A Combined MPI-CUDA Parallel Solution of Linear and Nonlinear Poisson-Boltzmann Equation
Directory of Open Access Journals (Sweden)
José Colmenares
2014-01-01
Full Text Available The Poisson-Boltzmann equation models the electrostatic potential generated by fixed charges on a polarizable solute immersed in an ionic solution. This approach is often used in computational structural biology to estimate the electrostatic energetic component of the assembly of molecular biological systems. In the last decades, the amount of data concerning proteins and other biological macromolecules has remarkably increased. To fruitfully exploit these data, a huge computational power is needed as well as software tools capable of exploiting it. It is therefore necessary to move towards high performance computing and to develop proper parallel implementations of already existing and of novel algorithms. Nowadays, workstations can provide an amazing computational power: up to 10 TFLOPS on a single machine equipped with multiple CPUs and accelerators such as Intel Xeon Phi or GPU devices. The actual obstacle to the full exploitation of modern heterogeneous resources is efficient parallel coding and porting of software on such architectures. In this paper, we propose the implementation of a full Poisson-Boltzmann solver based on a finite-difference scheme using different and combined parallel schemes and in particular a mixed MPI-CUDA implementation. Results show great speedups when using the two schemes, achieving an 18.9x speedup using three GPUs.
A combined MPI-CUDA parallel solution of linear and nonlinear Poisson-Boltzmann equation.
Colmenares, José; Galizia, Antonella; Ortiz, Jesús; Clematis, Andrea; Rocchia, Walter
2014-01-01
The Poisson-Boltzmann equation models the electrostatic potential generated by fixed charges on a polarizable solute immersed in an ionic solution. This approach is often used in computational structural biology to estimate the electrostatic energetic component of the assembly of molecular biological systems. In the last decades, the amount of data concerning proteins and other biological macromolecules has remarkably increased. To fruitfully exploit these data, a huge computational power is needed as well as software tools capable of exploiting it. It is therefore necessary to move towards high performance computing and to develop proper parallel implementations of already existing and of novel algorithms. Nowadays, workstations can provide an amazing computational power: up to 10 TFLOPS on a single machine equipped with multiple CPUs and accelerators such as Intel Xeon Phi or GPU devices. The actual obstacle to the full exploitation of modern heterogeneous resources is efficient parallel coding and porting of software on such architectures. In this paper, we propose the implementation of a full Poisson-Boltzmann solver based on a finite-difference scheme using different and combined parallel schemes and in particular a mixed MPI-CUDA implementation. Results show great speedups when using the two schemes, achieving an 18.9x speedup using three GPUs.
Simple and robust solver for the Poisson-Boltzmann equation
Baptista, M.; Schmitz, R.; Dünweg, B.
2009-07-01
A variational approach is used to develop a robust numerical procedure for solving the nonlinear Poisson-Boltzmann equation. Following Maggs and Rossetto [Phys. Rev. Lett. 88, 196402 (2002)], we construct an appropriate constrained free energy functional such that its Euler-Lagrange equations are equivalent to the Poisson-Boltzmann equation. This is a formulation that searches for a true minimum in function space, in contrast to previous variational approaches that rather searched for a saddle point. We then develop, implement, and test an algorithm for its numerical minimization, which is quite simple and unconditionally stable. The analytic solution for planar geometry is used for validation. Some results are presented for a charged colloidal sphere surrounded by counterions and optimizations based upon fast Fourier transforms and hierarchical preconditioning are briefly discussed.
Probabilistic Interpretation for the Nonlinear Poisson-Boltzmann Equation in Molecular Dynamics
Directory of Open Access Journals (Sweden)
Perrin Nicolas
2012-04-01
Full Text Available The Poisson-Boltzmann (PB equation describes the electrostatic potential of a biomolecular system composed by a molecule in a solvent. The electrostatic potential is involved in biomolecular models which are used in molecular simulation. In consequence, finding an efficient method to simulate the numerical solution of PB equation is very useful. As a first step, we establish in this paper a probabilistic interpretation of the nonlinear PB equation with Backward Stochastic Differential Equations (BSDEs. This interpretation requires an adaptation of existing results on BSDEs. En dynamique moléculaire, l’équation de Poisson-Boltzmann (PB permet de décrire le potentiel électrostatique d’un système moléculaire composé d’une molécule dans un solvant. Ce potentiel électrostatique intervient dans les modèles de simulation numérique permettant de comprendre la structure, la dynamique et le fonctionnement des protéines. La résolution numérique de l’équation de PB est donc une étape importante de ces simulations. Aussi, nous proposons dans un premier temps, une interprétation probabiliste de l’équation de PB non-linéaire à l’aide des Equations Différentielles Stochastiques Rétrogrades (EDSR. Cette interprétation nécessite une adaptation des résultats d’existence et d’unicité des solutions d’EDSR.
Surface Tension of Acid Solutions: Fluctuations beyond the Nonlinear Poisson-Boltzmann Theory.
Markovich, Tomer; Andelman, David; Podgornik, Rudi
2017-01-10
We extend our previous study of surface tension of ionic solutions and apply it to acids (and salts) with strong ion-surface interactions, as described by a single adhesivity parameter for the ionic species interacting with the interface. We derive the appropriate nonlinear boundary condition with an effective surface charge due to the adsorption of ions from the bulk onto the interface. The calculation is done using the loop-expansion technique, where the zero loop (mean field) corresponds of the full nonlinear Poisson-Boltzmann equation. The surface tension is obtained analytically to one-loop order, where the mean-field contribution is a modification of the Poisson-Boltzmann surface tension and the one-loop contribution gives a generalization of the Onsager-Samaras result. Adhesivity significantly affects both contributions to the surface tension, as can be seen from the dependence of surface tension on salt concentration for strongly absorbing ions. Comparison with available experimental data on a wide range of different acids and salts allows the fitting of the adhesivity parameter. In addition, it identifies the regime(s) where the hypotheses on which the theory is based are outside their range of validity.
Asymptotic analysis of the Poisson-Boltzmann equation describing electrokinetics in porous media
Allaire, Grégoire; Dufrêche, Jean-François; Mikelić, Andro; Piatnitski, Andrey
2013-03-01
We consider the Poisson-Boltzmann equation in a periodic cell, representative of a porous medium. It is a model for the electrostatic distribution of N chemical species diluted in a liquid at rest, occupying the pore space with charged solid boundaries. We study the asymptotic behaviour of its solution depending on a parameter β, which is the square of the ratio between a characteristic pore length and the Debye length. For small β we identify the limit problem which is still a nonlinear Poisson equation involving only one species with maximal valence, opposite to the average of the given surface charge density. This result justifies the Donnan effect, observing that the ions for which the charge is that of the solid phase are expelled from the pores. For large β we prove that the solution behaves like a boundary layer near the pore walls and is constant far away in the bulk. Our analysis is valid for Neumann boundary conditions (namely for imposed surface charge densities) and establishes rigorously that solid interfaces are uncoupled from the bulk fluid so that simplified additive theories, such as the popular Derjaguin, Landau, Verwey and Overbeek approach, can be used. We show that the asymptotic behaviour is completely different in the case of Dirichlet boundary conditions (namely for imposed surface potential).
ADAPTIVE FINITE ELEMENT MODELING TECHNIQUES FOR THE POISSON-BOLTZMANN EQUATION
HOLST, MICHAEL; MCCAMMON, JAMES ANDREW; YU, ZEYUN; ZHOU, YOUNGCHENG; ZHU, YUNRONG
2011-01-01
We consider the design of an effective and reliable adaptive finite element method (AFEM) for the nonlinear Poisson-Boltzmann equation (PBE). We first examine the two-term regularization technique for the continuous problem recently proposed by Chen, Holst, and Xu based on the removal of the singular electrostatic potential inside biomolecules; this technique made possible the development of the first complete solution and approximation theory for the Poisson-Boltzmann equation, the first provably convergent discretization, and also allowed for the development of a provably convergent AFEM. However, in practical implementation, this two-term regularization exhibits numerical instability. Therefore, we examine a variation of this regularization technique which can be shown to be less susceptible to such instability. We establish a priori estimates and other basic results for the continuous regularized problem, as well as for Galerkin finite element approximations. We show that the new approach produces regularized continuous and discrete problems with the same mathematical advantages of the original regularization. We then design an AFEM scheme for the new regularized problem, and show that the resulting AFEM scheme is accurate and reliable, by proving a contraction result for the error. This result, which is one of the first results of this type for nonlinear elliptic problems, is based on using continuous and discrete a priori L∞ estimates to establish quasi-orthogonality. To provide a high-quality geometric model as input to the AFEM algorithm, we also describe a class of feature-preserving adaptive mesh generation algorithms designed specifically for constructing meshes of biomolecular structures, based on the intrinsic local structure tensor of the molecular surface. All of the algorithms described in the article are implemented in the Finite Element Toolkit (FETK), developed and maintained at UCSD. The stability advantages of the new regularization scheme
Influence of Grid Spacing in Poisson-Boltzmann Equation Binding Energy Estimation.
Harris, Robert C; Boschitsch, Alexander H; Fenley, Marcia O
2013-08-13
Grid-based solvers of the Poisson-Boltzmann, PB, equation are routinely used to estimate electrostatic binding, ΔΔGel, and solvation, ΔGel, free energies. The accuracies of such estimates are subject to grid discretization errors from the finite difference approximation to the PB equation. Here, we show that the grid discretization errors in ΔΔGel are more significant than those in ΔGel, and can be divided into two parts: (i) errors associated with the relative positioning of the grid and (ii) systematic errors associated with grid spacing. The systematic error in particular is significant for methods, such as the molecular mechanics PB surface area, MM-PBSA, approach that predict electrostatic binding free energies by averaging over an ensemble of molecular conformations. Although averaging over multiple conformations can control for the error associated with grid placement, it will not eliminate the systematic error, which can only be controlled by reducing grid spacing. The present study indicates that the widely-used grid spacing of 0.5 Å produces unacceptable errors in ΔΔGel, even though its predictions of ΔGel are adequate for the cases considered here. Although both grid discretization errors generally increase with grid spacing, the relative sizes of these errors differ according to the solute-solvent dielectric boundary definition. The grid discretization errors are generally smaller on the Gaussian surface used in the present study than on either the solvent-excluded or van der Waals surfaces, which both contain more surface discontinuities (e.g., sharp edges and cusps). Additionally, all three molecular surfaces converge to very different estimates of ΔΔGel.
Sensitivities to parameterization in the size-modified Poisson-Boltzmann equation
Harris, Robert C.; Boschitsch, Alexander H.; Fenley, Marcia O.
2014-02-01
Experimental results have demonstrated that the numbers of counterions surrounding nucleic acids differ from those predicted by the nonlinear Poisson-Boltzmann equation, NLPBE. Some studies have fit these data against the ion size in the size-modified Poisson-Boltzmann equation, SMPBE, but the present study demonstrates that other parameters, such as the Stern layer thickness and the molecular surface definition, can change the number of bound ions by amounts comparable to varying the ion size. These parameters will therefore have to be fit simultaneously against experimental data. In addition, the data presented here demonstrate that the derivative, SK, of the electrostatic binding free energy, ΔGel, with respect to the logarithm of the salt concentration is sensitive to these parameters, and experimental measurements of SK could be used to parameterize the model. However, although better values for the Stern layer thickness and ion size and better molecular surface definitions could improve the model's predictions of the numbers of ions around biomolecules and SK, ΔGel itself is more sensitive to parameters, such as the interior dielectric constant, which in turn do not significantly affect the distributions of ions around biomolecules. Therefore, improved estimates of the ion size and Stern layer thickness to use in the SMPBE will not necessarily improve the model's predictions of ΔGel.
Sensitivities to parameterization in the size-modified Poisson-Boltzmann equation.
Harris, Robert C; Boschitsch, Alexander H; Fenley, Marcia O
2014-02-21
Experimental results have demonstrated that the numbers of counterions surrounding nucleic acids differ from those predicted by the nonlinear Poisson-Boltzmann equation, NLPBE. Some studies have fit these data against the ion size in the size-modified Poisson-Boltzmann equation, SMPBE, but the present study demonstrates that other parameters, such as the Stern layer thickness and the molecular surface definition, can change the number of bound ions by amounts comparable to varying the ion size. These parameters will therefore have to be fit simultaneously against experimental data. In addition, the data presented here demonstrate that the derivative, SK, of the electrostatic binding free energy, ΔGel, with respect to the logarithm of the salt concentration is sensitive to these parameters, and experimental measurements of SK could be used to parameterize the model. However, although better values for the Stern layer thickness and ion size and better molecular surface definitions could improve the model's predictions of the numbers of ions around biomolecules and SK, ΔGel itself is more sensitive to parameters, such as the interior dielectric constant, which in turn do not significantly affect the distributions of ions around biomolecules. Therefore, improved estimates of the ion size and Stern layer thickness to use in the SMPBE will not necessarily improve the model's predictions of ΔGel.
pK(A) in proteins solving the Poisson-Boltzmann equation with finite elements.
Sakalli, Ilkay; Knapp, Ernst-Walter
2015-11-05
Knowledge on pK(A) values is an eminent factor to understand the function of proteins in living systems. We present a novel approach demonstrating that the finite element (FE) method of solving the linearized Poisson-Boltzmann equation (lPBE) can successfully be used to compute pK(A) values in proteins with high accuracy as a possible replacement to finite difference (FD) method. For this purpose, we implemented the software molecular Finite Element Solver (mFES) in the framework of the Karlsberg+ program to compute pK(A) values. This work focuses on a comparison between pK(A) computations obtained with the well-established FD method and with the new developed FE method mFES, solving the lPBE using protein crystal structures without conformational changes. Accurate and coarse model systems are set up with mFES using a similar number of unknowns compared with the FD method. Our FE method delivers results for computations of pK(A) values and interaction energies of titratable groups, which are comparable in accuracy. We introduce different thermodynamic cycles to evaluate pK(A) values and we show for the FE method how different parameters influence the accuracy of computed pK(A) values. © 2015 Wiley Periodicals, Inc.
Xie, Yang; Ying, Jinyong; Xie, Dexuan
2017-03-30
SMPBS (Size Modified Poisson-Boltzmann Solvers) is a web server for computing biomolecular electrostatics using finite element solvers of the size modified Poisson-Boltzmann equation (SMPBE). SMPBE not only reflects ionic size effects but also includes the classic Poisson-Boltzmann equation (PBE) as a special case. Thus, its web server is expected to have a broader range of applications than a PBE web server. SMPBS is designed with a dynamic, mobile-friendly user interface, and features easily accessible help text, asynchronous data submission, and an interactive, hardware-accelerated molecular visualization viewer based on the 3Dmol.js library. In particular, the viewer allows computed electrostatics to be directly mapped onto an irregular triangular mesh of a molecular surface. Due to this functionality and the fast SMPBE finite element solvers, the web server is very efficient in the calculation and visualization of electrostatics. In addition, SMPBE is reconstructed using a new objective electrostatic free energy, clearly showing that the electrostatics and ionic concentrations predicted by SMPBE are optimal in the sense of minimizing the objective electrostatic free energy. SMPBS is available at the URL: smpbs.math.uwm.edu © 2017 Wiley Periodicals, Inc. © 2017 Wiley Periodicals, Inc.
Li, B O; Liu, Yuan
A phase-field free-energy functional for the solvation of charged molecules (e.g., proteins) in aqueous solvent (i.e., water or salted water) is constructed. The functional consists of the solute volumetric and solute-solvent interfacial energies, the solute-solvent van der Waals interaction energy, and the continuum electrostatic free energy described by the Poisson-Boltzmann theory. All these are expressed in terms of phase fields that, for low free-energy conformations, are close to one value in the solute phase and another in the solvent phase. A key property of the model is that the phase-field interpolation of dielectric coefficient has the vanishing derivative at both solute and solvent phases. The first variation of such an effective free-energy functional is derived. Matched asymptotic analysis is carried out for the resulting relaxation dynamics of the diffused solute-solvent interface. It is shown that the sharp-interface limit is exactly the variational implicit-solvent model that has successfully captured capillary evaporation in hydrophobic confinement and corresponding multiple equilibrium states of underlying biomolecular systems as found in experiment and molecular dynamics simulations. Our phase-field approach and analysis can be used to possibly couple the description of interfacial fluctuations for efficient numerical computations of biomolecular interactions.
Fogolari, Federico; Corazza, Alessandra; Esposito, Gennaro
2015-04-05
The generalized Born model in the Onufriev, Bashford, and Case (Onufriev et al., Proteins: Struct Funct Genet 2004, 55, 383) implementation has emerged as one of the best compromises between accuracy and speed of computation. For simulations of nucleic acids, however, a number of issues should be addressed: (1) the generalized Born model is based on a linear model and the linearization of the reference Poisson-Boltmann equation may be questioned for highly charged systems as nucleic acids; (2) although much attention has been given to potentials, solvation forces could be much less sensitive to linearization than the potentials; and (3) the accuracy of the Onufriev-Bashford-Case (OBC) model for nucleic acids depends on fine tuning of parameters. Here, we show that the linearization of the Poisson Boltzmann equation has mild effects on computed forces, and that with optimal choice of the OBC model parameters, solvation forces, essential for molecular dynamics simulations, agree well with those computed using the reference Poisson-Boltzmann model. © 2015 Wiley Periodicals, Inc.
Dielectric boundary force in numerical Poisson-Boltzmann methods: Theory and numerical strategies
Cai, Qin; Ye, Xiang; Wang, Jun; Luo, Ray
2011-10-01
Continuum modeling of electrostatic interactions based upon the numerical solutions of the Poisson-Boltzmann equation has been widely adopted in biomolecular applications. To extend their applications to molecular dynamics and energy minimization, robust and efficient methodologies to compute solvation forces must be developed. In this study, we have first reviewed the theory for the computation of dielectric boundary force based on the definition of the Maxwell stress tensor. This is followed by a new formulation of the dielectric boundary force suitable for the finite-difference Poisson-Boltzmann methods. We have validated the new formulation with idealized analytical systems and realistic molecular systems.
PB-AM: An open-source, fully analytical linear poisson-boltzmann solver
Energy Technology Data Exchange (ETDEWEB)
Felberg, Lisa E. [Department of Chemical and Biomolecular Engineering, University of California Berkeley, Berkeley California 94720; Brookes, David H. [Department of Chemistry, University of California Berkeley, Berkeley California 94720; Yap, Eng-Hui [Department of Systems and Computational Biology, Albert Einstein College of Medicine, Bronx New York 10461; Jurrus, Elizabeth [Division of Computational and Statistical Analytics, Pacific Northwest National Laboratory, Richland Washington 99352; Scientific Computing and Imaging Institute, University of Utah, Salt Lake City Utah 84112; Baker, Nathan A. [Advanced Computing, Mathematics, and Data Division, Pacific Northwest National Laboratory, Richland Washington 99352; Division of Applied Mathematics, Brown University, Providence Rhode Island 02912; Head-Gordon, Teresa [Department of Chemical and Biomolecular Engineering, University of California Berkeley, Berkeley California 94720; Department of Chemistry, University of California Berkeley, Berkeley California 94720; Department of Bioengineering, University of California Berkeley, Berkeley California 94720; Chemical Sciences Division, Lawrence Berkeley National Labs, Berkeley California 94720
2016-11-02
We present the open source distributed software package Poisson-Boltzmann Analytical Method (PB-AM), a fully analytical solution to the linearized Poisson Boltzmann equation. The PB-AM software package includes the generation of outputs files appropriate for visualization using VMD, a Brownian dynamics scheme that uses periodic boundary conditions to simulate dynamics, the ability to specify docking criteria, and offers two different kinetics schemes to evaluate biomolecular association rate constants. Given that PB-AM defines mutual polarization completely and accurately, it can be refactored as a many-body expansion to explore 2- and 3-body polarization. Additionally, the software has been integrated into the Adaptive Poisson-Boltzmann Solver (APBS) software package to make it more accessible to a larger group of scientists, educators and students that are more familiar with the APBS framework.
Large Time Behavior of the Vlasov-Poisson-Boltzmann System
Directory of Open Access Journals (Sweden)
Li Li
2013-01-01
Full Text Available The motion of dilute charged particles can be modeled by Vlasov-Poisson-Boltzmann system. We study the large time stability of the VPB system. To be precise, we prove that when time goes to infinity, the solution of VPB system tends to global Maxwellian state in a rate Ot−∞, by using a method developed for Boltzmann equation without force in the work of Desvillettes and Villani (2005. The improvement of the present paper is the removal of condition on parameter λ as in the work of Li (2008.
Poisson-Boltzmann versus Size-Modified Poisson-Boltzmann Electrostatics Applied to Lipid Bilayers.
Wang, Nuo; Zhou, Shenggao; Kekenes-Huskey, Peter M; Li, Bo; McCammon, J Andrew
2014-12-26
Mean-field methods, such as the Poisson-Boltzmann equation (PBE), are often used to calculate the electrostatic properties of molecular systems. In the past two decades, an enhancement of the PBE, the size-modified Poisson-Boltzmann equation (SMPBE), has been reported. Here, the PBE and the SMPBE are reevaluated for realistic molecular systems, namely, lipid bilayers, under eight different sets of input parameters. The SMPBE appears to reproduce the molecular dynamics simulation results better than the PBE only under specific parameter sets, but in general, it performs no better than the Stern layer correction of the PBE. These results emphasize the need for careful discussions of the accuracy of mean-field calculations on realistic systems with respect to the choice of parameters and call for reconsideration of the cost-efficiency and the significance of the current SMPBE formulation.
Poisson-Boltzmann Calculations: van der Waals or Molecular Surface?
Pang, Xiaodong; Zhou, Huan-Xiang
2012-01-01
The Poisson-Boltzmann equation is widely used for modeling the electrostatics of biomolecules, but the calculation results are sensitive to the choice of the boundary between the low solute dielectric and the high solvent dielectric. The default choice for the dielectric boundary has been the molecular surface, but the use of the van der Waals surface has also been advocated. Here we review recent studies in which the two choices are tested against experimental results and explicit-solvent calculations. The assignment of the solvent high dielectric constant to interstitial voids in the solute is often used as a criticism against the van der Waals surface. However, this assignment may not be as unrealistic as previously thought, since hydrogen exchange and other NMR experiments have firmly established that all interior parts of proteins are transiently accessible to the solvent. PMID:23293674
A generalized Poisson and Poisson-Boltzmann solver for electrostatic environments
Energy Technology Data Exchange (ETDEWEB)
Fisicaro, G., E-mail: giuseppe.fisicaro@unibas.ch; Goedecker, S. [Department of Physics, University of Basel, Klingelbergstrasse 82, 4056 Basel (Switzerland); Genovese, L. [University of Grenoble Alpes, CEA, INAC-SP2M, L-Sim, F-38000 Grenoble (France); Andreussi, O. [Institute of Computational Science, Università della Svizzera Italiana, Via Giuseppe Buffi 13, CH-6904 Lugano (Switzerland); Theory and Simulations of Materials (THEOS) and National Centre for Computational Design and Discovery of Novel Materials (MARVEL), École Polytechnique Fédérale de Lausanne, Station 12, CH-1015 Lausanne (Switzerland); Marzari, N. [Theory and Simulations of Materials (THEOS) and National Centre for Computational Design and Discovery of Novel Materials (MARVEL), École Polytechnique Fédérale de Lausanne, Station 12, CH-1015 Lausanne (Switzerland)
2016-01-07
The computational study of chemical reactions in complex, wet environments is critical for applications in many fields. It is often essential to study chemical reactions in the presence of applied electrochemical potentials, taking into account the non-trivial electrostatic screening coming from the solvent and the electrolytes. As a consequence, the electrostatic potential has to be found by solving the generalized Poisson and the Poisson-Boltzmann equations for neutral and ionic solutions, respectively. In the present work, solvers for both problems have been developed. A preconditioned conjugate gradient method has been implemented for the solution of the generalized Poisson equation and the linear regime of the Poisson-Boltzmann, allowing to solve iteratively the minimization problem with some ten iterations of the ordinary Poisson equation solver. In addition, a self-consistent procedure enables us to solve the non-linear Poisson-Boltzmann problem. Both solvers exhibit very high accuracy and parallel efficiency and allow for the treatment of periodic, free, and slab boundary conditions. The solver has been integrated into the BigDFT and Quantum-ESPRESSO electronic-structure packages and will be released as an independent program, suitable for integration in other codes.
Ionic size effects on the Poisson-Boltzmann theory.
Colla, Thiago; Nunes Lopes, Lucas; Dos Santos, Alexandre P
2017-07-07
In this paper, we develop a simple theory to study the effects of ionic size on ionic distributions around a charged spherical particle. We include a correction to the regular Poisson-Boltzmann equation in order to take into account the size of ions in a mean-field regime. The results are compared with Monte Carlo simulations and a density functional theory based on the fundamental measure approach and a second-order bulk expansion which accounts for electrostatic correlations. The agreement is very good even for multivalent ions. Our results show that the theory can be applied with very good accuracy in the description of ions with highly effective ionic radii and low concentration, interacting with a colloid or a nanoparticle in an electrolyte solution.
Gavryushov, Sergei
2009-02-19
Potentials of mean force acting between two ions in SPC/E water have been determined via molecular dynamics simulations using the spherical cavity approach ( J. Phys. Chem. B 2006 , 110 , 10878 ). The potentials were obtained for Me(2+)-Me(+) pairs, where Me(2+) means cations Mg(2+) and Ca(2+) and Me(+) denotes monovalent ions Li(+), Na(+), and K(+). The hard-core interaction distance for effective Me(2+)-Me(+) potentials appears to be of about 5 A that looks like a sum of the effective radii of a Me(2+) ion (3 A) and of an alkali metal ion Me(+) (about 2 A). These ion-ion interaction parameters were used in the epsilon-Modified Poisson-Boltzmann (epsilon-MPB) calculations ( J. Phys. Chem. B 2007 , 111 , 5264 ) of ionic distributions around DNA generalized for the arbitrary mixture of different ion species. Ionic distributions around an all-atom geometry model of B-DNA in solution of a mixture of NaCl and MgCl(2) were obtained. It was found that even a small fraction of ions Mg(2+) led to sharp condensation of Mg(2+) near the phosphate groups of DNA due to polarization deficiency of cluster [Mg(H(2)O)(6)](2+) in an external field. The epsilon-MPB calculations of the B-DNA-B-DNA interaction energies suggest that adding 1 mM of Mg(2+) to 50 mM solution of NaCl notably affects the force acting between the two macromolecules. Being compared to Poisson-Boltzmann results and to MPB calculations for the primitive model of ions, the epsilon-MPB results also indicate an important contribution of dielectric saturation effects to the mediating role of divalent cations in the DNA-DNA interaction energies.
Directory of Open Access Journals (Sweden)
Thereza A. Soares
2004-08-01
Full Text Available The ability of biomolecules to catalyze chemical reactions is due chiefly to their sensitivity to variations of the pH in the surrounding environment. The reason for this is that they are made up of chemical groups whose ionization states are modulated by pH changes that are of the order of 0.4 units. The determination of the protonation states of such chemical groups as a function of conformation of the biomolecule and the pH of the environment can be useful in the elucidation of important biological processes from enzymatic catalysis to protein folding and molecular recognition. In the past 15 years, the theory of Poisson-Boltzmann has been successfully used to estimate the pKa of ionizable sites in proteins yielding results, which may differ by 0.1 unit from the experimental values. In this study, we review the theory of Poisson-Boltzmann under the perspective of its application to the calculation of pKa in proteins.
Park, H M; Lee, J S; Kim, T W
2007-11-15
In the analysis of electroosmotic flows, the internal electric potential is usually modeled by the Poisson-Boltzmann equation. The Poisson-Boltzmann equation is derived from the assumption of thermodynamic equilibrium where the ionic distributions are not affected by fluid flows. Although this is a reasonable assumption for steady electroosmotic flows through straight microchannels, there are some important cases where convective transport of ions has nontrivial effects. In these cases, it is necessary to adopt the Nernst-Planck equation instead of the Poisson-Boltzmann equation to model the internal electric field. In the present work, the predictions of the Nernst-Planck equation are compared with those of the Poisson-Boltzmann equation for electroosmotic flows in various microchannels where the convective transport of ions is not negligible.
Polyelectrolyte Microcapsules: Ion Distributions from a Poisson-Boltzmann Model
Tang, Qiyun; Denton, Alan R.; Rozairo, Damith; Croll, Andrew B.
2014-03-01
Recent experiments have shown that polystyrene-polyacrylic-acid-polystyrene (PS-PAA-PS) triblock copolymers in a solvent mixture of water and toluene can self-assemble into spherical microcapsules. Suspended in water, the microcapsules have a toluene core surrounded by an elastomer triblock shell. The longer, hydrophilic PAA blocks remain near the outer surface of the shell, becoming charged through dissociation of OH functional groups in water, while the shorter, hydrophobic PS blocks form a networked (glass or gel) structure. Within a mean-field Poisson-Boltzmann theory, we model these polyelectrolyte microcapsules as spherical charged shells, assuming different dielectric constants inside and outside the capsule. By numerically solving the nonlinear Poisson-Boltzmann equation, we calculate the radial distribution of anions and cations and the osmotic pressure within the shell as a function of salt concentration. Our predictions, which can be tested by comparison with experiments, may guide the design of microcapsules for practical applications, such as drug delivery. This work was supported by the National Science Foundation under Grant No. DMR-1106331.
Lu, Benzhuo; Cheng, Xiaolin; Huang, Jingfang; McCammon, J. Andrew
2010-06-01
://www.fastmultipole.org/). Nature of problem: Numerical solution of the linearized Poisson-Boltzmann equation that describes electrostatic interactions of molecular systems in ionic solutions. Solution method: A novel node-patch scheme is used to discretize the well-conditioned boundary integral equation formulation of the linearized Poisson-Boltzmann equation. Various Krylov subspace solvers can be subsequently applied to solve the resulting linear system, with a bounded number of iterations independent of the number of discretized unknowns. The matrix-vector multiplication at each iteration is accelerated by the adaptive new versions of fast multipole methods. The AFMPB solver requires other stand-alone pre-processing tools for boundary mesh generation, post-processing tools for data analysis and visualization, and can be conveniently coupled with different time stepping methods for dynamics simulation. Restrictions: Only three or six significant digits options are provided in this version. Unusual features: Most of the codes are in Fortran77 style. Memory allocation functions from Fortran90 and above are used in a few subroutines. Additional comments: The current version of the codes is designed and written for single core/processor desktop machines. Check http://lsec.cc.ac.cn/~lubz/afmpb.html and http://mccammon.ucsd.edu/ for updates and changes. Running time: The running time varies with the number of discretized elements ( N) in the system and their distributions. In most cases, it scales linearly as a function of N.
Is Poisson-Boltzmann theory insufficient for protein folding simulations?
Lwin, Thu Zar; Zhou, Ruhong; Luo, Ray
2006-01-21
The Poisson-Boltzmann theory has been widely used in the studies of energetics and conformations of biological macromolecules. Recently, introduction of the efficient generalized Born approximation has greatly extended its applicability to areas such as protein folding simulations where highly efficient computation is crucial. However, limitations have been found in the folding simulations of a well-studied beta hairpin with several generalized Born implementations and different force fields. These studies have raised the question whether the underlining Poisson-Boltzmann theory, on which the generalized Born model is calibrated, is adequate in the treatment of polar interactions for the challenging protein folding simulations. To address the question whether the Poisson-Boltzmann theory in the current formalism might be insufficient, we directly tested our efficient numerical Poisson-Boltzmann implementation in the beta-hairpin folding simulation. Good agreement between simulation and experiment was found for the beta-hairpin equilibrium structures when the numerical Poisson-Boltzmann solvent and a recently improved generalized Born solvent were used. In addition simulated thermodynamic properties also agree well with experiment in both solvents. Finally, an overall agreement on the beta-hairpin folding mechanism was found between the current and previous studies. Thus, our simulations indicate that previously observed limitations are most likely due to imperfect calibration in previous generalized Born models but not due to the limitation of the Poisson-Boltzmann theory.
Free-energy functionals of the electrostatic potential for Poisson-Boltzmann theory.
