integral equation
Directory of Open Access Journals (Sweden)
E.A.A. Ziada
2013-04-01
Full Text Available We are concerned here with a nonlinear quadratic integral equation (QIE. The existence of a unique solution will be proved. Convergence analysis of Adomian decomposition method (ADM applied to these type of equations is discussed. Convergence analysis is reliable enough to estimate the maximum absolute truncated error of Adomian’s series solution. Two methods are used to solve these type of equations; ADM and repeated trapezoidal method. The obtained results are compared.
Iterative Solution for Systems of Nonlinear Two Binary Operator Equations
Institute of Scientific and Technical Information of China (English)
ZHANGZhi-hong; LIWen-feng
2004-01-01
Using the cone and partial ordering theory and mixed monotone operator theory, the existence and uniqueness of solutions for some classes of systems of nonlinear two binary operator equations in a Banach space with a partial ordering are discussed. And the error estimates that the iterative sequences converge to solutions are also given. Some relevant results of solvability of two binary operator equations and systems of operator equations are imnroved and generalized.
Scalable nonlinear iterative methods for partial differential equations
Energy Technology Data Exchange (ETDEWEB)
Cai, X-C
2000-10-29
We conducted a six-month investigation of the design, analysis, and software implementation of a class of singularity-insensitive, scalable, parallel nonlinear iterative methods for the numerical solution of nonlinear partial differential equations. The solutions of nonlinear PDEs are often nonsmooth and have local singularities, such as sharp fronts. Traditional nonlinear iterative methods, such as Newton-like methods, are capable of reducing the global smooth nonlinearities at a nearly quadratic convergence rate but may become very slow once the local singularities appear somewhere in the computational domain. Even with global strategies such as line search or trust region the methods often stagnate at local minima of {parallel}F{parallel}, especially for problems with unbalanced nonlinearities, because the methods do not have built-in machinery to deal with the unbalanced nonlinearities. To find the same solution u* of F(u) = 0, we solve, instead, an equivalent nonlinearly preconditioned system G(F(u*)) = 0 whose nonlinearities are more balanced. In this project, we proposed and studied a nonlinear additive Schwarz based parallel nonlinear preconditioner and showed numerically that the new method converges well even for some difficult problems, such as high Reynolds number flows, when a traditional inexact Newton method fails.
Very High Order $\\PNM$ Schemes on Unstructured Meshes for the Resistive Relativistic MHD Equations
Dumbser, Michael
2009-01-01
In this paper we propose the first better than second order accurate method in space and time for the numerical solution of the resistive relativistic magnetohydrodynamics (RRMHD) equations on unstructured meshes in multiple space dimensions. The nonlinear system under consideration is purely hyperbolic and contains a source term, the one for the evolution of the electric field, that becomes stiff for low values of the resistivity. For the spatial discretization we propose to use high order $\\PNM$ schemes as introduced in \\cite{Dumbser2008} for hyperbolic conservation laws and a high order accurate unsplit time discretization is achieved using the element-local space-time discontinuous Galerkin approach proposed in \\cite{DumbserEnauxToro} for one-dimensional balance laws with stiff source terms. The divergence free character of the magnetic field is accounted for through the divergence cleaning procedure of Dedner et al. \\cite{Dedneretal}. To validate our high order method we first solve some numerical test c...
Non-linear interaction between high energy ions and MHD-modes
Energy Technology Data Exchange (ETDEWEB)
Bergkvist, Tommy
2001-12-01
When heating a fusion plasma with ICRE or NBI a non-Maxwellian distribution function with high energy ions is created. Ions which are in resonance with a MHD mode will interact with the electric field from the mode and in some circumstances energy will flow from the particles to the mode or opposite. A quasi-linear model for the interaction between high energy ions and a MHD mode has been developed. To solve the time evolution of the MHD mode a module has been implemented into the Monte Carlo code FIDO, which is used for calculating a 3-dimensional distribution function. The model has been tested for an internal kink mode during fishbone oscillations.
Quantum theory of nonlocal nonlinear Schrodinger equation
Vyas, Vivek M
2015-01-01
Nonlocal nonlinear Schrodinger model is quantised and exactly solved using the canonical framework. It is found that the usual canonical quantisation of the model leads to a theory with pathological inner product. This problem is resolved by constructing another inner product over the vector space of the theory. The resultant theory is found to be identical to that of nonrelativistic bosons with delta function interaction potential, devoid of any nonlocality. The exact eigenstates are found using the Bethe ansatz technique.
Directory of Open Access Journals (Sweden)
Khaled A. Gepreel
2016-07-01
Full Text Available In this paper, we improve the extended trial equation method to construct the exact solutions for nonlinear coupled system of partial differential equations in mathematical physics. We use the extended trial equation method to find some different types of exact solutions such as the Jacobi elliptic function solutions, soliton solutions, trigonometric function solutions and rational, exact solutions to the nonlinear coupled Schrodinger Boussinesq equations when the balance number is a positive integer. The performance of this method is reliable, effective and powerful for solving more complicated nonlinear partial differential equations in mathematical physics. The balance number of this method is not constant as we have in other methods. This method allows us to construct many new types of exact solutions. By using the Maple software package we show that all obtained solutions satisfy the original partial differential equations.
Nonlinear Biharmonic Equations with Critical Potential
Institute of Scientific and Technical Information of China (English)
Hui XIONG; Yao Tian SHEN
2005-01-01
In this paper, we study two semilinear singular biharmonic equations: one with subcritical exponent and critical potential, another with sub-critical potential and critical exponent. By Pohozaev identity for singular solution, we prove there is no nontrivial solution for equations with critical exponent and critical potential. And by using the concentrate compactness principle and Mountain Pass theorem, respectively, we get two existence results for the two problems. Meanwhile,we have compared the changes of the critical dimensions in singular and non-singular cases, and we get an interesting result.
The Jeffcott equations in nonlinear rotordynamics
Zalik, R. A.
1989-01-01
The solutions of the Jeffcott equations describing the behavior of a rotating shaft are investigated analytically, with a focus on the case where deadband is taken into account. Bounds on the solutions are obtained from those for the linearized equations, and the onset of destructive vibrations is predicted by analyzing the Fourier transforms of the solutions; good agreement with numerical solutions and power-spectrum density plots is demonstrated. It is suggested that the present analytical approach could be applied to determine cryogenic-pump stability margins in flight and hot-fire ground testing of launch vehicles such as the Space Shuttle.
Energy Technology Data Exchange (ETDEWEB)
Zhang, Sheng [Department of Mathematics, Bohai University, Jinzhou 121000 (China)]. E-mail: zhshaeng@yahoo.com.cn; Xia, Tiecheng [Department of Mathematics, Bohai University, Jinzhou 121000 (China); Department of Mathematics, Shanghai University, Shanghai 200444 (China)
2007-04-09
In this Letter, a generalized new auxiliary equation method is proposed for constructing more general exact solutions of nonlinear partial differential equations. With the aid of symbolic computation, we choose the combined KdV-mKdV equation and the (2+1)-dimensional asymmetric Nizhnik-Novikov-Vesselov equations to illustrate the validity and advantages of the method. As a result, many new and more general exact solutions are obtained.
Institute of Scientific and Technical Information of China (English)
Zhi Hong-Yan; Zhao Xue-Qin; Zhang Hong-Qing
2005-01-01
Based on the study of tanh function method and the coupled projective Riccati equation method, we propose a new algorithm to search for explicit exact solutions of nonlinear evolution equations. We use the higher-order Schrodinger equation and mKdV equation to illustrate this algorithm. As a result, more new solutions are obtained, which include new solitary solutions, periodic solutions, and singular solutions. Some new solutions are illustrated in figures.
Backward stochastic differential equations from linear to fully nonlinear theory
Zhang, Jianfeng
2017-01-01
This book provides a systematic and accessible approach to stochastic differential equations, backward stochastic differential equations, and their connection with partial differential equations, as well as the recent development of the fully nonlinear theory, including nonlinear expectation, second order backward stochastic differential equations, and path dependent partial differential equations. Their main applications and numerical algorithms, as well as many exercises, are included. The book focuses on ideas and clarity, with most results having been solved from scratch and most theories being motivated from applications. It can be considered a starting point for junior researchers in the field, and can serve as a textbook for a two-semester graduate course in probability theory and stochastic analysis. It is also accessible for graduate students majoring in financial engineering.
Multi-diffusive nonlinear Fokker-Planck equation
Ribeiro, Mauricio S.; Casas, Gabriela A.; Nobre, Fernando D.
2017-02-01
Nonlinear Fokker-Planck equations, characterized by more than one diffusion term, have appeared recently in literature. Here, it is shown that these equations may be derived either from approximations in a master equation, or from a Langevin-type approach. An H-theorem is proven, relating these Fokker-Planck equations to an entropy composed by a sum of contributions, each of them associated with a given diffusion term. Moreover, the stationary state of the Fokker-Planck equation is shown to coincide with the equilibrium state, obtained by extremization of the entropy, in the sense that both procedures yield precisely the same equation. Due to the nonlinear character of this equation, the equilibrium probability may be obtained, in most cases, only by means of numerical approaches. Some examples are worked out, where the equilibrium probability distribution is computed for nonlinear Fokker-Planck equations presenting two diffusion terms, corresponding to an entropy characterized by a sum of two contributions. It is shown that the resulting equilibrium distribution, in general, presents a form that differs from a sum of the equilibrium distributions that maximizes each entropic contribution separately, although in some cases one may construct such a linear combination as a good approximation for the equilibrium distribution.
EXACT EXPLICIT SOLUTIONS OF THE NONLINEAR SCHR(O)DINGER EQUATION COUPLED TO THE BOUSSINESQ EQUATION
Institute of Scientific and Technical Information of China (English)
姚若侠; 李忠斌
2003-01-01
A system comprised of the nonlinear Schrodinger equation coupled to theBoussinesq equation (S-B equations) which dealing with the stationary propagation of cou-pled non-linear upper-hybrid and magnetosonic waves in magnetized plasma is proposed.To examine its solitary wave solutions, a reduced set of ordinary differential equations areconsidered by a simple traveling wave transformation. It is then shown that several newsolutions (either functional or parametrical) can be obtained systematically, in addition torederiving all known ones by means of our simple and direct algebra method with the helpof the computer algebra system Maple.
Residual models for nonlinear partial differential equations
Directory of Open Access Journals (Sweden)
Garry Pantelis
2005-11-01
Full Text Available Residual terms that appear in nonlinear PDEs that are constructed to generate filtered representations of the variables of the fully resolved system are examined by way of a consistency condition. It is shown that certain commonly used empirical gradient models for the residuals fail the test of consistency and therefore cannot be validated as approximations in any reliable sense. An alternate method is presented for computing the residuals. These residual models are independent of free or artificial parameters and there direct link with the functional form of the system of PDEs which describe the fully resolved system are established.
Directory of Open Access Journals (Sweden)
Khaled A. Gepreel
2012-01-01
Full Text Available We modified the rational Jacobi elliptic functions method to construct some new exact solutions for nonlinear differential difference equations in mathematical physics via the lattice equation, the discrete nonlinear Schrodinger equation with a saturable nonlinearity, the discrete nonlinear Klein-Gordon equation, and the quintic discrete nonlinear Schrodinger equation. Some new types of the Jacobi elliptic solutions are obtained for some nonlinear differential difference equations in mathematical physics. The proposed method is more effective and powerful to obtain the exact solutions for nonlinear differential difference equations.
Algebraic calculation of stroboscopic maps of ordinary, nonlinear differential equations
Energy Technology Data Exchange (ETDEWEB)
Wackerbauer, R. (Max-Planck-Institut fuer Extraterrestrische Physik, Garching (Germany)); Huebler, A. (Illinois Univ., Urbana, IL (United States). Center for Complex Systems Research); Mayer-Kress, G. (Los Alamos National Lab., NM (United States) California Univ., Santa Cruz, CA (United States). Dept. of Mathematics)
1991-07-25
The relation between the parameters of a differential equation and corresponding discrete maps are becoming increasingly important in the study of nonlinear dynamical systems. Maps are well adopted for numerical computation and several universal properties of them are known. Therefore some perturbation methods have been proposed to deduce them for physical systems, which can be modeled by an ordinary differential equation (ODE) with a small nonlinearity. A new iterative, rigorous algebraic method for the calculation of the coefficients of a Taylor expansion of a stroboscopic map from ODE's with not necessarily small nonlinearities is presented. It is shown analytically that most of the coefficients are small for a small integration time and grow slowly in the course of time if the flow vector field of the ODE is polynomial and if the ODE has fixed point in the origin. Approximations of different orders respectively of the rest term are investigated for several nonlinear systems. 31 refs., 16 figs.
Institute of Scientific and Technical Information of China (English)
2008-01-01
In this article, we consider the existence of local and global solution to the Cauchy problem of a doubly nonlinear equation. By introducing the norms |||f|||h and
Stochasticity in numerical solutions of the nonlinear Schroedinger equation
Shen, Mei-Mei; Nicholson, D. R.
1987-01-01
The cubically nonlinear Schroedinger equation is an important model of nonlinear phenomena in fluids and plasmas. Numerical solutions in a spatially periodic system commonly involve truncation to a finite number of Fourier modes. These solutions are found to be stochastic in the sense that the largest Liapunov exponent is positive. As the number of modes is increased, the size of this exponent appears to converge to zero, in agreement with the recent demonstration of the integrability of the spatially periodic case.
Energy Technology Data Exchange (ETDEWEB)
Belmonte-Beitia, Juan [Departamento de Matematicas, E. T. S. de Ingenieros Industriales and Instituto de Matematica Aplicada a la Ciencia y la IngenierIa (IMACI), E. T. S. I. Industriales, Avda. Camilo Jose Cela, s/n Universidad de Castilla-La Mancha 13071 Ciudad Real (Spain)
2009-01-23
We introduce a model of a Bose-Einstein condensate based on the one-dimensional nonlinear Schroedinger equation, in which the nonlinear term depends on the domain. The nonlinear term changes a cubic term into a quintic term, according to the domain considered. We study the existence, stability and bifurcation of solutions, and use the qualitative theory of dynamical systems to study certain properties of such solutions.
Indian Academy of Sciences (India)
R S Kaushal; Ranjit Kumar; Awadhesh Prasad
2006-08-01
Attempts have been made to look for the soliton content in the solutions of the recently studied nonlinear diffusion-reaction equations [R S Kaushal, J. Phys. 38, 3897 (2005)] involving quadratic or cubic nonlinearities in addition to the convective flux term which renders the system nonconservative and the corresponding Hamiltonian non-Hermitian.
Energy Technology Data Exchange (ETDEWEB)
Belmonte-Beitia, Juan [Departamento de Matematicas, E.T.S. de Ingenieros Industriales and Instituto de Matematica Aplicada a la Ciencia y la Ingenieria (IMACI), Avda. Camilo Jose Cela 3, Universidad de Castilla-La Mancha, 13071 Ciudad Real (Spain)], E-mail: juan.belmonte@uclm.es; Calvo, Gabriel F. [Departamento de Matematicas, E.T.S. de Ingenieros Industriales and Instituto de Matematica Aplicada a la Ciencia y la Ingenieria (IMACI), Avda. Camilo Jose Cela 3, Universidad de Castilla-La Mancha, 13071 Ciudad Real (Spain)], E-mail: gabriel.fernandez@uclm.es
2009-01-19
In this Letter, by means of similarity transformations, we construct explicit solutions to the quintic nonlinear Schroedinger equation with potentials and nonlinearities depending both on time and on the spatial coordinates. We present the general approach and use it to study some examples and find nontrivial explicit solutions such as periodic (breathers), quasiperiodic and bright and dark soliton solutions.
Analytic treatment of nonlinear evolution equations using ﬁrst integral method
Indian Academy of Sciences (India)
Ahmet Bekir; Ömer Ünsal
2012-07-01
In this paper, we show the applicability of the ﬁrst integral method to combined KdV-mKdV equation, Pochhammer–Chree equation and coupled nonlinear evolution equations. The power of this manageable method is conﬁrmed by applying it for three selected nonlinear evolution equations. This approach can also be applied to other nonlinear differential equations.
Solitonlike solutions of the generalized discrete nonlinear Schrödinger equation
DEFF Research Database (Denmark)
Rasmussen, Kim; Henning, D.; Gabriel, H.
1996-01-01
We investigate the solution properties oi. a generalized discrete nonlinear Schrodinger equation describing a nonlinear lattice chain. The generalized equation interpolates between the integrable discrete Ablowitz-Ladik equation and the nonintegrable discrete Schrodinger equation. Special interest...... nonlinear Schrodinger equation. In this way eve are able to construct coherent solitonlike structures of profile determined by the map parameters....
AD GALERKIN ANALYSIS FOR NONLINEAR PSEUDO-HYPERBOLIC EQUATIONS
Institute of Scientific and Technical Information of China (English)
Xia Cui
2003-01-01
AD (Alternating direction) Galerkin schemes for d-dimensional nonlinear pseudo-hyperbolic equations are studied. By using patch approximation technique, AD procedure is realized,and calculation work is simplified. By using Galerkin approach, highly computational accuracy is kept. By using various priori estimate techniques for differential equations,difficulty coming from non-linearity is treated, and optimal H1 and L2 convergence properties are demonstrated. Moreover, although all the existed AD Galerkin schemes using patch approximation are limited to have only one order accuracy in time increment, yet the schemes formulated in this paper have second order accuracy in it. This implies an essential advancement in AD Galerkin analysis.
New traveling wave solutions for nonlinear evolution equations
Energy Technology Data Exchange (ETDEWEB)
El-Wakil, S.A. [Theoretical Research Group, Physics Department, Faculty of Science, Mansoura University, 35516 Mansoura (Egypt); Madkour, M.A. [Theoretical Research Group, Physics Department, Faculty of Science, Mansoura University, 35516 Mansoura (Egypt); Abdou, M.A. [Theoretical Research Group, Physics Department, Faculty of Science, Mansoura University, 35516 Mansoura (Egypt)]. E-mail: m_abdou_eg@yahoo.com
2007-06-11
The generalized Jacobi elliptic function expansion method is used with a computerized symbolic computation for constructing the new exact traveling wave solutions. The validity and reliability of the method is tested by its applications on a class of nonlinear evolution equations of special interest in mathematical physics. As a result, many exact traveling wave solutions are obtained which include the kink-shaped solutions, bell-shaped solutions, singular solutions and periodic solutions. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations in mathematical physics.
THE MORTAR ELEMENT METHOD FOR A NONLINEAR BIHARMONIC EQUATION
Institute of Scientific and Technical Information of China (English)
Zhong-ci Shi; Xue-jun Xu
2005-01-01
The mortar element method is a new domain decomposition method(DDM) with nonoverlapping subdomains. It can handle the situation where the mesh on different subdomains need not align across interfaces, and the matching of discretizations on adjacent subdomains is only enforced weakly. But until now there has been very little work for nonlinear PDEs. In this paper, we will present a mortar-type Morley element method for a nonlinear biharmonic equation which is related to the well-known Navier-Stokes equation. Optimal energy and H1-norm estimates are obtained under a reasonable elliptic regularity assumption.
Numerical simulation of nonlinear continuity equations by evolving diffeomorphisms
Carrillo, José A.
2016-09-22
In this paper we present a numerical scheme for nonlinear continuity equations, which is based on the gradient flow formulation of an energy functional with respect to the quadratic transportation distance. It can be applied to a large class of nonlinear continuity equations, whose dynamics are driven by internal energies, given external potentials and/or interaction energies. The solver is based on its variational formulation as a gradient flow with respect to the Wasserstein distance. Positivity of solutions as well as energy decrease of the semi-discrete scheme are guaranteed by its construction. We illustrate this property with various examples in spatial dimension one and two.
Solving Nonlinear Partial Differential Equations with Maple and Mathematica
Shingareva, Inna K
2011-01-01
The emphasis of the book is given in how to construct different types of solutions (exact, approximate analytical, numerical, graphical) of numerous nonlinear PDEs correctly, easily, and quickly. The reader can learn a wide variety of techniques and solve numerous nonlinear PDEs included and many other differential equations, simplifying and transforming the equations and solutions, arbitrary functions and parameters, presented in the book). Numerous comparisons and relationships between various types of solutions, different methods and approaches are provided, the results obtained in Maple an
Critical Exponents for Fast Diffusion Equations with Nonlinear Boundary Sources
Institute of Scientific and Technical Information of China (English)
WANG LU-SHENG; WANG ZE-JIA
2011-01-01
In this paper, we study the large time behavior of solutions to a class of fast diffusion equations with nonlinear boundary sources on the exterior domain of the unit ball. We are interested in the critical global exponent q0 and the critical Fujita exponent qc for the problem considered, and show that q0 ＝ qc for the multidimensional Non-Newtonian polytropic filtration equation with nonlinear boundary sources, which is quite different from the known results that q0 ＜ qc for the onedimensional case; moreover, the value is different from the slow case.