Jadhao, Vikram; Solis, Francisco J; de la Cruz, Monica Olvera
2013-08-01
In simulating charged systems, it is often useful to treat some ionic components of the system at the mean-field level and solve the Poisson-Boltzmann (PB) equation to get their respective density profiles. The numerically intensive task of solving the PB equation at each step of the simulation can be bypassed using variational methods that treat the electrostatic potential as a dynamic variable. But such approaches require the access to a true free-energy functional: a functional that not only provides the correct solution of the PB equation upon extremization, but also evaluates to the true free energy of the system at its minimum. Moreover, the numerical efficiency of such procedures is further enhanced if the free-energy functional is local and is expressed in terms of the electrostatic potential. Existing PB functionals of the electrostatic potential, while possessing the local structure, are not free-energy functionals. We present a variational formulation with a local free-energy functional of the potential. In addition, we also construct a nonlocal free-energy functional of the electrostatic potential. These functionals are suited for employment in simulation schemes based on the ideas of dynamical optimization.
Li, Bo; Cheng, Xiaoliang; Zhang, Zhengfang
2011-01-01
In an implicit-solvent description of molecular solvation, the electrostatic free energy is given through the electrostatic potential. This potential solves a boundary-value problem of the Poisson-Boltzmann equation in which the dielectric coefficient changes across the solute-solvent interface-the dielectric boundary. The dielectric boundary force acting on such a boundary is the negative first variation of the electrostatic free energy with respect to the location change of the boundary. In this work, the concept of shape derivative is used to define such variations and formulas of the dielectric boundary force are derived. It is shown that such a force is always in the direction toward the charged solute molecules.
Poisson-Boltzmann model of electrolytes containing uniformly charged spherical nanoparticles.
Bohinc, Klemen; Volpe Bossa, Guilherme; Gavryushov, Sergei; May, Sylvio
2016-12-21
Like-charged macromolecules typically repel each other in aqueous solutions that contain small mobile ions. The interaction tends to turn attractive if mobile ions with spatially extended charge distributions are added. Such systems can be modeled within the mean-field Poisson-Boltzmann formalism by explicitly accounting for charge-charge correlations within the spatially extended ions. We consider an aqueous solution that contains a mixture of spherical nanoparticles with uniform surface charge density and small mobile salt ions, sandwiched between two like-charged planar surfaces. We perform the minimization of an appropriate free energy functional, which leads to a non-linear integral-differential equation for the electrostatic potential that we solve numerically and compare with predictions from Monte Carlo simulations. Nanoparticles with uniform surface charge density are contrasted with nanoparticles that have all their charges relocated at the center. Our mean-field model predicts that only the former (especially when large and highly charged particles) but not the latter are able to mediate attractive interactions between like-charged planar surfaces. We also demonstrate that at high salt concentration attractive interactions between like-charged planar surfaces turn into repulsion.
Lu, Benzhuo; Cheng, Xiaolin; Huang, Jingfang; McCammon, J. Andrew
2010-01-01
A Fortran program package is introduced for rapid evaluation of the electrostatic potentials and forces in biomolecular systems modeled by the linearized Poisson-Boltzmann equation. The numerical solver utilizes a well-conditioned boundary integral equation (BIE) formulation, a node-patch discretization scheme, a Krylov subspace iterative solver package with reverse communication protocols, and an adaptive new version of fast multipole method in which the exponential expansions are used to diagonalize the multipole to local translations. The program and its full description, as well as several closely related libraries and utility tools are available at http://lsec.cc.ac.cn/lubz/afmpb.html and a mirror site at http://mccammon.ucsd.edu/. This paper is a brief summary of the program: the algorithms, the implementation and the usage. PMID:20532187
A Continuum Poisson-Boltzmann Model for Membrane Channel Proteins.
Xiao, Li; Diao, Jianxiong; Greene, D'Artagnan; Wang, Junmei; Luo, Ray
2017-07-11
Membrane proteins constitute a large portion of the human proteome and perform a variety of important functions as membrane receptors, transport proteins, enzymes, signaling proteins, and more. Computational studies of membrane proteins are usually much more complicated than those of globular proteins. Here, we propose a new continuum model for Poisson-Boltzmann calculations of membrane channel proteins. Major improvements over the existing continuum slab model are as follows: (1) The location and thickness of the slab model are fine-tuned based on explicit-solvent MD simulations. (2) The highly different accessibilities in the membrane and water regions are addressed with a two-step, two-probe grid-labeling procedure. (3) The water pores/channels are automatically identified. The new continuum membrane model is optimized (by adjusting the membrane probe, as well as the slab thickness and center) to best reproduce the distributions of buried water molecules in the membrane region as sampled in explicit water simulations. Our optimization also shows that the widely adopted water probe of 1.4 Å for globular proteins is a very reasonable default value for membrane protein simulations. It gives the best compromise in reproducing the explicit water distributions in membrane channel proteins, at least in the water accessible pore/channel regions. Finally, we validate the new membrane model by carrying out binding affinity calculations for a potassium channel, and we observe good agreement with the experimental results.
Cooper, Christopher D
2015-01-01
Interactions between surfaces and proteins occur in many vital processes and are crucial in biotechnology: the ability to control specific interactions is essential in fields like biomaterials, biomedical implants and biosensors. In the latter case, biosensor sensitivity hinges on ligand proteins adsorbing on bioactive surfaces with a favorable orientation, exposing reaction sites to target molecules. Protein adsorption, being a free-energy-driven process, is difficult to study experimentally. This paper develops and evaluates a computational model to study electrostatic interactions of proteins and charged nanosurfaces, via the Poisson-Boltzmann equation. We extended the implicit-solvent model used in the open-source code PyGBe to include surfaces of imposed charge or potential. This code solves the boundary integral formulation of the Poisson-Boltzmann equation, discretized with surface elements. PyGBe has at its core a treecode-accelerated Krylov iterative solver, resulting in O(N log N) scaling, with furt...
The Vlasov-Poisson-Boltzmann System for a Disparate Mass Binary Mixture
Duan, Renjun; Liu, Shuangqian
2017-11-01
The Vlasov-Poisson-Boltzmann system is often used to govern the motion of plasmas consisting of electrons and ions with disparate masses when collisions of charged particles are described by the two-component Boltzmann collision operator. The perturbation theory of the system around global Maxwellians recently has been well established in Guo (Commun Pure Appl Math 55:1104-1135, 2002). It should be more interesting to further study the existence and stability of nontrivial large time asymptotic profiles for the system even with slab symmetry in space, particularly understanding the effect of the self-consistent potential on the non-trivial long-term dynamics of the binary system. In this paper, we consider the problem in the setting of rarefaction waves. The analytical tool is based on the macro-micro decomposition introduced in Liu et al. (Physica D 188(3-4):178-192, 2004) that we have been able to develop for the case of the two-component Boltzmann equations around local bi-Maxwellians. Our focus is to explore how the disparate masses and charges of particles play a role in the analysis of the approach of the complex coupling system time-asymptotically toward a non-constant equilibrium state whose macroscopic quantities satisfy the quasineutral nonisentropic Euler system.
The ionic atmosphere around A-RNA: Poisson-Boltzmann and molecular dynamics simulations.
Kirmizialtin, Serdal; Silalahi, Alexander R J; Elber, Ron; Fenley, Marcia O
2012-02-22
The distributions of different cations around A-RNA are computed by Poisson-Boltzmann (PB) equation and replica exchange molecular dynamics (MD). Both the nonlinear PB and size-modified PB theories are considered. The number of ions bound to A-RNA, which can be measured experimentally, is well reproduced in all methods. On the other hand, the radial ion distribution profiles show differences between MD and PB. We showed that PB results are sensitive to ion size and functional form of the solvent dielectric region but not the solvent dielectric boundary definition. Size-modified PB agrees with replica exchange molecular dynamics much better than nonlinear PB when the ion sizes are chosen from atomistic simulations. The distribution of ions 14 Å away from the RNA central axis are reasonably well reproduced by size-modified PB for all ion types with a uniform solvent dielectric model and a sharp dielectric boundary between solvent and RNA. However, this model does not agree with MD for shorter distances from the A-RNA. A distance-dependent solvent dielectric function proposed by another research group improves the agreement for sodium and strontium ions, even for shorter distances from the A-RNA. However, Mg(2+) distributions are still at significant variances for shorter distances. Copyright Â© 2012 Biophysical Society. Published by Elsevier Inc. All rights reserved.
Acceleration of Linear Finite-Difference Poisson-Boltzmann Methods on Graphics Processing Units.
Qi, Ruxi; Botello-Smith, Wesley M; Luo, Ray
2017-07-11
Electrostatic interactions play crucial roles in biophysical processes such as protein folding and molecular recognition. Poisson-Boltzmann equation (PBE)-based models have emerged as widely used in modeling these important processes. Though great efforts have been put into developing efficient PBE numerical models, challenges still remain due to the high dimensionality of typical biomolecular systems. In this study, we implemented and analyzed commonly used linear PBE solvers for the ever-improving graphics processing units (GPU) for biomolecular simulations, including both standard and preconditioned conjugate gradient (CG) solvers with several alternative preconditioners. Our implementation utilizes the standard Nvidia CUDA libraries cuSPARSE, cuBLAS, and CUSP. Extensive tests show that good numerical accuracy can be achieved given that the single precision is often used for numerical applications on GPU platforms. The optimal GPU performance was observed with the Jacobi-preconditioned CG solver, with a significant speedup over standard CG solver on CPU in our diversified test cases. Our analysis further shows that different matrix storage formats also considerably affect the efficiency of different linear PBE solvers on GPU, with the diagonal format best suited for our standard finite-difference linear systems. Further efficiency may be possible with matrix-free operations and integrated grid stencil setup specifically tailored for the banded matrices in PBE-specific linear systems.
Cooper, Christopher D.; Barba, Lorena A.
2016-05-01
Interactions between surfaces and proteins occur in many vital processes and are crucial in biotechnology: the ability to control specific interactions is essential in fields like biomaterials, biomedical implants and biosensors. In the latter case, biosensor sensitivity hinges on ligand proteins adsorbing on bioactive surfaces with a favorable orientation, exposing reaction sites to target molecules. Protein adsorption, being a free-energy-driven process, is difficult to study experimentally. This paper develops and evaluates a computational model to study electrostatic interactions of proteins and charged nanosurfaces, via the Poisson-Boltzmann equation. We extended the implicit-solvent model used in the open-source code PyGBe to include surfaces of imposed charge or potential. This code solves the boundary integral formulation of the Poisson-Boltzmann equation, discretized with surface elements. PyGBe has at its core a treecode-accelerated Krylov iterative solver, resulting in O(N log N) scaling, with further acceleration on hardware via multi-threaded execution on GPUs. It computes solvation and surface free energies, providing a framework for studying the effect of electrostatics on adsorption. We derived an analytical solution for a spherical charged surface interacting with a spherical dielectric cavity, and used it in a grid-convergence study to build evidence on the correctness of our approach. The study showed the error decaying with the average area of the boundary elements, i.e., the method is O(1 / N) , which is consistent with our previous verification studies using PyGBe. We also studied grid-convergence using a real molecular geometry (protein G B1 D4‧), in this case using Richardson extrapolation (in the absence of an analytical solution) and confirmed the O(1 / N) scaling. With this work, we can now access a completely new family of problems, which no other major bioelectrostatics solver, e.g. APBS, is capable of dealing with. PyGBe is open
Stability of Nonlinear Wave Patterns to the Bipolar Vlasov-Poisson-Boltzmann System
Li, Hailiang; Wang, Yi; Yang, Tong; Zhong, Mingying
2017-10-01
The main purpose of the present paper is to investigate the nonlinear stability of viscous shock waves and rarefaction waves for the bipolar Vlasov-Poisson-Boltzmann (VPB) system. To this end, motivated by the micro-macro decomposition to the Boltzmann equation in Liu and Yu (Commun Math Phys 246:133-179, 2004) and Liu et al. (Physica D 188:178-192, 2004), we first set up a new micro-macro decomposition around the local Maxwellian related to the bipolar VPB system and give a unified framework to study the nonlinear stability of the basic wave patterns to the system. Then, as applications of this new decomposition, the time-asymptotic stability of the two typical nonlinear wave patterns, viscous shock waves and rarefaction waves are proved for the 1D bipolar VPB system. More precisely, it is first proved that the linear superposition of two Boltzmann shock profiles in the first and third characteristic fields is nonlinearly stable to the 1D bipolar VPB system up to some suitable shifts without the zero macroscopic mass conditions on the initial perturbations. Then the time-asymptotic stability of the rarefaction wave fan to compressible Euler equations is proved for the 1D bipolar VPB system. These two results are concerned with the nonlinear stability of wave patterns for Boltzmann equation coupled with additional (electric) forces, which together with spectral analysis made in Li et al. (Indiana Univ Math J 65(2):665-725, 2016) sheds light on understanding the complicated dynamic behaviors around the wave patterns in the transportation of charged particles under the binary collisions, mutual interactions, and the effect of the electrostatic potential forces.
Formulation of a new and simple nonuniform size-modified Poisson-Boltzmann description.
Boschitsch, Alexander H; Danilov, Pavel V
2012-04-30
The nonlinear Poisson-Boltzmann equation (PBE) governing biomolecular electrostatics neglects ion size and ion correlation effects, and recent research activity has focused on accounting for these effects to achieve better physical modeling realism. Here, attention is focused on the comparatively simpler challenge of addressing ion size effects within a continuum-based solvent modeling framework. Prior works by Borukhov et al. (Phys. Rev. Lett. 1997, 79, 435; Electrochim. Acta 2000, 46, 221) have examined the case of uniform ion size in considerable detail. Generalizations to accommodate different species ion sizes have been performed by Li (Nonlinearity 2009, 22, 811; SIAM J. Math. Anal. 2009, 40, 2536) and Zhou et al. (Phys. Rev. E 2011, 84, 021901) using a variational principle, Chu et al. (Biophys. J. 2007, 93, 3202) using a lattice gas model, and Tresset (Phys. Rev. E 2008, 78, 061506) using a generalized Poisson-Fermi distribution. The current work provides an alternative derivation using simple statistical mechanics principles that place the ion size effects and energy distributions on a consistent statistical footing. The resulting expressions differ from the prior nonuniform ion-size developments. However, all treatments reduce to the same form in the cases of uniform ion-size and zero ion size (the PBE). Because of their importance to molecular modeling and salt-dependent behavior, expressions for the salt sensitivities and ionic forces are also derived using the nonuniform ion size description. Emphasis in this article is on formulation and numerically robust evaluation; results are presented for a simple sphere and a previously considered DNA structure for comparison and validation. More extensive application to biomolecular systems is deferred to a subsequent article. Copyright © 2012 Wiley Periodicals, Inc.
Nonextensive statistical mechanics of ionic solutions
Energy Technology Data Exchange (ETDEWEB)
Varela, L.M. [Grupo de Nanomateriales y Materia Blanda, Departamento de Fisica de la Materia Condensada, Universidad de Santiago de Compostela, E-15782 Santiago de Compostela (Spain)], E-mail: fmluis@usc.es; Carrete, J. [Grupo de Nanomateriales y Materia Blanda, Departamento de Fisica de la Materia Condensada, Universidad de Santiago de Compostela, E-15782 Santiago de Compostela (Spain); Munoz-Sola, R. [Departamento de Matematica Aplicada, Universidad de Santiago de Compostela, E-15782 Santiago de Compostela (Spain); Rodriguez, J.R.; Gallego, J. [Grupo de Nanomateriales y Materia Blanda, Departamento de Fisica de la Materia Condensada, Universidad de Santiago de Compostela, E-15782 Santiago de Compostela (Spain)
2007-10-29
Classical mean-field Poisson-Boltzmann theory of ionic solutions is revisited in the theoretical framework of nonextensive Tsallis statistics. The nonextensive equivalent of Poisson-Boltzmann equation is formulated revisiting the statistical mechanics of liquids and the Debye-Hueckel framework is shown to be valid for highly diluted solutions even under circumstances where nonextensive thermostatistics must be applied. The lowest order corrections associated to nonadditive effects are identified for both symmetric and asymmetric electrolytes and the behavior of the average electrostatic potential in a homogeneous system is analytically and numerically analyzed for various values of the complexity measurement nonextensive parameter q.
Robbins, Timothy J; Ziebarth, Jesse D; Wang, Yongmei
2014-08-01
The ion atmosphere created by monovalent (Na(+) ) or divalent (Mg(2+) ) cations surrounding a B-form DNA duplex were examined using atomistic molecular dynamics (MD) simulations and the nonlinear Poisson-Boltzmann (PB) equation. The ion distributions predicted by the two methods were compared using plots of radial and two-dimensional cation concentrations and by calculating the total number of cations and net solution charge surrounding the DNA. Na(+) ion distributions near the DNA were more diffuse in PB calculations than in corresponding MD simulations, with PB calculations predicting lower concentrations near DNA groove sites and phosphate groups and a higher concentration in the region between these locations. Other than this difference, the Na(+) distributions generated by the two methods largely agreed, as both predicted similar locations of high Na(+) concentration and nearly identical values of the number of cations and the net solution charge at all distances from the DNA. In contrast, there was greater disagreement between the two methods for Mg(2+) cation concentration profiles, as both the locations and magnitudes of peaks in Mg(2+) concentration were different. Despite experimental and simulation observations that Mg(2+) typically maintains its first solvation shell when interacting with nucleic acids, modeling Mg(2+) as an unsolvated ion during PB calculations improved the agreement of the Mg(2+) ion atmosphere predicted by the two methods and allowed for values of the number of bound ions and net solution charge surrounding the DNA from PB calculations that approached the values observed in MD simulations. © 2014 Wiley Periodicals, Inc.
Analytical estimation of effective charges at saturation in Poisson-Boltzmann cell models
Trizac, E; Bocquet, L
2003-01-01
We propose a simple approximation scheme for computing the effective charges of highly charged colloids (spherical or cylindrical with infinite length). Within non-linear Poisson-Boltzmann theory, we start from an expression for the effective charge in the infinite-dilution limit which is asymptotically valid for large salt concentrations; this result is then extended to finite colloidal concentration, approximating the salt partitioning effect which relates the salt content in the suspension to that of a dialysing reservoir. This leads to an analytical expression for the effective charge as a function of colloid volume fraction and salt concentration. These results compare favourably with the effective charges at saturation (i.e. in the limit of large bare charge) computed numerically following the standard prescription proposed by Alexander et al within the cell model.
Bu, Wei; Vaknin, David; Travesset, Alex
2005-12-01
Surface sensitive synchrotron-x-ray scattering studies reveal the distributions of monovalent ions next to highly charged interfaces. A lipid phosphate (dihexadecyl hydrogen phosphate) was spread as a monolayer at the air-water interface, containing CsI at various concentrations. Using anomalous reflectivity off and at the L3 Cs+ resonance, we provide spatial counterion distributions (Cs+) next to the negatively charged interface over a wide range of ionic concentrations. We argue that at low salt concentrations and for pure water the enhanced concentration of hydroniums H3O+ at the interface leads to proton transfer back to the phosphate group by a high contact potential, whereas high salt concentrations lower the contact potential resulting in proton release and increased surface charge density. The experimental ionic distributions are in excellent agreement with a renormalized-surface-charge Poisson-Boltzmann theory without fitting parameters or additional assumptions.
Sun, Hui; Wen, Jiayi; Zhao, Yanxiang; Li, Bo; McCammon, J Andrew
2015-12-28
Dielectric boundary based implicit-solvent models provide efficient descriptions of coarse-grained effects, particularly the electrostatic effect, of aqueous solvent. Recent years have seen the initial success of a new such model, variational implicit-solvent model (VISM) [Dzubiella, Swanson, and McCammon Phys. Rev. Lett. 96, 087802 (2006) and J. Chem. Phys. 124, 084905 (2006)], in capturing multiple dry and wet hydration states, describing the subtle electrostatic effect in hydrophobic interactions, and providing qualitatively good estimates of solvation free energies. Here, we develop a phase-field VISM to the solvation of charged molecules in aqueous solvent to include more flexibility. In this approach, a stable equilibrium molecular system is described by a phase field that takes one constant value in the solute region and a different constant value in the solvent region, and smoothly changes its value on a thin transition layer representing a smeared solute-solvent interface or dielectric boundary. Such a phase field minimizes an effective solvation free-energy functional that consists of the solute-solvent interfacial energy, solute-solvent van der Waals interaction energy, and electrostatic free energy described by the Poisson-Boltzmann theory. We apply our model and methods to the solvation of single ions, two parallel plates, and protein complexes BphC and p53/MDM2 to demonstrate the capability and efficiency of our approach at different levels. With a diffuse dielectric boundary, our new approach can describe the dielectric asymmetry in the solute-solvent interfacial region. Our theory is developed based on rigorous mathematical studies and is also connected to the Lum-Chandler-Weeks theory (1999). We discuss these connections and possible extensions of our theory and methods.
Electro-osmosis of non-Newtonian fluids in porous media using lattice Poisson-Boltzmann method.
Chen, Simeng; He, Xinting; Bertola, Volfango; Wang, Moran
2014-12-15
Electro-osmosis in porous media has many important applications in various areas such as oil and gas exploitation and biomedical detection. Very often, fluids relevant to these applications are non-Newtonian because of the shear-rate dependent viscosity. The purpose of this study was to investigate the behaviors and physical mechanism of electro-osmosis of non-Newtonian fluids in porous media. Model porous microstructures (granular, fibrous, and network) were created by a random generation-growth method. The nonlinear governing equations of electro-kinetic transport for a power-law fluid were solved by the lattice Poisson-Boltzmann method (LPBM). The model results indicate that: (i) the electro-osmosis of non-Newtonian fluids exhibits distinct nonlinear behaviors compared to that of Newtonian fluids; (ii) when the bulk ion concentration or zeta potential is high enough, shear-thinning fluids exhibit higher electro-osmotic permeability, while shear-thickening fluids lead to the higher electro-osmotic permeability for very low bulk ion concentration or zeta potential; (iii) the effect of the porous medium structure depends significantly on the constitutive parameters: for fluids with large constitutive coefficients strongly dependent on the power-law index, the network structure shows the highest electro-osmotic permeability while the granular structure exhibits the lowest permeability on the entire range of power law indices considered; when the dependence of the constitutive coefficient on the power law index is weaker, different behaviors can be observed especially in case of strong shear thinning. Copyright © 2014 Elsevier Inc. All rights reserved.
Duval, J.F.L.
2005-01-01
In a previous study (Langmuir 2004, 20, 10324), the electrokinetic properties of diffuse soft layers were theoretically investigated within the framework of the Debye-H¿ckel approximation valid in the limit of sufficiently low values for the Donnan potential. In the current paper, the
Le, Guigao; Zhang, Junfeng
2011-05-03
In this paper, we propose a general Poisson-Boltzmann model for electric double layer (EDL) analysis with the position dependence of dielectric permittivity considered. This model provides physically reasonable property profiles in the EDL region, and it is then utilized to investigate the depletion layer effect on EDL structure and interaction near hydrophobic surfaces. Our results show that both the electric potential and the interaction pressure between surfaces decrease due to the lower permittivity in the depletion layer. The reduction becomes more profound at larger variation magnitude and range. This trend is in general agreement with that observed from the previous stepwise model; however, that model has overestimated the influence of permittivity variation effect. For the thin depletion layer and the relative thick EDL, our calculation indicates that the permittivity variation effect on EDL usually can be neglected. Furthermore, our model can be readily extended to study the permittivity variation in EDL due to ion accumulation and hydration in the EDL region.
Solution of Finite Element Equations
DEFF Research Database (Denmark)
Krenk, Steen
An important step in solving any problem by the finite element method is the solution of the global equations. Numerical solution of linear equations is a subject covered in most courses in numerical analysis. However, the equations encountered in most finite element applications have some special...
Enclosing Solutions of Integral Equations
DEFF Research Database (Denmark)
Madsen, Kaj; NA NA NA Caprani, Ole; Stauning, Ole
1996-01-01
We present a method for enclosing the solution of an integral equation. It is assumed that a solution exists and that the corresponding integral operator T is a contraction near y. When solving the integral equation by iteration we obtain a result which is normally different from y because...
Solutions of Nonlocal -Laplacian Equations
Directory of Open Access Journals (Sweden)
Mustafa Avci
2013-01-01
Full Text Available In view of variational approach we discuss a nonlocal problem, that is, a Kirchhoff-type equation involving -Laplace operator. Establishing some suitable conditions, we prove the existence and multiplicity of solutions.
Graphical Solution of Polynomial Equations
Grishin, Anatole
2009-01-01
Graphing utilities, such as the ubiquitous graphing calculator, are often used in finding the approximate real roots of polynomial equations. In this paper the author offers a simple graphing technique that allows one to find all solutions of a polynomial equation (1) of arbitrary degree; (2) with real or complex coefficients; and (3) possessing…
Approximative solutions of difference equations
Directory of Open Access Journals (Sweden)
Janusz Migda
2014-03-01
\\Delta^m x_n=a_nf(n,x_{\\sigma(n}+b_n $$ are studied. Using the iterated remainder operator and fixed point theorems we obtain sufficient conditions under which for any solution $y$ of the equation $\\Delta^my=b$ and for any real $s\\leq 0$ there exists a solution $x$ of the above equation such that $\\Delta^kx=\\Delta^ky+\\mathrm{o}(n^{s-k}$ for any nonnegative integer $k\\leq m$. Using a discrete variant of the Bihari lemma and a certain new technique we give also sufficient conditions under which for a given real $s\\leq m-1$ all solutions $x$ of the equation satisfy the condition $x=y+\\mathrm{o}(n^s$ where $y$ is a solution of the equation $\\Delta^my=b$. Moreover, we give sufficient conditions under which for a given natural $k
Solutions of equations in languages
Hesselink, Wim H.
A context-free grammar corresponds to a system of equations in languages. The language generated by the grammar is the smallest solution of the system. We give a necessary and sufficient condition for an arbitrary solution to be the smallest one. We revive an old criterion to decide that a grammar
Numerical Solution of Parabolic Equations
DEFF Research Database (Denmark)
Østerby, Ole
These lecture notes are designed for a one-semester course on finite-difference methods for parabolic equations. These equations which traditionally are used for describing diffusion and heat-conduction problems in Geology, Physics, and Chemistry have recently found applications in Finance Theory...... approximations which depend on a step size, such as numerical integration and solution of ordinary and partial differential equations. An integral part of the error estimation is the estimation of the order of the method and can thus satisfy the inquisitive mind: Is the order what we expect it to be from theopry...
Numerical Solution of Heun Equation Via Linear Stochastic Differential Equation
Directory of Open Access Journals (Sweden)
Hamidreza Rezazadeh
2014-05-01
Full Text Available In this paper, we intend to solve special kind of ordinary differential equations which is called Heun equations, by converting to a corresponding stochastic differential equation(S.D.E.. So, we construct a stochastic linear equation system from this equation which its solution is based on computing fundamental matrix of this system and then, this S.D.E. is solved by numerically methods. Moreover, its asymptotic stability and statistical concepts like expectation and variance of solutions are discussed. Finally, the attained solutions of these S.D.E.s compared with exact solution of corresponding differential equations.
Tzoupis, Haralambos; Leonis, Georgios; Durdagi, Serdar; Mouchlis, Varnavas; Mavromoustakos, Thomas; Papadopoulos, Manthos G.
2011-10-01
The objectives of this study include the design of a series of novel fullerene-based inhibitors for HIV-1 protease (HIV-1 PR), by employing two strategies that can also be applied to the design of inhibitors for any other target. Additionally, the interactions which contribute to the observed exceptionally high binding free energies were analyzed. In particular, we investigated: (1) hydrogen bonding (H-bond) interactions between specific fullerene derivatives and the protease, (2) the regions of HIV-1 PR that play a significant role in binding, (3) protease changes upon binding and (4) various contributions to the binding free energy, in order to identify the most significant of them. This study has been performed by employing a docking technique, two 3D-QSAR models, molecular dynamics (MD) simulations and the molecular mechanics Poisson-Boltzmann surface area (MM-PBSA) method. Our computed binding free energies are in satisfactory agreement with the experimental results. The suitability of specific fullerene derivatives as drug candidates was further enhanced, after ADMET (absorption, distribution, metabolism, excretion and toxicity) properties have been estimated to be promising. The outcomes of this study revealed important protein-ligand interaction patterns that may lead towards the development of novel, potent HIV-1 PR inhibitors.
Prabhu, Ninad V; Zhu, Peijuan; Sharp, Kim A
2004-12-01
A fast stable finite difference Poisson-Boltzmann (FDPB) model for implicit solvation in molecular dynamics simulations was developed using the smooth permittivity FDPB method implemented in the OpenEye ZAP libraries. This was interfaced with two widely used molecular dynamics packages, AMBER and CHARMM. Using the CHARMM-ZAP software combination, the implicit solvent model was tested on eight proteins differing in size, structure, and cofactors: calmodulin, horseradish peroxidase (with and without substrate analogue bound), lipid carrier protein, flavodoxin, ubiquitin, cytochrome c, and a de novo designed 3-helix bundle. The stability and accuracy of the implicit solvent simulations was assessed by examining root-mean-squared deviations from crystal structure. This measure was compared with that of a standard explicit water solvent model. In addition we compared experimental and calculated NMR order parameters to obtain a residue level assessment of the accuracy of MD-ZAP for simulating dynamic quantities. Overall, the agreement of the implicit solvent model with experiment was as good as that of explicit water simulations. The implicit solvent method was up to eight times faster than the explicit water simulations, and approximately four times slower than a vacuum simulation (i.e., with no solvent treatment). (c) 2004 Wiley Periodicals, Inc.
Positive Integer Solutions of Certain Diophantine Equations
Indian Academy of Sciences (India)
29
An important area of number theory is devoted to finding solutions of equations where the solutions are restricted to the set of integers. Diophantine equations get their name from Diophantus of. Alexandria and they are algebraic equations for which rational or integer solutions are sought. Many researchers considered the ...
Singh, Randhir; Das, Nilima; Kumar, Jitendra
2017-06-01
An effective analytical technique is proposed for the solution of the Lane-Emden equations. The proposed technique is based on the variational iteration method (VIM) and the convergence control parameter h . In order to avoid solving a sequence of nonlinear algebraic or complicated integrals for the derivation of unknown constant, the boundary conditions are used before designing the recursive scheme for solution. The series solutions are found which converges rapidly to the exact solution. Convergence analysis and error bounds are discussed. Accuracy, applicability of the method is examined by solving three singular problems: i) nonlinear Poisson-Boltzmann equation, ii) distribution of heat sources in the human head, iii) second-kind Lane-Emden equation.
On exact solutions of the Bogoyavlenskii equation
Indian Academy of Sciences (India)
Abstract. Exact solutions for the Bogoyavlenskii equation are studied by the travelling wave method and the singular manifold method. It is found that the linear superposition of the shock wave solution and the complex solitary wave solution for the physical field is still a solution of the equation of interest, except for a ...
Liu, Fu-Feng; Liu, Zhen; Bai, Shu; Dong, Xiao-Yan; Sun, Yan
2012-04-01
Aggregation of amyloid-β (Aβ) peptides correlates with the pathology of Alzheimer's disease. However, the inter-molecular interactions between Aβ protofibril remain elusive. Herein, molecular mechanics Poisson-Boltzmann surface area analysis based on all-atom molecular dynamics simulations was performed to study the inter-molecular interactions in Aβ17-42 protofibril. It is found that the nonpolar interactions are the important forces to stabilize the Aβ17-42 protofibril, while electrostatic interactions play a minor role. Through free energy decomposition, 18 residues of the Aβ17-42 are identified to provide interaction energy lower than -2.5 kcal/mol. The nonpolar interactions are mainly provided by the main chain of the peptide and the side chains of nine hydrophobic residues (Leu17, Phe19, Phe20, Leu32, Leu34, Met35, Val36, Val40, and Ile41). However, the electrostatic interactions are mainly supplied by the main chains of six hydrophobic residues (Phe19, Phe20, Val24, Met35, Val36, and Val40) and the side chains of the charged residues (Glu22, Asp23, and Lys28). In the electrostatic interactions, the overwhelming majority of hydrogen bonds involve the main chains of Aβ as well as the guanidinium group of the charged side chain of Lys28. The work has thus elucidated the molecular mechanism of the inter-molecular interactions between Aβ monomers in Aβ17-42 protofibril, and the findings are considered critical for exploring effective agents for the inhibition of Aβ aggregation.
On oscillatory solutions of certain difference equations
Directory of Open Access Journals (Sweden)
Grzegorz Grzegorczyk
2006-01-01
Full Text Available Some difference equations with deviating arguments are discussed in the context of the oscillation problem. The aim of this paper is to present the sufficient conditions for oscillation of solutions of the equations discussed.
Exact solution to fractional logistic equation
West, Bruce J.
2015-07-01
The logistic equation is one of the most familiar nonlinear differential equations in the biological and social sciences. Herein we provide an exact solution to an extension of this equation to incorporate memory through the use of fractional derivatives in time. The solution to the fractional logistic equation (FLE) is obtained using the Carleman embedding technique that allows the nonlinear equation to be replaced by an infinite-order set of linear equations, which we then solve exactly. The formal series expansion for the initial value solution of the FLE is shown to be expressed in terms of a series of weighted Mittag-Leffler functions that reduces to the well known analytic solution in the limit where the fractional index for the derivative approaches unity. The numerical integration to the FLE provides an excellent fit to the analytic solution. We propose this approach as a general technique for solving a class of nonlinear fractional differential equations.
Geometrical and Graphical Solutions of Quadratic Equations.
Hornsby, E. John, Jr.
1990-01-01
Presented are several geometrical and graphical methods of solving quadratic equations. Discussed are Greek origins, Carlyle's method, von Staudt's method, fixed graph methods and imaginary solutions. (CW)
Explicit solutions of the Rand Equation
African Journals Online (AJOL)
user
Keywords: Nonlinear partial differential equations, evolution equations, symmetries, similarity solutions, Rand Equation. PACS-Code: ... Classical symmetry analysis - algebraic group properties ... The result is a well-defined system of eight linear homogeneous PDEs (describing the point symmetries) for the infinitesimals. ),(.
Geometrical Solutions of Quadratic Equations.
Grewal, A. S.; Godloza, L.
1999-01-01
Demonstrates that the equation of a circle (x-h)2 + (y-k)2 = r2 with center (h; k) and radius r reduces to a quadratic equation x2-2xh + (h2 + k2 -r2) = O at the intersection with the x-axis. Illustrates how to determine the center of a circle as well as a point on a circle. (Author/ASK)
The Dirac equation and its solutions
Energy Technology Data Exchange (ETDEWEB)
Bagrov, Vladislav G. [Tomsk State Univ., Tomsk (Russian Federation). Dept. of Quantum Field Theroy; Gitman, Dmitry [Sao Paulo Univ. (Brazil). Inst. de Fisica; P.N. Lebedev Physical Institute, Moscow (Russian Federation); Tomsk State Univ., Tomsk (Russian Federation). Faculty of Physics
2013-07-01
The Dirac equation is of fundamental importance for relativistic quantum mechanics and quantum electrodynamics. In relativistic quantum mechanics, the Dirac equation is referred to as one-particle wave equation of motion for electron in an external electromagnetic field. In quantum electrodynamics, exact solutions of this equation are needed to treat the interaction between the electron and the external field exactly. In particular, all propagators of a particle, i.e., the various Green's functions, are constructed in a certain way by using exact solutions of the Dirac equation.
Asymptotically periodic solutions of Volterra integral equations
Directory of Open Access Journals (Sweden)
Muhammad N. Islam
2016-03-01
Full Text Available We study the existence of asymptotically periodic solutions of a nonlinear Volterra integral equation. In the process, we obtain the existence of periodic solutions of an associated nonlinear integral equation with infinite delay. Schauder's fixed point theorem is used in the analysis.
Super Darboux-Egoroff equations and solutions
Kersten, P.H.M.; Martini, Ruud
1998-01-01
Super Darboux-Egoroff equations are discussed. First of all linearity of the potential $\\varphi$ with respect to odd variables is proved. Solutions of Darboux-Egoroff equations in dimension $(2|2)$ including flatness of the unit vector field are constructed. Moreover solutions of Darboux-Egoroff
New stiff matter solutions to Einstein equations
Energy Technology Data Exchange (ETDEWEB)
Hajj-Boutros, J.
1989-01-01
New exact solutions are presented to the Einstein field equations which are spherically symmetric and static, with a perfect fluid distribution of matter satisfying the equation of state /rho/ = p. One of the obtained solutions may only be used locally, the other represents the stellar interior globally and is singularity-free.
PARALLEL SOLUTION METHODS OF PARTIAL DIFFERENTIAL EQUATIONS
Directory of Open Access Journals (Sweden)
Korhan KARABULUT
1998-03-01
Full Text Available Partial differential equations arise in almost all fields of science and engineering. Computer time spent in solving partial differential equations is much more than that of in any other problem class. For this reason, partial differential equations are suitable to be solved on parallel computers that offer great computation power. In this study, parallel solution to partial differential equations with Jacobi, Gauss-Siedel, SOR (Succesive OverRelaxation and SSOR (Symmetric SOR algorithms is studied.
Solutions manual to accompany Ordinary differential equations
Greenberg, Michael D
2014-01-01
Features a balance between theory, proofs, and examples and provides applications across diverse fields of study Ordinary Differential Equations presents a thorough discussion of first-order differential equations and progresses to equations of higher order. The book transitions smoothly from first-order to higher-order equations, allowing readers to develop a complete understanding of the related theory. Featuring diverse and interesting applications from engineering, bioengineering, ecology, and biology, the book anticipates potential difficulties in understanding the various solution steps
Exact Solutions for Some Fractional Differential Equations
Sonmezoglu, Abdullah
2015-01-01
The extended Jacobi elliptic function expansion method is used for solving fractional differential equations in the sense of Jumarie’s modified Riemann-Liouville derivative. By means of this approach, a few fractional differential equations are successfully solved. As a result, some new Jacobi elliptic function solutions including solitary wave solutions and trigonometric function solutions are established. The proposed method can also be applied to other fractional differential e...
Symmetric solutions of evolutionary partial differential equations
Bruell, Gabriele; Ehrnström, Mats; Geyer, Anna; Pei, Long
2017-10-01
We show that for a large class of evolutionary nonlinear and nonlocal partial differential equations, symmetry of solutions implies very restrictive properties of the solutions and symmetry axes. These restrictions are formulated in terms of three principles, based on the structure of the equations. The first principle covers equations that allow for steady solutions and shows that any spatially symmetric solution is in fact steady with a speed determined by the motion of the axis of symmetry at the initial time. The second principle includes equations that admit breathers and steady waves, and therefore is less strong: it holds that the axes of symmetry are constant in time. The last principle is a mixed case, when the equation contains terms of the kind from both earlier principles, and there may be different outcomes; for a class of such equations one obtains that a spatially symmetric solution must be constant in both time and space. We list and give examples of more than 30 well-known equations and systems in one and several dimensions satisfying these principles; corresponding results for weak formulations of these equations may be attained using the same techniques. Our investigation is a generalisation of a local and one-dimensional version of the first principle from Ehrnström et al (2009 Int. Math. Res. Not. 2009 4578-96) to nonlocal equations, systems and higher dimensions, as well as a study of the standing and mixed cases.
Multicomponent integrable wave equations: II. Soliton solutions
Energy Technology Data Exchange (ETDEWEB)
Degasperis, A [Dipartimento di Fisica, Universita di Roma ' La Sapienza' , and Istituto Nazionale di Fisica Nucleare, Sezione di Roma, Rome (Italy); Lombardo, S [School of Mathematics, University of Manchester, Alan Turing Building, Upper Brook Street, Manchester M13 9EP (United Kingdom)], E-mail: antonio.degasperis@roma1.infn.it, E-mail: sara.lombardo@manchester.ac.uk, E-mail: sara@few.vu.nl
2009-09-25
The Darboux-dressing transformations developed in Degasperis and Lombardo (2007 J. Phys. A: Math. Theor. 40 961-77) are here applied to construct soliton solutions for a class of boomeronic-type equations. The vacuum (i.e. vanishing) solution and the generic plane wave solution are both dressed to yield one-soliton solutions. The formulae are specialized to the particularly interesting case of the resonant interaction of three waves, a well-known model which is of boomeronic type. For this equation a novel solution which describes three locked dark pulses (simulton) is introduced.
Solutions of the coupled Higgs field equations
Talukdar, Benoy; Ghosh, Swapan K.; Saha, Aparna; Pal, Debabrata
2013-07-01
By an appropriate choice for the phase of the complex nucleonic field and going over to the traveling coordinate, we reduce the coupled Higgs equations to the Hamiltonian form and treat the resulting equation using the dynamical system theory. We present a phase-space analysis of its stable points. The results of our study demonstrate that the equation can support both traveling- and standing-wave solutions. The traveling-wave solution appears in the form of a soliton and resides in the midst of doubly periodic standing-wave solutions.
Approximate solution for Fokker-Planck equation
Directory of Open Access Journals (Sweden)
M.T. Araujo
2015-12-01
Full Text Available In this paper, an approximate solution to a specific class of the Fokker-Planck equation is proposed. The solution is based on the relationship between the Schrödinger type equation with a partially confining and symmetrical potential. To estimate the accuracy of the solution, a function error obtained from the original Fokker-Planck equation is suggested. Two examples, a truncated harmonic potential and non-harmonic polynomial, are analyzed using the proposed method. For the truncated harmonic potential, the system behavior as a function of temperature is also discussed.
Almost periodic solutions of impulsive differential equations
Stamov, Gani T
2012-01-01
Impulsive differential equations are suitable for the mathematical simulation of evolutionary processes in which the parameters undergo relatively long periods of smooth variation followed by short-term rapid changes (that is, jumps) in their values. Processes of this type are often investigated in various fields of science and technology. The question of the existence and uniqueness of almost periodic solutions of differential equations is an age-old problem of great importance. The qualitative theory of impulsive differential equations is currently undergoing rapid development in relation to the investigation of various processes which are subject to impacts during their evolution, and many findings on the existence and uniqueness of almost periodic solutions of these equations are being made. This book systematically presents findings related to almost periodic solutions of impulsive differential equations and illustrates their potential applications.
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.
New exact wave solutions for Hirota equation
Indian Academy of Sciences (India)
Nonlinear partial differential equations (NPDEs) of mathematical physics are major sub- jects in physical science. With the development of soliton theory, many useful methods for obtaining exact solutions of NPDEs have been presented. Some of them are: the (G /G)- expansion method [1–4], the simplest equation method ...
Power Series Solution to the Pendulum Equation
Benacka, Jan
2009-01-01
This note gives a power series solution to the pendulum equation that enables to investigate the system in an analytical way only, i.e. to avoid numeric methods. A method of determining the number of the terms for getting a required relative error is presented that uses bigger and lesser geometric series. The solution is suitable for modelling the…
New exact wave solutions for Hirota equation
Indian Academy of Sciences (India)
... integrals in polynomial form with a high accuracy for two-dimensional plane autonomous systems. Exact soliton solution is constructed through the established first integrals. This method is a powerful tool for searching exact travelling solutions of nonlinear partial differential equations (NPDEs) in mathematical physics.
A numerical solution for a telegraph equation
Ashyralyev, Allaberen; Modanli, Mahmut
2014-08-01
In this study, the initial value problem for a telegraph equation in a Hilbert space is considered. The stability estimate for the solution of this problem is given. A first and a second order of approximation difference schemes approximately solving the initial value problem are presented. The stability estimates for the solution of these difference schemes are given. The theoretical statements for the solution of these difference schemes are supported by the results of numerical experiments.
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
Exact solutions of the time-fractional Fisher equation by using modified trial equation method
Tandogan, Yusuf Ali; Bildik, Necdet
2016-06-01
In this study, modified trial equation method has been proposed to obtain precise solutions of nonlinear fractional differential equation. Using the modified test equation method, we obtained some new exact solutions of the time fractional nonlinear Fisher equation. The obtained results are classified as a soliton solution, singular solutions, rational function solutions and periodic solutions.
Soliton solutions for a quasilinear Schrodinger equation
Directory of Open Access Journals (Sweden)
Duchao Liu
2013-12-01
Full Text Available In this article, critical point theory is used to show the existence of nontrivial weak solutions to the quasilinear Schrodinger equation $$ -\\Delta_p u-\\frac{p}{2^{p-1}}u\\Delta_p(u^2=f(x,u $$ in a bounded smooth domain $\\Omega\\subset\\mathbb{R}^{N}$ with Dirichlet boundary conditions.
Analytical solution of population balance equation involving ...
Indian Academy of Sciences (India)
Laplace transform obtained from literature. ... used in the literature. Keywords. Population balance; aggregation; breakage; auxiliary equation method; Laplace transform. PACS Nos 02.70.−c; 02.30.Mv; 02.30.Jr. 1. ...... assumptions proposed earlier, a more realistic representation of the solutions is obtained compared to the ...
Iterative solution of the semiconductor device equations
Energy Technology Data Exchange (ETDEWEB)
Bova, S.W.; Carey, G.F. [Univ. of Texas, Austin, TX (United States)
1996-12-31
Most semiconductor device models can be described by a nonlinear Poisson equation for the electrostatic potential coupled to a system of convection-reaction-diffusion equations for the transport of charge and energy. These equations are typically solved in a decoupled fashion and e.g. Newton`s method is used to obtain the resulting sequences of linear systems. The Poisson problem leads to a symmetric, positive definite system which we solve iteratively using conjugate gradient. The transport equations lead to nonsymmetric, indefinite systems, thereby complicating the selection of an appropriate iterative method. Moreover, their solutions exhibit steep layers and are subject to numerical oscillations and instabilities if standard Galerkin-type discretization strategies are used. In the present study, we use an upwind finite element technique for the transport equations. We also evaluate the performance of different iterative methods for the transport equations and investigate various preconditioners for a few generalized gradient methods. Numerical examples are given for a representative two-dimensional depletion MOSFET.
Nicasio-Collazo, Luz Adriana; Delgado-González, Alexandra; Hernández-Lemus, Enrique; Castañeda-Priego, Ramón
2017-04-01
The study of the effects associated with the electrostatic properties of DNA is of fundamental importance to understand both its molecular properties at the single molecule level, like the rigidity of the chain, and its interaction with other charged bio-molecules, including other DNA molecules; such interactions are crucial to maintain the thermodynamic stability of the intra-cellular medium. In the present work, we combine the Poisson-Boltzmann mean-field theory with an irreversible thermodynamic approximation to analyze the effects of counterion accumulation inside DNA on both the denaturation profile of the chain and the equation of state of the suspension. To this end, we model the DNA molecule as a porous charged cylinder immersed in an aqueous solution. These thermo-electrostatic effects are explicitly studied in the particular case of some genes for which damage in their sequence is associated with diffuse large B-cell lymphoma.
Method of lines solution of Richards` equation
Energy Technology Data Exchange (ETDEWEB)
Kelley, C.T.; Miller, C.T.; Tocci, M.D.
1996-12-31
We consider the method of lines solution of Richard`s equation, which models flow through porous media, as an example of a situation in which the method can give incorrect results because of premature termination of the nonlinear corrector iteration. This premature termination arises when the solution has a sharp moving front and the Jacobian is ill-conditioned. While this problem can be solved by tightening the tolerances provided to the ODE or DAE solver used for the temporal integration, it is more efficient to modify the termination criteria of the nonlinear solver and/or recompute the Jacobian more frequently. In this paper we continue previous work on this topic by analyzing the modifications in more detail and giving a strategy on how the modifications can be turned on and off in response to changes in the character of the solution.
Hypergeometric solutions to Schr\\"odinger equations for the quantum Painlev\\'e equations
Nagoya, Hajime
2011-01-01
We consider Schr\\"odinger equations for the quantum Painlev\\'e equations. We present hypergeometric solutions of the Schr\\"odinger equations for the quantum Painlev\\'e equations, as particular solutions. We also give a representation theoretic correspondence between Hamiltonians of the Schr\\"odinger equations for the quantum Painlev\\'e equations and those of the KZ equation or the confluent KZ equations.
Iterative solution of the Helmholtz equation
Energy Technology Data Exchange (ETDEWEB)
Larsson, E.; Otto, K. [Uppsala Univ. (Sweden)
1996-12-31
We have shown that the numerical solution of the two-dimensional Helmholtz equation can be obtained in a very efficient way by using a preconditioned iterative method. We discretize the equation with second-order accurate finite difference operators and take special care to obtain non-reflecting boundary conditions. We solve the large, sparse system of equations that arises with the preconditioned restarted GMRES iteration. The preconditioner is of {open_quotes}fast Poisson type{close_quotes}, and is derived as a direct solver for a modified PDE problem.The arithmetic complexity for the preconditioner is O(n log{sub 2} n), where n is the number of grid points. As a test problem we use the propagation of sound waves in water in a duct with curved bottom. Numerical experiments show that the preconditioned iterative method is very efficient for this type of problem. The convergence rate does not decrease dramatically when the frequency increases. Compared to banded Gaussian elimination, which is a standard solution method for this type of problems, the iterative method shows significant gain in both storage requirement and arithmetic complexity. Furthermore, the relative gain increases when the frequency increases.
Analyticity of solutions of singular fractional differential equations
Kangro, Urve
2016-06-01
We study singular fractional differential equations in spaces of analytic functions. We reformulate the equation as a cordial Volterra integral equation of the second kind and use results from the theory of cordial Volterra integral equations. This enables us to obtain conditions under which the equation has a unique analytic solution. Note that the smooth solution in this case is unique without any initial conditions; in fact, giving initial conditions usually results in nonsmooth solution. We also consider approximate solution of these equations and prove exponential convergence of approximate solutions to the exact solution.
Numerical solution of the bidomain equations.
Linge, S; Sundnes, J; Hanslien, M; Lines, G T; Tveito, A
2009-05-28
Knowledge of cardiac electrophysiology is efficiently formulated in terms of mathematical models. However, most of these models are very complex and thus defeat direct mathematical reasoning founded on classical and analytical considerations. This is particularly so for the celebrated bidomain model that was developed almost 40 years ago for the concurrent analysis of extra- and intracellular electrical activity. Numerical simulations based on this model represent an indispensable tool for studying electrophysiology. However, complex mathematical models, steep gradients in the solutions and complicated geometries lead to extremely challenging computational problems. The greatest achievement in scientific computing over the past 50 years has been to enable the solving of linear systems of algebraic equations that arise from discretizations of partial differential equations in an optimal manner, i.e. such that the central processing unit (CPU) effort increases linearly with the number of computational nodes. Over the past decade, such optimal methods have been introduced in the simulation of electrophysiology. This development, together with the development of affordable parallel computers, has enabled the solution of the bidomain model combined with accurate cellular models, on geometries resembling a human heart. However, in spite of recent progress, the full potential of modern computational methods has yet to be exploited for the solution of the bidomain model. This paper reviews the development of numerical methods for solving the bidomain model. However, the field is huge and we thus restrict our focus to developments that have been made since the year 2000.
Analytic Solutions of Special Functional Equations
Directory of Open Access Journals (Sweden)
Octav Olteanu
2013-07-01
Full Text Available We recall some of our earlier results on the construction of a mapping defined implicitly, without using the implicit function theorem. All these considerations work in the real case, for functions and operators. Then we consider the complex case, proving the analyticity of the function defined implicitly, under certain hypothesis. Some consequences are given. An approximating formula for the analytic form of the solution is also given. Finally, one illustrates the preceding results by an application to a concrete functional and operatorial equation. Some related examples are given.
Numerical methods for solution of singular integral equations
Boykov, I. V.
2016-01-01
This paper is devoted to overview of the authors works for numerical solution of singular integral equations (SIE), polysingular integral equations and multi-dimensional singular integral equations of the second kind. The authors investigated onsidered iterative - projective methods and parallel methods for solution of singular integral equations, polysingular integral equations and multi-dimensional singular integral equations. The paper is the second part of overview of the authors works de...
Integral solutions of fractional evolution equations with nondense domain
Directory of Open Access Journals (Sweden)
Haibo Gu
2017-06-01
Full Text Available In this article, we study the existence of integral solutions for two classes of fractional order evolution equations with nondensely defined linear operators. First, we consider the nonhomogeneous fractional order evolution equation and obtain its integral solution by Laplace transform and probability density function. Subsequently, based on the form of integral solution for nonhomogeneous fractional order evolution equation, we investigate the existence of integral solution for nonlinear fractional order evolution equation by noncompact measure method.
Fractional solutions of Bessel equation with N-method.
Bas, Erdal; Yilmazer, Resat; Panakhov, Etibar
2013-01-01
This paper deals with the design fractional solution of Bessel equation. We obtain explicit solutions of the equation with the help of fractional calculus techniques. Using the N-fractional calculus operator N(ν) method, we derive the fractional solutions of the equation.
The exact solutions of differential equation with delay
Hasebe, K; Sugiyama, Y
1998-01-01
The exact solutions of the first order differential equation with delay are derived. The equation has been introduced as a model of traffic flow. The solution describes the traveling cluster of jam, which is characterized by Jacobi's elliptic function. We also obtain the family of solutions of such type of equations.
Exact solution of the neutron transport equation in spherical geometry
Energy Technology Data Exchange (ETDEWEB)
Anli, Fikret; Akkurt, Abdullah; Yildirim, Hueseyin; Ates, Kemal [Kahramanmaras Suetcue Imam Univ. (Turkey). Faculty of Sciences and Letters
2017-03-15
Solution of the neutron transport equation in one dimensional slab geometry construct a basis for the solution of neutron transport equation in a curvilinear geometry. Therefore, in this work, we attempt to derive an exact analytical benchmark solution for both neutron transport equations in slab and spherical medium by using P{sub N} approximation which is widely used in neutron transport theory.
Exact traveling wave solutions of some nonlinear evolution equations
Kumar, Hitender; Chand, Fakir
2014-02-01
Using a traveling wave reduction technique, we have shown that Maccari equation, (2+1)-dimensional nonlinear Schrödinger equation, medium equal width equation, (3+1)-dimensional modified KdV-Zakharov-Kuznetsev equation, (2+1)-dimensional long wave-short wave resonance interaction equation, perturbed nonlinear Schrödinger equation can be reduced to the same family of auxiliary elliptic-like equations. Then using extended F-expansion and projective Riccati equation methods, many types of exact traveling wave solutions are obtained. With the aid of solutions of the elliptic-like equation, more explicit traveling wave solutions expressed by the hyperbolic functions, trigonometric functions and rational functions are found out. It is shown that these methods provide a powerful mathematical tool for solving nonlinear evolution equations in mathematical physics. A variety of structures of the exact solutions of the elliptic-like equation are illustrated.
Solution of diffusion equation in deformable spheroids
Energy Technology Data Exchange (ETDEWEB)
Ayyoubzadeh, Seyed Mohsen [Department of Energy Engineering, Sharif University of Technology, Tehran (Iran, Islamic Republic of); Safari, Mohammad Javad, E-mail: iFluka@gmail.com [Department of Energy Engineering, Sharif University of Technology, Tehran (Iran, Islamic Republic of); Vosoughi, Naser [Department of Energy Engineering, Sharif University of Technology, Tehran (Iran, Islamic Republic of)
2011-05-15
Research highlights: > Developing an explicit solution for the diffusion equation in spheroidal geometry. > Proving an orthogonality relation for spheroidal eigenfunctions. > Developing a relation for the extrapolation distance in spheroidal geometry. > Considering the sphere and slab as limiting cases for a spheroid. > Cross-validation of the analytical solution with Monte Carlo simulations. - Abstract: The time-dependent diffusion of neutrons in a spheroid as a function of the focal distance has been studied. The solution is based on an orthogonal basis and an extrapolation distanced related boundary condition for the spheroidal geometry. It has been shown that spheres and disks are two limiting cases for the spheroids, for which there is a smooth transition for the systems properties between these two limits. Furthermore, it is demonstrated that a slight deformation from a sphere does not affect the fundamental mode properties, to the first order. The calculations for both multiplying and non-multiplying media have been undertaken, showing good agreement with direct Monte Carlo simulations.
Minimal solution for inconsistent singular fuzzy matrix equations
Nikuie, M.; M.K. Mirnia
2013-01-01
The fuzzy matrix equations $Ailde{X}=ilde{Y}$ is called a singular fuzzy matrix equations while the coefficients matrix of its equivalent crisp matrix equations be a singular matrix. The singular fuzzy matrix equations are divided into two parts: consistent singular matrix equations and inconsistent fuzzy matrix equations. In this paper, the inconsistent singular fuzzy matrix equations is studied and the effect of generalized inverses in finding minimal solution of an inconsistent singular fu...
Soliton solutions of some nonlinear evolution equations with time ...
Indian Academy of Sciences (India)
solutions for the mKdV equation are also given. Keywords. Dark and bright soliton; KdV equation; nonlinear Schrödinger equation; G(m, n) equation. PACS Nos 42.81.Dp; 42.65.Tg; 05.45.Yv. 1. Introduction. To find exact solutions of the nonlinear evolution equations (NLEEs) is one of the cen- tral themes in mathematics and ...
Stability of Stationary Solutions of the Multifrequency Radiation Diffusion Equations
Energy Technology Data Exchange (ETDEWEB)
Hald, O H; Shestakov, A I
2004-01-20
A nondimensional model of the multifrequency radiation diffusion equation is derived. A single material, ideal gas, equation of state is assumed. Opacities are proportional to the inverse of the cube of the frequency. Inclusion of stimulated emission implies a Wien spectrum for the radiation source function. It is shown that the solutions are uniformly bounded in time and that stationary solutions are stable. The spatially independent solutions are asymptotically stable, while the spatially dependent solutions of the linearized equations approach zero.
Semianalytic Solution of Space-Time Fractional Diffusion Equation
Directory of Open Access Journals (Sweden)
A. Elsaid
2016-01-01
Full Text Available We study the space-time fractional diffusion equation with spatial Riesz-Feller fractional derivative and Caputo fractional time derivative. The continuation of the solution of this fractional equation to the solution of the corresponding integer order equation is proved. The series solution of this problem is obtained via the optimal homotopy analysis method (OHAM. Numerical simulations are presented to validate the method and to show the effect of changing the fractional derivative parameters on the solution behavior.
New exact travelling wave solutions of some complex nonlinear equations
Bekir, Ahmet
2009-04-01
In this paper, we establish exact solutions for complex nonlinear equations. The tanh-coth and the sine-cosine methods are used to construct exact periodic and soliton solutions of these equations. Many new families of exact travelling wave solutions of the coupled Higgs and Maccari equations are successfully obtained. These solutions may be important of significance for the explanation of some practical physical problems.
New Exact Solutions of the New Hamiltonian Amplitude-Equation and Fokas Lenells Equation
Directory of Open Access Journals (Sweden)
Seyma Tuluce Demiray
2015-08-01
Full Text Available In this paper, exact solutions of the new Hamiltonian amplitude equation and Fokas-Lenells equation are successfully obtained. The extended trial equation method (ETEM and generalized Kudryashov method (GKM are applied to find several exact solutions of the new Hamiltonian amplitude equation and Fokas-Lenells equation. Primarily, we seek some exact solutions of the new Hamiltonian amplitude equation and Fokas-Lenells equation by using ETEM. Then, we research dark soliton solutions of the new Hamiltonian amplitude equation and Fokas-Lenells equation by using GKM. Lastly, according to the values of some parameters, we draw two and three dimensional graphics of imaginary and real values of certain solutions found by utilizing both methods.
Nontrivial Periodic Solutions for Nonlinear Second-Order Difference Equations
Directory of Open Access Journals (Sweden)
Tieshan He
2011-01-01
Full Text Available This paper is concerned with the existence of nontrivial periodic solutions and positive periodic solutions to a nonlinear second-order difference equation. Under some conditions concerning the first positive eigenvalue of the linear equation corresponding to the nonlinear second-order equation, we establish the existence results by using the topological degree and fixed point index theories.
Intuitive Understanding of Solutions of Partially Differential Equations
Kobayashi, Y.
2008-01-01
This article uses diagrams that help the observer see how solutions of the wave equation and heat conduction equation are obtained. The analytical approach cannot necessarily show the mechanisms of the key to the solution without transforming the differential equation into a more convenient form by separation of variables. The visual clues based…
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.
Soliton solutions of Hirota equation and Hirota-Maccari system
Directory of Open Access Journals (Sweden)
Ahmed Arnous
2016-07-01
Full Text Available In this paper, the trial equation method is presented to seek the exact solutions of two nonlinear partial differential equations (NLPDEs, namely, the Hirota equation and the Hirota-Maccari system. The obtained solutions are solitary, topological, singular solitons and singular periodic waves. This method is powerful, effective and it can be extended to many NLPDEs.
On the Solutions of Some Linear Complex Quaternionic Equations
İpek, Ahmet
2014-01-01
Some complex quaternionic equations in the type AX − XB = C are investigated. For convenience, these equations were called generalized Sylvester-quaternion equations, which include the Sylvester equation as special cases. By the real matrix representations of complex quaternions, the necessary and sufficient conditions for the solvability and the general expressions of the solutions are obtained. PMID:25101318
On the Solutions of Some Linear Complex Quaternionic Equations
Directory of Open Access Journals (Sweden)
Cennet Bolat
2014-01-01
Full Text Available Some complex quaternionic equations in the type AX-XB=C are investigated. For convenience, these equations were called generalized Sylvester-quaternion equations, which include the Sylvester equation as special cases. By the real matrix representations of complex quaternions, the necessary and sufficient conditions for the solvability and the general expressions of the solutions are obtained.
Perturbation Solutions of the Quintic Duffing Equation with Strong Nonlinearities
Directory of Open Access Journals (Sweden)
Mehmet Pakdemirli
Full Text Available The quintic Duffing equation with strong nonlinearities is considered. Perturbation solutions are constructed using two different techniques: The classical multiple scales method (MS and the newly developed multiple scales Lindstedt Poincare method (MSLP. The validity criteria for admissible solutions are derived. Both approximate solutions are contrasted with the numerical solutions. It is found that MSLP provides compatible solution with the numerical solution for strong nonlinearities whereas MS solution fail to produce physically acceptable solution for large perturbation parameters.
Differential and difference equations a comparison of methods of solution
Maximon, Leonard C
2016-01-01
This book, intended for researchers and graduate students in physics, applied mathematics and engineering, presents a detailed comparison of the important methods of solution for linear differential and difference equations - variation of constants, reduction of order, Laplace transforms and generating functions - bringing out the similarities as well as the significant differences in the respective analyses. Equations of arbitrary order are studied, followed by a detailed analysis for equations of first and second order. Equations with polynomial coefficients are considered and explicit solutions for equations with linear coefficients are given, showing significant differences in the functional form of solutions of differential equations from those of difference equations. An alternative method of solution involving transformation of both the dependent and independent variables is given for both differential and difference equations. A comprehensive, detailed treatment of Green’s functions and the associat...
Analysis of solutions of a nonlinear scalar field differential equation
Muhamadiev, E. M.; Naimov, A. N.
2017-10-01
We consider a nonlinear differential equation arising in mathematical models of elementary particle theory. For this equation, we examine questions of the extendability of solutions, the boundedness of solutions at infinity, and the search for new conditions for the existence of a positive particle-like solution.
Exact solutions of some nonlinear partial differential equations using ...
Indian Academy of Sciences (India)
Research Articles Volume 81 Issue 2 August 2013 pp 225-236 ... Abstract. The functional variable method is a powerful solution method for obtaining exact solutions of some nonlinear partial differential equations. In this paper ... By using this useful method, we found some exact solutions of the above-mentioned equations.
Stationary solutions and Neumann boundary conditions in the Sivashinsky equation.
Denet, Bruno
2006-09-01
New stationary solutions of the (Michelson) Sivashinsky equation of premixed flames are obtained numerically in this paper. Some of these solutions, of the bicoalescent type recently described by Guidi and Marchetti, are stable with Neumann boundary conditions. With these boundary conditions, the time evolution of the Sivashinsky equation in the presence of a moderate white noise is controlled by jumps between stationary solutions.
Solutions to Class of Linear and Nonlinear Fractional Differential Equations
Abdel-Salam, Emad A.-B.; Hassan, Gamal F.
2016-02-01
In this paper, the fractional auxiliary sub-equation expansion method is proposed to solve nonlinear fractional differential equations. To illustrate the effectiveness of the method, we discuss the space-time fractional KdV equation, the space-time fractional RLW equation, the space-time fractional Boussinesq equation, and the (3+1)-space-time fractional ZK equation. The solutions are expressed in terms of fractional hyperbolic and fractional trigonometric functions. These solutions are useful to understand the mechanisms of the complicated nonlinear physical phenomena and fractional differential equations. Among these solutions, some are found for the first time. The analytical solution of homogenous linear FDEs with constant coefficients are obtained by using the series and the Mittag-Leffler function methods. The obtained results recover the well-know solutions when α = 1.
Analytical Solutions of Classical and Fractional KP-Burger Equation and Coupled KdV equation
Ghosh, Uttam; Sarkar, Susmita; Das, Shantanu
2016-01-01
Evaluation of analytical solutions of non-linear partial differential equations (both classical and fractional) is a rising subject in Applied Mathematics because its applications in Physical biological and social sciences. In this paper we have used generalized Tanh method to find the exact solution of KP-Burger equation and coupled KdV equation. The fractional Sub-equation method has been used to find the solution of fractional KP-Burger equation and fractional coupled KdV equations. The ex...