Properties of some nonlinear Schroedinger equations motivated through information theory
Energy Technology Data Exchange (ETDEWEB)
Yuan, Liew Ding; Parwani, Rajesh R, E-mail: parwani@nus.edu.s [Department of Physics, National University of Singapore, Kent Ridge (Singapore)
2009-06-01
We update our understanding of nonlinear Schroedinger equations motivated through information theory. In particular we show that a q-deformation of the basic nonlinear equation leads to a perturbative increase in the energy of a system, thus favouring the simplest q = 1 case. Furthermore the energy minimisation criterion is shown to be equivalent, at leading order, to an uncertainty maximisation argument. The special value eta = 1/4 for the interpolation parameter, where leading order energy shifts vanish, implies the preservation of existing supersymmetry in nonlinearised supersymmetric quantum mechanics. Physically, eta might be encoding relativistic effects.
Direct Perturbation Method for Derivative Nonlinear Schrodinger Equation
Institute of Scientific and Technical Information of China (English)
CHENG Xue-Ping; LIN Ji; HAN Ping
2008-01-01
We extend Lou's direct perturbation method for solving the nonlinear SchrSdinger equation to the case of the derivative nonlinear Schrodinger equation (DNLSE). By applying this method, different types of perturbation solutions axe obtained. Based on these approximate solutions, the analytical forms of soliton parameters, such as the velocity, the width and the initial position, are carried out and the effects of perturbation on solitons are analyzed at the same time. A numerical simulation of perturbed DNLSE finally verifies the results of the perturbation method.
A nonlinear viscoelastic constitutive equation - Yield predictions in multiaxial deformations
Shay, R. M., Jr.; Caruthers, J. M.
1987-01-01
Yield stress predictions of a nonlinear viscoelastic constitutive equation for amorphous polymer solids have been obtained and are compared with the phenomenological von Mises yield criterion. Linear viscoelasticity theory has been extended to include finite strains and a material timescale that depends on the instantaneous temperature, volume, and pressure. Results are presented for yield and the correct temperature and strain-rate dependence in a variety of multiaxial deformations. The present nonlinear viscoelastic constitutive equation can be formulated in terms of either a Cauchy or second Piola-Kirchhoff stress tensor, and in terms of either atmospheric or hydrostatic pressure.
BIHARMONIC EQUATIONS WITH ASYMPTOTICALLY LINEAR NONLINEARITIES
Institute of Scientific and Technical Information of China (English)
Liu Yue; Wang Zhengping
2007-01-01
This article considers the equation △2u = f(x, u)with boundary conditions either u|(a)Ω = (a)u/(a)n|(a)Ω = 0 or u|(a)Ω = △u|(a)Ω = 0, where f(x,t) is asymptotically linear with respect to t at infinity, and Ω is a smooth bounded domain in RN, N ＞ 4. By a variant version of Mountain Pass Theorem, it is proved that the above problems have a nontrivial solution under suitable assumptions of f(x, t).
Weakly nonlinear Schr\\"odinger equation with random initial data
Lukkarinen, Jani
2009-01-01
There is wide interest in weakly nonlinear wave equations with random initial data. A common approach is the approximation through a kinetic transport equation, which clearly poses the issue of understanding its validity in the kinetic limit. While for the general case a proof of the kinetic limit remains open, we report here on first progress. As wave equation we consider the nonlinear Schrodinger equation discretized on a hypercubic lattice. Since this is a Hamiltonian system, a natural choice of random initial data is distributing them according to a Gibbs measure with a chemical potential chosen so that the Gibbs field has exponential mixing. The solution psi_t(x) of the nonlinear Schrodinger equation yields then a stochastic process stationary in x in Z^d and t in R. If lambda denotes the strength of the nonlinearity, we prove that the space-time covariance of psi_t(x) has a limit as lambda -> 0 for t=lambda^{-2} tau, with tau fixed and |tau| sufficiently small. The limit agrees with the prediction from ...
Nonlinear electrostatic wave equations for magnetized plasmas - II
DEFF Research Database (Denmark)
Dysthe, K. B.; Mjølhus, E.; Pécseli, H. L.
1985-01-01
For pt.I see ibid., vol.26, p.443-7 (1984). The problem of extending the high frequency part of the Zakharov equations for nonlinear electrostatic waves to magnetized plasmas, is considered. Weak electromagnetic and thermal effects are retained on an equal footing. Direction dependent (electrosta......For pt.I see ibid., vol.26, p.443-7 (1984). The problem of extending the high frequency part of the Zakharov equations for nonlinear electrostatic waves to magnetized plasmas, is considered. Weak electromagnetic and thermal effects are retained on an equal footing. Direction dependent...... (electrostatic) cut-off implies that various cases must be considered separately, leading to equations with rather different properties. Various equations encountered previously in the literature are recovered as limiting cases....
Multisymplectic five-point scheme for the nonlinear wave equation
Institute of Scientific and Technical Information of China (English)
WANG Yushun; WANG Bin; YANG Hongwei; WANG Yunfeng
2003-01-01
In this paper, we introduce the multisymplectic structure of the nonlinear wave equation, and prove that the classical five-point scheme for the equation is multisymplectic. Numerical simulations of this multisymplectic scheme on highly oscillatory waves of the nonlinear Klein-Gordon equation and the collisions between kink and anti-kink solitons of the sine-Gordon equation are also provided. The multisymplectic schemes do not need to discrete PDEs in the space first as the symplectic schemes do and preserve not only the geometric structure of the PDEs accurately, but also their first integrals approximately such as the energy, the momentum and so on. Thus the multisymplectic schemes have better numerical stability and long-time numerical behavior than the energy-conserving scheme and the symplectic scheme.
Inverse Coefficient Problems for Nonlinear Parabolic Differential Equations
Institute of Scientific and Technical Information of China (English)
Yun Hua OU; Alemdar HASANOV; Zhen Hai LIU
2008-01-01
This paper is devoted to a class of inverse problems for a nonlinear parabolic differential equation.The unknown coefficient of the equation depends on the gradient of the solution and belongs to a set of admissible coefficients.It is proved that the convergence of solutions for the corresponding direct problems continuously depends on the coefficient convergence.Based on this result the existence of a quasisolution of the inverse problem is obtained in the appropriate class of admissible coefficients.
Intermittency and solitons in the driven dissipative nonlinear Schroedinger equation
Moon, H. T.; Goldman, M. V.
1984-01-01
The cubic nonlinear Schroedinger equation, in the presence of driving and Landau damping, is studied numerically. As the pump intensity is increased, the system exhibits a transition from intermittency to a two-torus to chaos. The laminar phase of the intermittency is also a two-torus motion which corresponds in physical space to two identical solitons of amplitude determined by a power-balance equation.
Conservation laws of inviscid Burgers equation with nonlinear damping
Abdulwahhab, Muhammad Alim
2014-06-01
In this paper, the new conservation theorem presented in Ibragimov (2007) [14] is used to find conservation laws of the inviscid Burgers equation with nonlinear damping ut+g(u)ux+λh(u)=0. We show that this equation is both quasi self-adjoint and self-adjoint, and use these concepts to simplify conserved quantities for various choices of g(u) and h(u).
Oscillation criteria for nonlinear fractional differential equation with damping term
Directory of Open Access Journals (Sweden)
Bayram Mustafa
2016-01-01
Full Text Available In this paper, we study the oscillation of solutions to a non-linear fractional differential equation with damping term. The fractional derivative is defined in the sense of the modified Riemann-Liouville derivative. By using a variable transformation, a generalized Riccati transformation, inequalities, and integration average techniquewe establish new oscillation criteria for the fractional differential equation. Several illustrative examples are also given.
Stochastic Cahn-Hilliard equation with singular nonlinearity and reflection
Goudenège, Ludovic
2008-01-01
International audience; We consider a stochastic partial differential equation with logarithmic (or negative power) nonlinearity, with one reflection at 0 and with a constraint of conservation of the space average. The equation, driven by the derivative in space of a space-time white noise, contains a bi-Laplacian in the drift. The lack of the maximum principle for the bi-Laplacian generates difficulties for the classical penalization method, which uses a crucial monotonicity property. Being ...
SINGULAR AND RAREFACTIVE SOLUTIONS TO A NONLINEAR VARIATIONAL WAVE EQUATION
Institute of Scientific and Technical Information of China (English)
无
2001-01-01
Following a recent paper of the authors in Communications in Partial Differential Equations, this paper establishes the global existence of weak solutions to a nonlinear variational wave equation under relaxed conditions on the initial data so that the solutions can contain singularities (blow-up). Propagation of local oscillations along one family of characteristics remains under control despite singularity formation in the other family of characteristics.
Asymptotic Behavior of Solutions for Nonlinear Volterra Discrete Equations
Directory of Open Access Journals (Sweden)
E. Messina
2008-01-01
Full Text Available We consider nonlinear difference equations of unbounded order of the form xi=bi−∑j=0iai,jfi−j(xj, i=0,1,2,…, where fj(x (j=0,…,i are suitable functions. We establish sufficient conditions for the boundedness and the convergence of xi as i→+∞. Some of these conditions are interesting mainly for studying stability of numerical methods for Volterra integral equations.
Institute of Scientific and Technical Information of China (English)
杨灵娥
2003-01-01
In this paper, we prove that in the inviscid limit the solution of the gen eralized derivative Ginzburg-Landau equations converges to the solution of derivative nonlinear Schrodinger equation, we also give the convergence rates for the difference of the solution.
The chaotic effects in a nonlinear QCD evolution equation
Zhu, Wei; Shen, Zhenqi; Ruan, Jianhong
2016-10-01
The corrections of gluon fusion to the DGLAP and BFKL equations are discussed in a united partonic framework. The resulting nonlinear evolution equations are the well-known GLR-MQ-ZRS equation and a new evolution equation. Using the available saturation models as input, we find that the new evolution equation has the chaos solution with positive Lyapunov exponents in the perturbative range. We predict a new kind of shadowing caused by chaos, which blocks the QCD evolution in a critical small x range. The blocking effect in the evolution equation may explain the Abelian gluon assumption and even influence our expectations to the projected Large Hadron Electron Collider (LHeC), Very Large Hadron Collider (VLHC) and the upgrade (CppC) in a circular e+e- collider (SppC).
The modified simple equation method for solving some fractional-order nonlinear equations
Indian Academy of Sciences (India)
KAPLAN MELIKE; BEKIR AHMET
2016-07-01
Nonlinear fractional differential equations are encountered in various fields of mathematics, physics, chemistry, biology, engineering and in numerous other applications. Exact solutions of these equations play a crucial role in the proper understanding of the qualitative features of many phenomena 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 method.
Elimination and nonlinear equations of Rees algebra
Busé, Laurent; Simis, Aron
2009-01-01
A new approach is established to computing the image of a rational map, whereby the use of approximation complexes is complemented with a detailed analysis of the torsion of the symmetric algebra in certain degrees. In the case the map is everywhere defined this analysis provides free resolutions of graded parts of the Rees algebra of the base ideal in degrees where it does not coincide with the corresponding symmetric algebra. A surprising fact is that the torsion in those degrees only contributes to the first free module in the resolution of the symmetric algebra modulo torsion. An additional point is that this contribution -- which of course corresponds to non linear equations of the Rees algebra -- can be described in these degrees in terms of non Koszul syzygies via certain upgrading maps in the vein of the ones introduced earlier by J. Herzog, the third named author and W. Vasconcelos. As a measure of the reach of this torsion analysis we could say that, in the case of a general everywhere defined map, ...
SOME DISCRETE NONLINEAR INEQUALITIES AND APPLICATIONS TO DIFFERENCE EQUATIONS
Institute of Scientific and Technical Information of China (English)
Cheung Wing-Sum; Ma Qing-Hua; Josip Pe(c)ari(c)
2008-01-01
In this article, the authors establish some new nonlinear difference inequalities in two independent variables, which generalize some existing results and can be used as handy tools in the study of qualitative as well as quantitative properties of solutions of certain classes of difference equations.
BOUNDARY LAYER AND VANISHING DIFFUSION LIMIT FOR NONLINEAR EVOLUTION EQUATIONS
Institute of Scientific and Technical Information of China (English)
彭艳
2014-01-01
In this paper, we consider an initial-boundary value problem for some nonlinear evolution equations with damping and diffusion. The main purpose is to investigate the boundary layer effect and the convergence rates as the diffusion parameterαgoes to zero.
The Peridic Wave Solutions for Two Nonlinear Evolution Equations
Institute of Scientific and Technical Information of China (English)
ZHANG Jin-Liang; WANG Ming-Liang; CHENG Dong-Ming; FANG Zong-De
2003-01-01
By using the F-expansion method proposed recently, the periodic wave solutions expressed by Jacobielliptic functions for two nonlinear evolution equations are derived. In the limit cases, the solitary wave solutions andthe other type of traveling wave solutions for the system are obtained.
The Local Stability of Solutions for a Nonlinear Equation
Directory of Open Access Journals (Sweden)
Haibo Yan
2014-01-01
Full Text Available The approach of Kruzkov’s device of doubling the variables is applied to establish the local stability of strong solutions for a nonlinear partial differential equation in the space L1(R by assuming that the initial value only lies in the space L1(R∩L∞(R.
Further studies of a simple gyrotron equation: nonlinear theory
Energy Technology Data Exchange (ETDEWEB)
Shi Meixuan, E-mail: meixuan@cims.nyu.ed [Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012-1185 (United States)
2010-11-05
A nonlinear version of a standard system of gyrotron model equations is studied using asymptotic analysis and variational methods. The condition for obtaining a high-amplitude wave is achieved in the study. A simple method for obtaining the patterns and amplitude of the wave based on the given free-space wave-number pattern is shown.
Oscillation criteria for first-order forced nonlinear difference equations
Grace Said R; Agarwal Ravi P.; Smith Tim
2006-01-01
Some new criteria for the oscillation of first-order forced nonlinear difference equations of the form Δx(n)+q1(n)xμ(n+1) = q2(n)xλ(n+1)+e(n), where λ, μ are the ratios of positive odd integers 0 <μ < 1 and λ > 1, are established.
Oscillation Theorems for Nonlinear Second Order Elliptic Equations
Institute of Scientific and Technical Information of China (English)
2007-01-01
Some oscillation theorems are given for the nonlinear second order elliptic equation N ∑i,j=1 Di[aij(x)Ψ(y)||(△)y||p-2Djy]+c(x)f(y)=0. The results are extensions of modified Riccati techniques and include recent results of Usami.
Exact controllability for a nonlinear stochastic wave equation
Directory of Open Access Journals (Sweden)
2006-01-01
Full Text Available The exact controllability for a semilinear stochastic wave equation with a boundary control is established. The target and initial spaces are L 2 ( G × H −1 ( G with G being a bounded open subset of R 3 and the nonlinear terms having at most a linear growth.
Exact periodic solution in coupled nonlinear Schrodinger equations
Institute of Scientific and Technical Information of China (English)
Li Qi-Liang; Chen Jun-Lang; Sun Li-Li; Yu Shu-Yi; Qian Sheng
2007-01-01
The coupled nonlinear Schrodinger equations (CNLSEs) of two symmetrical optical fibres are nonintegrable, however the transformed CNLSEs have integrability. Integrability of the transformed CNLSEs is proved by the Hamilton dynamics theory and Galilei transform. Making use of a transform for CNLSEs and using the ansatz with Jacobi elliptic function form, this paper obtains the exact optical pulse solutions.
EXISTENCE OF SOLUTIONS OF NONLINEAR FRACTIONAL PANTOGRAPH EQUATIONS
Institute of Scientific and Technical Information of China (English)
K. BALACHANDRAN; S. KIRUTHIKA; J.J. TRUJILLO
2013-01-01
This article deals with the existence of solutions of nonlinear fractional pantograph equations.Such model can be considered suitable to be applied when the corresponding process occurs through strongly anomalous media.The results are obtained using fractional calculus and fixed point theorems.An example is provided to illustrate the main result obtained in this article.
Advances in nonlinear partial differential equations and stochastics
Kawashima, S
1998-01-01
In the past two decades, there has been great progress in the theory of nonlinear partial differential equations. This book describes the progress, focusing on interesting topics in gas dynamics, fluid dynamics, elastodynamics etc. It contains ten articles, each of which discusses a very recent result obtained by the author. Some of these articles review related results.
Nonlocal Cauchy problem for nonlinear mixed integrodifferential equations
Directory of Open Access Journals (Sweden)
H.L. Tidke
2010-12-01
Full Text Available The present paper investigates the existence and uniqueness of mild and strong solutions of a nonlinear mixed Volterra-Fredholm integrodifferential equation with nonlocal condition. The results obtained by using Schauder fixed point theorem and the theory of semigroups.
An Orthogonal Residual Procedure for Nonlinear Finite Element Equations
DEFF Research Database (Denmark)
Krenk, S.
A general and robust solution procedure for nonlinear finite element equations with limit points is developed. At each equilibrium iteration the magnitude of the load is adjusted such that the residual force is orthogonal to the current displacement increment from the last equilibrium state...
Maximum Likelihood Estimation of Nonlinear Structural Equation Models.
Lee, Sik-Yum; Zhu, Hong-Tu
2002-01-01
Developed an EM type algorithm for maximum likelihood estimation of a general nonlinear structural equation model in which the E-step is completed by a Metropolis-Hastings algorithm. Illustrated the methodology with results from a simulation study and two real examples using data from previous studies. (SLD)
Case-Deletion Diagnostics for Nonlinear Structural Equation Models
Lee, Sik-Yum; Lu, Bin
2003-01-01
In this article, a case-deletion procedure is proposed to detect influential observations in a nonlinear structural equation model. The key idea is to develop the diagnostic measures based on the conditional expectation of the complete-data log-likelihood function in the EM algorithm. An one-step pseudo approximation is proposed to reduce the…
Local Influence Analysis of Nonlinear Structural Equation Models
Lee, Sik-Yum; Tang, Nian-Sheng
2004-01-01
By regarding the latent random vectors as hypothetical missing data and based on the conditional expectation of the complete-data log-likelihood function in the EM algorithm, we investigate assessment of local influence of various perturbation schemes in a nonlinear structural equation model. The basic building blocks of local influence analysis…
Probabilistic methods for discrete nonlinear Schr\\"odinger equations
Chatterjee, Sourav
2010-01-01
Using techniques from probability theory, we show that the thermodynamics of the focusing cubic discrete nonlinear Schrodinger equation (NLS) are exactly solvable in dimensions three and higher. A number of explicit formulas are derived. The probabilistic results, combined with dynamical information, prove the existence and typicality of solutions to the discrete NLS with highly stable localized modes that are sometimes called discrete breathers.
An inhomogeneous wave equation and non-linear Diophantine approximation
DEFF Research Database (Denmark)
Beresnevich, V.; Dodson, M. M.; Kristensen, S.;
2008-01-01
A non-linear Diophantine condition involving perfect squares and arising from an inhomogeneous wave equation on the torus guarantees the existence of a smooth solution. The exceptional set associated with the failure of the Diophantine condition and hence of the existence of a smooth solution...... is studied. Both the Lebesgue and Hausdorff measures of this set are obtained....
Tensor methods for large sparse systems of nonlinear equations
Energy Technology Data Exchange (ETDEWEB)
Bouaricha, A. [Argonne National Lab., IL (United States). Mathematics and Computer Science Div.; Schnabel, R.B. [Colorado Univ., Boulder, CO (United States). Dept. of Computer Science
1996-12-31
This paper introduces censor methods for solving, large sparse systems of nonlinear equations. Tensor methods for nonlinear equations were developed in the context of solving small to medium- sized dense problems. They base each iteration on a quadratic model of the nonlinear equations. where the second-order term is selected so that the model requires no more derivative or function information per iteration than standard linear model-based methods, and hardly more storage or arithmetic operations per iteration. Computational experiments on small to medium-sized problems have shown censor methods to be considerably more efficient than standard Newton-based methods, with a particularly large advantage on singular problems. This paper considers the extension of this approach to solve large sparse problems. The key issue that must be considered is how to make efficient use of sparsity in forming and solving the censor model problem at each iteration. Accomplishing this turns out to require an entirely new way of solving the tensor model that successfully exploits the sparsity of the Jacobian, whether the Jacobian is nonsingular or singular. We develop such an approach and, based upon it, an efficient tensor method for solving large sparse systems of nonlinear equations. Test results indicate that this tensor method is significantly more efficient and robust than an efficient sparse Newton-based method. in terms of iterations, function evaluations. and execution time.
Large Time Asymptotics for Solutions of Nonlinear Partial Differential Equations
Sachdev, PL
2010-01-01
A large number of physical phenomena are modeled by nonlinear partial differential equations, subject to appropriate initial/boundary conditions. This title presents the constructive mathematical techniques. It deals with the asymptotic methods which include self-similarity, balancing argument, and matched asymptotic expansions
A Dual Orthogonality Procedure for Nonlinear Finite Element Equations
DEFF Research Database (Denmark)
Krenk, S.; Hededal, O.
In the orthogonal residual procedure for solution of nonlinear finite element equations the load is adjusted in each equilibrium iteration to satisfy an orthogonality condition to the current displacement increment. It is here shown that the quasi-newton formulation of the orthogonal residual...