Stochastic solution to a time-fractional attenuated wave equation.
Meerschaert, Mark M; Straka, Peter; Zhou, Yuzhen; McGough, Robert J
2012-10-01
The power law wave equation uses two different fractional derivative terms to model wave propagation with power law attenuation. This equation averages complex nonlinear dynamics into a convenient, tractable form with an explicit analytical solution. This paper develops a random walk model to explain the appearance and meaning of the fractional derivative terms in that equation, and discusses an application to medical ultrasound. In the process, a new strictly causal solution to this fractional wave equation is developed.
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.
Exact solution of some linear matrix equations using algebraic methods
Djaferis, T. E.; Mitter, S. K.
1977-01-01
A study is done of solution methods for Linear Matrix Equations including Lyapunov's equation, using methods of modern algebra. The emphasis is on the use of finite algebraic procedures which are easily implemented on a digital computer and which lead to an explicit solution to the problem. The action f sub BA is introduced a Basic Lemma is proven. The equation PA + BP = -C as well as the Lyapunov equation are analyzed. Algorithms are given for the solution of the Lyapunov and comment is given on its arithmetic complexity. The equation P - A'PA = Q is studied and numerical examples are given.
New Numerical Solution of von Karman Equation of Lengthwise Rolling
Rudolf Pernis; Tibor Kvackaj
2015-01-01
The calculation of average material contact pressure to rolls base on mathematical theory of rolling process given by Karman equation was solved by many authors. The solutions reported by authors are used simplifications for solution of Karman equation. The simplifications are based on two cases for approximation of the circular arch: (a) by polygonal curve and (b) by parabola. The contribution of the present paper for solution of two-dimensional differential equation of rol...
Renormalized asymptotic solutions of the Burgers equation and the Korteweg-de Vries equation
Zakharov, Sergei V.
2015-01-01
The Cauchy problem for the Burgers equation and the Korteweg-de Vries equation is considered. Uniform renormalized asymptotic solutions are constructed in cases of a large initial gradient and a perturbed initial weak discontinuity.
Asymptotics of solutions to semilinear stochastic wave equations
Chow, Pao-Liu
2006-01-01
Large-time asymptotic properties of solutions to a class of semilinear stochastic wave equations with damping in a bounded domain are considered. First an energy inequality and the exponential bound for a linear stochastic equation are established. Under appropriate conditions, the existence theorem for a unique global solution is given. Next the questions of bounded solutions and the exponential stability of an equilibrium solution, in mean-square and the almost sure sense, are studied. Then...
Solutions of fractional diffusion equations by variation of parameters method
Directory of Open Access Journals (Sweden)
Mohyud-Din Syed Tauseef
2015-01-01
Full Text Available This article is devoted to establish a novel analytical solution scheme for the fractional diffusion equations. Caputo’s formulation followed by the variation of parameters method has been employed to obtain the analytical solutions. Following the derived analytical scheme, solution of the fractional diffusion equation for several initial functions has been obtained. Graphs are plotted to see the physical behavior of obtained solutions.
The Investigation of Solutions to the Coupled Schrödinger-Boussinesq Equations
Directory of Open Access Journals (Sweden)
Xin Huang
2013-01-01
equations. The hyperbolic function solutions, trigonometric function solutions, and rational function solutions to the equations are obtained. The decaying properties of several solutions are analyzed.
Graphical Solution of the Monic Quadratic Equation with Complex Coefficients
Laine, A. D.
2015-01-01
There are many geometrical approaches to the solution of the quadratic equation with real coefficients. In this article it is shown that the monic quadratic equation with complex coefficients can also be solved graphically, by the intersection of two hyperbolas; one hyperbola being derived from the real part of the quadratic equation and one from…
Probabilistic representations of solutions to the heat equation
Indian Academy of Sciences (India)
R. Narasimhan (Krishtel eMaging) 1461 1996 Oct 15 13:05:22
representation of solutions of the heat equation to initial conditions that are arbitrary tempered distributions. Keywords. Brownian motion; heat equation; translation operators; infinite dimen- sional stochastic differential equations. 1. Introduction. Let (Xt)t≥0 be a d-dimensional Brownian motion, with X0 ≡ 0. Let ϕ ∈ S (Rd), ...
Series solution for the complete golden dynamical equation of ...
African Journals Online (AJOL)
... Dynamical Equation of motion for photon in the gravitational field of a massive body was published. In this paper the series method is used to solve this equation for comparison with the solutions of Einstein Equation for the photon in the same gravitational field. A value of 1.875” was found as the total deflection angle.
207 series solution for the complete golden dynamical equation of ...
African Journals Online (AJOL)
DR. AMINU
ABSTRACTS. In a paper (Howusu, 2004) the complete Golden Dynamical Equation of motion for photon in the gravitational field of a massive body was published. In this paper the series method is used to solve this equation for comparison with the solutions of Einstein Equation for the photon in the same gravitational field.
Explicit Solutions and Conservation Laws of a Coupled Burgers' Equation
Xue, Bo; Li, Fang; Li, Yihao; Sun, Mingming
2017-08-01
Based on the gauge transformation between the corresponding 3×3 matrix spectral problems, N-fold Darboux transformation for a coupled Burgers' equation is constructed. Considering the N=1 case of the derived Darboux transformation, explicit solutions for the coupled Burgers' equation are given and their figures are plotted. Moreover, conservation laws of this integrable equation are deduced.
Unsteady Stokes equations: Some complete general solutions
Indian Academy of Sciences (India)
R. Narasimhan (Krishtel eMaging) 1461 1996 Oct 15 13:05:22
homogeneous unsteady Stokes equations are examined. A necessary and sufficient condition for a divergence-free vector to represent the velocity field of a possible unsteady Stokes flow in the absence of body forces is derived. Keywords. Complete ...
On Newton's method for Riccati equation solution
Sandell, N. R., Jr.
1974-01-01
It is shown that the assumptions of controllability and observability in two theorems of Kleinman (1968, 1970) concerning Newton's method for the Ricatti equation can be weakened to stabilizability and detectability. Empirically, this has been known for some time.
Approximate Solution of nth-Order Fuzzy Linear Differential Equations
Directory of Open Access Journals (Sweden)
Xiaobin Guo
2013-01-01
Full Text Available The approximate solution of nth-order fuzzy linear differential equations in which coefficient functions maintain the sign is investigated by the undetermined fuzzy coefficients method. The differential equations is converted to a crisp function system of linear equations according to the operations of fuzzy numbers. The fuzzy approximate solution of the fuzzy linear differential equation is obtained by solving the crisp linear equations. Some numerical examples are given to illustrate the proposed method. It is an extension of Allahviranloo's results.
Uniqueness of singular solution of semilinear elliptic equation
Indian Academy of Sciences (India)
Information Science, Henan University, Kaifeng 475004, People's Republic of China. E-mail: laibaishun@henu.edu.cn ... Keywords. Nonhomogeneous semilinear elliptic equation; positive solutions; asymptotic behavior; singular solutions. 1. Introduction. In this paper, we study the elliptic equation u + K(|x|)up + μf (|x|) = 0,.
Iterative estimate of the solution of nonlinear integral equations by ...
African Journals Online (AJOL)
The paper considered the application of Picard's iteration scheme in the approximation of solutions of operator equations in Banach spaces. Using Lipschitz continuity condition and the prescribed auxiliary scalar function, the location of existence of solution for a nonlinear integral equation of Fredholm type and second kind ...
Numerical solution of the one-dimensional Burgers' equation ...
Indian Academy of Sciences (India)
Numerical solution of the one-dimensional Burgers' equation: Implicit and fully implicit exponential finite difference methods ... Research Articles Volume 81 Issue 4 October 2013 pp 547-556 ... This paper describes two new techniques which give improved exponential finite difference solutions of Burgers' equation.
Existence Results for Solutions of Nonlinear Fractional Differential Equations
Directory of Open Access Journals (Sweden)
Ali Yakar
2012-01-01
which generate a closed set. The existence of solutions for nonlinear fractional differential equations involving Riemann-Liouville differential operator in a closed set is obtained by utilizing various types of coupled upper and lower solutions. Furthermore, these results are extended to the finite systems of nonlinear fractional differential equations leading to more general results.
Maximal saddle solution of a nonlinear elliptic equation involving the ...
Indian Academy of Sciences (India)
p-Laplacian, Nonlinear Diff. Equ. Appl. 18 (2011) 101–114. [9] Kowalczyk M and Liu Y, Nondegeneracy of the saddle solution of the Allen–Cahn equation, Proc. Amer. Math. Soc. 139(12) (2011) 4319–4329. [10] Schatzman M, On the stability of the saddle solution of Allen-Cahn's equation, Proc. Roy. Soc. Edinburgh Sect.
Existence of Positive Solutions of Neutral Differential Equations
Directory of Open Access Journals (Sweden)
B. Dorociaková
2012-01-01
Full Text Available The paper contains some suffcient conditions for the existence of positive solutions which are bounded below and above by positive functions for the nonlinear neutral differential equations of higher order. These equations can also support the existence of positive solutions approaching zero at infinity.
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.
Spreadsheet Solution of Systems of Nonlinear Differential Equations
Ali El-Hajj, A. et al
2004-01-01
This paper presents a method for obtaining numerical approximation to solutions of systems of nonlinear differential equations of one variable using spreadsheets. This solution method is simply based on effecting an integration formula in a block. The input of the block is equal to a given variable and its output is equal to the integral of that variable. The block allows the user to set the initial condition for differential equation. A nonlinear differential equation of order n is solved th...
Numerical Solution of Turbulence Problems by Solving Burgers’ Equation
Directory of Open Access Journals (Sweden)
Alicia Cordero
2015-05-01
Full Text Available In this work we generate the numerical solutions of Burgers’ equation by applying the Crank-Nicholson method and different schemes for solving nonlinear systems, instead of using Hopf-Cole transformation to reduce Burgers’ equation into the linear heat equation. The method is analyzed on two test problems in order to check its efficiency on different kinds of initial conditions. Numerical solutions as well as exact solutions for different values of viscosity are calculated, concluding that the numerical results are very close to the exact solution.
Numerical solutions of telegraph equations with the Dirichlet boundary condition
Ashyralyev, Allaberen; Turkcan, Kadriye Tuba; Koksal, Mehmet Emir
2016-08-01
In this study, the Cauchy problem for telegraph equations in a Hilbert space is considered. Stability estimates for the solution of this problem are presented. The third order of accuracy difference scheme is constructed for approximate solutions of the problem. Stability estimates for the solution of this difference scheme are established. As a test problem to support theoretical results, one-dimensional telegraph equation with the Dirichlet boundary condition is considered. Numerical solutions of this equation are obtained by first, second and third order of accuracy difference schemes.
The solution of the generalized Kepler's equation
López, Rosario; Hautesserres, Denis; San-Juan, Juan Félix
2018-01-01
In the context of general perturbation theories, the main problem of the artificial satellite analyses the motion of an orbiter around an Earth-like planet, only perturbed by its equatorial bulge or J2 effect. By means of a Lie transform and the Krylov-Bogoliubov-Mitropolsky method, a first-order theory in closed form of the eccentricity is produced. During the evaluation of the theory, it is necessary to solve a generalization of the classical Kepler's equation. In this work, the application of a numerical technique and three initial guesses to the generalized Kepler's equation are discussed.
Numerical Solution of Differential Algebraic Equations
DEFF Research Database (Denmark)
Thomsen, Per Grove; Bendtsen, Claus
1999-01-01
Lecture notes for a PhD-course on the numerical solution of DAE's. The course was held at IMM in the autumn of 1998 and the early spring of 1999.......Lecture notes for a PhD-course on the numerical solution of DAE's. The course was held at IMM in the autumn of 1998 and the early spring of 1999....
Topography retrieval using different solutions of the transport intensity equation.
Pinhasi, Shirly V; Alimi, Roger; Perelmutter, Lior; Eliezer, Shalom
2010-10-01
The topography of a phase plate is recovered from the phase reconstruction by solving the transport intensity equation (TIE). The TIE is solved using two different approaches: (a) the classical solution of solving the Poisson differential equation and (b) an algebraic approach with Zernike functions. In this paper we present and compare the topography reconstruction of a phase plate with these solution methods and justify why one solution is preferable over the other.
Numerical Solution of the Beltrami Equation
Porter, R. Michael
2008-01-01
An effective algorithm is presented for solving the Beltrami equation fzbar = mu fz in a planar disk. The algorithm involves no evaluation of singular integrals. The strategy, working in concentric rings, is to construct a piecewise linear mu-conformal mapping and then correct the image using a known algorithm for conformal mappings. Numerical examples are provided and the computational complexity is analyzed.
Equation Solution Figures of Merit, Metaheuristic Search, and the Schrodinger Equation
MacNeil, Paul
2014-03-01
This presentation deals with: a definition of ``equation error'' a consideration of equation solution figures of merit based on equation error, and on other measures; and the use of metaheuristic techniques in the search for approximate solutions. These considerations are illustrated by application to the Schrodinger equation for a simple system. Models suitable for computation are produced. Computation results are used to compare the consequences of selection of different figures of merit. ``Equation error'' is defined to be the quantity by which an approximate solution fails to satisfy an equation. ``Equation error variance'' is defined to be the squared modulus of the equation error summed/integrated over the domain of interest. (Generalization to sets of equations is straightforward.) In the example, equation error variance is a functional of the Schrodinger wave function. Possible figures of merit include: ground state energy, system geometry, and equation solution variance. The (derivative-free) metaheuristic used to solve the Schrodinger equation has been changed from a genetic algorithm, used in earlier versions of this research, to evolution strategy with covariance matrix adaptation.
Mallet, D. G.; McCue, S. W.
2009-01-01
The solution of linear ordinary differential equations (ODEs) is commonly taught in first-year undergraduate mathematics classrooms, but the understanding of the concept of a solution is not always grasped by students until much later. Recognizing what it is to be a solution of a linear ODE and how to postulate such solutions, without resorting to…
Computational Solution of a Fractional Integro-Differential Equation
Directory of Open Access Journals (Sweden)
Muhammet Kurulay
2013-01-01
Full Text Available Although differential transform method (DTM is a highly efficient technique in the approximate analytical solutions of fractional differential equations, applicability of this method to the system of fractional integro-differential equations in higher dimensions has not been studied in detail in the literature. The major goal of this paper is to investigate the applicability of this method to the system of two-dimensional fractional integral equations, in particular to the two-dimensional fractional integro-Volterra equations. We deal with two different types of systems of fractional integral equations having some initial conditions. Computational results indicate that the results obtained by DTM are quite close to the exact solutions, which proves the power of DTM in the solutions of these sorts of systems of fractional integral equations.
Tisdell, C. C.
2017-01-01
Solution methods to exact differential equations via integrating factors have a rich history dating back to Euler (1740) and the ideas enjoy applications to thermodynamics and electromagnetism. Recently, Azevedo and Valentino presented an analysis of the generalized Bernoulli equation, constructing a general solution by linearizing the problem…
Nonoscillatory half-linear difference equations and recessive solutions
Directory of Open Access Journals (Sweden)
Mauro Marini
2005-05-01
Full Text Available Recessive and dominant solutions for the nonoscillatory half-linear difference equation are investigated. By using a uniqueness result for the zero-convergent solutions satisfying a suitable final condition, we prove that recessive solutions are the Ã¢Â€Âœsmallest solutions in a neighborhood of infinity,Ã¢Â€Â like in the linear case. Other asymptotic properties of recessive and dominant solutions are treated too.
Periodic Solution of the Hematopoiesis Equation
Directory of Open Access Journals (Sweden)
Ji-Huan He
2013-01-01
Full Text Available Wu and Liu (2012 presented some results for the existence and uniqueness of the periodic solutions for the hematopoiesis model. This paper gives a simple approach to find an approximate period of the model.
Soliton-like solutions to the ordinary Schroedinger equation
Energy Technology Data Exchange (ETDEWEB)
Zamboni-Rached, Michel [Universidade Estadual de Campinas (DMO/FEEC/UNICAMP), Campinas, SP (Brazil). Fac. de Engenharia Eletrica e de Computacao. Dept. de Microondas e Optica; Recami, Erasmo, E-mail: recami@mi.infn.i [Universita Statale di Bergamo, Bergamo (Italy). Facolta di Ingegneria
2011-07-01
In recent times it has been paid attention to the fact that (linear) wave equations admit of soliton-like solutions, known as Localized Waves or Non-diffracting Waves, which propagate without distortion in one direction. Such Localized Solutions (existing also for K-G or Dirac equations) are a priori suitable, more than Gaussian's, for describing elementary particle motion. In this paper we show that, mutatis mutandis, Localized Solutions exist even for the ordinary Schroedinger equation within standard Quantum Mechanics; and we obtain both approximate and exact solutions, also setting forth for them particular examples. In the ideal case such solutions bear infinite energy, as well as plane or spherical waves: we show therefore how to obtain nite-energy solutions. At last, we briefly consider solutions for a particle moving in the presence of a potential. (author)
Positive Solutions for Coupled Nonlinear Fractional Differential Equations
Directory of Open Access Journals (Sweden)
Wenning Liu
2014-01-01
Full Text Available We consider the existence of positive solutions for a coupled system of nonlinear fractional differential equations with integral boundary values. Assume the nonlinear term is superlinear in one equation and sublinear in the other equation. By constructing two cones K1, K2 and computing the fixed point index in product cone K1×K2, we obtain that the system has a pair of positive solutions. It is remarkable that it is established on the Cartesian product of two cones, in which the feature of two equations can be opposite.
On solutions of variable-order fractional differential equations
Directory of Open Access Journals (Sweden)
Ali Akgül
2017-01-01
solutions to fractional differential equations are compelling to get in real applications, due to the nonlocality and complexity of the fractional differential operators, especially for variable-order fractional differential equations. Therefore, it is significant to enhanced numerical methods for fractional differential equations. In this work, we consider variable-order fractional differential equations by reproducing kernel method. There has been much attention in the use of reproducing kernels for the solutions to many problems in the recent years. We give two examples to demonstrate how efficiently our theory can be implemented in practice.
Asymptotic behavior of solutions to a system of Schrodinger equations
Directory of Open Access Journals (Sweden)
Xavier Carvajal
2017-07-01
Full Text Available This article concerns the behaviour of solutions to a coupled system of Schrodinger equations that has applications in many physical problems, especially in nonlinear optics. In particular, when the solution exists globally, we obtain the growth of the solutions in the energy space. Finally, some conditions are also obtained for having blow-up in this space.
Self-similar solutions for the foam drainage equation
Zitha, P.L.J.; Vermolen, F.J.
2003-01-01
The travelling wave solutions of the equation for foam drainage in porous media are developed taking into account the mass conservation criterion. The existence of traveling wave solutions is also discussed. Finally, numerical solutions are obtained using a finite difference scheme together with the
Static Solutions of Einstein's Equations with Cylindrical Symmetry
Trendafilova, C. S.; Fulling, S. A.
2011-01-01
In analogy with the standard derivation of the Schwarzschild solution, we find all static, cylindrically symmetric solutions of the Einstein field equations for vacuum. These include not only the well-known cone solution, which is locally flat, but others in which the metric coefficients are powers of the radial coordinate and the spacetime is…
A new solution of Einstein's vacuum field equations
Indian Academy of Sciences (India)
A new solution of Einstein's vacuum field equations is discovered which appears as a generalization of the well-known Ozsváth–Schücking solution and explains its source of curvature which has otherwise remained hidden. Curiously, the new solution has a vanishing Kretschmann scalar and is singularity-free despite ...
Existence of extremal periodic solutions for quasilinear parabolic equations
Directory of Open Access Journals (Sweden)
Siegfried Carl
1997-01-01
bounded domain under periodic Dirichlet boundary conditions. Our main goal is to prove the existence of extremal solutions among all solutions lying in a sector formed by appropriately defined upper and lower solutions. The main tools used in the proof of our result are recently obtained abstract results on nonlinear evolution equations, comparison and truncation techniques and suitably constructed special testfunction.
Solitary wave solutions to nonlinear evolution equations in ...
Indian Academy of Sciences (India)
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 ...
A new solution of Einstein's vacuum field equations
Indian Academy of Sciences (India)
Abstract. A new solution of Einstein's vacuum field equations is discovered which appears as a generalization of the well-known Ozsváth–Schücking solution and explains its source of curvature which has otherwise remained hidden. Curiously, the new solution has a vanishing Kretschmann scalar and is singularity-free ...
A general polynomial solution to convection–dispersion equation ...
Indian Academy of Sciences (India)
A number of models have been established to simulate the behaviour of solute transport due to chemical pollution, both in croplands and groundwater systems. An approximate polynomial solution to convection–dispersion equation (CDE) based on boundary layer theory has been verified for the use to describe solute ...
Conformal invariance and new exact solutions of the elastostatics equations
Chirkunov, Yu. A.
2017-03-01
We fulfilled a group foliation of the system of n-dimensional (n ≥ 2) Lame equations of the classical static theory of elasticity with respect to the infinite subgroup contained in normal subgroup of main group of this system. It permitted us to move from the Lame equations to the equivalent unification of two first-order systems: automorphic and resolving. We obtained a general solution of the automorphic system. This solution is an n-dimensional analogue of the Kolosov-Muskhelishvili formula. We found the main Lie group of transformations of the resolving system of this group foliation. It turned out that in the two-dimensional and three-dimensional cases, which have a physical meaning, this system is conformally invariant, while the Lame equations admit only a group of similarities of the Euclidean space. This is a big success, since in the method of group foliation, resolving equations usually inherit Lie symmetries subgroup of the full symmetry group that was not used for the foliation. In the three-dimensional case for the solutions of the resolving system, we found the general form of the transformations similar to the Kelvin transformation. These transformations are the consequence of the conformal invariance of the resolving system. In the three-dimensional case with a help of the complex dependent and independent variables, the resolving system is written as a simple complex system. This allowed us to find non-trivial exact solutions of the Lame equations, which direct for the Lame equations practically impossible to obtain. For this complex system, all the essentially distinct invariant solutions of the maximal rank we have found in explicit form, or we reduced the finding of those solutions to the solving of the classical one-dimensional equations of the mathematical physics: the heat equation, the telegraph equation, the Tricomi equation, the generalized Darboux equation, and other equations. For the resolving system, we obtained double wave of a
Abundant Interaction Solutions of Sine-Gordon Equation
Directory of Open Access Journals (Sweden)
DaZhao Lü
2012-01-01
Full Text Available With the help of computer symbolic computation software (e.g., Maple, abundant interaction solutions of sine-Gordon equation are obtained by means of a constructed Wronskian form expansion method. The method is based upon the forms and structures of Wronskian solutions of sine-Gordon equation, and the functions used in the Wronskian determinants do not satisfy linear partial differential equations. Such interaction solutions are difficultly obtained via other methods. And the method can be automatically carried out in computer.
Periodic solutions of Volterra integral equations
Directory of Open Access Journals (Sweden)
M. N. Islam
1988-01-01
Full Text Available Consider the system of equationsx(t=f(t+∫−∞tk(t,sx(sds, (1andx(t=f(t+∫−∞tk(t,sg(s,x(sds. (2Existence of continuous periodic solutions of (1 is shown using the resolvent function of the kernel k. Some important properties of the resolvent function including its uniqueness are obtained in the process. In obtaining periodic solutions of (1 it is necessary that the resolvent of k is integrable in some sense. For a scalar convolution kernel k some explicit conditions are derived to determine whether or not the resolvent of k is integrable. Finally, the existence and uniqueness of continuous periodic solutions of (1 and (2 are btained using the contraction mapping principle as the basic tool.
Self similar solutions of generalized Burgers equation
Abdelilah Gmira; Ahmed Hamydy; Salek Ouailal
2005-01-01
In this paper, we study the initial-value problem $$displaylines{ (|u'|^{p-2}u')'+eta r u'+alpha u-gamma |u|^{q-1}u|u'|^{p-2}u'=0, quad r greater than 0, cr u(0)=A,quad u'(0)=0, }$$ where $A$ greater than 0, $p$ greater than 2, $q greater than 1, $alpha greater than 0, $eta greater than 0 and $ gamma in{mathbb{R}}$. Existence and complete classification of solutions are established. Asymptotic behavior for nonnegative solutions is also presented.
Self similar solutions of generalized Burgers equation
Directory of Open Access Journals (Sweden)
Abdelilah Gmira
2005-07-01
Full Text Available In this paper, we study the initial-value problem $$displaylines{ (|u'|^{p-2}u''+eta r u'+alpha u-gamma |u|^{q-1}u|u'|^{p-2}u'=0, quad r greater than 0, cr u(0=A,quad u'(0=0, }$$ where $A$ greater than 0, $p$ greater than 2, $q greater than 1, $alpha greater than 0, $eta greater than 0 and $ gamma in{mathbb{R}}$. Existence and complete classification of solutions are established. Asymptotic behavior for nonnegative solutions is also presented.
Analytical solution of population balance equation involving ...
Indian Academy of Sciences (India)
For an initial proof-of-concept, a general case when the number of particles varies with respect to time is chosen. Three cases, i.e. (1) balanced aggregation ... The results are then compared with the available analytical solution, based on Laplace transform obtained from literature. In this communication, it is shown that the ...
Analytic solutions of a class of nonlinear partial differential equations
Directory of Open Access Journals (Sweden)
Eugenia N. Petropoulou
2015-08-01
Full Text Available We study a class of nonlinear partial differential equations, which can be connected with wave-type equations and Laplace-type equations, by using a functional-analytic technique. We establish primarily the existence and uniqueness of bounded solutions in the two-dimensional Hardy-Lebesque space of analytic functions with independent variables lying in the open unit disc. However these results can be modified to expand the domain of definition. The proofs have a constructive character enabling the determination of concrete and easily verifiable conditions, and the determination of the coefficients appearing in the power series solution. Illustrative examples are given related to the sine-Gordon equation, the Klein-Gordon equation, and to equations with nonlinear terms of algebraic, exponential and logistic type.
Soliton solutions for ABS lattice equations: I. Cauchy matrix approach
Nijhoff, Frank; Atkinson, James; Hietarinta, Jarmo
2009-10-01
In recent years there have been new insights into the integrability of quadrilateral lattice equations, i.e. partial difference equations which are the natural discrete analogues of integrable partial differential equations in 1+1 dimensions. In the scalar (i.e. single-field) case, there now exist classification results by Adler, Bobenko and Suris (ABS) leading to some new examples in addition to the lattice equations 'of KdV type' that were known since the late 1970s and early 1980s. In this paper, we review the construction of soliton solutions for the KdV-type lattice equations and use those results to construct N-soliton solutions for all lattice equations in the ABS list except for the elliptic case of Q4, which is left to a separate treatment.
Exact traveling wave solutions for system of nonlinear evolution equations.
Khan, Kamruzzaman; Akbar, M Ali; Arnous, Ahmed H
2016-01-01
In this work, recently deduced generalized Kudryashov method is applied to the variant Boussinesq equations, and the (2 + 1)-dimensional breaking soliton equations. As a result a range of qualitative explicit exact traveling wave solutions are deduced for these equations, which motivates us to develop, in the near future, a new approach to obtain unsteady solutions of autonomous nonlinear evolution equations those arise in mathematical physics and engineering fields. It is uncomplicated to extend this method to higher-order nonlinear evolution equations in mathematical physics. And it should be possible to apply the same method to nonlinear evolution equations having more general forms of nonlinearities by utilizing the traveling wave hypothesis.
Thin-Layer Solutions of the Helmholtz and Related Equations
Ockendon, J. R.
2012-01-01
This paper concerns a certain class of two-dimensional solutions to four generic partial differential equations-the Helmholtz, modified Helmholtz, and convection-diffusion equations, and the heat conduction equation in the frequency domain-and the connections between these equations for this particular class of solutions.S pecifically, we consider thin-layer solutions, valid in narrow regions across which there is rapid variation, in the singularly perturbed limit as the coefficient of the Laplacian tends to zero.F or the wellstudied Helmholtz equation, this is the high-frequency limit and the solutions in question underpin the conventional ray theory/WKB approach in that they provide descriptions valid in some of the regions where these classical techniques fail.E xamples are caustics, shadow boundaries, whispering gallery, and creeping waves and focusing and bouncing ball modes.It transpires that virtually all such thin-layer models reduce to a class of generalized parabolic wave equations, of which the heat conduction equation is a special case. Moreover, in most situations, we will find that the appropriate parabolic wave equation solutions can be derived as limits of exact solutions of the Helmholtz equation.W e also show how reasonably well-understood thin-layer phenomena associated with any one of the four generic equations may translate into less well-known effects associated with the others.In addition, our considerations also shed some light on the relationship between the methods of matched asymptotic, WKB, and multiple-scales expansions. © 2012 Society for Industrial and Applied Mathematics.
Lectures on the practical solution of differential equations
Energy Technology Data Exchange (ETDEWEB)
Dresner, L.
1979-11-01
This report comprises lectures on the practical solution of ordinary and partial differential equations given in the In-Hours Continuing Education Program for Scientific and Technical Personnel at Oak Ridge National Laboratory.
Symmetry of solutions of differential equations Mythily Ramaswamy ...
Indian Academy of Sciences (India)
2007-11-02
.. • crystals, plants, flowers, insects....... • yet, there are symmetry break ups ! • When is a profile symmetric ? • If a physical phenomenon is modelled by a differential equation, when is the solution symmetric? • Can we ...
Sign changing solutions of the p (x)-Laplacian equation
Indian Academy of Sciences (India)
Abstract. This paper deals with the variational and Nehari manifold method for the ()-Laplacian equations in a bounded domain or in the whole space. We prove existence of sign changing solutions under certain conditions.
On solutions of a Volterra integral equation with deviating arguments
Directory of Open Access Journals (Sweden)
M. Diana Julie
2009-04-01
Full Text Available In this article, we establish the existence and asymptotic characterization of solutions to a nonlinear Volterra integral equation with deviating arguments. Our proof is based on measure of noncompactness and the Schauder fixed point theorem.
Exact solutions for the differential equations in fractal heat transfer
Directory of Open Access Journals (Sweden)
Yang Chun-Yu
2016-01-01
Full Text Available In this article we consider the boundary value problems for differential equations in fractal heat transfer. The exact solutions of non-differentiable type are obtained by using the local fractional differential transform method.
Positive Solutions for Systems of Second-Order Difference Equations
Directory of Open Access Journals (Sweden)
Johnny Henderson
2015-01-01
Full Text Available We study the existence and nonexistence of positive solutions of some systems of nonlinear second-order difference equations subject to multipoint boundary conditions which contain some positive constants.
Existence of periodic solutions for a semilinear ordinary differential equation
Directory of Open Access Journals (Sweden)
Petr Girg
1998-11-01
Full Text Available Dancer [3] found a necessary and sufficient condition for the existence of periodic solutions to the equation $$ ddot x +g_1(dot x + g_0(x = f(t,.$$ His condition is based on a functional that depends on the solution to the above equation with $g_0=0$. However, that solution is not always explicitly known which makes the condition unverifiable in practical situations. As an alternative, we find computable bounds for the functional that provide a sufficient condition and a necessary condition for the existence of solutions.
Conditionally invariant solutions of the rotating shallow water wave equations
Energy Technology Data Exchange (ETDEWEB)
Huard, Benoit, E-mail: huard@dms.umontreal.c [Departement de mathematiques et de statistique, CP 6128, Succc. Centre-ville, Montreal, (QC) H3C 3J7 (Canada)
2010-06-11
This paper is devoted to the extension of the recently proposed conditional symmetry method to first-order nonhomogeneous quasilinear systems which are equivalent to homogeneous systems through a locally invertible point transformation. We perform a systematic analysis of the rank-1 and rank-2 solutions admitted by the shallow water wave equations in (2 + 1) dimensions and construct the corresponding solutions of the rotating shallow water wave equations. These solutions involve in general arbitrary functions depending on Riemann invariants, which allow us to construct new interesting classes of solutions.
Interacting localized solutions for the Zakharov-Kusnetsov equation
Energy Technology Data Exchange (ETDEWEB)
Maccari, Attilio E-mail: solitone@yahoo.it
2004-09-01
Analytical investigation of the Zakharov-Kusnetsov equation shows the existence of approximate interacting localized solutions. Using the asymptotic perturbation method, based on Fourier expansion and spatio-temporal rescaling, it is found that the amplitude slow modulation of Fourier modes is described by a C-integrable (solvable via an appropriate change of variables) system of non-linear evolution equations. It is demonstrated the existence of localized solutions (dromions, lumps, ring solitons and breathers) as well as of multiple instanton solutions. The interaction between the localized solutions are completely elastic, because they pass through each other and preserve their shape, the only change being a phase shift.
Stability and Boundedness of Solutions to Nonautonomous Parabolic Integrodifferential Equations
Directory of Open Access Journals (Sweden)
Michael Gil'
2016-01-01
Full Text Available We consider a class of linear nonautonomous parabolic integrodifferential equations. We will assume that the coefficients are slowly varying in time. Conditions for the boundedness and stability of solutions to the considered equations are suggested. Our results are based on a combined usage of the recent norm estimates for operator functions and theory of equations on the tensor product of Hilbert spaces.
Computing all integer solutions of a genus 1 equation
R.J. Stroeker (Roel); N. Tzanakis (Nikos)
2001-01-01
textabstractThe Elliptic Logarithm Method has been applied with great success to the problem of computing all integer solutions of equations of degree 3 and 4 defining elliptic curves. We extend this method to include any equation f(u,v)=0 that defines a curve of genus 1. Here f is a polynomial with
Modulation equations for spatially periodic systems: derivation and solutions
Schielen, R.; Doelman, A.
1996-01-01
We study a class of partial dierential equations in one spatial dimension, which can be seen as model equations for the analysis of pattern formation in physical systems dened on unbounded, weakly oscillating domains. We perform a linear and weakly nonlinear stability analysis for solutions that
Analytical Solution Of Complete Schwarzschild\\'s Planetary Equation
African Journals Online (AJOL)
It is well known how to solve the Einstein\\'s planetary equation of motion by the method of successive approximation for the corresponding orbit solution. In this paper, we solve the complete schwarzschild\\'s planetary equation of motion by an exact analytical method. The result reveals that there are actually eight exact ...
The solution of complete Schwarzschild's planetary equation with ...
African Journals Online (AJOL)
The Einstein's solution of the planetary equation of motion from Schwarzschild's line element is well known. In this paper, we solve the complete Schwarzschild's planetary equation with the method of successive approximation for the corresponding precession and compare the result with that from the line element.
Probabilistic representations of solutions to the heat equation
Indian Academy of Sciences (India)
In this paper we provide a new (probabilistic) proof of a classical result in partial differential equations, viz. if is a tempered distribution, then the solution of the heat equation for the Laplacian, with initial condition , is given by the convolution of with the heat kernel (Gaussian density). Our results also extend the ...
Viscosity solutions to delay differential equations in demo-economy
Fabbri, Giorgio
2007-01-01
Economic and demographic models governed by linear delay differential equations are expressed as optimal control problems in infinite dimensions. A general objective function is considered and the concavity of the Hamiltonian is not required. The value function is a viscosity solution of the Hamilton-Jacobi-Bellman (HJB) equation and a verification theorem is proved.
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.
The Numerical Solution of an Abelian Ordinary Differential Equation ...
African Journals Online (AJOL)
In this paper we present a relatively new technique call theNew Hybrid of Adomian decomposition method (ADM) for solution of an Abelian Differential equation. The numerical results of the equation have been obtained in terms of convergent series with easily computable component. These methods are applied to solve ...
Solution of a general pexiderized permanental functional equation
Indian Academy of Sciences (India)
49
is determined without any regularity assumptions. This equation arises from identities satisfied by the permanent of certain symmetric matrices. The solution so obtained are applied to deduce a number of existing related functional equations. Keywords. permanent; multiplicative function; exponential function; functional.
Traveling wave solutions of a highly nonlinear shallow water equation
Geyer, A.; Quirchmayr, Ronald
2018-01-01
Motivated by the question whether higher-order nonlinear model equations, which go beyond the Camassa-Holm regime of moderate amplitude waves, could point us to new types of waves profiles, we study the traveling wave solutions of a quasilinear evolution equation which models the propagation of
Moving Mesh for the Numerical Solution of Partial Differential Equations
Directory of Open Access Journals (Sweden)
OLIVEIRA, F. S.
2013-06-01
Full Text Available In this paper we present moving meshes for the numeric resolution of partial differential equations. We describe some important concepts on this topic and point to existing body of work for the solution of partial differential equations using the methods of finite volumes and finite elements, both with moving meshes.
Existence of solutions to quasilinear Schrodinger equations with indefinite potential
Directory of Open Access Journals (Sweden)
Zupei Shen
2015-04-01
Full Text Available In this article, we study the existence and multiplicity of solutions of the quasilinear Schrodinger equation $$ -u''+V(xu-(|u| ^2''u=f(u $$ on $\\mathbb{R}$, where the potential $V$ allows sign changing and the nonlinearity satisfies conditions weaker than the classical Ambrosetti-Rabinowitz condition. By a local linking theorem and the fountain theorem, we obtain the existence and multiplicity of solutions for the equation.
On singular solutions of a magnetohydrodynamic nonlinear boundary layer equation
Mohammed Guedda; Abdelilah Gmira; Mohammed Benlahsen
2007-01-01
This paper concerns the singular solutions of the equation $$ f''' +kappa ff''-eta {f'}^2 = 0, $$ where $eta < 0$ and $kappa = 0$ or 1. This equation arises when modelling heat transfer past a vertical flat plate embedded in a saturated porous medium with an applied magnetic field. After suitable normalization, $f'$ represents the velocity parallel to the surface or the non-dimensional fluid temperature. Our interest is in solutions which develop a singularity at some point (t...
Numerical solution of ordinary differential equations
Fox, L
1987-01-01
Nearly 20 years ago we produced a treatise (of about the same length as this book) entitled Computing methods for scientists and engineers. It was stated that most computation is performed by workers whose mathematical training stopped somewhere short of the 'professional' level, and that some books are therefore needed which use quite simple mathematics but which nevertheless communicate the essence of the 'numerical sense' which is exhibited by the real computing experts and which is surely needed, at least to some extent, by all who use modern computers and modern numerical software. In that book we treated, at no great length, a variety of computational problems in which the material on ordinary differential equations occupied about 50 pages. At that time it was quite common to find books on numerical analysis, with a little on each topic ofthat field, whereas today we are more likely to see similarly-sized books on each major topic: for example on numerical linear algebra, numerical approximation, numeri...
Large-time behavior of solutions of linear dispersive equations
Dix, Daniel B
1997-01-01
This book studies the large-time asymptotic behavior of solutions of the pure initial value problem for linear dispersive equations with constant coefficients and homogeneous symbols in one space dimension. Complete matched and uniformly-valid asymptotic expansions are obtained and sharp error estimates are proved. Using the method of steepest descent much new information on the regularity and spatial asymptotics of the solutions are also obtained. Applications to nonlinear dispersive equations are discussed. This monograph is intended for researchers and graduate students of partial differential equations. Familiarity with basic asymptotic, complex and Fourier analysis is assumed.
Solution of heat equation with variable coefficient using derive
CSIR Research Space (South Africa)
Lebelo, RS
2008-09-01
Full Text Available -reviewed Conference Proceedings, 22 – 26 September 2008 - 129 - Solution of heat equation with variable coefficient using derive RS Lebeloα, I Fedotov and M Shatalovβ Department of Mathematics and Statistics Tshwane University of Technology Pretoria... of algebraic and transcedental equations. Buffelspoort TIME2008 Peer-reviewed Confe- rence Proceedings, 22-26 September, South Africa, ISBN 978-3-901769- 82-5, pp. 162 – 173. [5] R.S. Lebelo (2008). Approximating solutions of partial differential equations...
Directory of Open Access Journals (Sweden)
Zehra Pınar
2013-01-01
Full Text Available It is well known that different types of exact solutions of an auxiliary equation produce new types of exact travelling wave solutions to nonlinear equations. In this paper, by means of symbolic computation, the new solutions of original auxiliary equation of first-order nonlinear ordinary differential equation with a sixth-degree nonlinear term are presented to obtain novel exact solutions of the Kawahara equation. By the aid of the solutions of the original auxiliary equation, some other physically important nonlinear equations can be solved to construct novel exact solutions.
Numerical solution to nonlinear Tricomi equation using WENO schemes
Directory of Open Access Journals (Sweden)
Adrian Sescu
2010-09-01
Full Text Available Nonlinear Tricomi equation is a hybrid (hyperbolic-elliptic second order partial differential equation, modelling the sonic boom focusing. In this paper, the Tricomi equation is transformed into a hyperbolic system of first order equations, in conservation law form. On the upper boundary, a new mixed boundary condition for the acoustic pressure is used to avoid the inclusion of the Dirac function in the numerical solution. Weighted Essentially Non-Oscillatory (WENO schemes are used for the spatial discretization, and the time marching is carried out using the second order accurate Runge-Kutta total-variation diminishing (TVD scheme.
On the Exact Solution of Wave Equations on Cantor Sets
Directory of Open Access Journals (Sweden)
Dumitru Baleanu
2015-09-01
Full Text Available The transfer of heat due to the emission of electromagnetic waves is called thermal radiations. In local fractional calculus, there are numerous contributions of scientists, like Mandelbrot, who described fractal geometry and its wide range of applications in many scientific fields. Christianto and Rahul gave the derivation of Proca equations on Cantor sets. Hao et al. investigated the Helmholtz and diffusion equations in Cantorian and Cantor-Type Cylindrical Coordinates. Carpinteri and Sapora studied diffusion problems in fractal media in Cantor sets. Zhang et al. studied local fractional wave equations under fixed entropy. In this paper, we are concerned with the exact solutions of wave equations by the help of local fractional Laplace variation iteration method (LFLVIM. We develop an iterative scheme for the exact solutions of local fractional wave equations (LFWEs. The efficiency of the scheme is examined by two illustrative examples.
Invariant Solutions for a Class of Perturbed Nonlinear Wave Equations
Directory of Open Access Journals (Sweden)
Waheed A. Ahmed
2017-11-01
Full Text Available Approximate symmetries of a class of perturbed nonlinear wave equations are computed using two newly-developed methods. Invariant solutions associated with the approximate symmetries are constructed for both methods. Symmetries and solutions are compared through discussing the advantages and disadvantages of each method.
On periodic solutions to second-order Duffing type equations
Czech Academy of Sciences Publication Activity Database
Lomtatidze, Alexander; Šremr, Jiří
2018-01-01
Roč. 40, April (2018), s. 215-242 ISSN 1468-1218 Institutional support: RVO:67985840 Keywords : periodic solution * Duffing type equation * positive solution Subject RIV: BA - General Mathematics Impact factor: 1.659, year: 2016 http://www.sciencedirect.com/science/ article /pii/S1468121817301335?via%3Dihub
Exponential decay for solutions to semilinear damped wave equation
Gerbi, Stéphane
2011-10-01
This paper is concerned with decay estimate of solutions to the semilinear wave equation with strong damping in a bounded domain. Intro- ducing an appropriate Lyapunov function, we prove that when the damping is linear, we can find initial data, for which the solution decays exponentially. This result improves an early one in [4].
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...
Traveling wave solutions of the BBM-like equations
Energy Technology Data Exchange (ETDEWEB)
Kuru, S [Department of Physics, Faculty of Science, Ankara University 06100 Ankara (Turkey)], E-mail: kuru@science.ankara.edu.tr
2009-09-18
In this work, we apply the factorization technique to the Benjamin-Bona-Mahony-like equations, B(m, n), in order to get traveling wave solutions. We will focus on some special cases for which m {ne} n, and we will obtain these solutions in terms of the special forms of Weierstrass functions.
Solutions of the telegrapher's equation in the presence of traps
Masoliver, Jaume, 1951-; Porrà i Rovira, Josep Maria; Weiss, George H. (George Herbert), 1930-
1992-01-01
Several problems in the theory of photon migration in a turbid medium suggest the utility of calculating solutions of the telegrapher¿s equation in the presence of traps. This paper contains two such solutions for the one-dimensional problem, the first being for a semi-infinite line terminated by a trap, and the second being for a finite line terminated by two traps. Because solutions to the telegrapher¿s equation represent an interpolation between wavelike and diffusive phenomena, they will ...
Integrable and continuous solutions of a nonlinear quadratic integral equation
Directory of Open Access Journals (Sweden)
Ahmed El-Sayed
2008-08-01
Full Text Available We are concerned here with a nonlinear quadratic integral equation of Volterra type. The existence of at least one $L_1-$ positive solution will be proved under the Carath\\`{e}odory condition. Secondly we will make a link between Peano condition and Carath\\`{e}odory condition to prove the existence of at least one positive continuous solution. Finally the existence of the maximal and minimal solutions will be proved.
Wang, J.; Fec̆kan, M.; Zhou, Y.
2013-09-01
In this paper, a class of impulsive fractional Langevin equations is firstly offered. Formula of solutions involving Mittag-Leffler functions and impulsive terms of such equations are successively derived by studying the corresponding linear Langevin equations with two different fractional derivatives. Meanwhile, existence results of solutions are established by utilizing boundedness, continuity, monotonicity and nonnegative of Mittag-Leffler functions and fixed point methods. Further, other existence results of nonlinear impulsive problems are also presented. Finally, an example is given to illustrate our theoretical results.
Shallow water equations: viscous solutions and inviscid limit
Chen, Gui-Qiang; Perepelitsa, Mikhail
2012-12-01
We establish the inviscid limit of the viscous shallow water equations to the Saint-Venant system. For the viscous equations, the viscosity terms are more degenerate when the shallow water is close to the bottom, in comparison with the classical Navier-Stokes equations for barotropic gases; thus, the analysis in our earlier work for the classical Navier-Stokes equations does not apply directly, which require new estimates to deal with the additional degeneracy. We first introduce a notion of entropy solutions to the viscous shallow water equations and develop an approach to establish the global existence of such solutions and their uniform energy-type estimates with respect to the viscosity coefficient. These uniform estimates yield the existence of measure-valued solutions to the Saint-Venant system generated by the viscous solutions. Based on the uniform energy-type estimates and the features of the Saint-Venant system, we further establish that the entropy dissipation measures of the viscous solutions for weak entropy-entropy flux pairs, generated by compactly supported C 2 test-functions, are confined in a compact set in H -1, which yields that the measure-valued solutions are confined by the Tartar-Murat commutator relation. Then, the reduction theorem established in Chen and Perepelitsa [5] for the measure-valued solutions with unbounded support leads to the convergence of the viscous solutions to a finite-energy entropy solution of the Saint-Venant system with finite-energy initial data, which is relative with respect to the different end-states of the bottom topography of the shallow water at infinity. The analysis also applies to the inviscid limit problem for the Saint-Venant system in the presence of friction.
Efficient traveltime solutions of the acoustic TI eikonal equation
Waheed, Umair bin
2015-02-01
Numerical solutions of the eikonal (Hamilton-Jacobi) equation for transversely isotropic (TI) media are essential for imaging and traveltime tomography applications. Such solutions, however, suffer from the inherent higher-order nonlinearity of the TI eikonal equation, which requires solving a quartic polynomial for every grid point. Analytical solutions of the quartic polynomial yield numerically unstable formulations. Thus, it requires a numerical root finding algorithm, adding significantly to the computational load. Using perturbation theory we approximate, in a first order discretized form, the TI eikonal equation with a series of simpler equations for the coefficients of a polynomial expansion of the eikonal solution, in terms of the anellipticity anisotropy parameter. Such perturbation, applied to the discretized form of the eikonal equation, does not impose any restrictions on the complexity of the perturbed parameter field. Therefore, it provides accurate traveltime solutions even for models with complex distribution of velocity and anisotropic anellipticity parameter, such as that for the complicated Marmousi model. The formulation allows for large cost reduction compared to using the direct TI eikonal solver. Furthermore, comparative tests with previously developed approximations illustrate remarkable gain in accuracy in the proposed algorithm, without any addition to the computational cost.
Li, Bo; Zhao, Yanxiang
2013-01-01
Central in a variational implicit-solvent description of biomolecular solvation is an effective free-energy functional of the solute atomic positions and the solute-solvent interface (i.e., the dielectric boundary). The free-energy functional couples together the solute molecular mechanical interaction energy, the solute-solvent interfacial energy, the solute-solvent van der Waals interaction energy, and the electrostatic energy. In recent years, the sharp-interface version of the variational implicit-solvent model has been developed and used for numerical computations of molecular solvation. In this work, we propose a diffuse-interface version of the variational implicit-solvent model with solute molecular mechanics. We also analyze both the sharp-interface and diffuse-interface models. We prove the existence of free-energy minimizers and obtain their bounds. We also prove the convergence of the diffuse-interface model to the sharp-interface model in the sense of Γ-convergence. We further discuss properties of sharp-interface free-energy minimizers, the boundary conditions and the coupling of the Poisson-Boltzmann equation in the diffuse-interface model, and the convergence of forces from diffuse-interface to sharp-interface descriptions. Our analysis relies on the previous works on the problem of minimizing surface areas and on our observations on the coupling between solute molecular mechanical interactions with the continuum solvent. Our studies justify rigorously the self consistency of the proposed diffuse-interface variational models of implicit solvation.
Soliton solution and other solutions to a nonlinear fractional differential equation
Guner, Ozkan; Unsal, Omer; Bekir, Ahmet; Kadem, Abdelouahab
2017-01-01
In this paper, the fractional derivatives in the sense of modified Riemann-Liouville derivative and the ansatz method and the functional variable method are used to construct exact solutions for (3+1)-dimensional time fractional KdV-Zakharov-Kuznetsov (KdV-ZK) equation. This fractional equation is turned into another nonlinear ordinary differential equation by fractional complex transform then these methods are applied to solve it. As a result, some new exact solutions obtained.
Approximation of entropy solutions to degenerate nonlinear parabolic equations
Abreu, Eduardo; Colombeau, Mathilde; Panov, Evgeny Yu
2017-12-01
We approximate the unique entropy solutions to general multidimensional degenerate parabolic equations with BV continuous flux and continuous nondecreasing diffusion function (including scalar conservation laws with BV continuous flux) in the periodic case. The approximation procedure reduces, by means of specific formulas, a system of PDEs to a family of systems of the same number of ODEs in the Banach space L^∞, whose solutions constitute a weak asymptotic solution of the original system of PDEs. We establish well posedness, monotonicity and L^1-stability. We prove that the sequence of approximate solutions is strongly L^1-precompact and that it converges to an entropy solution of the original equation in the sense of Carrillo. This result contributes to justify the use of this original method for the Cauchy problem to standard multidimensional systems of fluid dynamics for which a uniqueness result is lacking.
The semi-dynamical reflection equation: solutions and structure matrices
Energy Technology Data Exchange (ETDEWEB)
Avan, J; Zambon, C [Laboratoire de Physique Theorique et Modelisation, Universite de Cergy-Pontoise (CNRS UMR 8089), Saint-Martin 2 avenue Adolphe Chauvin, 95302 Cergy-Pontoise Cedex (France)], E-mail: avan@u-cergy.fr, E-mail: cristina.zambon@u-cergy.fr
2008-05-16
Explicit solutions of the non-constant semi-dynamical reflection equation are constructed, together with suitable parametrizations of their structure matrices. Considering the semi-dynamical reflection equation with rational non-constant Arutyunov-Chekhov-Frolov structure matrices, and a specific meromorphic ansatz, it is found that only two sets of the previously found constant solutions are extendible to the non-constant case. In order to simplify future constructions of spin-chain Hamiltonians, a parametrization procedure is applied explicitly to all elements of the semi-dynamical reflection equation available. Interesting expressions for 'twists' and R-matrices entering the parametrization procedure are found. In particular, some expressions for the R-matrices seem to appear here for the first time. In addition, a new set of consistent structure matrices for the semi-dynamical reflection equation is obtained.
On the solutions of fractional order of evolution equations
Morales-Delgado, V. F.; Taneco-Hernández, M. A.; Gómez-Aguilar, J. F.
2017-01-01
In this paper we present a discussion of generalized Cauchy problems in a diffusion wave process, we consider bi-fractional-order evolution equations in the Riemann-Liouville, Liouville-Caputo, and Caputo-Fabrizio sense. Through Fourier transforms and Laplace transform we derive closed-form solutions to the Cauchy problems mentioned above. Similarly, we establish fundamental solutions. Finally, we give an application of the above results to the determination of decompositions of Dirac type for bi-fractional-order equations and write a formula for the moments for the fractional vibration of a beam equation. This type of decomposition allows us to speak of internal degrees of freedom in the vibration of a beam equation.
Poisson-Fermi Formulation of Nonlocal Electrostatics in Electrolyte Solutions
Directory of Open Access Journals (Sweden)
Liu Jinn-Liang
2017-10-01
Full Text Available We present a nonlocal electrostatic formulation of nonuniform ions and water molecules with interstitial voids that uses a Fermi-like distribution to account for steric and correlation efects in electrolyte solutions. The formulation is based on the volume exclusion of hard spheres leading to a steric potential and Maxwell’s displacement field with Yukawa-type interactions resulting in a nonlocal electric potential. The classical Poisson-Boltzmann model fails to describe steric and correlation effects important in a variety of chemical and biological systems, especially in high field or large concentration conditions found in and near binding sites, ion channels, and electrodes. Steric effects and correlations are apparent when we compare nonlocal Poisson-Fermi results to Poisson-Boltzmann calculations in electric double layer and to experimental measurements on the selectivity of potassium channels for K+ over Na+.
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.
Positive solutions for fractional differential equations with variable coefficients
Directory of Open Access Journals (Sweden)
Yi Chen
2012-05-01
Full Text Available In this article, we study the existence of the positive solutions for a class of differential equations of fractional order with variable coefficients. The equation of this type plays an important role in the description and modeling of control systems, such as $PD^{mu}$-controller. The differential operator is taken in the Riemann-Liouville sense. Our analysis relies on the Leggett-Williams fixed point theorem.
Governing equations and solutions of anomalous random walk limits.
Meerschaert, Mark M; Benson, David A; Scheffler, Hans-Peter; Becker-Kern, Peter
2002-12-01
Continuous time random walks model anomalous diffusion. Coupling allows the magnitude of particle jumps to depend on the waiting time between jumps. Governing equations for the long-time scaling limits of these models are found to have fractional powers of coupled space and time differential operators. Explicit solutions and scaling properties are presented for these equations, which can be used to model flow in porous media and other physical systems.
Solution of the Bagley Torvik equation by fractional DTM
Arora, Geeta; Pratiksha
2017-07-01
In this paper, fractional differential transform method(DTM) is implemented on the Bagley Torvik equation. This equation models the viscoelastic behavior of geological strata, metals, glasses etc. It explains the motion of a rigid plate immersed in a Newtonian fluid. DTM is a simple, reliable and efficient method that gives a series solution. Caputo fractional derivative is considered throughout this work. Two examples are given to demonstrate the validity and applicability of the method and comparison is made with the existing results.
Global Solutions to the Coupled Chemotaxis-Fluid Equations
Duan, Renjun
2010-08-10
In this paper, we are concerned with a model arising from biology, which is a coupled system of the chemotaxis equations and the viscous incompressible fluid equations through transport and external forcing. The global existence of solutions to the Cauchy problem is investigated under certain conditions. Precisely, for the Chemotaxis-Navier-Stokes system over three space dimensions, we obtain global existence and rates of convergence on classical solutions near constant states. When the fluid motion is described by the simpler Stokes equations, we prove global existence of weak solutions in two space dimensions for cell density with finite mass, first-order spatial moment and entropy provided that the external forcing is weak or the substrate concentration is small. © Taylor & Francis Group, LLC.
Solutions of Navier-Stokes Equation with Coriolis Force
Directory of Open Access Journals (Sweden)
Sunggeun Lee
2017-01-01
Full Text Available We investigate the Navier-Stokes equation in the presence of Coriolis force in this article. First, the vortex equation with the Coriolis effect is discussed. It turns out that the vorticity can be generated due to a rotation coming from the Coriolis effect, Ω. In both steady state and two-dimensional flow, the vorticity vector ω gets shifted by the amount of -2Ω. Second, we consider the specific expression of the velocity vector of the Navier-Stokes equation in two dimensions. For the two-dimensional potential flow v→=∇→ϕ, the equation satisfied by ϕ is independent of Ω. The remaining Navier-Stokes equation reduces to the nonlinear partial differential equations with respect to the velocity and the corresponding exact solution is obtained. Finally, the steady convective diffusion equation is considered for the concentration c and can be solved with the help of Navier-Stokes equation for two-dimensional potential flow. The convective diffusion equation can be solved in three dimensions with a simple choice of c.
On Approximate Solutions of Functional Equations in Vector Lattices
Directory of Open Access Journals (Sweden)
Bogdan Batko
2014-01-01
Full Text Available We provide a method of approximation of approximate solutions of functional equations in the class of functions acting into a Riesz space (algebra. The main aim of the paper is to provide a general theorem that can act as a tool applicable to a possibly wide class of functional equations. The idea is based on the use of the Spectral Representation Theory for Riesz spaces. The main result will be applied to prove the stability of an alternative Cauchy functional equation F(x+y+F(x+F(y≠0⇒F(x+y=F(x+F(y in Riesz spaces, the Cauchy equation with squares F(x+y2=(F(x+F(y2 in f-algebras, and the quadratic functional equation F(x+y+F(x-y=2F(x+2F(y in Riesz spaces.
Integral representation of a solution of Heun's general equation
Directory of Open Access Journals (Sweden)
Jomo Batola
2007-06-01
Full Text Available We establish an integral representation for the Frobenius solution with an exponent zero at $z=0$ of the general Heun equation. First we present an extension of Mellin's lemma which provides a powerful method that takes into account differential equations which are not of the form studied by Mellin. That is the case for equations of Heun's type. It is that aspect which makes our work different from Valent's work. The method is powerful because it allows obtaining directly the nucleus equation of the representation. The integral representation formula obtained with this method leads quickly and naturally to already known results in the case of hypergeometric functions. The generalisation of this method gives a type of differential equations which form is a novelty and deserves to be studied further.
Efficient Traveltime Solutions of the TI Acoustic Eikonal Equation
Waheed, Umair bin
2014-10-22
Numerical solutions of the eikonal (Hamilton-Jacobi) equation for transversely isotropic (TI) media are essential for integral imaging and traveltime tomography applications. Such solutions, however, suffer from the inherent higher-order nonlinearity of the TI eikonal equation, which requires solving a quartic polynomial at each computational step. Using perturbation theory, we approximate the first-order discretized form of the TI eikonal equation with a series of simpler equations for the coefficients of a polynomial expansion of the eikonal solution in terms of the anellipticity anisotropy parameter. Such perturbation, applied to the discretized form of the eikonal equation, does not impose any restrictions on the complexity of the perturbed parameter field. Therefore, it provides accurate traveltime solutions even for the anisotropic Marmousi model, with complex distribution of velocity and anellipticity anisotropy parameter. The formulation allows tremendous cost reduction compared to using the exact TI eikonal solver. Furthermore, comparative tests with previously developed approximations illustrate remarkable gain in accuracy of the proposed approximation, without any addition to the computational cost.
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.
The numerical solution of the vorticity transport equation
Dennis, S C R
1973-01-01
A method of approximating the two-dimensional vorticity transport equation in which the matrix associated with the difference equations is diagonally dominant and the truncation error is the same as that of the fully central-difference approximation, is discussed. An example from boundary layer theory is given by calculating the viscous stagnation point flow at the nose of a cylinder. Some new solutions of the Navier-Stokes equations are obtained for symmetrical flow past a flat plate of finite length. (16 refs).
An integral equation solution for multistage turbomachinery design calculations
Mcfarland, Eric R.
1993-01-01
A method was developed to calculate flows in multistage turbomachinery. The method is an extension of quasi-three-dimensional blade-to-blade solution methods. Governing equations for steady compressible inviscid flow are linearized by introducing approximations. The linearized flow equations are solved using integral equation techniques. The flows through both stationary and rotating blade rows are determined in a single calculation. Multiple bodies can be modelled for each blade row, so that arbitrary blade counts can be analyzed. The method's benefits are its speed and versatility.
Analytical solutions for systems of partial differential-algebraic equations.
Benhammouda, Brahim; Vazquez-Leal, Hector
2014-01-01
This work presents the application of the power series method (PSM) to find solutions of partial differential-algebraic equations (PDAEs). Two systems of index-one and index-three are solved to show that PSM can provide analytical solutions of PDAEs in convergent series form. What is more, we present the post-treatment of the power series solutions with the Laplace-Padé (LP) resummation method as a useful strategy to find exact solutions. The main advantage of the proposed methodology is that the procedure is based on a few straightforward steps and it does not generate secular terms or depends of a perturbation parameter.
On the Analytic Solution for a Steady Magnetohydrodynamic Equation
Soltanalizadeh, Babak; Ghehsareh, Hadi Roohani; Yıldırım, Ahmet; Abbasbandy, Saeid
2013-07-01
The purpose of this study is to apply the Laplace-Adomian Decomposition Method (LADM) for obtaining the analytical and numerical solutions of a nonlinear differential equation that describes a magnetohydrodynamic (MHD) flow near the forward stagnation point of two-dimensional and axisymmetric bodies. By using this method, the similarity solutions of the problem are obtained for some typical values of the model parameters. For getting computational solutions, we combined the obtained series solutions by LADM with the Padé approximation. The method is easy to apply and gives high accurate results. The presented results through tables and figures show the efficiency and accuracy of the proposed technique.
Boundary properties of solutions of equations of minimal surface kind
Miklyukov, V. M.
2001-10-01
Generalized solutions of equations of minimal-surface type are studied. It is shown that a solution makes at most countably many jumps at the boundary. In particular, a solution defined in the exterior of a disc extends by continuity to the boundary circle everywhere outside a countable point set. An estimate of the sum of certain non-local characteristics of the jumps of a solution at the boundary is presented. A result similar to Fatou's theorem on angular boundary values is proved.
On singular solutions of a magnetohydrodynamic nonlinear boundary layer equation
Directory of Open Access Journals (Sweden)
Mohammed Guedda
2007-05-01
Full Text Available This paper concerns the singular solutions of the equation $$ f''' +kappa ff''-eta {f'}^2 = 0, $$ where $eta < 0$ and $kappa = 0$ or 1. This equation arises when modelling heat transfer past a vertical flat plate embedded in a saturated porous medium with an applied magnetic field. After suitable normalization, $f'$ represents the velocity parallel to the surface or the non-dimensional fluid temperature. Our interest is in solutions which develop a singularity at some point (the blow-up point. In particular, we shall examine in detail the behavior of $f$ near the blow-up point.
Radial selfsimilar solutions of a nonlinear Ornstein-Uhlenbeck equation
Arij Bouzelmate; Abdelilah Gmira; Guillermo Reyes
2007-01-01
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...
Iterative Solutions of Nonlinear Integral Equations of Hammerstein Type
Directory of Open Access Journals (Sweden)
Abebe R. Tufa
2015-11-01
Full Text Available Let H be a real Hilbert space. Let F,K : H → H be Lipschitz monotone mappings with Lipschtiz constants L1and L2, respectively. Suppose that the Hammerstein type equation u + KFu = 0 has a solution in H. It is our purpose in this paper to construct a new explicit iterative sequence and prove strong convergence of the sequence to a solution of the generalized Hammerstein type equation. The results obtained in this paper improve and extend known results in the literature.
Coincident Bifurcation of Equilibrium and Periodic Solutions of Evolution Equations.
1979-12-01
of Hoyle [7], involving the center manifold theory. Throughout, we refer to the papers of Crandall and Rabinowitz Ell, [21, for pre- liminary results...bw+ a2 + s2=a)" + b1w q+ p s =0 Solutions of (4.11) lie on secondary branches of solutions of equations (4.1), ponding to secondary branches of...8217. Comm. ’.ath. Ph-Ys., to appear. 6. Henry, D., Geometric theor of serilinear parabolic equations. Vniversity of Kentucky lecture notes, 1974. 7. Hoyle
Some Properties of Solutions to Weakly Hypoelliptic Equations
Directory of Open Access Journals (Sweden)
Christian Bär
2013-01-01
Full Text Available A linear different operator L is called weakly hypoelliptic if any local solution u of Lu=0 is smooth. We allow for systems, that is, the coefficients may be matrices, not necessarily of square size. This is a huge class of important operators which coverall elliptic, overdetermined elliptic, subelliptic, and parabolic equations. We extend several classical theorems from complex analysis to solutions of any weakly hypoelliptic equation: the Montel theorem providing convergent subsequences, the Vitali theorem ensuring convergence of a given sequence, and Riemann's first removable singularity theorem. In the case of constant coefficients, we show that Liouville's theorem holds, any bounded solution must be constant, and any Lp-solution must vanish.