An Efficient Numerical Approach for Nonlinear Fokker-Planck equations
Otten, Dustin; Vedula, Prakash
2009-03-01
Fokker-Planck equations which are nonlinear with respect to their probability densities that occur in many nonequilibrium systems relevant to mean field interaction models, plasmas, classical fermions and bosons can be challenging to solve numerically. To address some underlying challenges in obtaining numerical solutions, we propose a quadrature based moment method for efficient and accurate determination of transient (and stationary) solutions of nonlinear Fokker-Planck equations. In this approach the distribution function is represented as a collection of Dirac delta functions with corresponding quadrature weights and locations, that are in turn determined from constraints based on evolution of generalized moments. Properties of the distribution function can be obtained by solution of transport equations for quadrature weights and locations. We will apply this computational approach to study a wide range of problems, including the Desai-Zwanzig Model (for nonlinear muscular contraction) and multivariate nonlinear Fokker-Planck equations describing classical fermions and bosons, and will also demonstrate good agreement with results obtained from Monte Carlo and other standard numerical methods.
ON THE NONLINEAR TIMOSHENKO-KIRCHHOFF BEAM EQUATION
Institute of Scientific and Technical Information of China (English)
A.AROSIO
1999-01-01
When an elastic string with fixed ends is subjected to transverse vibrations, its length vaxiewith the time: this introduces chvages of the tension in the string. Thls induced Kirchoffto propose a nonlinear correction of the classical D'Alembert equation. Later on, Wolnowsky-
Numerical treatments for solving nonlinear mixed integral equation
Directory of Open Access Journals (Sweden)
M.A. Abdou
2016-12-01
Full Text Available We consider a mixed type of nonlinear integral equation (MNLIE of the second kind in the space C[0,T]×L2(Ω,T<1. The Volterra integral terms (VITs are considered in time with continuous kernels, while the Fredholm integral term (FIT is considered in position with singular general kernel. Using the quadratic method and separation of variables method, we obtain a nonlinear system of Fredholm integral equations (NLSFIEs with singular kernel. A Toeplitz matrix method, in each case, is then used to obtain a nonlinear algebraic system. Numerical results are calculated when the kernels take a logarithmic form or Carleman function. Moreover, the error estimates, in each case, are then computed.
Mapping deformation method and its application to nonlinear equations
Institute of Scientific and Technical Information of China (English)
李画眉
2002-01-01
An extended mapping deformation method is proposed for finding new exact travelling wave solutions of nonlinearpartial differential equations (PDEs). The key idea of this method is to take full advantage of the simple algebraicmapping relation between the solutions of the PDEs and those of the cubic nonlinear Klein-Gordon equation. This isapplied to solve a system of variant Boussinesq equations. As a result, many explicit and exact solutions are obtained,including solitary wave solutions, periodic wave solutions, Jacobian elliptic function solutions and other exact solutions.
A procedure to construct exact solutions of nonlinear evolution equations
Indian Academy of Sciences (India)
Adem Cengiz Çevikel; Ahmet Bekir; Mutlu Akar; Sait San
2012-09-01
In this paper, we implemented the functional variable method for the exact solutions of the Zakharov-Kuznetsov-modified equal-width (ZK-MEW), the modified Benjamin-Bona-Mohany (mBBM) and the modified kdV-Kadomtsev-Petviashvili (kdV-KP) equation. By using this scheme, we found some exact solutions of the above-mentioned equation. The obtained solutions include solitary wave solutions, periodic wave solutions and combined formal solutions. The functional variable method presents a wider-applicability for handling nonlinear wave equations.
An Efficient Series Solution for Nonlinear Multiterm Fractional Differential Equations
Directory of Open Access Journals (Sweden)
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.
A granular computing method for nonlinear convection-diffusion equation
Directory of Open Access Journals (Sweden)
Tian Ya Lan
2016-01-01
Full Text Available This paper introduces a method of solving nonlinear convection-diffusion equation (NCDE, based on the combination of granular computing (GrC and characteristics finite element method (CFEM. The key idea of the proposed method (denoted as GrC-CFEM is to reconstruct the solution from coarse-grained layer to fine-grained layer. It first gets the nonlinear solution on the coarse-grained layer, and then the function (Taylor expansion is applied to linearize the NCDE on the fine-grained layer. Switch to the fine-grained layer, the linear solution is directly derived from the nonlinear solution. The full nonlinear problem is solved only on the coarse-grained layer. Numerical experiments show that the GrC-CFEM can accelerate the convergence and improve the computational efficiency without sacrificing the accuracy.
The First Integral Method to the Nonlinear Schrodinger Equations in Higher Dimensions
Directory of Open Access Journals (Sweden)
Shoukry Ibrahim Atia El-Ganaini
2013-01-01
Full Text Available The first integral method introduced by Feng is adopted for solving some important nonlinear partial differential equations, including the (2 + 1-dimensional hyperbolic nonlinear Schrodinger (HNLS equation, the generalized nonlinear Schrodinger (GNLS equation with a source, and the higher-order nonlinear Schrodinger equation in nonlinear optical fibers. This method provides polynomial first integrals for autonomous planar systems. Through the established first integrals, exact traveling wave solutions are formally derived in a concise manner.
Grete, P; Schmidt, W; Schleicher, D R G
2016-01-01
Even though compressible plasma turbulence is encountered in many astrophysical phenomena, its effect is often not well understood. Furthermore, direct numerical simulations are typically not able to reach the extreme parameters of these processes. For this reason, large-eddy simulations (LES), which only simulate large and intermediate scales directly, are employed. The smallest, unresolved scales and the interactions between small and large scales are introduced by means of a subgrid-scale (SGS) model. We propose and verify a new set of nonlinear SGS closures for future application as an SGS model in LES of compressible magnetohydrodynamics (MHD). We use 15 simulations (without explicit SGS model) of forced, isotropic, homogeneous turbulence with varying sonic Mach number $\\mathrm{M_s} = 0.2$ to $20$ as reference data for the most extensive \\textit{a priori} tests performed so far in literature. In these tests we explicitly filter the reference data and compare the performance of the new closures against th...
A mixed finite element method for nonlinear diffusion equations
Burger, Martin
2010-01-01
We propose a mixed finite element method for a class of nonlinear diffusion equations, which is based on their interpretation as gradient flows in optimal transportation metrics. We introduce an appropriate linearization of the optimal transport problem, which leads to a mixed symmetric formulation. This formulation preserves the maximum principle in case of the semi-discrete scheme as well as the fully discrete scheme for a certain class of problems. In addition solutions of the mixed formulation maintain exponential convergence in the relative entropy towards the steady state in case of a nonlinear Fokker-Planck equation with uniformly convex potential. We demonstrate the behavior of the proposed scheme with 2D simulations of the porous medium equations and blow-up questions in the Patlak-Keller-Segel model. © American Institute of Mathematical Sciences.
Traveling wavefront solutions to nonlinear reaction-diffusion-convection equations
Indekeu, Joseph O.; Smets, Ruben
2017-08-01
Physically motivated modified Fisher equations are studied in which nonlinear convection and nonlinear diffusion is allowed for besides the usual growth and spread of a population. It is pointed out that in a large variety of cases separable functions in the form of exponentially decaying sharp wavefronts solve the differential equation exactly provided a co-moving point source or sink is active at the wavefront. The velocity dispersion and front steepness may differ from those of some previously studied exact smooth traveling wave solutions. For an extension of the reaction-diffusion-convection equation, featuring a memory effect in the form of a maturity delay for growth and spread, also smooth exact wavefront solutions are obtained. The stability of the solutions is verified analytically and numerically.
Modelling of nonlinear shoaling based on stochastic evolution equations
DEFF Research Database (Denmark)
Kofoed-Hansen, Henrik; Rasmussen, Jørgen Hvenekær
1998-01-01
A one-dimensional stochastic model is derived to simulate the transformation of wave spectra in shallow water including generation of bound sub- and super-harmonics, near-resonant triad wave interaction and wave breaking. Boussinesq type equations with improved linear dispersion characteristics...... are recast into evolution equations for the complex amplitudes, and serve as the underlying deterministic model. Next, a set of evolution equations for the cumulants is derived. By formally introducing the well-known Gaussian closure hypothesis, nonlinear evolution equations for the power spectrum...... and bispectrum are derived. A simple description of depth-induced wave breaking is incorporated in the model equations, assuming that the total rate of dissipation may be distributed in proportion to the spectral energy density on each discrete frequency. The proposed phase-averaged model is compared...
Stability of planar diffusion wave for nonlinear evolution equation
Institute of Scientific and Technical Information of China (English)
无
2012-01-01
It is known that the one-dimensional nonlinear heat equation ut = f(u)x1x1,f'(u) 0,u(±∞,t) = u±,u+ = u_ has a unique self-similar solution u(x1/1+t).In multi-dimensional space,u(x1/1+t) is called a planar diffusion wave.In the first part of the present paper,it is shown that under some smallness conditions,such a planar diffusion wave is nonlinearly stable for the nonlinear heat equation:ut-△f(u) = 0,x ∈ Rn.The optimal time decay rate is obtained.In the second part of this paper,it is further shown that this planar diffusion wave is still nonlinearly stable for the quasilinear wave equation with damping:utt + utt+ △f(u) = 0,x ∈ Rn.The time decay rate is also obtained.The proofs are given by an elementary energy method.
Thurgood, J O; 10.1051/0004-6361/201219850
2012-01-01
Context: Coronal magnetic null points have been implicated as possible locations for localised heating events in 2D models. We investigate this possibility about fully 3D null points. Aims: We investigate the nature of the fast magnetoacoustic wave about a fully 3D magnetic null point, with a specific interest in its propagation, and we look for evidence of MHD mode coupling and/or conversion to the Alfv\\'en mode. Methods: A special fieldline and flux-based coordinate system was constructed to permit the introduction of a pure fast magnetoacoustic wave in the vicinity of proper and improper 3D null points. We considered the ideal, {\\beta} = 0, MHD equations, which are solved using the LARE3D numerical code. The constituent modes of the resulting wave were isolated and identified using the special coordinate system. Numerical results were supported by analytical work derived from perturbation theory and a linear implementation of the WKB method. Results: An initially pure fast wave is found to be permanently d...
Modulational instability in fractional nonlinear Schrödinger equation
Zhang, Lifu; He, Zenghui; Conti, Claudio; Wang, Zhiteng; Hu, Yonghua; Lei, Dajun; Li, Ying; Fan, Dianyuan
2017-07-01
Fractional calculus is entering the field of nonlinear optics to describe unconventional regimes, as disorder biological media and soft-matter. Here we investigate spatiotemporal modulational instability (MI) in a fractional nonlinear Schrödinger equation. We derive the MI gain spectrum in terms of the Lévy indexes and a varying number of spatial dimensions. We show theoretically and numerically that the Lévy indexes affect fastest growth frequencies and MI bandwidth and gain. Our results unveil a very rich scenario that may occur in the propagation of ultrashort pulses in random media and metamaterials, and may sustain novel kinds of propagation invariant optical bullets.
On nonlinear equation of Schrödinger type
Soltanov, Kamal N.
2012-11-01
In this paper we study a mixed problem for the nonlinear Schrödinger equation that have a nonlinear adding, in which the coefficient is a generalized function. Here is proved a solvability theorem of the considered problem with use of the general solvability theorem of the article [28]. Furthermore here is investigated also the behaviour of the solution of the studied problem. aipproc class produce a paper with the correct layout for AIP Conference Proceedings 8.5in × 11in double column.
Quasi-periodic solutions of nonlinear beam equations with quintic quasi-periodic nonlinearities
Directory of Open Access Journals (Sweden)
Qiuju Tuo
2015-01-01
Full Text Available In this article, we consider the one-dimensional nonlinear beam equations with quasi-periodic quintic nonlinearities $$ u_{tt}+u_{xxxx}+(B+ \\varepsilon\\phi(tu^5=0 $$ under periodic boundary conditions, where B is a positive constant, $\\varepsilon$ is a small positive parameter, $\\phi(t$ is a real analytic quasi-periodic function in t with frequency vector $\\omega=(\\omega_1,\\omega_2,\\dots,\\omega_m$. It is proved that the above equation admits many quasi-periodic solutions by KAM theory and partial Birkhoff normal form.
Charged anisotropic matter with linear or nonlinear equation of state
Varela, Victor; Ray, Saibal; Chakraborty, Kaushik; Kalam, Mehedi
2010-01-01
Ivanov pointed out substantial analytical difficulties associated with self-gravitating, static, isotropic fluid spheres when pressure explicitly depends on matter density. Simplification achieved with the introduction of electric charge were noticed as well. We deal with self-gravitating, charged, anisotropic fluids and get even more flexibility in solving the Einstein-Maxwell equations. In order to discuss analytical solutions we extend Krori and Barua's method to include pressure anisotropy and linear or non-linear equations of state. The field equations are reduced to a system of three algebraic equations for the anisotropic pressures as well as matter and electrostatic energy densities. Attention is paid to compact sources characterized by positive matter density and positive radial pressure. Arising solutions satisfy the energy conditions of general relativity. Spheres with vanishing net charge contain fluid elements with unbounded proper charge density located at the fluid-vacuum interface. Notably the...
Nonlinear electromagnetic gyrokinetic equation for plasmas with large mean flows
Energy Technology Data Exchange (ETDEWEB)
Sugama, H. [National Inst. for Fusion Science, Toki, Gifu (Japan); Horton, W.
1998-02-01
A new nonlinear electromagnetic gyrokinetic equation is derived for plasmas with large flow velocities on the order of the ion thermal speed. The gyrokinetic equation derived here is given in the form which is valid for general magnetic geometries including the slab, cylindrical and toroidal configurations. The source term for the anomalous viscosity arising through the Reynolds stress is identified in the gyrokinetic equation. For the toroidally rotating plasma, particle, energy and momentum balance equations as well as the detailed definitions of the anomalous transport fluxes and the anomalous entropy production are shown. The quasilinear anomalous transport matrix connecting the conjugate pairs of the anomalous fluxes and the forces satisfies the Onsager symmetry. (author)
Generalized creation and annihilation operators via complex nonlinear Riccati equations
Schuch, Dieter; Castaños, Octavio; Rosas-Ortiz, Oscar
2013-06-01
Based on Gaussian wave packet solutions of the time-dependent Schrödinger equation, a generalization of the conventional creation and annihilation operators and the corresponding coherent states can be obtained. This generalization includes systems where also the width of the coherent states is time-dependent as they occur for harmonic oscillators with time-dependent frequency or systems in contact with a dissipative environment. The key point is the replacement of the frequency ω0 that occurs in the usual definition of the creation/annihilation operator by a complex time-dependent function that fulfils a nonlinear Riccati equation. This equation and its solutions depend on the system under consideration and on the (complex) initial conditions. Formal similarities also exist with supersymmetric quantum mechanics. The generalized creation and annihilation operators also allow to construct exact analytic solutions of the free motion Schrödinger equation in terms of Hermite polynomials with time-dependent variable.
Multi-soliton rational solutions for some nonlinear evolution equations
Directory of Open Access Journals (Sweden)
Osman Mohamed S.
2016-01-01
Full Text Available The Korteweg-de Vries equation (KdV and the (2+ 1-dimensional Nizhnik-Novikov-Veselov system (NNV are presented. Multi-soliton rational solutions of these equations are obtained via the generalized unified method. The analysis emphasizes the power of this method and its capability of handling completely (or partially integrable equations. Compared with Hirota’s method and the inverse scattering method, the proposed method gives more general exact multi-wave solutions without much additional effort. The results show that, by virtue of symbolic computation, the generalized unified method may provide us with a straightforward and effective mathematical tool for seeking multi-soliton rational solutions for solving many nonlinear evolution equations arising in different branches of sciences.
Numerical solution of control problems governed by nonlinear differential equations
Energy Technology Data Exchange (ETDEWEB)
Heinkenschloss, M. [Virginia Polytechnic Institute and State Univ., Blacksburg, VA (United States)
1994-12-31
In this presentation the author investigates an iterative method for the solution of optimal control problems. These problems are formulated as constrained optimization problems with constraints arising from the state equation and in the form of bound constraints on the control. The method for the solution of these problems uses the special structure of the problem arising from the bound constraint and the state equation. It is derived from SQP methods and projected Newton methods and combines the advantages of both methods. The bound constraint is satisfied by all iterates using a projection, the nonlinear state equation is satisfied in the limit. Only a linearized state equation has to be solved in every iteration. The solution of the linearized problems are done using multilevel methods and GMRES.
Radial selfsimilar solutions of a nonlinear Ornstein-Uhlenbeck equation
Directory of Open Access Journals (Sweden)
Arij Bouzelmate
2007-05-01
Full Text Available This paper concerns the existence, uniqueness and asymptotic properties (as $r=|x|oinfty$ of radial self-similar solutions to the nonlinear Ornstein-Uhlenbeck equation [ v_t=Delta_p v+xcdot abla (|v|^{q-1}v ] in $mathbb{R}^Nimes (0, +infty$. Here $q>p-1>1$, $Ngeq 1$, and $Delta_p$ denotes the $p$-Laplacian operator. These solutions are of the form [ v(x,t=t^{-gamma} U(cxt^{-sigma}, ] where $gamma$ and $sigma$ are fixed powers given by the invariance properties of differential equation, while $U$ is a radial function, $U(y=u(r$, $r=|y|$. With the choice $c=(q-1^{-1/p}$, the radial profile $u$ satisfies the nonlinear ordinary differential equation $$ (|u'|^{p-2}u''+frac{N-1}r |u'|^{p-2}u'+frac{q+1-p}{p} r u'+(q-1 r(|u|^{q-1}u'+u=0 $$in $mathbb{R}_+$. We carry out a careful analysis of this equation anddeduce the corresponding consequences for the Ornstein-Uhlenbeck equation.
On the Amplitude Equations for Weakly Nonlinear Surface Waves
Benzoni-Gavage, Sylvie; Coulombel, Jean-François
2012-09-01
Nonlocal generalizations of Burgers' equation were derived in earlier work by Hunter (Contemp Math, vol 100, pp 185-202. AMS, 1989), and more recently by Benzoni-Gavage and Rosini (Comput Math Appl 57(3-4):1463-1484, 2009), as weakly nonlinear amplitude equations for hyperbolic boundary value problems admitting linear surface waves. The local-in-time well-posedness of such equations in Sobolev spaces was proved by Benzoni-Gavage (Differ Integr Equ 22(3-4):303-320, 2009) under an appropriate stability condition originally pointed out by Hunter. The same stability condition has also been shown to be necessary for well-posedness in Sobolev spaces in a previous work of the authors in collaboration with Tzvetkov (Benzoni-Gavage et al. in Adv Math 227(6):2220-2240, 2011). In this article, we show how the verification of Hunter's stability condition follows from natural stability assumptions on the original hyperbolic boundary value problem, thus avoiding lengthy computations in each particular situation. We also show that the resulting amplitude equation has a Hamiltonian structure when the original boundary value problem has a variational origin. Our analysis encompasses previous equations derived for nonlinear Rayleigh waves in elasticity.
Singular solutions of fully nonlinear elliptic equations and applications
Armstrong, Scott N; Smart, Charles K
2011-01-01
We study the properties of solutions of fully nonlinear, positively homogeneous elliptic equations near boundary points of Lipschitz domains at which the solution may be singular. We show that these equations have two positive solutions in each cone of $\\mathbb{R}^n$, and the solutions are unique in an appropriate sense. We introduce a new method for analyzing the behavior of solutions near certain Lipschitz boundary points, which permits us to classify isolated boundary singularities of solutions which are bounded from either above or below. We also obtain a sharp Phragm\\'en-Lindel\\"of result as well as a principle of positive singularities in certain Lipschitz domains.
Critical exponent for damped wave equations with nonlinear memory
Fino, Ahmad
2010-01-01
We consider the Cauchy problem in $\\mathbb{R}^n,$ $n\\geq 1,$ for a semilinear damped wave equation with nonlinear memory. Global existence and asymptotic behavior as $t\\to\\infty$ of small data solutions have been established in the case when $1\\leq n\\leq3.$ Moreover, we derive a blow-up result under some positive data for in any dimensional space. It turns out that the critical exponent indeed coincides with the one to the corresponding semilinear heat equation.
Nonzero solutions of nonlinear integral equations modeling infectious disease
Energy Technology Data Exchange (ETDEWEB)
Williams, L.R. (Indiana Univ., South Bend); Leggett, R.W.
1982-01-01
Sufficient conditions to insure the existence of periodic solutions to the nonlinear integral equation, x(t) = ..integral../sup t//sub t-tau/f(s,x(s))ds, are given in terms of simple product and product integral inequalities. The equation can be interpreted as a model for the spread of infectious diseases (e.g., gonorrhea or any of the rhinovirus viruses) if x(t) is the proportion of infectives at time t and f(t,x(t)) is the proportion of new infectives per unit time.
Various Newton-type iterative methods for solving nonlinear equations
Directory of Open Access Journals (Sweden)
Manoj Kumar
2013-10-01
Full Text Available The aim of the present paper is to introduce and investigate new ninth and seventh order convergent Newton-type iterative methods for solving nonlinear equations. The ninth order convergent Newton-type iterative method is made derivative free to obtain seventh-order convergent Newton-type iterative method. These new with and without derivative methods have efficiency indices 1.5518 and 1.6266, respectively. The error equations are used to establish the order of convergence of these proposed iterative methods. Finally, various numerical comparisons are implemented by MATLAB to demonstrate the performance of the developed methods.