A Liapounov bound for solutions of the Poisson equation
Glynn, Peter W.; Meyn, Sean P.
1996-01-01
In this paper we consider $\\psi$-irreducible Markov processes evolving in discrete or continuous time on a general state space. We develop a Liapounov function criterion that permits one to obtain explicit bounds on the solution to the Poisson equation and, in particular, obtain conditions under which the solution is square integrable. ¶ These results are applied to obtain sufficient conditions that guarantee the validity of a functional central limit theorem for the Markov ...
Explicit solutions of Fisher's equation with three zeros
Directory of Open Access Journals (Sweden)
M. F. K. Abur-Robb
1990-01-01
Full Text Available Explicit traveling wave solutions of Fisher's equation with three simple zeros ut=uxx+u(1−u(u−a, a∈(0,1, are obtained for the wave speeds C=±2(12−a suggested by pure analytic considerations. Two types of solutions are obtained: one type is of a permanent wave form whereas the other is not.
Symmetries and exact solutions of fractional filtration equations
Gazizov, Rafail K.; Kasatkin, Alexey A.; Lukashchuk, Stanislav Yu.
2017-11-01
Few fractional differential models of fluid flow through porous medium are considered. We use several modifications of Darcy's law that contain time-and space-fractional derivatives corresponding to memory or non-local effects in filtration. Symmetry properties of the resulting nonlinear anomalous diffusion-type equations are analyzed and new group-invariant solutions are constructed. In particular, we obtain fractional analogues of so-called blow-up solutions.
Oyelami, Benjamin Oyediran
2013-01-01
In this paper, criteria for the existence of weak solutions and uniformly weak bounded solution of impulsive heat equation containing maximum temperature are investigated and results obtained. An example is given for heat flow system with impulsive temperature using maximum temperature simulator and criteria for the uniformly weak bounded of solutions of the system are obtained.
Directory of Open Access Journals (Sweden)
Oyelami, Benjamin Oyediran
2013-09-01
Full Text Available In this paper, criteria for the existence of weak solutions and uniformly weak bounded solution of impulsive heat equation containing maximum temperature are investigated and results obtained. An example is given for heat flow system with impulsive temperature using maximum temperature simulator and criteria for the uniformly weak bounded of solutions of the system are obtained.
TRAVELING WAVE SOLUTIONS OF SOME FRACTIONAL DIFFERENTIAL EQUATIONS
Directory of Open Access Journals (Sweden)
SERIFE MUGE EGE
2016-07-01
Full Text Available The modified Kudryashov method is powerful, efficient and can be used as an alternative to establish new solutions of different type of fractional differential equations applied in mathematical physics. In this article, we’ve constructed new traveling wave solutions including symmetrical Fibonacci function solutions, hyperbolic function solutions and rational solutions of the space-time fractional Cahn Hillihard equation D_t^α u − γD_x^α u − 6u(D_x^α u^2 − (3u^2 − 1D_x^α (D_x^α u + D_x^α(D_x^α(D_x^α(D_x^α u = 0 and the space-time fractional symmetric regularized long wave (SRLW equation D_t^α(D_t^α u + D_x^α(D_x^α u + uD_t^α(D_x^α u + D_x^α u D_t^α u + D_t^α(D_t^α(D_x^α(D_x^α u = 0 via modified Kudryashov method. In addition, some of the solutions are described in the figures with the help of Mathematica.
Spatial local solutions of the Navier-Stokes equations
Garipov, R. M.
2009-03-01
This paper considers solutions of the Navier-Stokes equations polynomial in the coordinates, which. are called local solutions. For an incompressible fluid, all higher-order terms (sums of higher-order. monomials) of degree 2 are found and it is proved that nontrivial axisymmetric higher-order terms. of degree higher than 2 do not exist. Nonsolenoidal axisymmetric solutions are listed, which can be. treated as steady-state barotropic gas flows in a potential external-force field. All elliptic vortices. generalizing the well-known Kirchhoff solution are calculated. All solutions of degree 3 with the. higher-order term of partial form are found. Some of these solutions break down in a finite time. regardless of the value and sign of viscosity.
Numerical Comparison of Solutions of Kinetic Model Equations
Directory of Open Access Journals (Sweden)
A. A. Frolova
2015-01-01
Full Text Available The collision integral approximation by different model equations has created a whole new trend in the theory of rarefied gas. One widely used model is the Shakhov model (S-model obtained by expansion of inverse collisions integral in a series of Hermite polynomials up to the third order. Using the same expansion with another value of free parameters leads to a linearized ellipsoidal statistical model (ESL.Both model equations (S and ESL have the same properties, as they give the correct relaxation of non-equilibrium stress tensor components and heat flux vector, the correct Prandtl number at the transition to the hydrodynamic regime and do not guarantee the positivity of the distribution function.The article presents numerical comparison of solutions of Shakhov equation, ESL- model and full Boltzmann equation in the four Riemann problems for molecules of hard spheres.We have considered the expansion of two gas flows, contact discontinuity, the problem of the gas counter-flows and the problem of the shock wave structure. For the numerical solution of the kinetic equations the method of discrete ordinates is used.The comparison shows that solution has a weak sensitivity to the form of collision operator in the problem of expansions of two gas flows and results obtained by the model and the kinetic Boltzmann equations coincide.In the problem of the contact discontinuity the solution of model equations differs from full kinetic solutions at the point of the initial discontinuity. The non-equilibrium stress tensor has the maximum errors, the error of the heat flux is much smaller, and the ESL - model gives the exact value of the extremum of heat flux.In the problems of gas counter-flows and shock wave structure the model equations give significant distortion profiles of heat flux and non-equilibrium stress tensor components in front of the shock waves. This behavior is due to fact that in the models under consideration there is no dependency of the
Mild Solutions of Neutral Stochastic Partial Functional Differential Equations
Directory of Open Access Journals (Sweden)
T. E. Govindan
2011-01-01
Full Text Available This paper studies the existence and uniqueness of a mild solution for a neutral stochastic partial functional differential equation using a local Lipschitz condition. When the neutral term is zero and even in the deterministic special case, the result obtained here appears to be new. An example is included to illustrate the theory.
An algebraic solution of Lindblad-type master equations
Energy Technology Data Exchange (ETDEWEB)
Klimov, A B; Romero, J L [Departamento de Fisica, Universidad de Guadalajara, Revolucion 1500, 44410, Guadalajara, Jal. (Mexico)
2003-06-01
We propose an algebraic solution for a wide class of Lindblad-type master equations. Examples of dissipation in free field evolution, field evolution in the Kerr medium, two-photon field dissipation, atomic dissipation and two-mode field dissipation are given.
Asymptotic behaviour of solutions of a nonlinear transport equation
C.J. van Duijn (Hans); M.A. Peletier (Mark)
1996-01-01
textabstractWe investigate the asymptotic behaviour of solutions of the convection- diffusion equation $$ b(u)_t + divleft( u q - n u right) = 0 qquad hbox{for r = |x| > e quadhbox{andquad t>0, $$ where $q=l/r, er $, $l>0$. The asymptotic limits that we consider are $ttoinfty$ and $e downto0$. We
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.
Analysis of Quadratic Diophantine Equations with Fibonacci Number Solutions
Leyendekkers, J. V.; Shannon, A. G.
2004-01-01
An analysis is made of the role of Fibonacci numbers in some quadratic Diophantine equations. A general solution is obtained for finding factors in sums of Fibonacci numbers. Interpretation of the results is facilitated by the use of a modular ring which also permits extension of the analysis.
About Global Stable of Solutions of Logistic Equation with Delay
Kaschenko, S. A.; Loginov, D. O.
2017-12-01
The article is devoted to the definition of all the arguments for which all positive solutions of logistic equation with delay tend to zero for t → ∞. The authors have proved the acquainted Wright’s conjecture on evaluation of a multitude of such arguments. An approach that enables subsequent refinement of this evaluation has been developed.
Numerical Solution of Differential Algebraic Equations and Applications
DEFF Research Database (Denmark)
Thomsen, Per Grove
2005-01-01
These lecture notes have been written as part of a special course on the numerical solution of Differential Algebraic Equations and applications . The course was held at IMM in the spring of 2005. The authors of the different chapters have all taken part in the course and the chapters are written...
Solution of fractional differential equations via coupled fixed point
Directory of Open Access Journals (Sweden)
Hojjat Afshari
2015-11-01
Full Text Available In this article, we investigate the existence and uniqueness of a solution for the fractional differential equation by introducing some new coupled fixed point theorems for the class of mixed monotone operators with perturbations in the context of partially ordered complete metric space.
Operator solutions for fractional Fokker-Planck equations.
Górska, K; Penson, K A; Babusci, D; Dattoli, G; Duchamp, G H E
2012-03-01
We obtain exact results for fractional equations of Fokker-Planck type using the evolution operator method. We employ exact forms of one-sided Lévy stable distributions to generate a set of self-reproducing solutions. Explicit cases are reported and studied for various fractional order of derivatives, different initial conditions, and for different versions of Fokker-Planck operators.
Numerical solution of the one-dimensional Burgers' equation ...
Indian Academy of Sciences (India)
travelling in a viscous fluid [1]. In literature, many numerical methods have been proposed and implemented for approximating solution of the Burgers' equation. Many authors have used numerical techniques based on finite difference [1–8], finite element [9–13] and boundary element. [14] methods in attempting to solve the ...
Properties of Solutions to the Irving-Mullineux Oscillator Equation
Mickens, Ronald E.
2002-10-01
A nonlinear differential equation is given in the book by Irving and Mullineux to model certain oscillatory phenomena.^1 They use a regular perturbation method^2 to obtain a first-approximation to the assumed periodic solution. However, their result is not uniformly valid and this means that the obtained solution is not periodic because of the presence of secular terms. We show their way of proceeding is not only incorrect, but that in fact the actual solution to this differential equation is a damped oscillatory function. Our proof uses the method of averaging^2,3 and the qualitative theory of differential equations for 2-dim systems. A nonstandard finite-difference scheme is used to calculate numerical solutions for the trajectories in phase-space. References: ^1J. Irving and N. Mullineux, Mathematics in Physics and Engineering (Academic, 1959); section 14.1. ^2R. E. Mickens, Nonlinear Oscillations (Cambridge University Press, 1981). ^3D. W. Jordan and P. Smith, Nonlinear Ordinary Differential Equations (Oxford, 1987).
An Analytical Method For The Solution Of Reactor Dynamic Equations
African Journals Online (AJOL)
In this paper, an analytical method for the solution of nuclear reactor dynamic equations is presented. The method is applied to a linearised high-order deterministic model of a pressurised water reactor plant driven by step-reactivity insertion. A comparison of this method with two other techniques (the matrix exponential and ...
Solitary wave solutions to nonlinear evolution equations in ...
Indian Academy of Sciences (India)
Solitary wave solutions to nonlinear evolution equations in mathematical physics. ANWAR JA'AFAR MOHAMAD JAWAD1, M MIRZAZADEH2,∗ and. ANJAN BISWAS3,4. 1Computer Engineering Technique Department, Al-Rafidain University College, Baghdad, Iraq. 2Department of Engineering Sciences, Faculty of ...
Numerical Solution of Hamilton-Jacobi Equations in High Dimension
2012-11-23
high dimension FA9550-10-1-0029 Maurizio Falcone Dipartimento di Matematica SAPIENZA-Universita di Roma P. Aldo Moro, 2 00185 ROMA AH930...solution of Hamilton-Jacobi equations in high dimension AFOSR contract n. FA9550-10-1-0029 Maurizio Falcone Dipartimento di Matematica SAPIENZA
On the practical solution of the Thue equation
Tzanakis, N.; de Weger, B.M.M.
1989-01-01
This paper gives in detail a practical general method for the explicit determination of all solutions of any Thue equation. It uses a combination of Baker's theory of linear forms in logarithms and recent computational diophantine approximation techniques. An elaborated example is presented.
A series solution of fuzzy integro-differential equations
Directory of Open Access Journals (Sweden)
S. Abbasbandy
2012-11-01
Full Text Available In this work we have used the homotopy analysis method (HAM to obtain solution of fuzzy integro-differential equation (FIDE under Hukuhara differentiability. In this paper for first time, $hbar$-mesh curve introduced for solving FIDE. Also some examples illustrate high efficiency and precision of HAM.
Special solutions of the quantum Yang-Baxter equation
N.W. van den Hijligenberg
1996-01-01
textabstractWe present solutions of the Quantum Yang-Baxter Equation that satisfy the condition [ R_{cd^{ab neq 0 Rightarrow ({ a,b = { c,d ) quad mbox{or quad (b=sigma(a) quad hbox{ and ; d= sigma (c)), ] where $sigma$ denotes the involution on ${ 1, ldots ,n $ given by $sigma (i)=n+1-i$.
Bifurcation of solutions of separable parameterized equations into lines
Directory of Open Access Journals (Sweden)
Yun-Qiu Shen
2010-09-01
Full Text Available Many applications give rise to separable parameterized equations of the form $A(y, muz+b(y, mu=0$, where $y in mathbb{R}^n$, $z in mathbb{R}^N$ and the parameter $mu in mathbb{R}$; here $A(y, mu$ is an $(N+n imes N$ matrix and $b(y, mu in mathbb{R}^{N+n}$. Under the assumption that $A(y,mu$ has full rank we showed in [21] that bifurcation points can be located by solving a reduced equation of the form $f(y, mu=0$. In this paper we extend that method to the case that $A(y,mu$ has rank deficiency one at the bifurcation point. At such a point the solution curve $(y,mu,z$ branches into infinitely many additional solutions, which form a straight line. A numerical method for reducing the problem to a smaller space and locating such a bifurcation point is given. Applications to equilibrium solutions of nonlinear ordinary equations and solutions of discretized partial differential equations are provided.
Radial solutions to semilinear elliptic equations via linearized operators
Directory of Open Access Journals (Sweden)
Phuong Le
2017-04-01
Full Text Available Let $u$ be a classical solution of semilinear elliptic equations in a ball or an annulus in $\\mathbb{R}^N$ with zero Dirichlet boundary condition where the nonlinearity has a convex first derivative. In this note, we prove that if the $N$-th eigenvalue of the linearized operator at $u$ is positive, then $u$ must be radially symmetric.
Positive Solutions for Some Beam Equation Boundary Value Problems
Directory of Open Access Journals (Sweden)
Xu Weiya
2009-01-01
Full Text Available A new fixed point theorem in a cone is applied to obtain the existence of positive solutions of some fourth-order beam equation boundary value problems with dependence on the first-order derivative where is continuous.
Calculation of Volterra kernels for solutions of nonlinear differential equations
van Hemmen, JL; Kistler, WM; Thomas, EGF
2000-01-01
We consider vector-valued autonomous differential equations of the form x' = f(x) + phi with analytic f and investigate the nonanticipative solution operator phi bar right arrow A(phi) in terms of its Volterra series. We show that Volterra kernels of order > 1 occurring in the series expansion of
Explosive solutions of elliptic equations with absorption and non ...
Indian Academy of Sciences (India)
We prove the existence of positive large solutions to the equation u + q ( x ) | ∇ u | a = p ( x ) f ( u ) in a smooth bounded domain ⊂ R N , provided that , are non-negative continuous functions so that any zero of is surrounded by a surface strictly included in on which is positive. Under additional hypotheses on ...
On the Study of Global Solutions for a Nonlinear Equation
Directory of Open Access Journals (Sweden)
Haibo Yan
2014-01-01
Full Text Available The well-posedness of global strong solutions for a nonlinear partial differential equation including the Novikov equation is established provided that its initial value v0(x satisfies a sign condition and v0(x∈Hs(R with s>3/2. If the initial value v0(x∈Hs(R (1≤s≤3/2 and the mean function of (1-∂x2v0(x satisfies the sign condition, it is proved that there exists at least one global weak solution to the equation in the space v(t,x∈L2([0,+∞,Hs(R in the sense of distribution and vx∈L∞([0,+∞×R.
Precise asymptotic behavior of solutions to damped simple pendulum equations
Directory of Open Access Journals (Sweden)
Tetsutaro Shibata
2009-11-01
Full Text Available We consider the simple pendulum equation $$displaylines{ -u''(t + epsilon f(u'(t = lambdasin u(t, quad t in I:=(-1, 1,cr u(t > 0, quad t in I, quad u(pm 1 = 0, }$$ where $0 < epsilon le 1$, $lambda > 0$, and the friction term is either $f(y = pm|y|$ or $f(y = -y$. Note that when $f(y = -y$ and $epsilon = 1$, we have well known original damped simple pendulum equation. To understand the dependance of solutions, to the damped simple pendulum equation with $lambda gg 1$, upon the term $f(u'(t$, we present asymptotic formulas for the maximum norm of the solutions. Also we present an asymptotic formula for the time at which maximum occurs, for the case $f(u = -u$.
Solution of internal erosion equations by asymptotic expansion
Directory of Open Access Journals (Sweden)
Dubujet P.
2012-07-01
Full Text Available One dimensional coupled soil internal erosion and consolidation equations are considered in this work for the special case of well determined sand and clay mixtures with a small proportion of clay phase. An enhanced modelling of the effect of erosion on elastic soil behavior was introduced through damage mechanics concepts. A modified erosion law was proposed. The erosion phenomenon taking place inside the soil was shown to act like a perturbation affecting the classical soil consolidation equation. This interpretation has enabled considering an asymptotic expansion of the coupled erosion consolidation equations in terms of a perturbation parameter linked to the maximum expected internal erosion. A robust analytical solution was obtained via direct integration of equations at order zero and an adequate finite difference scheme that was applied at order one.
New Approximate Analytical Solutions of the Falkner-Skan Equation
Directory of Open Access Journals (Sweden)
Beong In Yun
2012-01-01
Full Text Available We propose an iterative method for solving the Falkner-Skan equation. The method provides approximate analytical solutions which consist of coefficients of the previous iterate solution. By some examples, we show that the presented method with a small number of iterations is competitive with the existing method such as Adomian decomposition method. Furthermore, to improve the accuracy of the proposed method, we suggest an efficient correction method. In practice, for some examples one can observe that the correction method results in highly improved approximate solutions.
Approximate solution of fourth order differential equation in Neumann problem
Directory of Open Access Journals (Sweden)
Jalil Rashidinia
2014-07-01
Full Text Available Generalized solution on Neumann problem of the fourth order ordinary differential equation in space $ W^{2}_{\\alpha} (0, b $ has been discussed , we obtain the condition on B.V.P when the solution is in classical form. Formulation of Quintic Spline Function has been derived and the consistency relations are given.Numerical method,based on Quintic spline approximation has been developed .Spline solution of the given problem has been considered for a certain value of $\\alpha.$ Error analysis of the spline method is given and it has been tested by an example
Boolean delay equations. II. Periodic and aperiodic solutions
Ghil, M.; Mullhaupt, A.
1985-10-01
Boolean delay equations (BDEs) are evolution equations for a vector of discrete variables x (t). The value of each component X i ( t), 0 or 1. depends on previous values of all components x j (t- t ij ), x i (t)=f i (x1( t- t i1),..., x n ( t - t in )). BDEs model the evolution of biological and physical systems with threshold behavior and nonlinear feedbacks. The delays model distinct interaction times between pairs of variables. In this paper, BDEs are studied by algebraic, analytic, and numerical methods. It is shown that solutions depend continuously on the initial data and on the delays. BDEs are classified into conservative and dissipative. All BDEs with rational delays only have periodic solutions only. But conservative BDEs with rationally unrelated delays have aperiodic solutions of increasing complexity. These solutions can be approximated arbitrarily well by periodic solutions of increasing period. Self-similarity and intermittency of aperiodic solutions is studied as a function of delay values, and certain number-theoretic questions related to resonances and diophantine approximation are raised. Period length is shown to be a lower semicontinuous function of the delays for a given BDE, and can be evaluated explicitly for linear equations. We prove that a BDE is structurable stable if and only if it has eventually periodic solutions of bounded period, and if the length of initial transients is bounded. It is shown that, for dissipative BDEs, asymptotic solution behavior is typically governed by a reduced BDE. Applications to climate dynamics and other problems are outlined.
Periodic Solutions and S-Asymptotically Periodic Solutions to Fractional Evolution Equations
Directory of Open Access Journals (Sweden)
Jia Mu
2017-01-01
Full Text Available This paper deals with the existence and uniqueness of periodic solutions, S-asymptotically periodic solutions, and other types of bounded solutions for some fractional evolution equations with the Weyl-Liouville fractional derivative defined for periodic functions. Applying Fourier transform we give reasonable definitions of mild solutions. Then we accurately estimate the spectral radius of resolvent operator and obtain some existence and uniqueness results.
Numerical approximation of random periodic solutions of stochastic differential equations
Feng, Chunrong; Liu, Yu; Zhao, Huaizhong
2017-10-01
In this paper, we discuss the numerical approximation of random periodic solutions of stochastic differential equations (SDEs) with multiplicative noise. We prove the existence of the random periodic solution as the limit of the pull-back flow when the starting time tends to -∞ along the multiple integrals of the period. As the random periodic solution is not explicitly constructible, it is useful to study the numerical approximation. We discretise the SDE using the Euler-Maruyama scheme and modified Milstein scheme. Subsequently, we obtain the existence of the random periodic solution as the limit of the pull-back of the discretised SDE. We prove that the latter is an approximated random periodic solution with an error to the exact one at the rate of √{Δ t} in the mean square sense in Euler-Maruyama method and Δ t in the Milstein method. We also obtain the weak convergence result for the approximation of the periodic measure.
Numerical solution of the space fractional Fokker-Planck equation
Liu, F.; Anh, V.; Turner, I.
2004-04-01
The traditional second-order Fokker-Planck equation may not adequately describe the movement of solute in an aquifer because of large deviation from the dynamics of Brownian motion. Densities of α-stable type have been used to describe the probability distribution of these motions. The resulting governing equation of these motions is similar to the traditional Fokker-Planck equation except that the order α of the highest derivative is fractional. In this paper, a space fractional Fokker-Planck equation (SFFPE) with instantaneous source is considered. A numerical scheme for solving SFFPE is presented. Using the Riemann-Liouville and Grunwald-Letnikov definitions of fractional derivatives, the SFFPE is transformed into a system of ordinary differential equations (ODE). Then the ODE system is solved by a method of lines. Numerical results for SFFPE with a constant diffusion coefficient are evaluated for comparison with the known analytical solution. The numerical approximation of SFFPE with a time-dependent diffusion coefficient is also used to simulate Levy motion with α-stable densities. We will show that the numerical method of SFFPE is able to more accurately model these heavy-tailed motions.
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.
New Numerical Solution of von Karman Equation of Lengthwise Rolling
Directory of Open Access Journals (Sweden)
Rudolf Pernis
2015-01-01
Full Text Available The calculation of average material contact pressure to rolls base on mathematical theory of rolling process given by Karman equation was solved by many authors. The solutions reported by authors are used simplifications for solution of Karman equation. The simplifications are based on two cases for approximation of the circular arch: (a by polygonal curve and (b by parabola. The contribution of the present paper for solution of two-dimensional differential equation of rolling is based on description of the circular arch by equation of a circle. The new term relative stress as nondimensional variable was defined. The result from derived mathematical models can be calculated following variables: normal contact stress distribution, front and back tensions, angle of neutral point, coefficient of the arm of rolling force, rolling force, and rolling torque during rolling process. Laboratory cold rolled experiment of CuZn30 brass material was performed. Work hardening during brass processing was calculated. Comparison of theoretical values of normal contact stress with values of normal contact stress obtained from cold rolling experiment was performed. The calculations were not concluded with roll flattening.
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.)
Weighted Asymptotically Periodic Solutions of Linear Volterra Difference Equations
Directory of Open Access Journals (Sweden)
Josef Diblík
2011-01-01
Full Text Available A linear Volterra difference equation of the form x(n+1=a(n+b(nx(n+∑i=0nK(n,ix(i, where x:N0→R, a:N0→R, K:N0×N0→R and b:N0→R∖{0} is ω-periodic, is considered. Sufficient conditions for the existence of weighted asymptotically periodic solutions of this equation are obtained. Unlike previous investigations, no restriction on ∏j=0ω-1b(j is assumed. The results generalize some of the recent results.
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.
Global solutions of nonlinear Schrödinger equations
Bourgain, J
1999-01-01
This volume presents recent progress in the theory of nonlinear dispersive equations, primarily the nonlinear Schrödinger (NLS) equation. The Cauchy problem for defocusing NLS with critical nonlinearity is discussed. New techniques and results are described on global existence and properties of solutions with large Cauchy data. Current research in harmonic analysis around Strichartz's inequalities and its relevance to nonlinear PDE is presented. Several topics in NLS theory on bounded domains are reviewed. Using the NLS as an example, the book offers comprehensive insight on current research r
On the Partial Analytical Solution of the Kirchhoff Equation
Michels, Dominik L.
2015-09-01
We derive a combined analytical and numerical scheme to solve the (1+1)-dimensional differential Kirchhoff system. Here the object is to obtain an accurate as well as an efficient solution process. Purely numerical algorithms typically have the disadvantage that the quality of solutions decreases enormously with increasing temporal step sizes, which results from the numerical stiffness of the underlying partial differential equations. To prevent that, we apply a differential Thomas decomposition and a Lie symmetry analysis to derive explicit analytical solutions to specific parts of the Kirchhoff system. These solutions are general and depend on arbitrary functions, which we set up according to the numerical solution of the remaining parts. In contrast to a purely numerical handling, this reduces the numerical solution space and prevents the system from becoming unstable. The differential Kirchhoff equation describes the dynamic equilibrium of one-dimensional continua, i.e. slender structures like fibers. We evaluate the advantage of our method by simulating a cilia carpet.
Zielinski, Michal W; McGann, Locksley E; Nychka, John A; Elliott, Janet A W
2017-11-22
The prediction of nonideal chemical potentials in aqueous solutions is important in fields such as cryobiology, where models of water and solute transport-that is, osmotic transport-are used to help develop cryopreservation protocols and where solutions contain many varied solutes and are generally highly concentrated and thus thermodynamically nonideal. In this work, we further the development of a nonideal multisolute solution theory that has found application across a broad range of aqueous systems. This theory is based on the osmotic virial equation and does not depend on multisolute data. Specifically, we derive herein a novel solute chemical potential equation that is thermodynamically consistent with the existing model, and we establish the validity of a grouped solute model for the intracellular space. With this updated solution theory, it is now possible to model cellular osmotic behavior in nonideal solutions containing multiple permeating solutes, such as those commonly encountered by cells during cryopreservation. In addition, because we show here that for the osmotic virial equation the grouped solute approach is mathematically equivalent to treating each solute separately, multisolute solutions in other applications with fixed solute mass ratios can now be treated rigorously with such a model, even when all of the solutes cannot be enumerated.
Parker, A.
1995-07-01
In this second of two articles (designated I and II), the bilinear transformation method is used to obtain stationary periodic solutions of the partially integrable regularized long-wave (RLW) equation. These solutions are expressed in terms of Riemann theta functions, and this approach leads to a new and compact expression for the important dispersion relation. The periodic solution (or cnoidal wave) can be represented as an infinite sum of sech2 ``solitary waves'': this remarkable property may be interpreted in the context of a nonlinear superposition principle. The RLW cnoidal wave approximates to a sinusoidal wave and a solitary wave in the limits of small and large amplitudes, respectively. Analytic approximations and error estimates are given which shed light on the character of the cnoidal wave in the different parameter regimes. Similar results are presented in brief for the related RLW Boussinesq (RLWB) equation.
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 interes...
Approximate Solutions of Fisher's Type Equations with Variable Coefficients
Directory of Open Access Journals (Sweden)
A. H. Bhrawy
2013-01-01
Full Text Available The spectral collocation approximations based on Legendre polynomials are used to compute the numerical solution of time-dependent Fisher’s type problems. The spatial derivatives are collocated at a Legendre-Gauss-Lobatto interpolation nodes. The proposed method has the advantage of reducing the problem to a system of ordinary differential equations in time. The four-stage A-stable implicit Runge-Kutta scheme is applied to solve the resulted system of first order in time. Numerical results show that the Legendre-Gauss-Lobatto collocation method is of high accuracy and is efficient for solving the Fisher’s type equations. Also the results demonstrate that the proposed method is powerful algorithm for solving the nonlinear partial differential equations.
Numerical solutions of the complete Navier-Stokes equations
Hassan, H. A.
1993-01-01
The objective of this study is to compare the use of assumed pdf (probability density function) approaches for modeling supersonic turbulent reacting flowfields with the more elaborate approach where the pdf evolution equation is solved. Assumed pdf approaches for averaging the chemical source terms require modest increases in CPU time typically of the order of 20 percent above treating the source terms as 'laminar.' However, it is difficult to assume a form for these pdf's a priori that correctly mimics the behavior of the actual pdf governing the flow. Solving the evolution equation for the pdf is a theoretically sound approach, but because of the large dimensionality of this function, its solution requires a Monte Carlo method which is computationally expensive and slow to coverage. Preliminary results show both pdf approaches to yield similar solutions for the mean flow variables.
Numerical Solution of Stochastic Nonlinear Fractional Differential Equations
El-Beltagy, Mohamed A.
2015-01-07
Using Wiener-Hermite expansion (WHE) technique in the solution of the stochastic partial differential equations (SPDEs) has the advantage of converting the problem to a system of deterministic equations that can be solved efficiently using the standard deterministic numerical methods [1]. WHE is the only known expansion that handles the white/colored noise exactly. This work introduces a numerical estimation of the stochastic response of the Duffing oscillator with fractional or variable order damping and driven by white noise. The WHE technique is integrated with the Grunwald-Letnikov approximation in case of fractional order and with Coimbra approximation in case of variable-order damping. The numerical solver was tested with the analytic solution and with Monte-Carlo simulations. The developed mixed technique was shown to be efficient in simulating SPDEs.
A Solution to the Fundamental Linear Fractional Order Differential Equation
Hartley, Tom T.; Lorenzo, Carl F.
1998-01-01
This paper provides a solution to the fundamental linear fractional order differential equation, namely, (sub c)d(sup q, sub t) + ax(t) = bu(t). The impulse response solution is shown to be a series, named the F-function, which generalizes the normal exponential function. The F-function provides the basis for a qth order "fractional pole". Complex plane behavior is elucidated and a simple example, the inductor terminated semi- infinite lossy line, is used to demonstrate the theory.
Analytical Solution for the Time-Fractional Telegraph Equation
Directory of Open Access Journals (Sweden)
F. Huang
2009-01-01
Full Text Available We discuss and derive the analytical solution for three basic problems of the so-called time-fractional telegraph equation. The Cauchy and Signaling problems are solved by means of juxtaposition of transforms of the Laplace and Fourier transforms in variable t and x, respectively. the appropriate structures and negative prosperities for their Green functions are provided. The boundary problem in a bounded space domain is also solved by the spatial Sine transform and temporal Laplace transform, whose solution is given in the form of a series.
Radial solutions of equations and inequalities involving the -Laplacian
Directory of Open Access Journals (Sweden)
Reichel Wolfgang
1997-01-01
Full Text Available Several problems for the differential equation are considered. For , the operator is the radially symmetric -Laplacian in . For the initial value problem with given data various uniqueness conditions and counterexamples to uniqueness are given. For the case where is increasing in , a sharp comparison theorem is established; it leads to maximal solutions, nonuniqueness and uniqueness results, among others. Using these results, a strong comparison principle for the boundary value problem and a number of properties of blow-up solutions are proved under weak assumptions on the nonlinearity .
Numerical Solution of a Model Equation of Price Formation
Chernogorova, T.; Vulkov, L.
2009-10-01
The paper [2] is devoted to the effect of reconciling the classical Black-Sholes theory of option pricing and hedging with various phenomena observed in the markets such as the influence of trading and hedging on the dynamics of an asset. Here we will discuss the numerical solution of initial boundary-value problems to a model equation of the theory. The lack of regularity in the solution as a result from Dirac delta coefficient reduces the accuracy in the numerical computations. First, we apply the finite volume method to discretize the differential problem. Second, we implement a technique of local regularization introduced by A-K. Tornberg and B. Engquist [7] for handling this equation. We derived the numerical regularization process into two steps: the Dirac delta function is regularized and then the regularized differential equation is discretized by difference schemes. Using the discrete maximum principle a priori bounds are obtained for the difference equations that imply stability and convergence of difference schemes for the problem under consideration. Numerical experiments are discussed.