Comparison Criteria for Nonlinear Functional Dynamic Equations of Higher Order
Directory of Open Access Journals (Sweden)
Taher S. Hassan
2016-01-01
Full Text Available We will consider the higher order functional dynamic equations with mixed nonlinearities of the form xnt+∑j=0Npjtϕγjxφjt=0, on an above-unbounded time scale T, where n≥2, xi(t≔ri(tϕαixi-1Δ(t, i=1,…,n-1, with x0=x, ϕβ(u≔uβsgnu, and α[i,j]≔αi⋯αj. The function φi:T→T is a rd-continuous function such that limt→∞φi(t=∞ for j=0,1,…,N. The results extend and improve some known results in the literature on higher order nonlinear dynamic equations.
Quasi-periodic Solutions of the General Nonlinear Beam Equations
Institute of Scientific and Technical Information of China (English)
GAO YI-XIAN
2012-01-01
In this paper,one-dimensional (1D) nonlinear beam equations of the form utt - uxx + uxxxx + mu = f(u)with Dirichlet boundary conditions are considered,where the nonlinearity f is an analytic,odd function and f(u) = O(u3).It is proved that for all m ∈ (0,M*] (∈) R(M* is a fixed large number),but a set of small Lebesgue measure,the above equations admit small-amplitude quasi-periodic solutions corresponding to finite dimensional invariant tori for an associated infinite dimensional dynamical system.The proof is based on an infinite dimensional KAM theory and a partial Birkhoff normal form technique.
An Explicit High Resolution Scheme for Nonlinear Shallow Water Equations
Institute of Scientific and Technical Information of China (English)
FANG Ke-zhao; ZOU Zhi-li; WANG Yan
2005-01-01
The present study develops a numerical model of the two-dimensional fully nonlinear shallow water equations (NSWE) for the wave run-up on a beach. The finite volume method (FVM) is used to solve the equations, and a second-order explicit scheme is developed to improve the computation efficiency. The numerical fluxes are obtained by the two dimensional Roe's flux function to overcome the errors caused by the use of one dimensional fluxes in dimension splitting methods. The high-resolution Godunov-type TVD upwind scheme is employed and a second-order accuracy is achieved based on monotonic upstream schemes for conservation laws (MUSCL) variable extrapolation; a nonlinear limiter is applied to prevent unwanted spurious oscillation. A simple but efficient technique is adopted to deal with the moving shoreline boundary. The verification of the solution technique is carried out by comparing the model output with documented results and it shows that the solution technique is robust.
RESTRICTED NONLINEAR APPROXIMATION AND SINGULAR SOLUTIONS OF BOUNDARY INTEGRAL EQUATIONS
Institute of Scientific and Technical Information of China (English)
Reinhard Hochmuth
2002-01-01
This paper studies several problems, which are potentially relevant for the construction of adaptive numerical schemes. First, biorthogonal spline wavelets on [0,1 ] are chosen as a starting point for characterizations of functions in Besov spaces B , (0,1) with 0＜σ＜∞ and (1+σ)-1＜τ＜∞. Such function spaces are known to be related to nonlinear approximation. Then so called restricted nonlinear approximation procedures with respect to Sobolev space norms are considered. Besides characterization results Jackson type estimates for various tree-type and tresholding algorithms are investigated. Finally known approximation results for geometry induced singularity functions of boundary integeral equations are combined with the characterization results for restricted nonlinear approximation to show Besov space regularity results.
Nonlinear partial differential equations for scientists and engineers
Debnath, Lokenath
1997-01-01
"An exceptionally complete overview. There are numerous examples and the emphasis is on applications to almost all areas of science and engineering. There is truly something for everyone here. This reviewer feels that it is a very hard act to follow, and recommends it strongly. [This book] is a jewel." ---Applied Mechanics Review (Review of First Edition) This expanded and revised second edition is a comprehensive and systematic treatment of linear and nonlinear partial differential equations and their varied applications. Building upon the successful material of the first book, this edition contains updated modern examples and applications from areas of fluid dynamics, gas dynamics, plasma physics, nonlinear dynamics, quantum mechanics, nonlinear optics, acoustics, and wave propagation. Methods and properties of solutions are presented, along with their physical significance, making the book more useful for a diverse readership. Topics and key features: * Thorough coverage of derivation and methods of soluti...
Nonlinear Self-Adjoint Classification of a Burgers-KdV Family of Equations
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Júlio Cesar Santos Sampaio
2014-01-01
Full Text Available The concepts of strictly, quasi, weak, and nonlinearly self-adjoint differential equations are revisited. A nonlinear self-adjoint classification of a class of equations with second and third order is carried out.
Periodic Wave Solutions of Generalized Derivative Nonlinear Schr(o)dinger Equation
Institute of Scientific and Technical Information of China (English)
ZHA Qi-Lao; LI Zhi-Bin
2008-01-01
A Darboux transformation of the generalized derivative nonlinear Schr(o)dinger equation is derived. As an application, some new periodic wave solutions of the generalized derivative nonlinear Schr(o)dinger equation are explicitly given.
A Hierarchy of New Nonlinear Evolution Equations Associated with a 3 × 3 Matrix Spectral Problem
Institute of Scientific and Technical Information of China (English)
GENG Xian-Guo; LI Fang
2009-01-01
A 3 × 3 matrix spectral problem with three potentials and the corresponding hierarchy of new nonlinear evolution equations are proposed. Generalized Hamiltonian structures for the hierarchy of nonlinear evolution equations are derived with the aid of trace identity.
Local H\\"older continuity for doubly nonlinear parabolic equations
Kuusi, Tuomo; Urbano, José Miguel
2010-01-01
We give a proof of the H\\"older continuity of weak solutions of certain degenerate doubly nonlinear parabolic equations in measure spaces. We only assume the measure to be a doubling non-trivial Borel measure which supports a Poincar\\'e inequality. The proof discriminates between large scales, for which a Harnack inequality is used, and small scales, that require intrinsic scaling methods.
Solitary wave solutions to nonlinear evolution equations in mathematical physics
Indian Academy of Sciences (India)
Anwar Ja’afar Mohamad Jawad; M Mirzazadeh; Anjan Biswas
2014-10-01
This paper obtains solitons as well as other solutions to a few nonlinear evolution equations that appear in various areas of mathematical physics. The two analytical integrators that are applied to extract solutions are tan–cot method and functional variable approaches. The soliton solutions can be used in the further study of shallow water waves in (1+1) as well as (2+1) dimensions.
Parallel Evolutionary Modeling for Nonlinear Ordinary Differential Equations
Institute of Scientific and Technical Information of China (English)
无
2001-01-01
We introduce a new parallel evolutionary algorithm in modeling dynamic systems by nonlinear higher-order ordinary differential equations (NHODEs). The NHODEs models are much more universal than the traditional linear models. In order to accelerate the modeling process, we propose and realize a parallel evolutionary algorithm using distributed CORBA object on the heterogeneous networking. Some numerical experiments show that the new algorithm is feasible and efficient.
Symmetries and Similarity Reductions of Nonlinear Diffusion Equation
Institute of Scientific and Technical Information of China (English)
LI Hui-Jun; RUAN Hang-Yu
2004-01-01
The inverse recursion operator, three new sets of symmetries, and infinite-dimensional Lie algebras for the nonlinear diffusion equation are given. Some nonlocal symmetries related to eigenvectors of the recursion operator Ф with the eigenvalue λi are also obtained with the help of the recursion operator Фi = Ф - λi. Using a part of these symmetries we get twelve types of nontrivial new similarity reduction.
Symmetries and Similarity Reductions of Nonlinear Diffusion Equation
Institute of Scientific and Technical Information of China (English)
LIHui-Jun; RUANHang-Yu
2004-01-01
The inverse recursion operator, three new sets of symmetries, and infinite-dimensional Lie algebras for the nonlinear diffusion equation are given. Some nonlocal symmetries related to eigenvectors of the recursion operator with the eigenvalue λi are also obtained with the help of the recursion operator φi=φ-λi. Using a part of these symmetries we get twelve types of nontrivial new similarity reduction.
ANALYTIC INVARIANT CURVES OF A NONLINEAR SECOND ORDER DIFFERENCE EQUATION
Institute of Scientific and Technical Information of China (English)
Wang Wusheng
2009-01-01
This article studies the existence of analytic invariant curves for a nonlinear second order difference equation which was modeled from macroeconomics of the business cycle. The author not only discusses the case of the eigenvalue off the unit circle S1 and the case on S1 with the Diophantine condition but also considers the case of the eigenvalue at a root of the unity, which obviously violates the Diophantine condition.
Symposium on Nonlinear Semigroups, Partial Differential Equations and Attractors
Zachary, Woodford
1987-01-01
The original idea of the organizers of the Washington Symposium was to span a fairly narrow range of topics on some recent techniques developed for the investigation of nonlinear partial differential equations and discuss these in a forum of experts. It soon became clear, however, that the dynamical systems approach interfaced significantly with many important branches of applied mathematics. As a consequence, the scope of this resulting proceedings volume is an enlarged one with coverage of a wider range of research topics.
New Efficient Fourth Order Method for Solving Nonlinear Equations
Directory of Open Access Journals (Sweden)
Farooq Ahmad
2013-12-01
Full Text Available In a paper [Appl. Math. Comput., 188 (2 (2007 1587--1591], authors have suggested and analyzed a method for solving nonlinear equations. In the present work, we modified this method by using the finite difference scheme, which has a quintic convergence. We have compared this modified Halley method with some other iterative of fifth-orders convergence methods, which shows that this new method having convergence of fourth order, is efficient.
Chaoticons described by nonlocal nonlinear Schrödinger equation
Zhong, Lanhua; Li, Yuqi; Chen, Yong; Hong, Weiyi; Hu, Wei; Guo, Qi
2017-01-01
It is shown that the unstable evolutions of the Hermite-Gauss-type stationary solutions for the nonlocal nonlinear Schrödinger equation with the exponential-decay response function can evolve into chaotic states. This new kind of entities are referred to as chaoticons because they exhibit not only chaotic properties (with positive Lyapunov exponents and spatial decoherence) but also soliton-like properties (with invariant statistic width and interaction of quasi-elastic collisions). PMID:28134268
Kumar, Rakesh
2015-01-01
This investigation deals with the analysis of stagnation point heat transfer and corresponding flow features of hydromagnetic viscous incompressible fluid over a vertical shrinking sheet. The considered sheet is assumed to be permeable and subject to addition of stagnation point to control the generated vorticity in the boundary layer. The sheet is placed on the right side of the fluid saturated porous medium which is having permeability of specified form. Nonlinear convection waves in the flow field are realized due to the envisaged nonlinear relation between density and temperature. The equations governing the nonlinear convection boundary layer flow are modeled and simplified using similarity transformations. The economized equations are solved for numerical solutions by employing the implicit finite difference scheme also known as Keller-box method. The influence of the associated parameters of the problem on velocity and temperature distributions, skin friction and rate of heat transfer are presented thr...
DEFF Research Database (Denmark)
Oster, Michael; Gaididei, Yuri B.; Johansson, Magnus
2004-01-01
We study the continuum limit of a nonlinear Schrodinger lattice model with both on-site and inter-site nonlinearities, describing weakly coupled optical waveguides or Bose-Einstein condensates. The resulting continuum nonlinear Schrodinger-type equation includes both nonlocal and nonlinear...
Lu, Bin
2012-06-01
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.
Solution behaviors in coupled Schrödinger equations with full-modulated nonlinearities
Pınar, Zehra; Deliktaş, Ekin
2017-02-01
The nonlinear partial differential equations have an important role in real life problems. To obtain the exact solutions of the nonlinear partial differential equations, a number of approximate methods are known in the literature. In this work, a time- space modulated nonlinearities of coupled Schrödinger equations are considered. We provide a large class of Jacobi-elliptic solutions via the auxiliary equation method with sixth order nonlinearity and the Chebyshev approximation.
Periodic and Chaotic Breathers in the Nonlinear Schr(o)dinger Equation
Institute of Scientific and Technical Information of China (English)
LIU Xue-Shen; QI Yue-Ying; DING Pei-Zhu
2004-01-01
@@ The breathers in the cubic nonlinear Schrodinger equation are investigated numerically by using the symplectic method. We show that the solitonlike wave, the periodic, quasiperiodic and chaotic breathers can be observed with the increase of cubic nonlinear perturbation. Finally, we discuss the breathers in the cubic-quintic nonlinear Schrodinger equation with the increase of quintic nonlinear perturbation.
Institute of Scientific and Technical Information of China (English)
无
2005-01-01
In this paper, we extend the mapping transformation method through introducing variable coefficients.By means of the extended mapping transformation method, many explicit and exact general solutions with arbitrary functions for some nonlinear partial differential equations, which contain solitary wave solutions, trigonometric function solutions, and rational solutions, are obtained.
Extended Elliptic Mild Slope Equation Incorporating the Nonlinear Shoaling Effect
Directory of Open Access Journals (Sweden)
Xiao Qian-lu
2016-10-01
Full Text Available The transformation during wave propagation is significantly important for the calculations of hydraulic and coastal engineering, as well as the sediment transport. The exact wave height deformation calculation on the coasts is essential to near-shore hydrodynamics research and the structure design of coastal engineering. According to the wave shoaling results gained from the elliptical cosine wave theory, the nonlinear wave dispersion relation is adopted to develop the expression of the corresponding nonlinear wave shoaling coefficient. Based on the extended elliptic mild slope equation, an efficient wave numerical model is presented in this paper for predicting wave deformation across the complex topography and the surf zone, incorporating the nonlinear wave dispersion relation, the nonlinear wave shoaling coefficient and other energy dissipation factors. Especially, the phenomenon of wave recovery and second breaking could be shown by the present model. The classical Berkhoff single elliptic topography wave tests, the sinusoidal varying topography experiment, and complex composite slopes wave flume experiments are applied to verify the accuracy of the calculation of wave heights. Compared with experimental data, good agreements are found upon single elliptical topography and one-dimensional beach profiles, including uniform slope and step-type profiles. The results indicate that the newly-developed nonlinear wave shoaling coefficient improves the calculated accuracy of wave transformation in the surf zone efficiently, and the wave breaking is the key factor affecting the wave characteristics and need to be considered in the nearshore wave simulations.
Focusing of Spherical Nonlinear Pulses for Nonlinear Wave Equations Ⅲ. Subcritical Case
Institute of Scientific and Technical Information of China (English)
无
2005-01-01
This paper studied spherical pulses of solutions of the system of semilinear wave equations with the pulses focusing at a point in three space variables. It is shown that there is no nonlinear effect at leading terms of pulses, when the initial data is subcritical.
Algebraic dynamics solution to and algebraic dynamics algorithm for nonlinear advection equation
Institute of Scientific and Technical Information of China (English)
2008-01-01
Algebraic dynamics approach and algebraic dynamics algorithm for the solution of nonlinear partial differential equations are applied to the nonlinear advection equa-tion. The results show that the approach is effective for the exact analytical solu-tion and the algorithm has higher precision than other existing algorithms in nu-merical computation for the nonlinear advection equation.
On localization in the discrete nonlinear Schrödinger equation
DEFF Research Database (Denmark)
Bang, O.; Juul Rasmussen, J.; Christiansen, P.L.
1993-01-01
For some values of the grid resolution, depending on the nonlinearity, the discrete nonlinear Schrodinger equation with arbitrary power nonlinearity can be approximated by the corresponding continuum version of the equation. When the discretization becomes too coarse, the discrete equation exhibits...
An efficient algorithm for solving nonlinear system of differential equations and applications
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Mustafa GÜLSU
2015-10-01
Full Text Available In this article, we apply Chebyshev collocation method to obtain the numerical solutions of nonlinear systems of differential equations. This method transforms the nonlinear systems of differential equation to nonlinear systems of algebraic equations. The convergence of the numerical method are given and their applicability is illustrated with some examples.
Energy Technology Data Exchange (ETDEWEB)
Xie, Xi-Yang; Tian, Bo, E-mail: tian_bupt@163.com; Wang, Yu-Feng; Sun, Ya; Jiang, Yan
2015-11-15
In this paper, we investigate a generalized nonautonomous nonlinear equation which describes the ultrashort optical pulse propagating in a nonlinear inhomogeneous fiber. By virtue of the generalized Darboux transformation, the first- and second-order rogue-wave solutions for the generalized nonautonomous nonlinear equation are obtained, under some variable–coefficient constraints. Properties of the first- and second-order rogue waves are graphically presented and analyzed: When the coefficients are all chosen as the constants, we can observe the some functions, the shapes of wave crests and troughs for the first- and second-order rogue waves change. Oscillating behaviors of the first- and second-order rogue waves are observed when the coefficients are the trigonometric functions.
Nonlinear Schrodinger equations on the half-line with nonlinear boundary conditions
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Ahmet Batal
2016-08-01
Full Text Available In this article, we study the initial boundary value problem for nonlinear Schrodinger equations on the half-line with nonlinear boundary conditions $$ u_x(0,t+\\lambda|u(0,t|^ru(0,t=0,\\quad \\lambda\\in\\mathbb{R}-\\{0\\},\\; r> 0. $$ We discuss the local well-posedness when the initial data $u_0=u(x,0$ belongs to an $L^2$-based inhomogeneous Sobolev space $H^s(\\mathbb{R}_+$ with $s\\in (\\frac{1}{2},\\frac{7}{2}-\\{\\frac{3}{2}\\}$. We deal with the nonlinear boundary condition by first studying the linear Schrodinger equation with a time-dependent inhomogeneous Neumann boundary condition $u_x(0,t=h(t$ where $h\\in H^{\\frac{2s-1}{4}}(0,T$.
Energy Technology Data Exchange (ETDEWEB)
Moawad, S. M., E-mail: smmoawad@hotmail.com [Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef (Egypt)
2015-02-15
In this paper, we present a solution method for constructing exact analytic solutions to magnetohydrodynamics (MHD) equations. The method is constructed via all the trigonometric and hyperbolic functions. The method is applied to MHD equilibria with mass flow. Applications to a solar system concerned with the properties of coronal mass ejections that affect the heliosphere are presented. Some examples of the constructed solutions which describe magnetic structures of solar eruptions are investigated. Moreover, the constructed method can be applied to a variety classes of elliptic partial differential equations which arise in plasma physics.
Dynamic behavior of a nonlinear rational difference equation and generalization
Directory of Open Access Journals (Sweden)
Shi Qihong
2011-01-01
Full Text Available Abstract This paper is concerned about the dynamic behavior for the following high order nonlinear difference equation x n = (x n-k + x n-m + x n-l /(x n-k x n-m + x n-m x n-l +1 with the initial data { x - l , x - l + 1 , … , x - 1 } ∈ ℝ + l and 1 ≤ k ≤ m ≤ l. The convergence of solution to this equation is investigated by introducing a new sequence, which extends and includes corresponding results obtained in the references (Li in J Math Anal Appl 312:103-111, 2005; Berenhaut et al. Appl. Math. Lett. 20:54-58, 2007; Papaschinopoulos and Schinas J Math Anal Appl 294:614-620, 2004 to a large extent. In addition, some propositions for generalized equations are reported.
Ullah, Imran; Khan, Ilyas; Shafie, Sharidan
2016-11-01
In the present work, the effects of chemical reaction on hydromagnetic natural convection flow of Casson nanofluid induced due to nonlinearly stretching sheet immersed in a porous medium under the influence of thermal radiation and convective boundary condition are performed numerically. Moreover, the effects of velocity slip at stretching sheet wall are also examined in this study. The highly nonlinear-coupled governing equations are converted to nonlinear ordinary differential equations via similarity transformations. The transformed governing equations are then solved numerically using the Keller box method and graphical results for velocity, temperature, and nanoparticle concentration as well as wall shear stress, heat, and mass transfer rate are achieved through MATLAB software. Numerical results for the wall shear stress and heat transfer rate are presented in tabular form and compared with previously published work. Comparison reveals that the results are in good agreement. Findings of this work demonstrate that Casson fluids are better to control the temperature and nanoparticle concentration as compared to Newtonian fluid when the sheet is stretched in a nonlinear way. Also, the presence of suspended nanoparticles effectively promotes the heat transfer mechanism in the base fluid.
Ullah, Imran; Khan, Ilyas; Shafie, Sharidan
2016-12-01
In the present work, the effects of chemical reaction on hydromagnetic natural convection flow of Casson nanofluid induced due to nonlinearly stretching sheet immersed in a porous medium under the influence of thermal radiation and convective boundary condition are performed numerically. Moreover, the effects of velocity slip at stretching sheet wall are also examined in this study. The highly nonlinear-coupled governing equations are converted to nonlinear ordinary differential equations via similarity transformations. The transformed governing equations are then solved numerically using the Keller box method and graphical results for velocity, temperature, and nanoparticle concentration as well as wall shear stress, heat, and mass transfer rate are achieved through MATLAB software. Numerical results for the wall shear stress and heat transfer rate are presented in tabular form and compared with previously published work. Comparison reveals that the results are in good agreement. Findings of this work demonstrate that Casson fluids are better to control the temperature and nanoparticle concentration as compared to Newtonian fluid when the sheet is stretched in a nonlinear way. Also, the presence of suspended nanoparticles effectively promotes the heat transfer mechanism in the base fluid.