Periodic solutions for the Landau-Lifshitz-Gilbert equation
Huber, Alexander
2010-01-01
Ferromagnetic materials tend to develop very complex magnetization patterns whose time evolution is modeled by the so-called Landau-Lifshitz-Gilbert equation (LLG). In this paper, we construct time-periodic solutions for LLG in the regime of soft and small ferromagnetic particles which satisfy a certain shape condition. Roughly speaking, it is assumed that the length of the particle is greater than its hight and its width. The approach is based on a perturbation argument and the spectral anal...
Differential invariants and exact solutions of the Einstein equations
Lychagin, Valentin; Yumaguzhin, Valeriy
2017-06-01
In this paper (cf. Lychagin and Yumaguzhin, in Anal Math Phys, 2016) a class of totally geodesics solutions for the vacuum Einstein equations is introduced. It consists of Einstein metrics of signature (1,3) such that 2-dimensional distributions, defined by the Weyl tensor, are completely integrable and totally geodesic. The complete and explicit description of metrics from these class is given. It is shown that these metrics depend on two functions in one variable and one harmonic function.
Upper and lower bounds of solutions for fractional integral equations
Directory of Open Access Journals (Sweden)
Shaher Momani
2008-03-01
Full Text Available In this paper we consider the integral equation offractional order in sense of Riemann-Liouville operatorum(t = a(t Iα [b(tu(t]+f(twith m ≥ 1, t ∈ [0, T], T < ∞ and 0< α <1. We discuss the existence, uniqueness, maximal, minimal and the upper and lower bounds of the solutions. Also we illustrate our results with examples.
Ground state solutions for non-local fractional Schrodinger equations
Directory of Open Access Journals (Sweden)
Yang Pu
2015-08-01
Full Text Available In this article, we study a time-independent fractional Schrodinger equation with non-local (regional diffusion $$ (-\\Delta^{\\alpha}_{\\rho}u + V(xu = f(x,u \\quad \\text{in }\\mathbb{R}^{N}, $$ where $\\alpha \\in (0,1$, $N > 2\\alpha$. We establish the existence of a non-negative ground state solution by variational methods.
Global regular solutions for the nonhomogeneous Carrier equation
N. A. Larkin
2002-01-01
We study in a n + 1 -dimensional cylinder Q global solvability of the mixed problem for the nonhomogeneous Carrier equation u t t − M ( x , t , || u ( t ) || 2 ) Δ u + g ( x , t , u t ) = f ( x , t ) without restrictions on a size of initial data and f ( x , t ) . For any natural n, we prove existence, uniqueness and the exponential decay of the energy for global generalized solutions. When n=2 , we pro...
Analytical Solution of Generalized Space-Time Fractional Cable Equation
Ram K. Saxena; Zivorad Tomovski; Trifce Sandev
2015-01-01
In this paper, we consider generalized space-time fractional cable equation in presence of external source. By using the Fourier-Laplace transform we obtain the Green function in terms of infinite series in H-functions. The fractional moments of the fundamental solution are derived and their asymptotic behavior in the short and long time limit is analyzed. Some previously obtained results are compared with those presented in this paper. By using the Bernstein characterization theorem we find ...
Nonstatic plane-symmetric solutions for Einstein-Maxwell equations
Energy Technology Data Exchange (ETDEWEB)
Hajj-Boutros, J.; Sfeila, J.
1985-11-16
The general solution of the Einstein-Maxwell field equations is obtained under the assumptions that 1) the source of the gravitational field is a charged dust, 2) the space-time is plane-symmetric, 3) the metric is of the form ds/sup 2/ = dt/sup 2/ - exp (2u(t, z)) dz/sup 2/ - Z/sup 2/(z) T/sup 2/(t)(dx/sup 2/ + dy/sup 2/). (orig.).
A polynomial bound on solutions of quadratic equations in free groups
Lysenok, Igor; Myasnikov, Alexei
2011-01-01
We provide polynomial upper bounds on the size of a shortest solution for quadratic equations in a free group. A similar bound is given for parametric solutions in the description of solutions sets of quadratic equations in a free group.
Tariq, Hira; Akram, Ghazala
2017-05-01
In this article, new exact analytical solutions of some nonlinear evolution equations (NLEEs) arising in science, engineering and mathematical physics, namely time fractional Cahn-Allen equation and time fractional Phi-4 equation are developed using tanh method by means of fractional complex transform. The obtained results are demonstrated by graphs for the new solutions.
The number of polynomial solutions of polynomial Riccati equations
Gasull, Armengol; Torregrosa, Joan; Zhang, Xiang
2016-11-01
Consider real or complex polynomial Riccati differential equations a (x) y ˙ =b0 (x) +b1 (x) y +b2 (x)y2 with all the involved functions being polynomials of degree at most η. We prove that the maximum number of polynomial solutions is η + 1 (resp. 2) when η ≥ 1 (resp. η = 0) and that these bounds are sharp. For real trigonometric polynomial Riccati differential equations with all the functions being trigonometric polynomials of degree at most η ≥ 1 we prove a similar result. In this case, the maximum number of trigonometric polynomial solutions is 2η (resp. 3) when η ≥ 2 (resp. η = 1) and, again, these bounds are sharp. Although the proof of both results has the same starting point, the classical result that asserts that the cross ratio of four different solutions of a Riccati differential equation is constant, the trigonometric case is much more involved. The main reason is that the ring of trigonometric polynomials is not a unique factorization domain.
Infinite hierarchy of nonlinear Schrödinger equations and their solutions.
Ankiewicz, A; Kedziora, D J; Chowdury, A; Bandelow, U; Akhmediev, N
2016-01-01
We study the infinite integrable nonlinear Schrödinger equation hierarchy beyond the Lakshmanan-Porsezian-Daniel equation which is a particular (fourth-order) case of the hierarchy. In particular, we present the generalized Lax pair and generalized soliton solutions, plane wave solutions, Akhmediev breathers, Kuznetsov-Ma breathers, periodic solutions, and rogue wave solutions for this infinite-order hierarchy. We find that "even- order" equations in the set affect phase and "stretching factors" in the solutions, while "odd-order" equations affect the velocities. Hence odd-order equation solutions can be real functions, while even-order equation solutions are always complex.
Homoclinic and quasi-homoclinic solutions for damped differential equations
Directory of Open Access Journals (Sweden)
Chuan-Fang Zhang
2015-01-01
Full Text Available We study the existence and multiplicity of homoclinic solutions for the second-order damped differential equation $$ \\ddot{u}+c\\dot{u}-L(tu+W_u(t,u=0, $$ where L(t and W(t,u are neither autonomous nor periodic in t. Under certain assumptions on L and W, we obtain infinitely many homoclinic solutions when the nonlinearity W(t,u is sub-quadratic or super-quadratic by using critical point theorems. Some recent results in the literature are generalized, and the open problem proposed by Zhang and Yuan is solved. In addition, with the help of the Nehari manifold, we consider the case where W(t,u is indefinite and prove the existence of at least one nontrivial quasi-homoclinic solution.
On the General Analytical Solution of the Kinematic Cosserat Equations
Michels, Dominik L.
2016-09-01
Based on a Lie symmetry analysis, we construct a closed form solution to the kinematic part of the (partial differential) Cosserat equations describing the mechanical behavior of elastic rods. The solution depends on two arbitrary analytical vector functions and is analytical everywhere except a certain domain of the independent variables in which one of the arbitrary vector functions satisfies a simple explicitly given algebraic relation. As our main theoretical result, in addition to the construction of the solution, we proof its generality. Based on this observation, a hybrid semi-analytical solver for highly viscous two-way coupled fluid-rod problems is developed which allows for the interactive high-fidelity simulations of flagellated microswimmers as a result of a substantial reduction of the numerical stiffness.
A numerical solution for the diffusion equation in hydrogeologic systems
Ishii, A.L.; Healy, R.W.; Striegl, R.G.
1989-01-01
The documentation of a computer code for the numerical solution of the linear diffusion equation in one or two dimensions in Cartesian or cylindrical coordinates is presented. Applications of the program include molecular diffusion, heat conduction, and fluid flow in confined systems. The flow media may be anisotropic and heterogeneous. The model is formulated by replacing the continuous linear diffusion equation by discrete finite-difference approximations at each node in a block-centered grid. The resulting matrix equation is solved by the method of preconditioned conjugate gradients. The conjugate gradient method does not require the estimation of iteration parameters and is guaranteed convergent in the absence of rounding error. The matrixes are preconditioned to decrease the steps to convergence. The model allows the specification of any number of boundary conditions for any number of stress periods, and the output of a summary table for selected nodes showing flux and the concentration of the flux quantity for each time step. The model is written in a modular format for ease of modification. The model was verified by comparison of numerical and analytical solutions for cases of molecular diffusion, two-dimensional heat transfer, and axisymmetric radial saturated fluid flow. Application of the model to a hypothetical two-dimensional field situation of gas diffusion in the unsaturated zone is demonstrated. The input and output files are included as a check on program installation. The definition of variables, input requirements, flow chart, and program listing are included in the attachments. (USGS)
Solution of the gap equation in neutron matter
Energy Technology Data Exchange (ETDEWEB)
Khodel, V.A. [Washington Univ., St. Louis, MO (United States). Dept. of Physics; Khodel, V.V. [Washington Univ., St. Louis, MO (United States). Dept. of Physics; Clark, J.W. [Washington Univ., St. Louis, MO (United States). Dept. of Physics
1996-03-04
The problem of solving the gap equation for S-wave pairing in pure neutron matter is considered for the case that the pairing matrix elements V(p,p`) are calculated directly from a realistic bare neutron-neutron potential containing a strong short-range repulsion. The original gap equation is replaced identically by a coupled set of equations: a non-singular quasilinear integral equation for the dimensionless gap function {chi}(p) defined by {Delta}(p)={Delta}{sub F}{chi}(p) and a non-linear algebraic equation for the gap magnitude {Delta}{sub F}={Delta}(p{sub F}) at the Fermi surface. This reformulation admits a robust and rapidly convergent iteration procedure for the determination of the gap function. The treatment may be extended to singlet or triplet pairing in non-zero angular momentum states. S-wave pairing is investigated numerically for the Reid-soft-core interaction. Although the pairing matrix elements of this potential are everywhere positive, non-trivial solutions of the gap equation are obtained on the range 0 < p{sub F} < p{sub c}=1.7496.. fm{sup -1} of Fermi momenta, with the gap parameter {Delta}{sub F} reaching a maximum of some 3 MeV near p{sub F}=0.85 fm{sup -1}. Numerical results are also provided for the highly realistic Argonne v{sub 14} and v{sub 18} interactions. Within the context of the new computational scheme, a condition for closure of the gap is derived in terms of the first zero p{sub 0} of the gap function {Delta}(p). It is shown that {Delta}{sub F} vanishes exponentially not only in the low-density limit p{sub F}{yields}0, but also as the Fermi momentum rises and approaches the upper critical value p{sub c} specified by p{sub F}=p{sub 0}(p{sub F}), beyond which there exists no non-trivial solution of the gap equation. The numerical results for the function {Delta}(p) in neutron matter display a remarkable universality of structure, visible especially in the stability of p{sub 0} under variation of density. (orig./WL).
A Note on the Relationship Between Solutions of Einstein, Ramanujan and Chazy Equations
Esmakhanova, Kuralay; Myrzakulov, Yerlan; Nugmanova, Gulgasyl; Myrzakulov, Ratbay
2012-04-01
The Einstein equation for the Friedmann-Robertson-Walker metric plays a fundamental role in cosmology. The direct search of the exact solutions of the Einstein equation even in this simple metric case is sometime a hard job. Therefore, it is useful to construct solutions of the Einstein equation using a known solutions of some other equations which are equivalent or related to the Einstein equation. In this work, we establish the relationship the Einstein equation with two other famous equations namely the Ramanujan equation and the Chazy equation. Both these two equations play an important role in the number theory. Using the known solutions of the Ramanujan and Chazy equations, we find the corresponding solutions of the Einstein equation.
Random ordinary differential equations and their numerical solution
Han, Xiaoying
2017-01-01
This book is intended to make recent results on the derivation of higher order numerical schemes for random ordinary differential equations (RODEs) available to a broader readership, and to familiarize readers with RODEs themselves as well as the closely associated theory of random dynamical systems. In addition, it demonstrates how RODEs are being used in the biological sciences, where non-Gaussian and bounded noise are often more realistic than the Gaussian white noise in stochastic differential equations (SODEs). RODEs are used in many important applications and play a fundamental role in the theory of random dynamical systems. They can be analyzed pathwise with deterministic calculus, but require further treatment beyond that of classical ODE theory due to the lack of smoothness in their time variable. Although classical numerical schemes for ODEs can be used pathwise for RODEs, they rarely attain their traditional order since the solutions of RODEs do not have sufficient smoothness to have Taylor ...
Reduction of the Sharma-Tasso-Olver equation and series solutions
Zhou, Yuqian; Yang, Fuchun; Liu, Qian
2011-02-01
This paper considers series solutions of the Sharma-Tasso-Olver (STO) equation. By using the extended homogenous balance method, we reduce the STO equation to a linear PDE and obtain Bäcklund transformation of it. Furthermore, the self-transformation of solutions for the STO equation is obtained. By the Bäcklund transformation and various series solutions of the PDE, abundant exact solutions of the STO equation are obtained including the multi-solitary wave solution, trigonometric function series solution, rational series solution and solution consisting of the three types of solutions.
Oscillation of solutions of some higher order linear differential equations
Directory of Open Access Journals (Sweden)
Hong-Yan Xu
2009-11-01
Full Text Available In this paper, we deal with the order of growth and the hyper order of solutions of higher order linear differential equations $$f^{(k}+B_{k-1}f^{(k-1}+\\cdots+B_1f'+B_0f=F$$ where $B_j(z (j=0,1,\\ldots,k-1$ and $F$ are entire functions or polynomials. Some results are obtained which improve and extend previous results given by Z.-X. Chen, J. Wang, T.-B. Cao and C.-H. Li.
Algebro-Geometric Solutions for a Discrete Integrable Equation
Directory of Open Access Journals (Sweden)
Mengshuang Tao
2017-01-01
Full Text Available With the assistance of a Lie algebra whose element is a matrix, we introduce a discrete spectral problem. By means of discrete zero curvature equation, we obtain a discrete integrable hierarchy. According to decomposition of the discrete systems, the new differential-difference integrable systems with two-potential functions are derived. By constructing the Abel-Jacobi coordinates to straighten the continuous and discrete flows, the Riemann theta functions are proposed. Based on the Riemann theta functions, the algebro-geometric solutions for the discrete integrable systems are obtained.
Periodic solutions for the Landau-Lifshitz-Gilbert equation
Huber, Alexander
2011-03-01
Ferromagnetic materials tend to develop very complex magnetization patterns whose time evolution is modeled by the so-called Landau-Lifshitz-Gilbert equation (LLG). In this paper, we construct time-periodic solutions for LLG in the regime of soft and small ferromagnetic particles which satisfy a certain shape condition. Roughly speaking, it is assumed that the length of the particle is greater than its hight and its width. The approach is based on a perturbation argument and the spectral analysis of the corresponding linearized problem as well as the theory of sectorial operators.
Existence of infinitely many radial solutions for quasilinear Schrodinger equations
Directory of Open Access Journals (Sweden)
Gui Bao
2014-10-01
Full Text Available In this article we prove the existence of radial solutions with arbitrarily many sign changes for quasilinear Schrodinger equation $$ -\\sum_{i,j=1}^{N}\\partial_j(a_{ij}(u\\partial_iu +\\frac{1}{2}\\sum_{i,j=1}^{N}a'_{ij}(u\\partial_iu\\partial_ju+V(xu =|u|^{p-1}u,~x\\in\\mathbb{R}^N, $$ where $N\\geq3$, $p\\in(1,\\frac{3N+2}{N-2}$. The proof is accomplished by using minimization under a constraint.
Analytical Solution of Generalized Space-Time Fractional Cable Equation
Directory of Open Access Journals (Sweden)
Ram K. Saxena
2015-04-01
Full Text Available In this paper, we consider generalized space-time fractional cable equation in presence of external source. By using the Fourier-Laplace transform we obtain the Green function in terms of infinite series in H-functions. The fractional moments of the fundamental solution are derived and their asymptotic behavior in the short and long time limit is analyzed. Some previously obtained results are compared with those presented in this paper. By using the Bernstein characterization theorem we find the conditions under which the even moments are non-negative.
A new method for the solution of the Schroedinger equation
Energy Technology Data Exchange (ETDEWEB)
Amore, Paolo [Facultad de Ciencias, Universidad de Colima, Bernal DIaz del Castillo 340, Colima, Colima (Mexico); Aranda, Alfredo [Facultad de Ciencias, Universidad de Colima, Bernal DIaz del Castillo 340, Colima, Colima (Mexico); De Pace, Arturo [Istituto Nazionale di Fisica Nucleare, Sezione di Torino, Via P Giuria 1, I-10125, Torino (Italy)
2004-03-12
We present a new method for the solution of the Schroedinger equation applicable to problems of a non-perturbative nature. The method works by identifying three different scales in the problem, which then are treated independently: an asymptotic scale, which depends uniquely on the form of the potential at large distances; an intermediate scale, still characterized by an exponential decay of the wavefunction; and, finally, a short distance scale, in which the wavefunction is sizable. The notion of optimized perturbation is then used in the last two regimes. We apply the method to the quantum anharmonic oscillator and find it suitable to treat both energy eigenvalues and wavefunctions, even for strong couplings.
On the solutions of fractional reaction-diffusion equations
Directory of Open Access Journals (Sweden)
Jagdev Singh
2013-05-01
Full Text Available In this paper, we obtain the solution of a fractional reaction-diffusion equation associated with the generalized Riemann-Liouville fractional derivative as the time derivative and Riesz-Feller fractional derivative as the space-derivative. The results are derived by the application of the Laplace and Fourier transforms in compact and elegant form in terms of Mittag-Leffler function and H-function. The results obtained here are of general nature and include the results investigated earlier by many authors.
Asymptotics of weakly collapsing solutions of nonlinear Schroedinger equation
Ovchinnikov, Yu N
2001-01-01
One studied possible types of asymptotic behavior of weakly collapsing solution of the 3-rd nonlinear Schroedinger equation. It is shown that within left brace A, C sub 1 right brace parameter space there are two neighboring lines along which the amplitude of oscillation terms is exponentially small as to C sub 1 parameter. The same lines locates values of left brace A, C sub 1 right brace parameters at which the energy is equal to zero. With increase of C sub 1 parameter the accuracy of numerical determination of points with zero energy drops abruptly
Sign-changing solutions for non-local elliptic equations
Directory of Open Access Journals (Sweden)
Huxiao Luo
2017-07-01
Full Text Available This article concerns the existence of sign-changing solutions for equations driven by a non-local integrodifferential operator with homogeneous Dirichlet boundary conditions, $$\\displaylines{ -\\mathcal{L}_Ku=f(x,u,\\quad x\\in \\Omega, \\cr u=0,\\quad x\\in \\mathbb{R}^n\\setminus\\Omega, }$$ where $\\Omega\\subset\\mathbb{R}^n\\; (n\\geq2$ is a bounded, smooth domain and the nonlinear term f satisfies suitable growth assumptions. By using Brouwer's degree theory and Deformation Lemma and arguing as in [2], we prove that there exists a least energy sign-changing solution. Our results generalize and improve some results obtained in [27
Energy Technology Data Exchange (ETDEWEB)
Kalla, C, E-mail: Caroline.Kalla@u-bourgogne.fr [Institut de Mathematiques de Bourgogne, Universite de Bourgogne, 9 avenue Alain Savary, 21078 Dijon (France)
2011-08-19
We present new solutions in terms of elementary functions of the multi-component nonlinear Schroedinger equations and known solutions of the Davey-Stewartson equations such as multi-soliton, breather, dromion and lump solutions. These solutions are given in a simple determinantal form and are obtained as limiting cases in suitable degenerations of previously derived algebro-geometric solutions. In particular, we present for the first time breather and rational breather solutions of the multi-component nonlinear Schroedinger equations.
Runkel, Robert L.; Chapra, Steven C.
1993-01-01
Several investigators have proposed solute transport models that incorporate the effects of transient storage. Transient storage occurs in small streams when portions of the transported solute become isolated in zones of water that are immobile relative to water in the main channel (e.g., pools, gravel beds). Transient storage is modeled by adding a storage term to the advection-dispersion equation describing conservation of mass for the main channel. In addition, a separate mass balance equation is written for the storage zone. Although numerous applications of the transient storage equations may be found in the literature, little attention has been paid to the numerical aspects of the approach. Of particular interest is the coupled nature of the equations describing mass conservation for the main channel and the storage zone. In the work described herein, an implicit finite difference technique is developed that allows for a decoupling of the governing differential equations. This decoupling method may be applied to other sets of coupled equations such as those describing sediment-water interactions for toxic contaminants. For the case at hand, decoupling leads to a 50% reduction in simulation run time. Computational costs may be further reduced through efficient application of the Thomas algorithm. These techniques may be easily incorporated into existing codes and new applications in which simulation run time is of concern.
Symmetry reductions and exact solutions of Shallow water wave equations
Clarkson, P A
1994-01-01
In this paper we study symmetry reductions and exact solutions of the shallow water wave (SWW) equation u_{xxxt} + \\alpha u_x u_{xt} + \\beta u_t u_{xx} - u_{xt} - u_{xx} = 0,\\eqno(1) where \\alpha and \\beta are arbitrary, nonzero, constants, which is derivable using the so-called Boussinesq approximation. Two special cases of this equation, or the equivalent nonlocal equation obtained by setting u_x=U, have been discussed in the literature. The case \\alpha=2\\beta was discussed by Ablowitz, Kaup, Newell and Segur [{\\it Stud.\\ Appl.\\ Math.}, {\\bf53} (1974) 249], who showed that this case was solvable by inverse scattering through a second order linear problem. This case and the case \\alpha=\\beta were studied by Hirota and Satsuma [{\\it J.\\ Phys.\\ Soc.\\ Japan}, {\\bf40} (1976) 611] using Hirota's bi-linear technique. Further the case \\alpha=\\beta is solvable by inverse scattering through a third order linear problem. In this paper a catalogue of symmetry reductions is obtained using the classical Lie method and th...
Numerical solution of the Fokker-Planck equation
Energy Technology Data Exchange (ETDEWEB)
Shoucri, M. [Institut de Recherche Hydro-Quebec (IREQ), Varennes Quebec (Canada); Peysson, Y. [Association Euratom-CEA Cadarache, CEA/DSM/DRFC, 13 - Saint-Paul-lez-Durance (France); Shkarofsky, I. [MPB Technologies Inc., Quebec (Canada)
2006-06-15
A code to solve the Fokker-Planck kinetic equation for electrons and ions is presented. The electrons are treated with a relativistic collision operator. The importance of the numerical approach associated with an exact relativistic treatment for the solution of the electrons dynamic is emphasized in order to study accurately all the physics associated with the electron distribution function, as for instance the physics associated with the hot tail, the fast electron transport and the shape of the distribution function. Accurate relativistic treatment of the hot tail helps make the study of these problems less phenomenological and more physical. The pertinent equations for two different ions species, a majority ion and a minority ion population, are included in the code. The ions are treated with non-relativistic equations. The code also includes the appropriate quasi-linear operator for lower hybrid current drive, electron cyclotron heating and current drive, ion cyclotron heating for each of the ion species. This allows accurate study of synergy effects between the different sources for plasma heating and current drive. The exchange terms between the different species are included. The code includes also the option of a variable grid size for electrons and ions. (authors)
Localized solutions for a nonlocal discrete NLS equation
Energy Technology Data Exchange (ETDEWEB)
Ben, Roberto I. [Instituto de Desarrollo Humano, Universidad Nacional de General Sarmiento, J.M. Gutiérrez 1150, 1613 Los Polvorines (Argentina); Cisneros Ake, Luís [Department of Mathematics, ESFM, Instituto Politécnico Nacional, Unidad Profesional Adolfo López Mateos Edificio 9, 07738 México D.F. (Mexico); Minzoni, A.A. [Depto. Matemáticas y Mecánica, I.I.M.A.S.-U.N.A.M., Apdo. Postal 20-726, 01000 México D.F. (Mexico); Panayotaros, Panayotis, E-mail: panos@mym.iimas.unam.mx [Depto. Matemáticas y Mecánica, I.I.M.A.S.-U.N.A.M., Apdo. Postal 20-726, 01000 México D.F. (Mexico)
2015-09-04
We study spatially localized time-periodic solutions of breather type for a cubic discrete NLS equation with a nonlocal nonlinearity that models light propagation in a liquid crystal waveguide array. We show the existence of breather solutions in the limit where both linear and nonlinear intersite couplings vanish, and in the limit where the linear coupling vanishes with arbitrary nonlinear intersite coupling. Breathers of this nonlocal regime exhibit some interesting features that depart from what is seen in the NLS breathers with power nonlinearity. One property we see theoretically is the presence of higher amplitude at interfaces between sites with zero and nonzero amplitude in the vanishing linear coupling limit. A numerical study also suggests the presence of internal modes of orbitally stable localized modes. - Highlights: • Show existence of spatially localized solutions in nonlocal discrete NLS model. • Study spatial properties of localized solutions for arbitrary nonlinear nonlocal coupling. • Present numerical evidence that nonlocality leads to internal modes around stable breathers. • Present theoretical and numerical evidence for amplitude maxima at interfaces.
Non-Intrusive Solution of Stochastic and Parametric Equations
Matthies, Hermann
2015-01-07
Many problems depend on parameters, which may be a finite set of numerical values, or mathematically more complicated objects like for example processes or fields. We address the situation where we have an equation which depends on parameters; stochastic equations are a special case of such parametric problems where the parameters are elements from a probability space. One common way to represent this dependability on parameters is by evaluating the state (or solution) of the system under investigation for different values of the parameters. But often one wants to evaluate the solution quickly for a new set of parameters where it has not been sampled. In this situation it may be advantageous to express the parameter dependent solution with an approximation which allows for rapid evaluation of the solution. Such approximations are also called proxy or surrogate models, response functions, or emulators. All these methods may be seen as functional approximations—representations of the solution by an “easily computable” function of the parameters, as opposed to pure samples. The most obvious methods of approximation used are based on interpolation, in this context often labelled as collocation. In the frequent situation where one has a “solver” for the equation for a given parameter value, i.e. a software component or a program, it is evident that this can be used to independently—if desired in parallel—solve for all the parameter values which subsequently may be used either for the interpolation or in the quadrature for the projection. Such methods are therefore uncoupled for each parameter value, and they additionally often carry the label “non-intrusive”. Without much argument all other methods— which produce a coupled system of equations–are almost always labelled as “intrusive”, meaning that one cannot use the original solver. We want to show here that this not necessarily the case. Another approach is to choose some other projection onto
Directory of Open Access Journals (Sweden)
Emad A.-B. Abdel-Salam
2013-01-01
Full Text Available The fractional Riccati expansion method is proposed to solve fractional differential equations. To illustrate the effectiveness of the method, space-time fractional Korteweg-de Vries equation, regularized long-wave equation, Boussinesq equation, and Klein-Gordon equation are considered. As a result, abundant types of exact analytical solutions are obtained. These solutions include generalized trigonometric and hyperbolic functions solutions which may be useful for further understanding of the mechanisms of the complicated nonlinear physical phenomena and fractional differential equations. Among these solutions, some are found for the first time. The periodic and kink solutions are founded as special case.
Singularly perturbed Burger-Huxley equation: Analytical solution ...
African Journals Online (AJOL)
user
The parameter ε in this equation defines the thermal diffusivity in the medium. In mathematics, heat equation is the prototypical parabolic partial differential equation. In statistics, the heat equation is connected with the study of Brownian motion via the Fokker-Planck equation. The heat equation is used in probability and.
Rapidly decaying solutions of the nonlinear Schroedinger equation
Energy Technology Data Exchange (ETDEWEB)
Cazenave, T. (Paris-6 Univ., 75 (France). Lab. d' Analyse Numerique); Weissler, F.B. (ENS, 94 - Cachan (France). Centre de Mathematiques Paris-7 Univ., 94 - Creteil (France). UFR de Sciences)
1992-06-01
We consider global solutions of the nonlinear Schroedinger equation iu{sub t}+{Delta}u={lambda}vertical strokeuvertical stroke{sup {alpha}}u, in R{sup N}, (NLS) where {lambda}{epsilon}R and 0<{alpha}< 4/N-2. In particular, for {alpha}>{alpha}{sub 0}=(2-N+{radical}(N{sup 2}+12N+4))/2N, we show that for every ({phi}{epsilon}H{sup 1}(R{sup N}) such that x{phi}(x){epsilon}L{sup 2}(R{sup N}), the solution of (NLS) with initial value {phi}(x)e{sup i(bvertical} {sup strokexvertical} {sup stroke2/4)} is global and rapidly decaying as t{yields}{infinity} if b is large enough. Furthermore, by applying the pseudo-conformal transformation and studying the resulting nonautonomous nonlinear Schroedinger equation, we obtain both new results and simpler proofs of some known results concerning the scattering theory. In particular, we construct the wave operators for 4/N+2<{alpha}<4/N-2. Also, we establish a low energy scattering theory for the same range of {alpha} and show that, at least for {lambda}<0, the lower bound on {alpha} is optimal. Finally, if {lambda}>0, we prove asymptotic completeness for {alpha}{sub 0}{<=}{alpha}<4/N-2. (orig.).
Exact unsteady solutions to the Navier-Stokes and viscous MHD equations
Energy Technology Data Exchange (ETDEWEB)
Bogoyavlenskij, Oleg I
2003-02-10
Infinite-dimensional families of exact solutions that depend on all four variables t,x,y,z are derived for the Navier-Stokes equations and for viscous magnetohydrodynamics equations. Soliton-like solutions--viscons--are introduced.
Solution of the Master Equation for Quantum Brownian Motion Given by the Schrödinger Equation
Directory of Open Access Journals (Sweden)
R. Sinuvasan
2016-12-01
Full Text Available We consider the master equation of quantum Brownian motion, and with the application of the group invariant transformation, we show that there exists a surface on which the solution of the master equation is given by an autonomous one-dimensional Schrödinger Equation.
Complex Singular Solutions of the 3-d Navier-Stokes Equations and Related Real Solutions
Boldrighini, Carlo; Li, Dong; Sinai, Yakov G.
2017-04-01
By applying methods of statistical physics Li and Sinai (J Eur Math Soc 10:267-313, 2008) proved that there are complex solutions of the Navier-Stokes equations in the whole space R3 which blow up at a finite time. We present a review of the results obtained so far, by theoretical work and computer simulations, for the singular complex solutions, and compare with the behavior of related real solutions. We also discuss the possible application of the techniques introduced in (J Eur Math Soc 10:267-313, 2008) to the study of the real ones.
Soliton solutions for a quasilinear Schrödinger equation via Morse ...
Indian Academy of Sciences (India)
Soliton solutions for a quasilinear Schrödinger equation via Morse theory ... Quasilinear Schrödinger equation; soliton solution; critical point; Morse theory; local linking. Abstract. In this paper, Morse theory is used to show the existence of nontrivial weak solutions to a class of quasilinear Schrödinger equation of the form.
Exact Solutions of Space-time Fractional EW and modified EW equations
Korkmaz, Alper
2016-01-01
The bright soliton solutions and singular solutions are constructed for space-time fractional EW and modified EW equations. Both equations are reduced to ordinary differential equations by the use of fractional complex transform and properties of modified Riemann-Liouville derivative. Then, implementation of ansatz method the solutions are constructed.
Exact solutions of space-time fractional EW and modified EW equations
Korkmaz, Alper
2017-03-01
The bright soliton solutions and singular solutions are constructed for space-time fractional EW and modified EW equations. Both equations are reduced to ordinary differential equations by the use of fractional complex transform and properties of modified Riemann-Liouville derivative. Then, implementation of ansatz method the solutions are constructed.
Directory of Open Access Journals (Sweden)
Olaniyi Samuel Iyiola
2014-09-01
Full Text Available In this paper, we obtain analytical solutions of homogeneous time-fractional Gardner equation and non-homogeneous time-fractional models (including Buck-master equation using q-Homotopy Analysis Method (q-HAM. Our work displays the elegant nature of the application of q-HAM not only to solve homogeneous non-linear fractional differential equations but also to solve the non-homogeneous fractional differential equations. The presence of the auxiliary parameter h helps in an effective way to obtain better approximation comparable to exact solutions. The fraction-factor in this method gives it an edge over other existing analytical methods for non-linear differential equations. Comparisons are made upon the existence of exact solutions to these models. The analysis shows that our analytical solutions converge very rapidly to the exact solutions.