Directory of Open Access Journals (Sweden)
Elsayed M.E. Zayed
2016-02-01
Full Text Available In this article, the modified extended tanh-function method is employed to solve fractional partial differential equations in the sense of the modified Riemann–Liouville derivative. Based on a nonlinear fractional complex transformation, certain fractional partial differential equations can be turned into nonlinear ordinary differential equations of integer orders. For illustrating the validity of this method, we apply it to four nonlinear equations namely, the space–time fractional generalized nonlinear Hirota–Satsuma coupled KdV equations, the space–time fractional nonlinear Whitham–Broer–Kaup equations, the space–time fractional nonlinear coupled Burgers equations and the space–time fractional nonlinear coupled mKdV equations.
MHD three-dimensional flow of nanofluid with velocity slip and nonlinear thermal radiation
Hayat, Tasawar; Imtiaz, Maria; Alsaedi, Ahmed; Kutbi, Marwan A.
2015-12-01
An analysis has been carried out for the three dimensional flow of viscous nanofluid in the presence of partial slip and thermal radiation effects. The flow is induced by a permeable stretching surface. Water is treated as a base fluid and alumina as a nanoparticle. Fluid is electrically conducting in the presence of applied magnetic field. Entire different concept of nonlinear thermal radiation is utilized in the heat transfer process. Different from the previous literature, the nonlinear system for temperature distribution is solved and analyzed. Appropriate transformations reduce the nonlinear partial differential system to ordinary differential system. Convergent series solutions are computed for the velocity and temperature. Effects of different parameters on the velocity, temperature, skin friction coefficient and Nusselt number are computed and examined. It is concluded that heat transfer rate increases when temperature and radiation parameters are increased.
BLOW-UP CRITERION OF SMOOTH SOLUTIONS TO THE MHD EQUATIONS IN BESOV SPACES
Institute of Scientific and Technical Information of China (English)
YUAN Baoquan
2005-01-01
In this paper we discuss the logarithmic Sobolev inequalities in Besov spaces,and show their applications to the blow-up criterion of smooth solutions to the incompressible magneto-hydrodynamics equations.
Jacobi Elliptic Solutions for Nonlinear Differential Difference Equations in Mathematical Physics
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Khaled A. Gepreel
2012-01-01
Full Text Available We put a direct new method to construct the rational Jacobi elliptic solutions for nonlinear differential difference equations which may be called the rational Jacobi elliptic functions method. We use the rational Jacobi elliptic function method to construct many new exact solutions for some nonlinear differential difference equations in mathematical physics via the lattice equation and the discrete nonlinear Schrodinger equation with a saturable nonlinearity. The proposed method is more effective and powerful to obtain the exact solutions for nonlinear differential difference equations.
Evaluation of model fit in nonlinear multilevel structural equation modeling
Directory of Open Access Journals (Sweden)
Karin eSchermelleh-Engel
2014-03-01
Full Text Available Evaluating model fit in nonlinear multilevel structural equation models (MSEM presents a challenge as no adequate test statistic is available. Nevertheless, using a product indicator approach a likelihood ratio test for linear models is provided which may also be useful for nonlinear MSEM. The main problem with nonlinear models is that product variables are nonnormally distributed. Although robust test statistics have been developed for linear SEM to ensure valid results under the condition of nonnormality, they were not yet investigated for nonlinear MSEM. In a Monte Carlo study, the performance of the robust likelihood ratio test was investigated for models with single-level latent interaction effects using the unconstrained product indicator approach. As overall model fit evaluation has a potential limitation in detecting the lack of fit at a single level even for linear models, level-specific model fit evaluation was also investigated using partially saturated models. Four population models were considered: a model with interaction effects at both levels, an interaction effect at the within-group level, an interaction effect at the between-group level, and a model with no interaction effects at both levels. For these models the number of groups, predictor correlation, and model misspecification was varied. The results indicate that the robust test statistic performed sufficiently well. Advantages of level-specific model fit evaluation for the detection of model misfit are demonstrated.
Evaluation of model fit in nonlinear multilevel structural equation modeling.
Schermelleh-Engel, Karin; Kerwer, Martin; Klein, Andreas G
2014-01-01
Evaluating model fit in nonlinear multilevel structural equation models (MSEM) presents a challenge as no adequate test statistic is available. Nevertheless, using a product indicator approach a likelihood ratio test for linear models is provided which may also be useful for nonlinear MSEM. The main problem with nonlinear models is that product variables are non-normally distributed. Although robust test statistics have been developed for linear SEM to ensure valid results under the condition of non-normality, they have not yet been investigated for nonlinear MSEM. In a Monte Carlo study, the performance of the robust likelihood ratio test was investigated for models with single-level latent interaction effects using the unconstrained product indicator approach. As overall model fit evaluation has a potential limitation in detecting the lack of fit at a single level even for linear models, level-specific model fit evaluation was also investigated using partially saturated models. Four population models were considered: a model with interaction effects at both levels, an interaction effect at the within-group level, an interaction effect at the between-group level, and a model with no interaction effects at both levels. For these models the number of groups, predictor correlation, and model misspecification was varied. The results indicate that the robust test statistic performed sufficiently well. Advantages of level-specific model fit evaluation for the detection of model misfit are demonstrated.
Domain decomposition solvers for nonlinear multiharmonic finite element equations
Copeland, D. M.
2010-01-01
In many practical applications, for instance, in computational electromagnetics, the excitation is time-harmonic. Switching from the time domain to the frequency domain allows us to replace the expensive time-integration procedure by the solution of a simple elliptic equation for the amplitude. This is true for linear problems, but not for nonlinear problems. However, due to the periodicity of the solution, we can expand the solution in a Fourier series. Truncating this Fourier series and approximating the Fourier coefficients by finite elements, we arrive at a large-scale coupled nonlinear system for determining the finite element approximation to the Fourier coefficients. The construction of fast solvers for such systems is very crucial for the efficiency of this multiharmonic approach. In this paper we look at nonlinear, time-harmonic potential problems as simple model problems. We construct and analyze almost optimal solvers for the Jacobi systems arising from the Newton linearization of the large-scale coupled nonlinear system that one has to solve instead of performing the expensive time-integration procedure. © 2010 de Gruyter.
The constructive technique and its application in solving a nonlinear reaction diffusion equation
Institute of Scientific and Technical Information of China (English)
Lai Shao-Yong; Guo Yun-Xi; Qing Yin; Wu Yong-Hong
2009-01-01
A mathematical technique based on the consideration of a nonlinear partial differential equation together with an additional condition in the form of an ordinary differential equation is employed to study a nonlinear reaction diffusion equation which describes a real process in physics and in chemistry. Several exact solutions for the equation are acquired under certain circumstances.
A Master Equation for Multi-Dimensional Non-Linear Field Theories
Park, Q H
1992-01-01
A master equation ( $n$ dimensional non--Abelian current conservation law with mutually commuting current components ) is introduced for multi-dimensional non-linear field theories. It is shown that the master equation provides a systematic way to understand 2-d integrable non-linear equations as well as 4-d self-dual equations and, more importantly, their generalizations to higher dimensions.
MHD three-dimensional flow of nanofluid with velocity slip and nonlinear thermal radiation
Energy Technology Data Exchange (ETDEWEB)
Hayat, Tasawar [Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000 (Pakistan); Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589 (Saudi Arabia); Imtiaz, Maria, E-mail: mi_qau@yahoo.com [Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000 (Pakistan); Alsaedi, Ahmed; Kutbi, Marwan A. [Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589 (Saudi Arabia)
2015-12-15
An analysis has been carried out for the three dimensional flow of viscous nanofluid in the presence of partial slip and thermal radiation effects. The flow is induced by a permeable stretching surface. Water is treated as a base fluid and alumina as a nanoparticle. Fluid is electrically conducting in the presence of applied magnetic field. Entire different concept of nonlinear thermal radiation is utilized in the heat transfer process. Different from the previous literature, the nonlinear system for temperature distribution is solved and analyzed. Appropriate transformations reduce the nonlinear partial differential system to ordinary differential system. Convergent series solutions are computed for the velocity and temperature. Effects of different parameters on the velocity, temperature, skin friction coefficient and Nusselt number are computed and examined. It is concluded that heat transfer rate increases when temperature and radiation parameters are increased. - Highlights: • Three-dimensional nanofluid flow with partial slip and nonlinear thermal radiation is studied. • Increasing values of velocity slip parameter decrease the velocity profiles. • The temperature increases via larger nanoparticle volume fraction. • Surface temperature gradient increases for higher temperature and radiation parameters.
Nonlinear evolution operators and semigroups applications to partial differential equations
Pavel, Nicolae H
1987-01-01
This research monograph deals with nonlinear evolution operators and semigroups generated by dissipative (accretive), possibly multivalued operators, as well as with the application of this theory to partial differential equations. It shows that a large class of PDE's can be studied via the semigroup approach. This theory is not available otherwise in the self-contained form provided by these Notes and moreover a considerable part of the results, proofs and methods are not to be found in other books. The exponential formula of Crandall and Liggett, some simple estimates due to Kobayashi and others, the characterization of compact semigroups due to Brézis, the proof of a fundamental property due to Ursescu and the author and some applications to PDE are of particular interest. Assuming only basic knowledge of functional analysis, the book will be of interest to researchers and graduate students in nonlinear analysis and PDE, and to mathematical physicists.
Caffarelli, Luis; Nirenberg, Louis
2011-01-01
The paper concerns singular solutions of nonlinear elliptic equations, which include removable singularities for viscosity solutions, a strengthening of the Hopf Lemma including parabolic equations, Strong maximum principle and Hopf Lemma for viscosity solutions including also parabolic equations.
Picone-type inequalities for nonlinear elliptic equations and their applications
Directory of Open Access Journals (Sweden)
Takaŝi Kusano
2001-01-01
Full Text Available Picone-type inequalities are derived for nonlinear elliptic equations, and Sturmian comparison theorems are established as applications. Oscillation theorems for forced super-linear elliptic equations and superlinear-sublinear elliptic equations are also obtained.
Method of the Logistic Function for Finding Analytical Solutions of Nonlinear Differential Equations
Kudryashov, N. A.
2015-01-01
The method of the logistic function is presented for finding exact solutions of nonlinear differential equations. The application of the method is illustrated by using the nonlinear ordinary differential equation of the fourth order. Analytical solutions obtained by this method are presented. These solutions are expressed via exponential functions.logistic function, nonlinear wave, nonlinear ordinary differential equation, Painlev´e test, exact solution
Directory of Open Access Journals (Sweden)
Xia Liu
2017-02-01
Full Text Available The discrete nonlinear Schrodinger equation is a nonlinear lattice system that appears in many areas of physics such as nonlinear optics, biomolecular chains and Bose-Einstein condensates. In this article, we consider a class of discrete nonlinear Schrodinger equations with unbounded potentials. We obtain some new sufficient conditions on the multiplicity results of ground state solutions for the equations by using the symmetric mountain pass lemma. Recent results in the literature are greatly improved.
Modified extended tanh-function method for solving nonlinear partial differential equations
Energy Technology Data Exchange (ETDEWEB)
El-Wakil, S.A. [Department of Physics, Faculty of Science, Theoretical Research Group, Mansoura University, 35516 Mansoura (Egypt); Abdou, M.A. [Department of Physics, Faculty of Science, Theoretical Research Group, Mansoura University, 35516 Mansoura (Egypt)]. E-mail: m_abdou_eg@yahoo.com
2007-03-15
Based on computerized symbolic computation, modified extended tanh-method for constructing multiple travelling wave solutions of nonlinear evolution equations is presented and implemented in a computer algebraic system. Applying this method, with the aid of Maple, we consider some nonlinear evolution equations in mathematical physics such as the nonlinear partial differential equation, nonlinear Fisher-type equation, ZK-BBM equation, generalized Burgers-Fisher equation and Drinfeld-Sokolov system. As a result, we can successfully recover the previously known solitary wave solutions that had been found by the extended tanh-function method and other more sophisticated methods.
Superposition of elliptic functions as solutions for a large number of nonlinear equations
Energy Technology Data Exchange (ETDEWEB)
Khare, Avinash [Raja Ramanna Fellow, Indian Institute of Science Education and Research (IISER), Pune 411021 (India); Saxena, Avadh [Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 (United States)
2014-03-15
For a large number of nonlinear equations, both discrete and continuum, we demonstrate a kind of linear superposition. We show that whenever a nonlinear equation admits solutions in terms of both Jacobi elliptic functions cn(x, m) and dn(x, m) with modulus m, then it also admits solutions in terms of their sum as well as difference. We have checked this in the case of several nonlinear equations such as the nonlinear Schrödinger equation, MKdV, a mixed KdV-MKdV system, a mixed quadratic-cubic nonlinear Schrödinger equation, the Ablowitz-Ladik equation, the saturable nonlinear Schrödinger equation, λϕ{sup 4}, the discrete MKdV as well as for several coupled field equations. Further, for a large number of nonlinear equations, we show that whenever a nonlinear equation admits a periodic solution in terms of dn{sup 2}(x, m), it also admits solutions in terms of dn {sup 2}(x,m)±√(m) cn (x,m) dn (x,m), even though cn(x, m)dn(x, m) is not a solution of these nonlinear equations. Finally, we also obtain superposed solutions of various forms for several coupled nonlinear equations.
Nardon, E.; Fil, A.; Hoelzl, M.; Huijsmans, G.; contributors, JET
2017-01-01
3D non-linear MHD simulations of a D 2 massive gas injection (MGI) triggered disruption in JET with the JOREK code provide results which are qualitatively consistent with experimental observations and shed light on the physics at play. In particular, it is observed that the gas destabilizes a large m/n = 2/1 tearing mode, with the island O-point coinciding with the gas deposition region, by enhancing the plasma resistivity via cooling. When the 2/1 island gets so large that its inner side reaches the q = 3/2 surface, a 3/2 tearing mode grows. Simulations suggest that this is due to a steepening of the current profile right inside q = 3/2. Magnetic field stochastization over a large fraction of the minor radius as well as the growth of higher n modes ensue rapidly, leading to the thermal quench (TQ). The role of the 1/1 internal kink mode is discussed. An I p spike at the TQ is obtained in the simulations but with a smaller amplitude than in the experiment. Possible reasons are discussed.
Nonlinear Allometric Equation for Crop Response to Soil Salinity
Directory of Open Access Journals (Sweden)
E. Misle
2015-06-01
Full Text Available Crop response to soil salinity has been extensively studied, from empirical works to modelling approach, being described by different equations, first as a piecewise linear model. The equation employed can differ with actual response, causing miscalculation in practical situations, particularly at the higher extremes of the curve. The aim of this work is to propose a new equation, which allows determining the full response to salinity of plant species and to provide a verification using different experimental data sets. A new nonlinear equation is exposed supported by the allometric approach, in which the allometric exponent is salinity-dependent and decreases with the increase in relative salinity. A conversion procedure of parameters of the threshold-slope model is presented; also, a simple procedure for estimating the maximum salinity (zero-yield point when data sets are incomplete is exposed. The equation was tested in a wide range of experimental situations, using data sets from published works, as well as new measurements on seed germination. The statistical indicators of quality (R2, absolute sum of squares and standard deviation of residuals showed that the equation accurately fits the tested empirical results. The new equation for determining crop response to soil salinity is able to follow the response curve of any crop with remarkable accuracy and flexibility. Remarkable characteristics are: a maximum at minimum salinity, a maximum salinity point can be found (zero-yield depending on the data sets, and a meaningful inflection point, as well as the two points at which the slope of the curve equals unity, can be found.
Computational uncertainty principle in nonlinear ordinary differential equations
Institute of Scientific and Technical Information of China (English)
LI; Jianping
2001-01-01
［1］Li Jianping, Zeng Qingcun, Chou Jifan, Computational Uncertainty Principle in Nonlinear Ordinary Differential Equations I. Numerical Results, Science in China, Ser. E, 2000, 43(5): 449［2］Henrici, P., Discrete Variable Methods in Ordinary Differential Equations, New York: John Wiley, 1962, 1; 187.［3］Henrici, P., Error Propagation for Difference Methods, New York: John Whiley, 1963.［4］Gear, C. W., Numerical Initial Value Problems in Ordinary Differential Equations, Englewood Cliffs, NJ: Prentice-Hall, 1971, 1; 72.［5］Hairer, E., Nrsett, S. P., Wanner, G., Solving Ordinary Differential Equations I. Nonstiff Problems, 2nd ed., Berlin-Heidelberg-New York: Springer-Verlag, 1993, 130.［6］Stoer, J., Bulirsch, R., Introduction to Numerical Analysis, 2nd ed., Vol. 1, Berlin-Heidelberg-New York: Springer-Verlag (reprinted in China by Beijing Wold Publishing Corporation), 1998, 428.［7］Li Qingyang, Numerical Methods in Ordinary Differential Equations (Stiff Problems and Boundary Value Problems), in Chinese Beijing: Higher Education Press, 1991, 1.［8］Li Ronghua, Weng Guochen, Numerical Methods in Differential Equations (in Chinese), 3rd ed., Beijing: Higher Education Press, 1996, 1.［9］Dahlquist, G., Convergence and stability in the numerical integration of ordinary differential equations, Math. Scandinavica, 1956, 4: 33.［10］Dahlquist, G., 33 years of numerical instability, Part I, BIT, 1985, 25: 188.［11］Heisenberg, W., The Physical Principles of Quantum Theory, Chicago: University of Chicago Press, 1930.［12］McMurry, S. M., Quantum Mechanics, London: Addison-Wesley Longman Ltd (reprined in China by Beijing World Publishing Corporation), 1998.
Some existence results on nonlinear fractional differential equations.
Baleanu, Dumitru; Rezapour, Shahram; Mohammadi, Hakimeh
2013-05-13
In this paper, by using fixed-point methods, we study the existence and uniqueness of a solution for the nonlinear fractional differential equation boundary-value problem D(α)u(t)=f(t,u(t)) with a Riemann-Liouville fractional derivative via the different boundary-value problems u(0)=u(T), and the three-point boundary condition u(0)=β(1)u(η) and u(T)=β(2)u(η), where T>0, t∈I=[0,T], 0<α<1, 0<η
On a Nonlinear Partial Integro-Differential Equation
Abergel, Frederic
2009-01-01
Consistently fitting vanilla option surfaces is an important issue when it comes to modelling in finance. Local volatility models introduced by Dupire in 1994 are widely used to price and manage the risks of structured products. However, the inconsistencies observed between the dynamics of the smile in those models and in real markets motivate researches for stochastic volatility modelling. Combining both those ideas to form Local and Stochastic Volatility models is of interest for practitioners. In this paper, we study the calibration of the vanillas in those models. This problem can be written as a nonlinear and nonlocal partial differential equation, for which we prove short-time existence of solutions.
Fourth order wave equations with nonlinear strain and source terms
Liu, Yacheng; Xu, Runzhang
2007-07-01
In this paper we study the initial boundary value problem for fourth order wave equations with nonlinear strain and source terms. First we introduce a family of potential wells and prove the invariance of some sets and vacuum isolating of solutions. Then we obtain a threshold result of global existence and nonexistence. Finally we discuss the global existence of solutions for the problem with critical initial condition I(u0)[greater-or-equal, slanted]0, E(0)=d. So the Esquivel-Avila's results are generalized and improved.
TRAVELING WAVE SOLUTIONS FOR A CLASS OF NONLINEAR DISPERSIVE EQUATIONS
Institute of Scientific and Technical Information of China (English)
无
2002-01-01
The method of the phase plane is emploied to investigate the solitary and periodic traveling waves for a class of nonlinear dispersive partial differential equations.By using the bifurcation theory of dynamical systems to do qualitative analysis,all possible phase portraits in the parametric space for the traveling wave systems are obtained.It can be shown that the existence of a singular straight line in the traveling wave system is the reason why smooth solitary wave solutions converge to solitary cusp wave solution when parameters are varied.The different parameter conditions for the existence of solitary and periodic wave solutions of different kinds are rigorously determined.
LOCAL DISCONTINUOUS GALERKIN METHODS FOR THREE CLASSES OF NONLINEAR WAVE EQUATIONS
Institute of Scientific and Technical Information of China (English)
Yan Xu; Chi-wang Shu
2004-01-01
In this paper, we further develop the local discontinuous Galerkin method to solve three classes of nonlinear wave equations formulated by the general KdV-Burgers type equations, the general fifth-order KdV type equations and the fully nonlinear K(n, n, n)equations, and prove their stability for these general classes of nonlinear equations. The schemes we present extend the previous work of Yan and Shu [30, 31] and of Levy, Shu and Yan [24] on local discontinuous Galerkin method solving partial differential equations with higher spatial derivatives. Numerical examples for nonlinear problems are shown to illustrate the accuracy and capability of the methods. The numerical experiments include stationary solitons, soliton interactions and oscillatory solitary wave solutions.The numerical experiments also include the compacton solutions of a generalized fifthorder KdV equation in which the highest order derivative term is nonlinear and the fully nonlinear K(n, n, n) equations.
Multiple scales analysis and travelling wave solutions for KdV type nonlinear evolution equations
Ayhan, Burcu; Ozer, M. Naci; Bekir, Ahmet
2017-01-01
Nonlinear evolution equations are the mathematical models of problems that arise in many field of science. These equations has become an important field of study in applied mathematics in recent years. We apply exact solution methods and multiple scale method which is known as a perturbation method to nonlinear evolution equations. Using exact solution methods we get travelling wave solutions expressed by hyperbolic functions, trigonometric functions and rational functions. Also we derive Nonlinear Schrödinger (NLS) type equations from Korteweg-de Vries (KdV) type nonlinear evolution equations and we get approximate solutions for KdV type equations using multiple scale method. The proposed methods are direct and effective and can be used for many nonlinear evolution equations. It is shown that these methods provide a powerful mathematical tool to solve nonlinear evolution equations in mathematical physics.
A Class of Traveling Wave Solutions to Some Nonlinear Partial Differential Equations
Institute of Scientific and Technical Information of China (English)
BAI Cheng-Lin
2003-01-01
For the Noyes-Fields equations, two-dimensional hyperbolic equations of conversation laws, and theBurgers-KdV equation, a class of traveling wave solutions has been obtained by constructing appropriate functiontransformations. The main idea of solving the equations is that nonlinear partial differential equations are changed intosolving algebraic equations. This method has a wide-rangingpracticability.
A Class of Traveling Wave Solutions to Some Nonlinear Partial Differential Equations
Institute of Scientific and Technical Information of China (English)
BAICheng-Lin
2003-01-01
For the Noyes-Fields equations, two-dimenslonal hyperbolic equations of conversation laww and the Burgers-KdV equation, a class of travellng wave solutions has been obtained by constructhag appropriate function transformations. The main idea of solving the equations is that nonlinear partial differential equations are changed into solving algebraic equations. This method has a wide-ranging practicability.
Energy Technology Data Exchange (ETDEWEB)
Zhang Huiqun [College of Mathematical Science, Qingdao University, Qingdao, Shandong 266071 (China)], E-mail: hellozhq@yahoo.com.cn
2009-02-15
By using some exact solutions of an auxiliary ordinary differential equation, a direct algebraic method is described to construct the exact complex solutions for nonlinear partial differential equations. The method is implemented for the NLS equation, a new Hamiltonian amplitude equation, the coupled Schrodinger-KdV equations and the Hirota-Maccari equations. New exact complex solutions are obtained.
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.
On invariant analysis of some time fractional nonlinear systems of partial differential equations. I
Singla, Komal; Gupta, R. K.
2016-10-01
An investigation of Lie point symmetries for systems of time fractional partial differential equations including Ito system, coupled Burgers equations, coupled Korteweg de Vries equations, Hirota-Satsuma coupled KdV equations, and coupled nonlinear Hirota equations has been done. Using the obtained symmetries, each one of the systems is reduced to the nonlinear system of fractional ordinary differential equations involving Erdélyi-Kober fractional differential operator depending on a parameter α.
Institute of Scientific and Technical Information of China (English)
HUANGYi; ZHANGYin-ke
2003-01-01
The nonlinear constitutive equations and field equations of unsaturated soils were constructed on the basis of mixture theory.The soils were treated as the mixture composed of three constituents.First ,from the researches of soil mechanics,some basic assumptions about the unsaturated soil mixture were mode,and the entropy inequality unsaturated soil mixture was derived.Then,with the common method usually used to deal with the constitutive problems in mixture theory,the nonlinear constitutive equations were obtained.Finally,putting the constiutive equtions of constituents into the balance equations of momentum,the nonlinear field equations of constitutents into the balance equations of momentum,the nonliear field equations of constitutents were set up.The balance equation of energy of unsaturated soil was also given,and thus the complete equations for solving the thermodynamic process of unsaturated soil was formed.
Generalized multiscale finite element methods. nonlinear elliptic equations
Efendiev, Yalchin R.
2013-01-01
In this paper we use the Generalized Multiscale Finite Element Method (GMsFEM) framework, introduced in [26], in order to solve nonlinear elliptic equations with high-contrast coefficients. The proposed solution method involves linearizing the equation so that coarse-grid quantities of previous solution iterates can be regarded as auxiliary parameters within the problem formulation. With this convention, we systematically construct respective coarse solution spaces that lend themselves to either continuous Galerkin (CG) or discontinuous Galerkin (DG) global formulations. Here, we use Symmetric Interior Penalty Discontinuous Galerkin approach. Both methods yield a predictable error decline that depends on the respective coarse space dimension, and we illustrate the effectiveness of the CG and DG formulations by offering a variety of numerical examples. © 2014 Global-Science Press.
New variable separation solutions for the generalized nonlinear diffusion equations
Fei-Yu, Ji; Shun-Li, Zhang
2016-03-01
The functionally generalized variable separation of the generalized nonlinear diffusion equations ut = A(u,ux)uxx + B(u,ux) is studied by using the conditional Lie-Bäcklund symmetry method. The variant forms of the considered equations, which admit the corresponding conditional Lie-Bäcklund symmetries, are characterized. To construct functionally generalized separable solutions, several concrete examples defined on the exponential and trigonometric invariant subspaces are provided. Project supported by the National Natural Science Foundation of China (Grant Nos. 11371293, 11401458, and 11501438), the National Natural Science Foundation of China, Tian Yuan Special Foundation (Grant No. 11426169), and the Natural Science Basic Research Plan in Shaanxi Province of China (Grant No. 2015JQ1014).
Stochastic Computational Approach for Complex Nonlinear Ordinary Differential Equations
Institute of Scientific and Technical Information of China (English)
Junaid Ali Khan; Muhammad Asif Zahoor Raja; Ijaz Mansoor Qureshi
2011-01-01
@@ We present an evolutionary computational approach for the solution of nonlinear ordinary differential equations (NLODEs).The mathematical modeling is performed by a feed-forward artificial neural network that defines an unsupervised error.The training of these networks is achieved by a hybrid intelligent algorithm, a combination of global search with genetic algorithm and local search by pattern search technique.The applicability of this approach ranges from single order NLODEs, to systems of coupled differential equations.We illustrate the method by solving a variety of model problems and present comparisons with solutions obtained by exact methods and classical numerical methods.The solution is provided on a continuous finite time interval unlike the other numerical techniques with comparable accuracy.With the advent of neuroprocessors and digital signal processors the method becomes particularly interesting due to the expected essential gains in the execution speed.%We present an evolutionary computational approach for the solution of nonlinear ordinary differential equations (NLODEs). The mathematical modeling is performed by a feed-forward artificial neural network that defines an unsupervised error. The training of these networks is achieved by a hybrid intelligent algorithm, a combination of global search with genetic algorithm and local search by pattern search technique. The applicability of this approach ranges from single order NLODEs, to systems of coupled differential equations. We illustrate the method by solving a variety of model problems and present comparisons with solutions obtained by exact methods and classical numerical methods. The solution is provided on a continuous finite time interval unlike the other numerical techniques with comparable accuracy. With the advent of neuroprocessors and digital signal processors the method becomes particularly interesting due to the expected essential gains in the execution speed.
DEFF Research Database (Denmark)
Miansari, Mo; Miansari, Me; Barari, Amin
2009-01-01
In this article, He’s variational iteration method (VIM), is implemented to solve the linear Helmholtz partial differential equation and some nonlinear fifth-order Korteweg-de Vries (FKdV) partial differential equations with specified initial conditions. The initial approximations can be freely...... and nonlinear problems. It is predicted that VIM can be widely applied in engineering....
Sahadevan, R.; Prakash, P.
2017-01-01
We show how invariant subspace method can be extended to time fractional coupled nonlinear partial differential equations and construct their exact solutions. Effectiveness of the method has been illustrated through time fractional Hunter-Saxton equation, time fractional coupled nonlinear diffusion system, time fractional coupled Boussinesq equation and time fractional Whitman-Broer-Kaup system. Also we explain how maximal dimension of the time fractional coupled nonlinear partial differential equations can be estimated.
Institute of Scientific and Technical Information of China (English)
Chang Jing; Gao Yi-xian; Cai Hua
2014-01-01
In this paper, the generalized extended tanh-function method is used for constructing the traveling wave solutions of nonlinear evolution equations. We choose Fisher’s equation, the nonlinear schr¨odinger equation to illustrate the validity and ad-vantages of the method. Many new and more general traveling wave solutions are obtained. Furthermore, this method can also be applied to other nonlinear equations in physics.
Indian Academy of Sciences (India)
Edmund Chadwick; Ali Hatam; Saeed Kazem
2016-02-01
A new approach, named the exponential function method (EFM) is used to obtain solutions to nonlinear ordinary differential equations with constant coefficients in a semi-infinite domain. The form of the solutions of these problems is considered to be an expansion of exponential functions with unknown coefficients. The derivative and product operational matrices arising from substituting in the proposed functions convert the solutions of these problems into an iterative method for finding the unknown coefficients. The method is applied to two problems: viscous flow due to a stretching sheet with surface slip and suction; and mageto hydrodynamic (MHD) flow of an incompressible viscous fluid over a stretching sheet. The two resulting solutions are compared against some standard methods which demonstrates the validity and applicability of the new approach.
Non-linear equation: energy conservation and impact parameter dependence
Kormilitzin, Andrey
2010-01-01
In this paper we address two questions: how energy conservation affects the solution to the non-linear equation, and how impact parameter dependence influences the inclusive production. Answering the first question we solve the modified BK equation which takes into account energy conservation. In spite of the fact that we used the simplified kernel, we believe that the main result of the paper: the small ($\\leq 40%$) suppression of the inclusive productiondue to energy conservation, reflects a general feature. This result leads us to believe that the small value of the nuclear modification factor is of a non-perturbative nature. In the solution a new scale appears $Q_{fr} = Q_s \\exp(-1/(2 \\bas))$ and the production of dipoles with the size larger than $2/Q_{fr}$ is suppressed. Therefore, we can expect that the typical temperature for hadron production is about $Q_{fr}$ ($ T \\approx Q_{fr}$). The simplified equation allows us to obtain a solution to Balitsky-Kovchegov equation taking into account the impact pa...
Energy Technology Data Exchange (ETDEWEB)
Huang Dingjiang [Department of Applied Mathematics, Dalian University of Technology, Dalian 116024 (China)]. E-mail: hdj8116@163.com; Zhang Hongqing [Department of Applied Mathematics, Dalian University of Technology, Dalian 116024 (China)
2006-08-15
Many travelling wave solutions of nonlinear evolution equations can be written as a polynomial in several elementary or special functions which satisfy a first order nonlinear ordinary differential equation with a sixth-degree nonlinear term. From that property, we deduce an algebraic method for constructing those solutions by determining only a finite number of coefficients. Being concise and straightforward, the method is applied to three nonlinear evolution equations. As a result, many exact travelling wave solutions are obtained which include new bell and kink profile solitary wave solutions, triangular periodic wave solutions and singular solutions.
Energy Technology Data Exchange (ETDEWEB)
Alvarez-Estrada, R.F.
1979-08-01
A comprehensive review of the inverse scattering solution of certain non-linear evolution equations of physical interest in one space dimension is presented. We explain in some detail the interrelated techniques which allow to linearize exactly the following equations: (1) the Korteweg and de Vries equation; (2) the non-linear Schrodinger equation; (3) the modified Korteweg and de Vries equation; (4) the Sine-Gordon equation. We concentrate in discussing the pairs of linear operators which accomplish such an exact linearization and the solution of the associated initial value problem. The application of the method to other non-linear evolution equations is reviewed very briefly.
Asymptotic behavior for a quadratic nonlinear Schrodinger equation
Directory of Open Access Journals (Sweden)
Pavel I. Naumkin
2008-02-01
Full Text Available We study the initial-value problem for the quadratic nonlinear Schrodinger equation $$displaylines{ iu_{t}+frac{1}{2}u_{xx}=partial _{x}overline{u}^{2},quad xin mathbb{R},; t>1, cr u(1,x=u_{1}(x,quad xin mathbb{R}. }$$ For small initial data $u_{1}in mathbf{H}^{2,2}$ we prove that there exists a unique global solution $uin mathbf{C}([1,infty ;mathbf{H}^{2,2}$ of this Cauchy problem. Moreover we show that the large time asymptotic behavior of the solution is defined in the region $|x|leq Csqrt{t}$ by the self-similar solution $frac{1}{sqrt{t}}MS(frac{x}{sqrt{t}}$ such that the total mass $$ frac{1}{sqrt{t}}int_{mathbb{R}}MS(frac{x}{sqrt{t}} dx=int_{mathbb{R}}u_{1}(xdx, $$ and in the far region $|x|>sqrt{t}$ the asymptotic behavior of solutions has rapidly oscillating structure similar to that of the cubic nonlinear Schrodinger equations.
Nonlinear subelliptic Schrodinger equations with external magnetic field
Directory of Open Access Journals (Sweden)
Kyril Tintarev
2004-10-01
Full Text Available To account for an external magnetic field in a Hamiltonian of a quantum system on a manifold (modelled here by a subelliptic Dirichlet form, one replaces the the momentum operator $frac 1i d$ in the subelliptic symbol by $frac 1i d-alpha$, where $alphain TM^*$ is called a magnetic potential for the magnetic field $eta=dalpha$. We prove existence of ground state solutions (Sobolev minimizers for nonlinear Schrodinger equation associated with such Hamiltonian on a generally, non-compact Riemannian manifold, generalizing the existence result of Esteban-Lions [5] for the nonlinear Schrödinger equation with a constant magnetic field on $mathbb{R}^N$ and the existence result of [6] for a similar problem on manifolds without a magnetic field. The counterpart of a constant magnetic field is the magnetic field, invariant with respect to a subgroup of isometries. As an example to the general statement we calculate the invariant magnetic fields in the Hamiltonians associated with the Kohn Laplacian and for the Laplace-Beltrami operator on the Heisenberg group.
Local conservative regularizations of compressible MHD and neutral flows
Krishnaswami, Govind S; Thyagaraja, Anantanarayanan
2016-01-01
Ideal systems like MHD and Euler flow may develop singularities in vorticity (w = curl v). Viscosity and resistivity provide dissipative regularizations of the singularities. In this paper we propose a minimal, local, conservative, nonlinear, dispersive regularization of compressible flow and ideal MHD, in analogy with the KdV regularization of the 1D kinematic wave equation. This work extends and significantly generalizes earlier work on incompressible Euler and ideal MHD. It involves a micro-scale cutoff length lambda which is a function of density, unlike in the incompressible case. In MHD, it can be taken to be of order the electron collisionless skin depth c/omega_pe. Our regularization preserves the symmetries of the original systems, and with appropriate boundary conditions, leads to associated conservation laws. Energy and enstrophy are subject to a priori bounds determined by initial data in contrast to the unregularized systems. A Hamiltonian and Poisson bracket formulation is developed and applied ...
Nonlinear differential equations with exact solutions expressed via the Weierstrass function
Kudryashov, NA
2004-01-01
A new problem is studied, that is to find nonlinear differential equations with special solutions expressed via the Weierstrass function. A method is discussed to construct nonlinear ordinary differential equations with exact solutions. The main step of our method is the assumption that nonlinear di
Institute of Scientific and Technical Information of China (English)
黄义; 张引科
2003-01-01
The nonlinear constitutive equations and field equations of unsaturated soils were cons tructed on the basis of mixture theory. The soils were treated as the mixture composed of three constituents. First, from the researches of soil mechanics, some basic assumptions about the unsaturated soil mixture were made, and the entropy inequality of unsaturated soil mixture was derived. Then, with the common method usually used to deal with the constitutive problems in mixture theory, the nonlinear constitutive equations were obtained. Finally, putting the constitutive equations of constituents into the balance equations of momentum, the nonlinear field equations of constituents were set up. The balance equation of energy of unsaturated soil was also given, and thus the complete equations for solving the thermodynamic process of unsaturated soil was formed.
Numerical solution of the nonlinear Schrödinger equation with wave operator on unbounded domains.
Li, Hongwei; Wu, Xiaonan; Zhang, Jiwei
2014-09-01
In this paper, we generalize the unified approach proposed in Zhang et al. [J. Zhang, Z. Xu, and X. Wu, Phys. Rev. E 78, 026709 (2008)] to design the nonlinear local absorbing boundary conditions (LABCs) for the nonlinear Schrödinger equation with wave operator on unbounded domains. In fact, based on the methodology underlying the unified approach, we first split the original equation into two parts-the linear equation and the nonlinear equation-then achieve a one-way operator to approximate the linear equation to make the wave outgoing, and finally combine the one-way operator with the nonlinear equation to achieve the nonlinear LABCs. The stability of the equation with the nonlinear LABCs is also analyzed by introducing some auxiliary variables, and some numerical examples are presented to verify the accuracy and effectiveness of our proposed method.
An effective analytic approach for solving nonlinear fractional partial differential equations
Ma, Junchi; Zhang, Xiaolong; Liang, Songxin
2016-08-01
Nonlinear fractional differential equations are widely used for modelling problems in applied mathematics. A new analytic approach with two parameters c1 and c2 is first proposed for solving nonlinear fractional partial differential equations. These parameters are used to improve the accuracy of the resulting series approximations. It turns out that much more accurate series approximations are obtained by choosing proper values of c1 and c2. To demonstrate the applicability and effectiveness of the new method, two typical fractional partial differential equations, the nonlinear gas dynamics equation and the nonlinear KdV-Burgers equation, are solved.
Nonpoint Symmetry and Reduction of Nonlinear Evolution and Wave Type Equations
Directory of Open Access Journals (Sweden)
Ivan Tsyfra
2015-01-01
Full Text Available We study the symmetry reduction of nonlinear partial differential equations with two independent variables. We propose new ansätze reducing nonlinear evolution equations to system of ordinary differential equations. The ansätze are constructed by using operators of nonpoint classical and conditional symmetry. Then we find solution to nonlinear heat equation which cannot be obtained in the framework of the classical Lie approach. By using operators of Lie-Bäcklund symmetries we construct the solutions of nonlinear hyperbolic equations depending on arbitrary smooth function of one variable too.
Energy Technology Data Exchange (ETDEWEB)
Belmonte-Beitia, J [Departamento de Matematicas, E T S de Ingenieros Industriales and Instituto de Matematica Aplicada a la Ciencia y la IngenierIa (IMACI), Avda Camilo Jose Cela, 3 Universidad de Castilla-La Mancha 13071 Ciudad Real (Spain); Cuevas, J [Grupo de Fisica No Lineal, Departamento de Fisica Aplicada I, Escuela Universitaria Politecnica, C/Virgen de Africa, 7, 41011 Sevilla (Spain)], E-mail: juan.belmonte@uclm.es, E-mail: jcuevas@us.es
2009-04-24
In this paper, we construct, by means of similarity transformations, explicit solutions to the cubic-quintic nonlinear Schroedinger equation with potentials and nonlinearities depending on both time and spatial coordinates. We present the general approach and use it to calculate bright and dark soliton solutions for nonlinearities and potentials of physical interest in applications to Bose-Einstein condensates and nonlinear optics.
Chaos in the fractional order nonlinear Bloch equation with delay
Baleanu, Dumitru; Magin, Richard L.; Bhalekar, Sachin; Daftardar-Gejji, Varsha
2015-08-01
The Bloch equation describes the dynamics of nuclear magnetization in the presence of static and time-varying magnetic fields. In this paper we extend a nonlinear model of the Bloch equation to include both fractional derivatives and time delays. The Caputo fractional time derivative (α) in the range from 0.85 to 1.00 is introduced on the left side of the Bloch equation in a commensurate manner in increments of 0.01 to provide an adjustable degree of system memory. Time delays for the z component of magnetization are inserted on the right side of the Bloch equation with values of 0, 10 and 100 ms to balance the fractional derivative with delay terms that also express the history of an earlier state. In the absence of delay, τ = 0 , we obtained results consistent with the previously published bifurcation diagram, with two cycles appearing at α = 0.8548 with subsequent period doubling that leads to chaos at α = 0.9436 . A periodic window is observed for the range 0.962 chaos arising again as α nears 1.00. The bifurcation diagram for the case with a 10 ms delay is similar: two cycles appear at the value α = 0.8532 , and the transition from two to four cycles at α = 0.9259 . With further increases in the fractional order, period doubling continues until at α = 0.9449 chaos ensues. In the case of a 100 millisecond delay the transitions from one cycle to two cycles and two cycles to four cycles are observed at α = 0.8441 , and α = 0.8635 , respectively. However, the system exhibits chaos at much lower values of α (α = 0.8635). A periodic window is observed in the interval 0.897 chaos again appearing for larger values of α . In general, as the value of α decreased the system showed transitions from chaos to transient chaos, and then to stability. Delays naturally appear in many NMR systems, and pulse programming allows the user control over the process. By including both the fractional derivative and time delays in the Bloch equation, we have developed a
OSCILLATION FOR NONLINEAR SECOND-ORDER DYNAMIC EQUATIONS ON TIME SCALES
Institute of Scientific and Technical Information of China (English)
无
2008-01-01
Through the use of generalized Riccati transformation techniques, we establish some oscillation criteria for one type of nonlinear dynamic equation on time scales. Several examples, including a semilinear dynamic equation and a nonlinear Emden-Fowler dynamic equation, are also given to illustrate these criteria and to improve the results obtained in some references.
Energy Technology Data Exchange (ETDEWEB)
Zhao Xiqiang [Department of Mathematics, Ocean University of China, Qingdao Shandong 266071 (China)] e-mail: zhaodss@yahoo.com.cn; Wang Limin [Shandong University of Technology, Zibo Shandong 255049 (China); Sun Weijun [Shandong University of Technology, Zibo Shandong 255049 (China)
2006-04-01
In this letter, a new method, called the repeated homogeneous balance method, is proposed for seeking the traveling wave solutions of nonlinear partial differential equations. The Burgers-KdV equation is chosen to illustrate our method. It has been confirmed that more traveling wave solutions of nonlinear partial differential equations can be effectively obtained by using the repeated homogeneous balance method.
Institute of Scientific and Technical Information of China (English)
Wan-sheng WANG; Shou-fu LI; Run-sheng YANG
2012-01-01
A series of contractivity and exponential stability results for the solutions to nonlinear neutral functional differential equations (NFDEs) in Banach spaces are obtained,which provide unified theoretical foundation for the contractivity analysis of solutions to nonlinear problems in functional differential equations (FDEs),neutral delay differential equations (NDDEs) and NFDEs of other types which appear in practice.
Institute of Scientific and Technical Information of China (English)
Chuan Qiang CHEN; Bo Wen HU
2013-01-01
We study microscopic spacetime convexity properties of fully nonlinear parabolic partial differential equations.Under certain general structure condition,we establish a constant rank theorem for the spacetime convex solutions of fully nonlinear parabolic equations.At last,we consider the parabolic convexity of solutions to parabolic equations and the convexity of the spacetime second fundamental form of geometric flows.
Deterministic and stochastic evolution equations for fully dispersive and weakly nonlinear waves
DEFF Research Database (Denmark)
Eldeberky, Y.; Madsen, Per A.
1999-01-01
This paper presents a new and more accurate set of deterministic evolution equations for the propagation of fully dispersive, weakly nonlinear, irregular, multidirectional waves. The equations are derived directly from the Laplace equation with leading order nonlinearity in the surface boundary c...
Kengne, Emmanuel; Saydé, Michel; Ben Hamouda, Fathi; Lakhssassi, Ahmed
2013-11-01
Analytical entire traveling wave solutions to the 1+1 density-dependent nonlinear reaction-diffusion equation via the extended generalized Riccati equation mapping method are presented in this paper. This equation can be regarded as an extension case of the Fisher-Kolmogoroff equation, which is used for studying insect and animal dispersal with growth dynamics. The analytical solutions are then used to investigate the effect of equation parameters on the population distribution.
Nonlinear unified equations for water waves propagating over uneven bottoms in the nearshore region
Institute of Scientific and Technical Information of China (English)
无
2001-01-01
Considering the continuous characteristics for water waves propagating over complex topography in the nearshore region, the unified nonlinear equations, based on the hypothesis for a typical uneven bottom, are presented by employing the Hamiltonian variational principle for water waves. It is verified that the equations include the following special cases: the extension of Airy's nonlinear shallow-water equations, the generalized mild-slope equation, the dispersion relation for the second-order Stokes waves and the higher order Boussinesq-type equations.
An almost symmetric Strang splitting scheme for nonlinear evolution equations.
Einkemmer, Lukas; Ostermann, Alexander
2014-07-01
In this paper we consider splitting methods for the time integration of parabolic and certain classes of hyperbolic partial differential equations, where one partial flow cannot be computed exactly. Instead, we use a numerical approximation based on the linearization of the vector field. This is of interest in applications as it allows us to apply splitting methods to a wider class of problems from the sciences. However, in the situation described, the classic Strang splitting scheme, while still being a method of second order, is not longer symmetric. This, in turn, implies that the construction of higher order methods by composition is limited to order three only. To remedy this situation, based on previous work in the context of ordinary differential equations, we construct a class of Strang splitting schemes that are symmetric up to a desired order. We show rigorously that, under suitable assumptions on the nonlinearity, these methods are of second order and can then be used to construct higher order methods by composition. In addition, we illustrate the theoretical results by conducting numerical experiments for the Brusselator system and the KdV equation.
Exponential Polynomials as Solutions of Certain Nonlinear Difference Equations
Institute of Scientific and Technical Information of China (English)
Zhi Tao WEN; Janne HEITTOKANGAS; Ilpo LAINE
2012-01-01
Recently,C.-C.Yang and I.Laine have investigated finite order entire solutions f of nonlinear differential-difference equations of the form fn+L(z,f)=h,where n ≥ 2 is an integer.In particular,it is known that the equation f(z)2+q(z)f(z+1) =p(z),where p(z),q(z) are polynomials,has no transcendental entire solutions of finite order.Assuming that Q(z) is also a polynomial and c ∈ C,equations of the form f(z)n + q(z)eQ(z)f(z + c) =p(z) do posses finite order entire solutions.A classification of these solutions in terms of growth and zero distribution will be given.In particular,it is shown that any exponential polynomial solution must reduce to a rather specific form.This reasoning relies on an earlier paper due to N.Steinmetz.
A new method for parameter estimation in nonlinear dynamical equations
Wang, Liu; He, Wen-Ping; Liao, Le-Jian; Wan, Shi-Quan; He, Tao
2015-01-01
Parameter estimation is an important scientific problem in various fields such as chaos control, chaos synchronization and other mathematical models. In this paper, a new method for parameter estimation in nonlinear dynamical equations is proposed based on evolutionary modelling (EM). This will be achieved by utilizing the following characteristics of EM which includes self-organizing, adaptive and self-learning features which are inspired by biological natural selection, and mutation and genetic inheritance. The performance of the new method is demonstrated by using various numerical tests on the classic chaos model—Lorenz equation (Lorenz 1963). The results indicate that the new method can be used for fast and effective parameter estimation irrespective of whether partial parameters or all parameters are unknown in the Lorenz equation. Moreover, the new method has a good convergence rate. Noises are inevitable in observational data. The influence of observational noises on the performance of the presented method has been investigated. The results indicate that the strong noises, such as signal noise ratio (SNR) of 10 dB, have a larger influence on parameter estimation than the relatively weak noises. However, it is found that the precision of the parameter estimation remains acceptable for the relatively weak noises, e.g. SNR is 20 or 30 dB. It indicates that the presented method also has some anti-noise performance.
Toward 3D MHD modeling of neoclassical tearing mode suppression by ECCD
Directory of Open Access Journals (Sweden)
Westerhof E.
2012-09-01
Full Text Available We propose a framework to extend the magnetohydrodynamic (MHD equations to include electron cyclotron current drive (ECCD and discuss previous models proposed by Giruzzi et al. [2] and by Hegna and Callen [3]. To model neoclassical tearing mode (NTM instabilities and study the growth of magnetic islands as NTMs evolve, we employ the nonlinear reduced-MHD simulation JOREK. We present tearing-mode growth-rate calculations from JOREK simulations.
On global attraction to stationary states for wave equations with concentrated nonlinearities
Kopylova, E.
2016-01-01
The global attraction to stationary states is established for solutions to 3D wave equations with concentrated nonlinearities: each finite energy solution converges as $t\\to\\pm\\infty$ to stationary states. The attraction is caused by nonlinear energy radiation.
Solitons and periodic solutions to a couple of fractional nonlinear evolution equations
Indian Academy of Sciences (India)
M Mirzazadeh; M Eslami; Anjan Biswas
2014-03-01
This paper studies a couple of fractional nonlinear evolution equations using first integral method. These evolution equations are foam drainage equation and Klein–Gordon equation (KGE), the latter of which is considered in (2 + 1) dimensions. For the fractional evolution, the Jumarie’s modified Riemann–Liouville derivative is considered. Exact solutions to these equations are obtained.
Note on Nonlinear Schr\\"odinger Equation, KdV Equation and 2D Topological Yang-Mills-Higgs Theory
Nian, Jun
2016-01-01
In this paper we discuss the relation between the (1+1)D nonlinear Schr\\"odinger equation and the KdV equation. By applying the boson/vortex duality, we can map the classical nonlinear Schr\\"odinger equation into the classical KdV equation in the small coupling limit, which corresponds to the UV regime of the theory. At quantum level, the two theories satisfy the Bethe Ansatz equations of the spin-$\\frac{1}{2}$ XXX chain and the XXZ chain in the continuous limit respectively. Combining these relations with the dualities discussed previously in the literature, we propose a duality web in the UV regime among the nonlinear Schr\\"odinger equation, the KdV equation and the 2D $\\mathcal{N}=(2,2)^*$ topological Yang-Mills-Higgs theory.
Energy Technology Data Exchange (ETDEWEB)
Lu, Bin, E-mail: lubinhb@163.com [School of Mathematical Sciences, Anhui University, Hefei 230601 (China)
2012-06-04
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.
Islam, Md Shafiqul; Khan, Kamruzzaman; Akbar, M Ali; Mastroberardino, Antonio
2014-10-01
The purpose of this article is to present an analytical method, namely the improved F-expansion method combined with the Riccati equation, for finding exact solutions of nonlinear evolution equations. The present method is capable of calculating all branches of solutions simultaneously, even if multiple solutions are very close and thus difficult to distinguish with numerical techniques. To verify the computational efficiency, we consider the modified Benjamin-Bona-Mahony equation and the modified Korteweg-de Vries equation. Our results reveal that the method is a very effective and straightforward way of formulating the exact travelling wave solutions of nonlinear wave equations arising in mathematical physics and engineering.
Nonlinear Evolution of Alfvenic Wave Packets
Buti, B.; Jayanti, V.; Vinas, A. F.; Ghosh, S.; Goldstein, M. L.; Roberts, D. A.; Lakhina, G. S.; Tsurutani, B. T.
1998-01-01
Alfven waves are a ubiquitous feature of the solar wind. One approach to studying the evolution of such waves has been to study exact solutions to approximate evolution equations. Here we compare soliton solutions of the Derivative Nonlinear Schrodinger evolution equation (DNLS) to solutions of the compressible MHD equations.
Dynamic properties of the cubic nonlinear Schr(o)dinger equation by symplectic method
Institute of Scientific and Technical Information of China (English)
Liu Xue-Shen; Wei Jia-Yu; Ding Pei-Zhu
2005-01-01
The dynamic properties of a cubic nonlinear Schrodinger equation are investigated numerically by using the symplectic method with different space approximations. The behaviours of the cubic nonlinear Schrodinger equation are discussed with different cubic nonlinear parameters in the harmonically modulated initial condition. We show that the conserved quantities will be preserved for long-time computation but the system will exhibit different dynamic behaviours in space difference approximation for the strong cubic nonlinearity.
Nonlinear equations on controlling interface patterns during solidification of a dilute binary alloy
Institute of Scientific and Technical Information of China (English)
王自东; 周永利; 常国威; 胡汉起
1999-01-01
In nonequilibrium nonlinear region, by assuming that there is local equilibrium at the solid/liquid interface, and considering that curvature, temperature and composition at the solid/liquid interface which are related to perturbation amplitude are nonlinear, nonlinear equations of the time dependence of the perturbation amplitude of the solid/liquid interface during solidification of a dilute binary alloy are established. Crystal growth from nonsteady state to steady state can be controlled by these nonlinear equations.
Soliton solution for nonlinear partial differential equations by cosine-function method
Energy Technology Data Exchange (ETDEWEB)
Ali, A.H.A. [Mathematics Department, Faculty of Science, Menoufia University, Shebein El-Koom (Egypt); Soliman, A.A. [Department of Mathematics, Faculty of Education (AL-Arish), Suez Canal University, AL-Arish 45111 (Egypt)], E-mail: asoliman_99@yahoo.com; Raslan, K.R. [Mathematics Department, Faculty of Science, Al-Azhar University, Nasr-City, Cairo (Egypt)
2007-08-20
In this Letter, we established a traveling wave solution by using Cosine-function algorithm for nonlinear partial differential equations. The method is used to obtain the exact solutions for five different types of nonlinear partial differential equations such as, general equal width wave equation (GEWE), general regularized long wave equation (GRLW), general Korteweg-de Vries equation (GKdV), general improved Korteweg-de Vries equation (GIKdV), and Coupled equal width wave equations (CEWE), which are the important soliton equations.
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.
Canonical equations of Hamilton for the nonlinear Schr\\"{o}dinger equation
Liang, Guo; Ren, Zhanmei
2013-01-01
We define two different systems of mathematical physics: the second-order differential system (SODS) and the first-order differential system (FODS). The Newton's second law of motion and the nonlinear Schr\\"{o}dinger equation (NLSE) are the exemplary SODS and FODS, respectively. We obtain a new kind of canonical equations of Hamilton (CEH), which are of some kind of symmetry in form and are formally different with the conventional CEH without symmetry [H. Goldstein, C. Poole, J. Safko, Classical Mechanics, third ed., Addison-Wesley, 2001]. We also prove that the number of the CEHs is equal to the number of the generalized coordinates for the FODS, but twice the number of the generalized coordinates for the SODS. We show that the FODS can only be expressed by the new CEH, but do not by the conventional CEH, while the SODS can be done by both the new and the conventional CEHs. As an example, we prove that the nonlinear Schr\\"{o}dinger equation can be expressed with the new CEH in a consistent way.
Canonical equations of Hamilton for the nonlinear Schrödinger equation
Liang, Guo; Guo, Qi; Ren, Zhanmei
2015-09-01
We define two different systems of mathematical physics: the second order differential system (SODS) and the first order differential system (FODS). The Newton's second law of motion and the nonlinear Schrödinger equation (NLSE) are the exemplary SODS and FODS, respectively. We obtain a new kind of canonical equations of Hamilton (CEH), which exhibit some kind of symmetry in form and are formally different from the conventional CEH without symmetry [H. Goldstein, C. Poole, J. Safko, Classical Mechanics, third ed., Addison- Wesley, 2001]. We also prove that the number of the CEHs is equal to the number of the generalized coordinates for the FODS, but twice the number of the generalized coordinates for the SODS. We show that the FODS can only be expressed by the new CEH, but not introduced by the conventional CEH, while the SODS can be done by both the new and the conventional CEHs. As an example, we prove that the nonlinear Schrödinger equation can be expressed with the new CEH in a consistent way.
On the so called rogue waves in nonlinear Schrodinger equations
Directory of Open Access Journals (Sweden)
Y. Charles Li
2016-04-01
Full Text Available The mechanism of a rogue water wave is still unknown. One popular conjecture is that the Peregrine wave solution of the nonlinear Schrodinger equation (NLS provides a mechanism. A Peregrine wave solution can be obtained by taking the infinite spatial period limit to the homoclinic solutions. In this article, from the perspective of the phase space structure of these homoclinic orbits in the infinite dimensional phase space where the NLS defines a dynamical system, we examine the observability of these homoclinic orbits (and their approximations. Our conclusion is that these approximate homoclinic orbits are the most observable solutions, and they should correspond to the most common deep ocean waves rather than the rare rogue waves. We also discuss other possibilities for the mechanism of a rogue wave: rough dependence on initial data or finite time blow up.
THREE POINT BOUNDARY VALUE PROBLEMS FOR NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS
Institute of Scientific and Technical Information of China (English)
Mujeeb ur Rehman; Rahmat Ali Khan; Naseer Ahmad Asif
2011-01-01
In this paper,we study existence and uniqueness of solutions to nonlinear three point boundary value problems for fractional differential equation of the type cDδ0+u(t) =f(t,u(t),cDσ0+u(t)),t ∈[0,T],u(0) =αu(η),u(T) =βu(η),where1 ＜δ＜2,0＜σ＜ 1,α,β∈R,η∈(0,T),αη(1-β)+(1-α)(T-βη) ≠0 and cDoδ+,cDσ0+ are the Caputo fractional derivatives.We use Schauder fixed point theorem and contraction mapping principle to obtain existence and uniqueness results.Examples are also included to show the applicability of our results.
A Unified and Explicit Construction of N-Soliton Solutions for the Nonlinear Schrfdinger Equation
Institute of Scientific and Technical Information of China (English)
FAN En-Gui
2001-01-01
An explicit N-fold Darboux transformation with multiparameters for nonlinear Schrodinger equation is constructed with the help of its Lax pairs and a reduction technique. According to this Darboux transformation, the solutions of the nonlinear Schrfdinger equation are reduced to solving a linear algebraic system, from which a unified and explicit formulation of N-soliton solutions with multiparameters for the nonlinear Schrfdinger equation is given.``
Approach in Theory of Nonlinear Evolution Equations: The Vakhnenko-Parkes Equation
Directory of Open Access Journals (Sweden)
V. O. Vakhnenko
2016-01-01
Full Text Available A variety of methods for examining the properties and solutions of nonlinear evolution equations are explored by using the Vakhnenko equation (VE as an example. The VE, which arises in modelling the propagation of high-frequency waves in a relaxing medium, has periodic and solitary traveling wave solutions some of which are loop-like in nature. The VE can be written in an alternative form, known as the Vakhnenko-Parkes equation (VPE, by a change of independent variables. The VPE has an N-soliton solution which is discussed in detail. Individual solitons are hump-like in nature whereas the corresponding solution to the VE comprises N-loop-like solitons. Aspects of the inverse scattering transform (IST method, as applied originally to the KdV equation, are used to find one- and two-soliton solutions to the VPE even though the VPE’s spectral equation is third-order and not second-order. A Bäcklund transformation for the VPE is used to construct conservation laws. The standard IST method for third-order spectral problems is used to investigate solutions corresponding to bound states of the spectrum and to a continuous spectrum. This leads to N-soliton solutions and M-mode periodic solutions, respectively. Interactions between these types of solutions are investigated.
DEFF Research Database (Denmark)
Christiansen, Peter Leth; Gaididei, Yuri Borisovich; Rasmussen, Kim
1996-01-01
We study the effect of adding noise and nonlinear damping in the two-dimensional nonlinear Schrodinger equation (NLS). Using a collective approach, we find that for initial conditions where total collapse occurs in the unperturbed NLS, the presence of the damping term will instead in an exponenti......We study the effect of adding noise and nonlinear damping in the two-dimensional nonlinear Schrodinger equation (NLS). Using a collective approach, we find that for initial conditions where total collapse occurs in the unperturbed NLS, the presence of the damping term will instead...
Energy Technology Data Exchange (ETDEWEB)
Aslan, İsmail, E-mail: ismailaslan@iyte.edu.tr [Department of Mathematics, Izmir Institute of Technology, Urla, İzmir 35430 (Turkey)
2011-11-14
We analyze the discrete nonlinear Schrödinger equation with a saturable nonlinearity through the (G{sup ′}/G)-expansion method to present some improved results. Three types of analytic solutions with arbitrary parameters are constructed; hyperbolic, trigonometric, and rational which have not been explicitly computed before. -- Highlights: ► Discrete nonlinear Schrödinger equation with a saturable nonlinearity. ► We confirm that the model supports three types of solutions with arbitrary parameters. ► A new application of the (G{sup ′}/G)-expansion method presented.
Directory of Open Access Journals (Sweden)
Juan Belmonte-Beitia
2008-01-01
Full Text Available We give a proof of the existence of stationary bright soliton solutions of the cubic-quintic nonlinear Schrödinger equation with inhomogeneous nonlinearity. By using bifurcation theory, we prove that the norm of the positive solution goes to zero as the parameter λ, called chemical potential in the Bose-Einstein condensates' literature, tends to zero. Moreover, we solve the time-dependent cubic-quintic nonlinear Schrödinger equation with inhomogeneous nonlinearities by using a numerical method.
Chaos control in the nonlinear Schrödinger equation with Kerr law nonlinearity
Yin, Jiu-Li; Zhao, Liu-Wei; Tian, Li-Xin
2014-02-01
The nonlinear Schrödinger equation with Kerr law nonlinearity in the two-frequency interference is studied by the numerical method. Chaos occurs easily due to the absence of damping. This phenomenon will cause the distortion in the process of information transmission. We find that fiber-optic transmit signals still present chaotic phenomena if the control intensity is smaller. With the increase of intensity, the fiber-optic signal can stay in a stable state in some regions. When the strength is suppressed to a certain value, an unstable phenomenon of the fiber-optic signal occurs. Moreover we discuss the sensitivities of the parameters to be controlled. The results show that the linear term coefficient and the environment of two quite different frequences have less effects on the fiber-optic transmission. Meanwhile the phenomena of vibration, attenuation and escape occur in some regions.
Directory of Open Access Journals (Sweden)
M. RAHIMI EOSBOEE,
2010-12-01
Full Text Available In this study magnetohydrodynamics (MHD boundary layer flow of an upper-convected Maxwell fluid has been investigated. Similarity transformation has been used to reduce the governing differential equations into an ordinary non-linear differential equation. homotopy perturbation Method (HPM has applied to solve this developed nonlinear equation. In this article firstly, the basic idea of the HPM for solving nonlinear differential equations is briefly ntroduced and then it is employed to derive solution of nonlinear governing equation of MHD boundary layer flow with highly nonlinear term. The obtained results from HPM have been compared with numerical Boundary Value problem Method (BVP to verify the accuracy of the proposed method. The effects of the Hartman number (M and Deborah number (β for various conditions have been shown through graphs.
MHD squeezing flow between two infinite plates
Directory of Open Access Journals (Sweden)
Umar Khan
2014-03-01
Full Text Available Magneto hydrodynamic (MHD squeezing flow of a viscous fluid has been discussed. Conservation laws combined with similarity transformations have been used to formulate the flow mathematically that leads to a highly nonlinear ordinary differential equation. Analytical solution to the resulting differential equation is determined by employing Variation of Parameters Method (VPM. Runge–Kutta order-4 method is also used to solve the same problem for the sake of comparison. It is found that solution using VPM reduces the computational work yet maintains a very high level of accuracy. The influence of different parameters is also discussed and demonstrated graphically.
Self-similar solutions for some nonlinear evolution equations: KdV, mKdV and Burgers equations
Directory of Open Access Journals (Sweden)
S.A. El-Wakil
2016-02-01
Full Text Available A method for solving three types of nonlinear evolution equations namely KdV, modified KdV and Burgers equations, with self-similar solutions is presented. The method employs ideas from symmetry reduction to space and time variables and similarity reductions for nonlinear evolution equations are performed. The obtained self-similar solutions of KdV and mKdV equations are related to Bessel and Airy functions whereas those of Burgers equation are related to the error and Hermite functions. These solutions appear as new types of solitary, shock and periodic waves. Also, the method can be applied to other nonlinear evolution equations in mathematical physics.
The nonlinear Schrödinger equation singular solutions and optical collapse
Fibich, Gadi
2015-01-01
This book is an interdisciplinary introduction to optical collapse of laser beams, which is modelled by singular (blow-up) solutions of the nonlinear Schrödinger equation. With great care and detail, it develops the subject including the mathematical and physical background and the history of the subject. It combines rigorous analysis, asymptotic analysis, informal arguments, numerical simulations, physical modelling, and physical experiments. It repeatedly emphasizes the relations between these approaches, and the intuition behind the results. The Nonlinear Schrödinger Equation will be useful to graduate students and researchers in applied mathematics who are interested in singular solutions of partial differential equations, nonlinear optics and nonlinear waves, and to graduate students and researchers in physics and engineering who are interested in nonlinear optics and Bose-Einstein condensates. It can be used for courses on partial differential equations, nonlinear waves, and nonlinear optics. Gadi Fib...
A New Expanded Method for Solving Nonlinear Differential-difference Equation
Institute of Scientific and Technical Information of China (English)
ZHANG Shan-qing
2008-01-01
A new expanded approach is presented to find exact solutions of nonlinear differential-difference equations. As its application, the soliton solutions and periodic solutions of a lattice equation are obtained.
Generalized Dromion Structures of New (2 + 1)-Dimensional Nonlinear EvolutionEquation
Institute of Scientific and Technical Information of China (English)
ZHANG Jie-Fang
2001-01-01
We derive the generalized dromions of the new (2 + 1)-dimensional nonlinear evolution equation by the arbitrary function presented in the bilinearized linear equations. The rich soliton and dromion structures for this system are released.
NONLINEAR GALERKIN METHODS FOR SOLVING TWO DIMENSIONAL NEWTON-BOUSSINESQ EQUATIONS
Institute of Scientific and Technical Information of China (English)
GUOBOLING
1995-01-01
The nonlinear Galerkin methods for solving two-dimensional Newton-Boussinesq equations are proposed. The existence and uniqueness of global generalized solution of these equations,and the convergence of approximate solutions are also obtained.
Single and multi-solitary wave solutions to a class of nonlinear evolution equations
Wang, Deng-Shan; Li, Hongbo
2008-07-01
In this paper, an effective discrimination algorithm is presented to deal with equations arising from physical problems. The aim of the algorithm is to discriminate and derive the single traveling wave solutions of a large class of nonlinear evolution equations. Many examples are given to illustrate the algorithm. At the same time, some factorization technique are presented to construct the traveling wave solutions of nonlinear evolution equations, such as Camassa-Holm equation, Kolmogorov-Petrovskii-Piskunov equation, and so on. Then a direct constructive method called multi-auxiliary equations expansion method is described to derive the multi-solitary wave solutions of nonlinear evolution equations. Finally, a class of novel multi-solitary wave solutions of the (2+1)-dimensional asymmetric version of the Nizhnik-Novikov-Veselov equation are given by three direct methods. The algorithm proposed in this paper can be steadily applied to some other nonlinear problems.
A Direct Algebraic Method in Finding Particular Solutions to Some Nonlinear Evolution Equations
Institute of Scientific and Technical Information of China (English)
LIUChun-Ping; CHENJian-Kang; CAIFan
2004-01-01
Firstly, a direct algebraic method and a routine way in finding traveling wave solutions to nonlinear evolution equations are explained. And then some new exact solutions for some evolution equations are obtained by using the method.
Directory of Open Access Journals (Sweden)
Shi Jing
2014-01-01
Full Text Available The solving processes of the homogeneous balance method, Jacobi elliptic function expansion method, fixed point method, and modified mapping method are introduced in this paper. By using four different methods, the exact solutions of nonlinear wave equation of a finite deformation elastic circular rod, Boussinesq equations and dispersive long wave equations are studied. In the discussion, the more physical specifications of these nonlinear equations, have been identified and the results indicated that these methods (especially the fixed point method can be used to solve other similar nonlinear wave equations.
ANTI-PERIODIC SOLUTIONS FOR FIRST AND SECOND ORDER NONLINEAR EVOLUTION EQUATIONS IN BANACH SPACES
Institute of Scientific and Technical Information of China (English)
WEI Wei; XIANG Xiaoling
2004-01-01
In this paper, a new existence theorem of anti-periodic solutions for a class ofstrongly nonlinear evolution equations in Banach spaces is presentedThe equations con-tain nonlinear monotone operators and a nonmonotone perturbationMoreover, throughan appropriate transformation, the existence of anti-periodic solutions for a class of second-order nonlinear evolution equations is verifiedOur abstract results are illustrated by anexample from quasi-linear partial differential equations with time anti-periodic conditionsand an example from quasi-linear anti-periodic hyperbolic differential equations.
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Shaheed N. Huseen
2013-01-01
Full Text Available A modified q-homotopy analysis method (mq-HAM was proposed for solving nth-order nonlinear differential equations. This method improves the convergence of the series solution in the nHAM which was proposed in (see Hassan and El-Tawil 2011, 2012. The proposed method provides an approximate solution by rewriting the nth-order nonlinear differential equation in the form of n first-order differential equations. The solution of these n differential equations is obtained as a power series solution. This scheme is tested on two nonlinear exactly solvable differential equations. The results demonstrate the reliability and efficiency of the algorithm developed.
Regarding on the exact solutions for the nonlinear fractional differential equations
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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.
Carasso, Alfred S
2013-01-01
Identifying sources of ground water pollution, and deblurring nanoscale imagery as well as astronomical galaxy images, are two important applications involving numerical computation of parabolic equations backward in time. Surprisingly, very little is known about backward continuation in nonlinear parabolic equations. In this paper, an iterative procedure originating in spectroscopy in the 1930's, is adapted into a useful tool for solving a wide class of 2D nonlinear backward parabolic equations. In addition, previously unsuspected difficulties are uncovered that may preclude useful backward continuation in parabolic equations deviating too strongly from the linear, autonomous, self adjoint, canonical model. This paper explores backward continuation in selected 2D nonlinear equations, by creating fictitious blurred images obtained by using several sharp images as initial data in these equations, and capturing the corresponding solutions at some positive time T. Successful backward continuation from t=T to t = 0, would recover the original sharp image. Visual recognition provides meaningful evaluation of the degree of success or failure in the reconstructed solutions. Instructive examples are developed, illustrating the unexpected influence of certain types of nonlinearities. Visually and statistically indistinguishable blurred images are presented, with vastly different deblurring results. These examples indicate that how an image is nonlinearly blurred is critical, in addition to the amount of blur. The equations studied represent nonlinear generalizations of Brownian motion, and the blurred images may be interpreted as visually expressing the results of novel stochastic processes.
Institute of Scientific and Technical Information of China (English)
LI Shoufu
2005-01-01
A series of stability, contractivity and asymptotic stability results of the solutions to nonlinear stiff Volterra functional differential equations (VFDEs) in Banach spaces is obtained, which provides the unified theoretical foundation for the stability analysis of solutions to nonlinear stiff problems in ordinary differential equations(ODEs), delay differential equations(DDEs), integro-differential equations(IDEs) and VFDEs of other type which appear in practice.
Generalized Jacobi Elliptic Function Solution to a Class of Nonlinear Schrödinger-Type Equations
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Zeid I. A. Al-Muhiameed
2011-01-01
Full Text Available With the help of the generalized Jacobi elliptic function, an improved Jacobi elliptic function method is used to construct exact traveling wave solutions of the nonlinear partial differential equations in a unified way. A class of nonlinear Schrödinger-type equations including the generalized Zakharov system, the Rangwala-Rao equation, and the Chen-Lee-Lin equation are investigated, and the exact solutions are derived with the aid of the homogenous balance principle.
Analytic study of solutions for the Born-Infeld equation in nonlinear electrodynamics
Gao, Hui; Xu, Tianzhou; Fan, Tianyou; Wang, Gangwei
2017-03-01
The Born-Infeld equation is an important nonlinear partial differential equation in theoretical and mathematical physics. The Lie group method is used for simplifying the nonlinear partial differential equation, which is partly solved, in which there are some difficulties; to overcome the difficulties, we develop a power series method, and find the solutions in analytic form. In the mean time, a wave propagation (traveling wave) method is developed for solving the equation, and analytic solutions are also constructed.
A new integrable discrete generalized nonlinear Schrodinger equation and its reductions
Li, Hongmin; Li, Yuqi; Chen, Yong
2013-01-01
A new integrable discrete system is constructed and studied, based on the algebraization of the difference operator. The model is named the discrete generalized nonlinear Schrodinger (GNLS) equation for which can be reduced to classical discrete nonlinear Schrodinger (NLS) equation. To show the complete integrability of the discrete GNLS equation, the recursion operator, symmetries and conservation quantities are obtained. Furthermore, all of reductions for the discrete GNLS equation are give...
Special Conditional Similarity Reduction Solutions for Two Nonlinear Partial Differential Equations
Institute of Scientific and Technical Information of China (English)
无
2007-01-01
We present a method of special conditional similarity reduction solutions for nonlinear partial differential equations. As concrete examples of its application, we apply this method to the (2+1)-dimensional modified Broer-Kaup equations and the variable coefficient KdV-mKdV equation, which have extensive physics backgrounds, and obtain abundant exact solutions derived from some reduction equations.
Energy Technology Data Exchange (ETDEWEB)
Zhou Yubin; Wang Mingliang; Miao Tiande
2004-03-15
The periodic wave solutions for a class of nonlinear partial differential equations, including the Davey-Stewartson equations and the generalized Zakharov equations, are obtained by using the F-expansion method, which can be regarded as an overall generalization of the Jacobi elliptic function expansion method recently proposed. In the limit cases the solitary wave solutions of the equations are also obtained.
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Yusuf Pandir
2012-01-01
Full Text Available We obtain the classification of exact solutions, including soliton, rational, and elliptic solutions, to the one-dimensional general improved Camassa Holm KP equation and KdV equation by the complete discrimination system for polynomial method. In discussion, we propose a more general trial equation method for nonlinear partial differential equations with generalized evolution.
Indian Academy of Sciences (India)
HONG-YU WU; LI-HONG JIANG
2017-09-01
From a generic transformation, a $(3+1)$-dimensional nonlinear Schrödinger equation with spatially modulated nonlinearity is studied and exact spatiotemporal soliton cluster solutions are derived. When the azimuthal parameter $m = 0$, Gaussian solitons are constructed. For the modulation depth $q = 1$ and the azimuthal parameter $m \
Wu, Hong-Yu; Jiang, Li-Hong
2017-09-01
From a generic transformation, a (3+1)-dimensional nonlinear Schrödinger equation with spatially modulated nonlinearity is studied and exact spatiotemporal soliton cluster solutions are derived. When the azimuthal parameter m = 0, Gaussian solitons are constructed. For the modulation depth q = 1 and the azimuthal parameter m\
Singular solitons and other solutions to a couple of nonlinear wave equations
Institute of Scientific and Technical Information of China (English)
Mustafa Inc; Esma Uluta(s); Anjan Biswas
2013-01-01
This paper addresses the extended (G′/G)-expansion method and applies it to a couple of nonlinear wave equations.These equations are modified the Benjamin-Bona-Mahoney equation and the Boussinesq equation.This extended method reveals several solutions to these equations.Additionally,the singular soliton solutions are revealed,for these two equations,with the aid of the ansatz method.
Exact solutions to a class of nonlinear Schrödinger-type equations
Indian Academy of Sciences (India)
Jin-Liang Zhang; Ming-Liang Wing
2006-12-01
A class of nonlinear Schrödinger-type equations, including the Rangwala–Rao equation, the Gerdjikov–Ivanov equation, the Chen–Lee–Lin equation and the Ablowitz–Ramani–Segur equation are investigated, and the exact solutions are derived with the aid of the homogeneous balance principle, and a set of subsidiary higher order ordinary differential equations (sub-ODEs for short).
Higher-order nonlinear Schrodinger equations for simulations of surface wavetrains
Slunyaev, Alexey
2016-04-01
Numerous recent results of numerical and laboratory simulations of waves on the water surface claim that solutions of the weakly nonlinear theory for weakly modulated waves in many cases allow a smooth generalization to the conditions of strong nonlinearity and dispersion, even when the 'envelope' is difficult to determine. The conditionally 'strongly nonlinear' high-order asymptotic equations still imply the smallness of the parameter employed in the asymptotic series. Thus at some (unknown a priori) level of nonlinearity and / or dispersion the asymptotic theory breaks down; then the higher-order corrections become useless and may even make the description worse. In this paper we use the higher-order nonlinear Schrodinger (NLS) equation, derived in [1] (the fifth-order NLS equation, or next-order beyond the classic Dysthe equation [2]), for simulations of modulated deep-water wave trains, which attain very large steepness (below or beyond the breaking limit) due to the Benjamin - Feir instability. The results are compared with fully nonlinear simulations of the potential Euler equations as well as with the weakly nonlinear theories represented by the nonlinear Schrodinger equation and the classic Dysthe equation with full linear dispersion [2]. We show that the next-order Dysthe equation can significantly improve the description of strongly nonlinear wave dynamics compared with the lower-order asymptotic models. [1] A.V. Slunyaev, A high-order nonlinear envelope equation for gravity waves in finite-depth water. JETP 101, 926-941 (2005). [2] K. Trulsen, K.B. Dysthe, A modified nonlinear Schrödinger equation for broader bandwidth gravity waves on deep water. Wave Motion 24, 281-289 (1996).
Directory of Open Access Journals (Sweden)
Md. Nur Alam
2016-06-01
Full Text Available In this article, we apply the exp(-Φ(ξ-expansion method to construct many families of exact solutions of nonlinear evolution equations (NLEEs via the nonlinear diffusive predator–prey system and the Bogoyavlenskii equations. These equations can be transformed to nonlinear ordinary differential equations. As a result, some new exact solutions are obtained through the hyperbolic function, the trigonometric function, the exponential functions and the rational forms. If the parameters take specific values, then the solitary waves are derived from the traveling waves. Also, we draw 2D and 3D graphics of exact solutions for the special diffusive predator–prey system and the Bogoyavlenskii equations by the help of programming language Maple.
Institute of Scientific and Technical Information of China (English)
Long Shuyao; Zhang Qin
2000-01-01
In this paper the dual reciprocity boundary element method is employed to solve nonlinear differential equation 2 u + u + εu3 = b. Results obtained in an example have a good agreement with those by FEM and show the applicability and simplicity of dual reciprocity method(DRM) in solving nonlinear dif ferential equations.
A new method to solve the damped nonlinear Klein-Gordon equation
Institute of Scientific and Technical Information of China (English)
2008-01-01
This paper discusses a damped nonlinear Klein-Gordon equation in the reproducing kernel space and provides a new method for solving the damped nonlinear Klein-Gordon equation based on the reproducing kernel space.Two numerical examples are given for illustrating the feasibility and accuracy of the method.
Initial-boundary value problems for a class of nonlinear thermoelastic plate equations
Institute of Scientific and Technical Information of China (English)
Zhang Jian-Wen; Rong Xiao-Liang; Wu Run-Heng
2009-01-01
This paper studies initial-boundary value problems for a class of nonlinear thermoelastic plate equations. Under some certain initial data and boundary conditions,it obtains an existence and uniqueness theorem of global weak solutions of the nonlinear thermoelstic plate equations,by means of the Galerkin method. Moreover,it also proves the existence of strong and classical solutions.
Stabilization of solutions to higher-order nonlinear Schrodinger equation with localized damping
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Eleni Bisognin
2007-01-01
Full Text Available We study the stabilization of solutions to higher-order nonlinear Schrodinger equations in a bounded interval under the effect of a localized damping mechanism. We use multiplier techniques to obtain exponential decay in time of the solutions of the linear and nonlinear equations.
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Bin Lu
2014-01-01
Full Text Available The Bäcklund transformation of fractional Riccati equation with nonlinear superposition principle of solutions is employed to establish the infinite sequence solutions of nonlinear fractional partial differential equations in the sense of modified Riemann-Liouville derivative. To illustrate the reliability of the method, some examples are provided.
Nonlinear Localized Dissipative Structures for Long-Time Solution of Wave Equation
2009-07-01
interesting that the role of the second order term (£0) in equation (2.11) is different from typical nonlinear pde’s studied, such as KdV , that harbor...the commonly used form of the CH equation. An important point is that other nonlinear pde’s like Kdv , which can successfully propagate localized
Applications of algebraic method to exactly solve some nonlinear partial differential equations
Energy Technology Data Exchange (ETDEWEB)
Darwish, A.A. [Department of Mathematics, Faculty of Science, Helwan University (Egypt)]. E-mail: profdarwish@yahoo.com; Ramady, A. [Department of Mathematics, Faculty of Science, Beni-Suef University (Egypt)]. E-mail: aramady@yahoo.com
2007-08-15
A direct and unified algebraic method for constructing multiple travelling wave solutions of nonlinear evolution equations is used and implemented in a computer algebraic system. New solutions for some nonlinear partial differential equations (NLPDE's) are obtained. Graphs of the solutions are displayed.
Similarity Reduction and Integrability for the Nonlinear Wave Equations from EPM Model
Institute of Scientific and Technical Information of China (English)
YAN ZhenYa
2001-01-01
Four types of similarity reductions are obtained for the nonlinear wave equation arising in the elasto-plasticmicrostructure model by using both the direct method due to Clarkson and Kruskal and the improved direct method due to Lou. As a result, the nonlinear wave equation is not integrable.``
Maximum Likelihood Analysis of Nonlinear Structural Equation Models with Dichotomous Variables
Song, Xin-Yuan; Lee, Sik-Yum
2005-01-01
In this article, a maximum likelihood approach is developed to analyze structural equation models with dichotomous variables that are common in behavioral, psychological and social research. To assess nonlinear causal effects among the latent variables, the structural equation in the model is defined by a nonlinear function. The basic idea of the…
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.
ON THE INSTABILITY OF SOLUTIONS TO A NONLINEAR VECTOR DIFFERENTIAL EQUATION OF FOURTH ORDER
Institute of Scientific and Technical Information of China (English)
无
2011-01-01
This paper presents a new result related to the instability of the zero solution to a nonlinear vector differential equation of fourth order.Our result includes and improves an instability result in the previous literature,which is related to the instability of the zero solution to a nonlinear scalar differential equation of fourth order.
NONLOCAL INITIAL PROBLEM FOR NONLINEAR NONAUTONOMOUS DIFFERENTIAL EQUATIONS IN A BANACH SPACE
Institute of Scientific and Technical Information of China (English)
M.I.Gil＇
2004-01-01
The nonlocal initial problem for nonlinear nonautonomous evolution equations in a Banach space is considered. It is assumed that the nonlinearities have the local Lipschitz properties. The existence and uniqueness of mild solutions are proved. Applications to integro-differential equations are discussed. The main tool in the paper is the normalizing mapping (the generalized norm).
New Exact Travelling Wave and Periodic Solutions of Discrete Nonlinear Schr(o)dinger Equation
Institute of Scientific and Technical Information of China (English)
YANG Qin; DAI Chao-Qing; ZHANG Jie-Fang
2005-01-01
Some new exact travelling wave and period solutions of discrete nonlinear Schrodinger equation are found by using a hyperbolic tangent function approach, which was usually presented to find exact travelling wave solutions of certain nonlinear partial differential models. Now we can further extend the new algorithm to other nonlinear differentialdifferent models.
An inverse problem of determining a nonlinear term in an ordinary differential equation
Kamimura, Yutaka
1998-01-01
An inverse problem for a nonlinear ordinary differential equation is discussed. We prove an existence theorem of a nonlinear term with which a boundary value problem admits a solution. This is an improvement of earlier work by A. Lorenzi. We also prove a uniqueness theorem of the nonlinear term.