ACCURATE CHEMICAL MASTER EQUATION SOLUTION USING MULTI-FINITE BUFFERS
Cao, Youfang; Terebus, Anna; Liang, Jie
2016-01-01
The discrete chemical master equation (dCME) provides a fundamental framework for studying stochasticity in mesoscopic networks. Because of the multi-scale nature of many networks where reaction rates have large disparity, directly solving dCMEs is intractable due to the exploding size of the state space. It is important to truncate the state space effectively with quantified errors, so accurate solutions can be computed. It is also important to know if all major probabilistic peaks have been computed. Here we introduce the Accurate CME (ACME) algorithm for obtaining direct solutions to dCMEs. With multi-finite buffers for reducing the state space by O(n!), exact steady-state and time-evolving network probability landscapes can be computed. We further describe a theoretical framework of aggregating microstates into a smaller number of macrostates by decomposing a network into independent aggregated birth and death processes, and give an a priori method for rapidly determining steady-state truncation errors. The maximal sizes of the finite buffers for a given error tolerance can also be pre-computed without costly trial solutions of dCMEs. We show exactly computed probability landscapes of three multi-scale networks, namely, a 6-node toggle switch, 11-node phage-lambda epigenetic circuit, and 16-node MAPK cascade network, the latter two with no known solutions. We also show how probabilities of rare events can be computed from first-passage times, another class of unsolved problems challenging for simulation-based techniques due to large separations in time scales. Overall, the ACME method enables accurate and efficient solutions of the dCME for a large class of networks. PMID:27761104
Accelerating numerical solution of stochastic differential equations with CUDA
Januszewski, M.; Kostur, M.
2010-01-01
hundreds of threads simultaneously makes it possible to speed up the computation by over two orders of magnitude, compared to a typical modern CPU. Solution method: The stochastic Runge-Kutta method of the second order is applied to integrate the equation of motion. Ensemble-averaged quantities of interest are obtained through averaging over multiple independent realizations of the system. Unusual features: The numerical solution of the stochastic differential equations in question is performed on a GPU using the CUDA environment. Running time: < 1 minute
Positive solutions of fractional differential equations with derivative terms
Directory of Open Access Journals (Sweden)
Cuiping Cheng
2012-11-01
Full Text Available In this article, we are concerned with the existence of positive solutions for nonlinear fractional differential equation whose nonlinearity contains the first-order derivative, $$displaylines{ D_{0^+}^{alpha}u(t+f(t,u(t,u'(t=0,quad tin (0,1,; n-1
Exact solutions to a class of nonlinear Schrödinger-type equations
Indian Academy of Sciences (India)
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 ...
Exact solutions to a class of nonlinear Schrödinger-type equations
Indian Academy of Sciences (India)
Abstract. 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 ...
Directory of Open Access Journals (Sweden)
Hassan A. Zedan
2017-01-01
Full Text Available Chebyshev spectral method based on operational matrix is applied to both systems of fractional integro-differential equations and Abel’s integral equations. Some test problems, for which the exact solution is known, are considered. Numerical results with comparisons are made to confirm the reliability of the method. Chebyshev spectral method may be considered as alternative and efficient technique for finding the approximation of system of fractional integro-differential equations and Abel’s integral equations.
Analytic Solutions of the Space-Time Fractional Combined KdV-mKdV Equation
Directory of Open Access Journals (Sweden)
Emad A.-B. Abdel-Salam
2015-01-01
Full Text Available The fractional mapping method is proposed to solve fractional differential equations. To illustrate the effectiveness of the method, we discuss the space-time fractional combined KdV-mKdV equation. Many types of exact analytical solutions are obtained. The solutions include generalized trigonometric and hyperbolic functions solutions. These solutions are useful to understand the mechanisms of the complicated nonlinear physical phenomena and fractional differential equations. Among these solutions, some are found for the first time.
Approximate explicit analytic solution of the Elenbaas-Heller equation
Liao, Meng-Ran; Li, Hui; Xia, Wei-Dong
2016-08-01
The Elenbaas-Heller equation describing the temperature field of a cylindrically symmetrical non-radiative electric arc has been solved, and approximate explicit analytic solutions are obtained. The radial distributions of the heat-flux potential and the electrical conductivity have been figured out briefly by using some special simplification techniques. The relations between both the core heat-flux potential and the electric field with the total arc current have also been given in several easy explicit formulas. Besides, the special voltage-ampere characteristic of electric arcs is explained intuitionally by a simple expression involving the Lambert W-function. The analyses also provide a preliminary estimation of the Joule heating per unit length, which has been verified in previous investigations. Helium arc is used to examine the theories, and the results agree well with the numerical computations.
Multiple solutions for perturbed non-local fractional Laplacian equations
Directory of Open Access Journals (Sweden)
Massimiliano Ferrara
2013-11-01
Full Text Available In article we consider problems modeled by the non-local fractional Laplacian equation $$\\displaylines{ (-\\Delta^s u=\\lambda f(x,u+\\mu g(x,u \\quad\\text{in } \\Omega\\cr u=0 \\quad\\text{in } \\mathbb{R}^n\\setminus \\Omega, }$$ where $s\\in (0,1$ is fixed, $(-\\Delta ^s$ is the fractional Laplace operator, $\\lambda,\\mu$ are real parameters, $\\Omega$ is an open bounded subset of $\\mathbb{R}^n$ ($n>2s$ with Lipschitz boundary $\\partial \\Omega$ and $f,g:\\Omega\\times\\mathbb{R}\\to\\mathbb{R}$ are two suitable Caratheodory functions. By using variational methods in an appropriate abstract framework developed by Servadei and Valdinoci [17] we prove the existence of at least three weak solutions for certain values of the parameters.
Semiclassical solution to the BFKL equation with massive gluons
Energy Technology Data Exchange (ETDEWEB)
Levin, Eugene [Tel Aviv University, Department of Particle Physics, School of Physics and Astronomy, Tel Aviv (Israel); Universidad Tecnica Federico Santa Maria and Centro Cientifico-Tecnologico de Valparaiso, Departamento de Fisica, Valparaiso (Chile); Lipatov, Lev [Petersburg Nuclear Physics Institute, Theoretical Physics Department, St. Petersburg (Russian Federation); Siddikov, Marat [Universidad Tecnica Federico Santa Maria and Centro Cientifico-Tecnologico de Valparaiso, Departamento de Fisica, Valparaiso (Chile)
2015-11-15
In this paper we proceed to study the high energy behavior of scattering amplitudes in a simple field model, with the Higgs mechanism for the gauge boson mass. The spectrum of the j-plane singularities of the t-channel partial waves and the corresponding eigenfunctions of the BFKL equation in leading log(1/x) approximation were previously calculated numerically. Here we develop a semiclassical approach to investigate the influence of the exponential decrease of the impact parameter dependence existing in this model, on the high energy asymptotic behavior of the scattering amplitude. This approach is much simpler than our earlier numerical calculations, and it reproduces those results. The analytical (semi-analytical) solutions which have been found in the approximation can be used to incorporate correctly the large impact parameter behavior in the framework of CGC/saturation approach. This behavior is interesting as it provides the high energy amplitude for the electroweak theory, which can be measured experimentally. (orig.)
Explicit solution of Calderon preconditioned time domain integral equations
Ulku, Huseyin Arda
2013-07-01
An explicit marching on-in-time (MOT) scheme for solving Calderon-preconditioned time domain integral equations is proposed. The scheme uses Rao-Wilton-Glisson and Buffa-Christiansen functions to discretize the domain and range of the integral operators and a PE(CE)m type linear multistep to march on in time. Unlike its implicit counterpart, the proposed explicit solver requires the solution of an MOT system with a Gram matrix that is sparse and well-conditioned independent of the time step size. Numerical results demonstrate that the explicit solver maintains its accuracy and stability even when the time step size is chosen as large as that typically used by an implicit solver. © 2013 IEEE.
Similarity and generalized finite-difference solutions of parabolic partial differential equations.
Clausing, A. M.
1971-01-01
Techniques are presented for obtaining generalized finite-difference solutions to partial differential equations of the parabolic type. It is shown that the advantages of similarity in the solution of similar problems are generally not lost if the solution to the original partial differential equations is effected in the physical plane by finite-difference methods. The analysis results in a considerable saving in computational effort in the solution of both similar and nonsimilar problems. Several examples, including both the heat-conduction equation and the boundary-layer equations, are given. The analysis also provides a practical means of estimating the accuracy of finite-difference solutions to parabolic equations.
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.
Fundamental Solutions for the Klein-Gordon Equation in de Sitter Spacetime
Yagdjian, Karen; Galstian, Anahit
2009-01-01
In this article we construct the fundamental solutions for the Klein-Gordon equation in de Sitter spacetime. We use these fundamental solutions to represent solutions of the Cauchy problem and to prove L p - L q estimates for the solutions of the equation with and without a source term.
Directory of Open Access Journals (Sweden)
Mostafa M.A. Khater
Full Text Available In this article and for the first time, we introduce and describe Khater method which is a new technique for solving nonlinear partial differential equations (PDEs.. We apply this method for each of the following models Bogoyavlenskii equation, couple Boiti-Leon-Pempinelli system and Time-fractional Cahn-Allen equation. Khater method is very powerful, Effective, felicitous and fabulous method to get exact and solitary wave solution of (PDEs.. Not only just like that but it considers too one of the general methods for solving that kind of equations since it involves some methods as we will see in our discuss of the results. We make a comparison between the results of this new method and another method. Keywords: Bogoyavlenskii equations system, Couple Boiti-Leon-Pempinelli equations system, Time-fractional Cahn-Allen equation, Khater method, Traveling wave solutions, Solitary wave solutions
Shallal, Muhannad A.; Jabbar, Hawraz N.; Ali, Khalid K.
2018-03-01
In this paper, we constructed a travelling wave solution for space-time fractional nonlinear partial differential equations by using the modified extended Tanh method with Riccati equation. The method is used to obtain analytic solutions for the space-time fractional Klein-Gordon and coupled conformable space-time fractional Boussinesq equations. The fractional complex transforms and the properties of modified Riemann-Liouville derivative have been used to convert these equations into nonlinear ordinary differential equations.
Local and global nonexistence of solutions to semilinear evolution equations
Directory of Open Access Journals (Sweden)
Mohammed Guedda
2002-12-01
Full Text Available For a fixed $ p $ and $ sigma > -1 $, such that $ p >max{1,sigma+1}$, one main concern of this paper is to find sufficient conditions for non solvability of [ u_t = -(-Delta^{frac{beta}{2}}u - V(xu + t^sigma h(xu^p + W(x,t, ] posed in $ S_T:=mathbb{R}^Nimes(0,T$, where $ 0 < T <+infty$, $(-Delta^{frac{beta}{2}}$ with $ 0 < beta leq 2$ is the $beta/2$ fractional power of the $ -Delta$, and $ W(x,t = t^gamma w(x geq 0$. The potential $ V $ satisfies $ limsup_{| x|o +infty }| V(x| | x|^{a} < +infty$, for some positive $ a$. We shall see that the existence of solutions depends on the behavior at infinity of both initial data and the function $h$ or of both $ w$ and $ h$. The non-global existence is also discussed. We prove, among other things, that if $ u_0(x $ satisfies [ lim_{| x|o+infty}u_0^{p-1}(x h(x| x|^{(1+sigmainf{beta,a}} = +infty, ] any possible local solution blows up at a finite time for any locally integrable function $W$. The situation is then extended to nonlinear hyperbolic equations.
On the Solution of Elliptic Partial Differential Equations on Regions with Corners
2015-07-09
In this report we investigate the solution of boundary value problems on polygonal domains for elliptic partial differential equations . We observe...efficient numerical algorithms. The results are illustrated by a number of numerical examples. On the solution of elliptic partial differential equations on...Solutions On the Solution of Elliptic Partial Differential Equations on Regions with Corners Kirill Serkh and Vladimir Rokhlin July 9, 2015 Contents 1
Exact Traveling Wave Solutions for Wick-Type Stochastic Schamel KdV Equation
Directory of Open Access Journals (Sweden)
Hossam A. Ghany
2014-01-01
Full Text Available F-expansion method is proposed to seek exact solutions of nonlinear partial differential equations. By means of Hermite transform, inverse Hermite transform, and white noise analysis, the variable coefficients and Wick-type stochastic Schamel KdV equations are completely described. Abundant exact traveling wave solutions for variable coefficients Schamel KdV equations are given. These solutions include exact stochastic Jacobi elliptic functions, trigonometric functions, and hyperbolic functions solutions.
Solution of Moving Boundary Space-Time Fractional Burger’s Equation
Directory of Open Access Journals (Sweden)
E. A.-B. Abdel-Salam
2014-01-01
Full Text Available The fractional Riccati expansion method is used to solve fractional differential equations with variable coefficients. To illustrate the effectiveness of the method, the moving boundary space-time fractional Burger’s equation is studied. The obtained solutions include generalized trigonometric and hyperbolic function solutions. Among these solutions, some are found for the first time. The linear and periodic moving boundaries for the kink solution of the Burger’s equation are presented graphically and discussed.
Directory of Open Access Journals (Sweden)
Mark A Lau
2016-09-01
Full Text Available This paper presents the implementation of numerical and analytical solutions of some of the classical partial differential equations using Excel spreadsheets. In particular, the heat equation, wave equation, and Laplace’s equation are presented herein since these equations have well known analytical solutions. The numerical solutions can be easily obtained once the differential equations are discretized via finite differences and then using cell formulas to implement the resulting recursive algorithms and other iterative methods such as the successive over-relaxation (SOR method. The graphing capabilities of spreadsheets can be exploited to enhance the visualization of the solutions to these equations. Furthermore, using Visual Basic for Applications (VBA can greatly facilitate the implementation of the analytical solutions to these equations, and in the process, one obtains Fourier series approximations to functions governing initial and/or boundary conditions.
Exact solutions to sourceless charged massive scalar field equation on Kerr-Newman background
Wu, S. Q.; Cai, X.
1999-09-01
The covariant Klein-Gordon equation in the Kerr-Newman black hole geometry is separated into a radial part and an angular part. It is discovered that in the nonextreme case, these two equations belong to a generalized spin-weighted spheroidal wave equation. Then general exact solutions in integral forms and several special solutions with physical interest are given. While in the extreme case, the radial equation can be transformed into a generalized Whittaker-Hill equation. In both cases, five-term recurrence relations between coefficients in power series expansion of general solutions are presented. Finally, the connection between the radial equations in both cases is discussed.
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.
Wu, Fuke; Yin, George; Mei, Hongwei
2017-02-01
This work is devoted to stochastic functional differential equations (SFDEs) with infinite delay. First, existence and uniqueness of the solutions of such equations are examined. Because the solutions of the delay equations are not Markov, a viable alternative for studying further asymptotic properties is to use solution maps or segment processes. By examining solution maps, this work investigates the Markov properties as well as the strong Markov properties. Also obtained are adaptivity and continuity, mean-square boundedness, and convergence of solution maps from different initial data. This paper then examines the ergodicity of underlying processes and establishes existence of the invariant measure for SFDEs with infinite delay under suitable conditions.
Directory of Open Access Journals (Sweden)
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.
Discrete Symmetries Analysis and Exact Solutions of the Inviscid Burgers Equation
Directory of Open Access Journals (Sweden)
Hongwei Yang
2012-01-01
Full Text Available We discuss the Lie point symmetries and discrete symmetries of the inviscid Burgers equation. By employing the Lie group method of infinitesimal transformations, symmetry reductions and similarity solutions of the governing equation are given. Based on discrete symmetries analysis, two groups of discrete symmetries are obtained, which lead to new exact solutions of the inviscid Burgers equation.
Directory of Open Access Journals (Sweden)
Qazi Mahmood Ul Hassan
2014-01-01
Full Text Available We use the fractional derivatives in Caputo’s sense to construct exact solutions to fractional fifth order nonlinear evolution equations. A generalized fractional complex transform is appropriately used to convert this equation to ordinary differential equation which subsequently resulted in a number of exact solutions.
Bound and periodic solutions of the Riccati equation in Banach space
Directory of Open Access Journals (Sweden)
A. Ya. Dorogovtsev
1995-01-01
Full Text Available An abstract, nonlinear, differential equation in Banach space is considered. Conditions are presented for the existence of bounded solutions of this equation with a bounded right side, and also for the existence of stationary (periodic solutions of this equation with a stationary (periodic process in the right side.
On the invariant solutions of space/time-fractional diffusion equations
Bahrami, Fariba; Najafi, Ramin; Hashemi, Mir Sajjad
2017-12-01
This paper is concerned with the space/time-fractional diffusion equations using Lie symmetry analysis. We introduce a generalized nonclassical method that is applied to differential equations with fractional order. The existing methods give some classical symmetries while the nonclassical approach will retrieve other symmetries to these equations. New exact solutions to the fractional diffusion equations are found.
Solution of Lie' nard Equations using Modified Initial Guess ...
African Journals Online (AJOL)
We also demonstrate the superiority of MIGVIM over the decomposition method and the variational iteration method for this type of equations by providing numerical comparisons. Keywords: Variational Iteration, Lagrange multiplier, Lie' nard equations, Adomian decomposition, Modified initial guess variational iteration.
Grid generation for the solution of partial differential equations
Eiseman, Peter R.; Erlebacher, Gordon
1989-01-01
A general survey of grid generators is presented with a concern for understanding why grids are necessary, how they are applied, and how they are generated. After an examination of the need for meshes, the overall applications setting is established with a categorization of the various connectivity patterns. This is split between structured grids and unstructured meshes. Altogether, the categorization establishes the foundation upon which grid generation techniques are developed. The two primary categories are algebraic techniques and partial differential equation techniques. These are each split into basic parts, and accordingly are individually examined in some detail. In the process, the interrelations between the various parts are accented. From the established background in the primary techniques, consideration is shifted to the topic of interactive grid generation and then to adaptive meshes. The setting for adaptivity is established with a suitable means to monitor severe solution behavior. Adaptive grids are considered first and are followed by adaptive triangular meshes. Then the consideration shifts to the temporal coupling between grid generators and PDE-solvers. To conclude, a reflection upon the discussion, herein, is given.
Directory of Open Access Journals (Sweden)
Ji Juan-Juan
2017-01-01
Full Text Available A table lookup method for solving nonlinear fractional partial differential equations (fPDEs is proposed in this paper. Looking up the corresponding tables, we can quickly obtain the exact analytical solutions of fPDEs by using this method. To illustrate the validity of the method, we apply it to construct the exact analytical solutions of four nonlinear fPDEs, namely, the time fractional simplified MCH equation, the space-time fractional combined KdV-mKdV equation, the (2+1-dimensional time fractional Zoomeron equation, and the space-time fractional ZKBBM equation. As a result, many new types of exact analytical solutions are obtained including triangular periodic solution, hyperbolic function solution, singular solution, multiple solitary wave solution, and Jacobi elliptic function solution.
Existence of high-energy solutions for supercritical fractional Schrodinger equations in R^N
Directory of Open Access Journals (Sweden)
Lu Gan
2016-12-01
Full Text Available In this article, we study supercritical fractional Schr\\"odinger equations. Applying the finite-dimensional reduction method and the penalization method, we obtain the high-energy solutions for this equation.
Exact Solutions for Some Fractional Partial Differential Equations by the Method
Directory of Open Access Journals (Sweden)
Bin Zheng
2013-01-01
derivative. Based on a certain variable transformation, these fractional partial differential equations are transformed into ordinary differential equations of integer order. With the aid of mathematical software, a variety of exact solutions for them are obtained.
National Research Council Canada - National Science Library
K. Issa; F. Salehi
2017-01-01
In this work, we obtain the approximate solution for the integrodifferential equations by adding perturbation terms to the right hand side of integrodifferential equation and then solve the resulting...
On vanishing at infinity solutions of higher order linear hyperbolic equations
Directory of Open Access Journals (Sweden)
Stavroulakis IP
2002-01-01
Full Text Available In the half strip the linear hyperbolic equation with coefficients and is considered. Sufficient conditions of existence of solutions to this equation satisfying the conditions are established, where and are the integral parts of and .
Solution of underdetermined systems of equations with gridded a priori constraints
National Research Council Canada - National Science Library
Stiros, Stathis C; Saltogianni, Vasso
2014-01-01
... of equations, can solve a wide range of underdetermined systems of non-linear equations. This approach is a generalization of a previous conclusion that this algorithm can be used for the solution of certain integer ambiguity problems...
Fundamental solutions of the wave equation in Robertson-Walker spaces
Yagdjian, Karen; Galstian, Anahit
2008-10-01
In this article we construct the fundamental solutions for the wave equation in the Robertson-Walker spaces arising in the de Sitter model of the universe. We then use these fundamental solutions to represent solutions of the Cauchy problem for the equation with and without a source term.
The solutions of three dimensional Fredholm integral equations using Adomian decomposition method
Almousa, Mohammad
2016-06-01
This paper presents the solutions of three dimensional Fredholm integral equations by using Adomian decomposition method (ADM). Some examples of these types of equations are tested to show the reliability of the technique. The solutions obtained by ADM give an excellent agreement with exact solution.
Solution of Grad-Shafranov equation by the method of fundamental solutions
Nath, D.; Kalra, M. S.; Kalra
2014-06-01
In this paper we have used the Method of Fundamental Solutions (MFS) to solve the Grad-Shafranov (GS) equation for the axisymmetric equilibria of tokamak plasmas with monomial sources. These monomials are the individual terms appearing on the right-hand side of the GS equation if one expands the nonlinear terms into polynomials. Unlike the Boundary Element Method (BEM), the MFS does not involve any singular integrals and is a meshless boundary-alone method. Its basic idea is to create a fictitious boundary around the actual physical boundary of the computational domain. This automatically removes the involvement of singular integrals. The results obtained by the MFS match well with the earlier results obtained using the BEM. The method is also applied to Solov'ev profiles and it is found that the results are in good agreement with analytical results.
Allison, Stuart A; Wu, Hengfu; Moyher, Avery; Soegiarto, Linda; Truong, Bi; Nguyen, Duy; Nguyen, Tam; Park, Donghyun
2014-03-20
The coarse-grained continuum primitive model is developed and used to characterize the titration and electrical conductance behavior of aqueous solutions of fullerene hexa malonic acid (FHMA). The spherical FHMA molecule, a highly charged electrolyte with an absolute valence charge as large as 12, is modeled as a dielectric sphere in Newtonian fluid, and electrostatics are treated numerically at the level of the non-linear Poisson-Boltzmann equation. Transport properties (electrophoretic mobilities and conductances) of the various charge states of FHMA are numerically computed using established numerical algorithms. For reasonable choices of the model parameters, good agreement between experiment (published literature) and modeling is achieved. In order to accomplish this, however, a moderate degree of specific binding of principal counterion and FHMA must be included in the modeling. It should be emphasized, however, that alternative explanations are possible. This comparison is made at 25 °C for both Na(+) and Ca(2+) principal counterions. The model is also used to characterize the different charge states and degree of counterion binding to those charge states as a function of pH.
On global attractivity of solutions of a functional-integral equation
Directory of Open Access Journals (Sweden)
Mohamed Darwish
2007-10-01
Full Text Available We prove an existence theorem for a quadratic functional-integral equation of mixed type. The functional-integral equation studied below contains as special cases numerous integral equations encountered in nonlinear analysis. With help of a suitable measure of noncompactness, we show that the functional integral equation of mixed type has solutions being continuous and bounded on the interval $[0,\\infty$ and those solutions are globally attractive.
N-soliton solutions for the Vakhnenko equation and its generalized forms
Energy Technology Data Exchange (ETDEWEB)
Wazwaz, Abdul-Majid, E-mail: wazwaz@sxu.ed [Department of Mathematics, Saint Xavier University, Chicago, IL 60655 (United States)
2010-12-15
In this paper, we present an analytic study for the nonlinear Vakhnenko equation, a generalized Vakhnenko equation and a modified generalized Vakhnenko equation. The simplified form of the bilinear method, established by Hereman and Nuseir (1997 Math. Comput. Simul. 43 13-27), will be used to formally derive multiple soliton solutions and multiple singular soliton solutions for each equation. The resonance phenomenon is examined for each model.
General soliton solutions of an n-dimensional nonlinear Schroedinger equation
Energy Technology Data Exchange (ETDEWEB)
Khater, A. H.; Seadawy, A. R. [Cairo Univ., Cairo (Egypt). Faculty of Science, Mathematics Dept.; Helal, M. A. [Cairo Univ., Cairo (Egypt). Faculty of Science, Mathematics Dept.
2000-11-01
Applying the function transformation method, an n-dimensional nonlinear Schroedinger (NDNLS) equation is transformed into a sinh-Gordon equation and other equations, which depend only on one function {zeta} leads to a general soliton solution of the NDNLS equation. It contains some interesting specific solutions such as the N multiple solitons, the propagational breathers and the quadric solitons. Their properties are simply discussed.
Operational matrix approach for approximate solution of fractional model of Bloch equation
Directory of Open Access Journals (Sweden)
Harendra Singh
2017-04-01
Full Text Available In present paper operational matrix of integration for Laguerre polynomial is used to solve fractional model of Bloch equation in nuclear magnetic resonance (NMR. The operational matrix converts the Bloch equation in a system of linear algebraic equations. Solving system we obtain the approximate solutions for fractional Bloch equation. Results are compared with existing methods and exact solution. Graphs are plotted for different fractional values of time derivatives.
Chen, Cheng; Jiang, Yao-Lin
2017-09-01
On the basis of Lie group theory, (1 + N)-dimensional time-fractional partial differential equations are studied and the expression of {η }α 0 is given. As applications, two special forms of nonlinear time-fractional diffusion-convection equations are investigated by Lie group analysis method. Then the equations are reduced into fractional ordinary differential equations under group transformations. Therefore, the invariant solutions and some exact solutions are obtained.
Zajkowski, Konrad
This paper presents an algorithm for solving N-equations of N-unknowns. This algorithm allows to determine the solution in a situation where coefficients Ai in equations are burdened with measurement errors. For some values of Ai (where i = 1,…, N), there is no inverse function of input equations. In this case, it is impossible to determine the solution of equations of classical methods.
On construction of solutions of linear fractional differential equations with constant coefficients
Borikhanov, Meiirkhan B.; Turmetov, Batirkhan Kh.
2016-08-01
One of the effective methods for finding exact solutions of differential equations is the method based on the operator representation of solutions. The essence of this method is to construct a series, whose members are the relevant iteration operators acting to some classes of sufficiently smooth functions. This method is widely used in the papers of Bondarenko for construction of solutions of differential equations of the integer order. In this paper, the operator method is applied to construct solutions of linear differential equations with constant coefficients and generalized Riemann-Liouville fractional derivative of order α and type γ. Then fundamental solutions are used to obtain the unique solution of the Cauchy problem.
Jacobi Elliptic Solutions for Nonlinear Differential Difference Equations in Mathematical Physics
Directory of Open Access Journals (Sweden)
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.
Solution of the two- dimensional heat equation for a rectangular plate
Directory of Open Access Journals (Sweden)
Nurcan BAYKUŞ SAVAŞANERİL
2015-11-01
Full Text Available Laplace equation is a fundamental equation of applied mathematics. Important phenomena in engineering and physics, such as steady-state temperature distribution, electrostatic potential and fluid flow, are modeled by means of this equation. The Laplace equation which satisfies boundary values is known as the Dirichlet problem. The solutions to the Dirichlet problem form one of the most celebrated topics in the area of applied mathematics. In this study, a novel method is presented for the solution of two-dimensional heat equation for a rectangular plate. In this alternative method, the solution function of the problem is based on the Green function, and therefore on elliptic functions.
Study of coupled nonlinear partial differential equations for finding exact analytical solutions.
Khan, Kamruzzaman; Akbar, M Ali; Koppelaar, H
2015-07-01
Exact solutions of nonlinear partial differential equations (NPDEs) are obtained via the enhanced (G'/G)-expansion method. The method is subsequently applied to find exact solutions of the Drinfel'd-Sokolov-Wilson (DSW) equation and the (2+1)-dimensional Painlevé integrable Burgers (PIB) equation. The efficiency of this method for finding these exact solutions is demonstrated. The method is effective and applicable for many other NPDEs in mathematical physics.
New exact solutions of the non-homogeneous Burgers equation in (1+1) dimensions
Energy Technology Data Exchange (ETDEWEB)
Schulze-Halberg, Axel [Department of Science, University of Colima, Bernal Diaz del Castillo 340, Colima Villas San Sebastian, C P 28045, Colima (Mexico)
2007-04-15
We construct an invertible transformation between the non-homogeneous Burgers equation (NBE) and the stationary Schroedinger equation in (1+1) dimensions. By means of this transformation, each solution of the stationary Schroedinger equation generates a fully time-dependent solution of the NBE. As applications we derive exact solutions of the NBE for general power-law nonhomogeneities, generalizing former results on the linear case.
New interaction solutions to the combined KdV–mKdV equation from CTE method
Directory of Open Access Journals (Sweden)
Hengchun Hu
2016-10-01
Full Text Available The consistent tanh expansion (CTE method is developed for the combined KdV–mKdV equation. The combined KdV–mKdV equation is proved to be CTE solvable. New exact interaction solutions such as soliton–cnoidal wave solutions, soliton–periodic wave solutions for the combined KdV–mKdV equation are given out analytically and graphically.
Stable subharmonic solutions and asymptotic behavior in reaction-diffusion equations
Directory of Open Access Journals (Sweden)
P. Polacik
2000-01-01
Full Text Available Time-periodic reaction-diffusion equations can be discussed in the context of discrete-time strongly monotone dynamical systems. It follows from the general theory that typical trajectories approach stable periodic solutions. Among these periodic solutions, there are some that have the same period as the equation, but, possibly, there might be others with larger minimal periods (these are called subharmonic solutions. The problem of existence of stable subharmonic solutions is therefore of fundamental importance in the study of the behavior of solutions. We address this problem for two classes of reaction diffusion equations under Neumann boundary conditions. Namely, we consider spatially inhomogeneous equations, which can have stable subharmonic solutions on any domain, and spatially homogeneous equations, which can have such solutions on some (necessarily non-convex domains.
Directory of Open Access Journals (Sweden)
Andreas Ruffing
2001-01-01
Full Text Available We revise the interrelations between the classical Black Scholes equation, the diffusion equation and Burgers equation. Some of the algebraic properties the diffusion equation shows are elaborated and qualitatively presented. The related numerical elementary recipes are briefly elucidated in context of the diffusion equation. The quality of the approximations to the exact solutions is compared throughout the visualizations. The article mainly is based on the pedagogical style of the presentations to the Novacella Easter School 2000 on Financial Mathematics.
National Research Council Canada - National Science Library
I. V. Makeev; I. Y. Popov; I. V. Blinova
2016-01-01
.... We suggest exact particular solutions of Stokes and continuity equations with variable viscosity and density in spherical coordinates for the case of spherically symmetric viscosity and density distributions...
Generalized Sturmian Solutions for Many-Particle Schrödinger Equations
DEFF Research Database (Denmark)
Avery, John; Avery, James Emil
2004-01-01
The generalized Sturmian method for obtaining solutions to the many-particle Schrodinger equation is reviewed. The method makes use of basis functions that are solutions of an approximate Schrodinger equation with a weighted zeroth-order potential. The weighting factors are especially chosen so......-the calculation of atomic spectra. Avery, J.: Generalized Sturmian Solutions for Many-Particle Schrödinger Equations. J. Phys. Chem. A 108, 8848. Available from: http://www.researchgate.net/publication/230557991_Avery_J._Generalized_Sturmian_Solutions_for_Many-Particle_Schrdinger_Equations._J._Phys._Chem._A_108...
Exact Travelling Wave Solutions of two Important Nonlinear Partial Differential Equations
Kim, Hyunsoo; Bae, Jae-Hyeong; Sakthivel, Rathinasamy
2014-04-01
Coupled nonlinear partial differential equations describing the spatio-temporal dynamics of predator-prey systems and nonlinear telegraph equations have been widely applied in many real world problems. So, finding exact solutions of such equations is very helpful in the theories and numerical studies. In this paper, the Kudryashov method is implemented to obtain exact travelling wave solutions of such physical models. Further, graphic illustrations in two and three dimensional plots of some of the obtained solutions are also given to predict their behaviour. The results reveal that the Kudryashov method is very simple, reliable, and effective, and can be used for finding exact solution of many other nonlinear evolution equations.
Ground state solutions for asymptotically periodic Schrodinger equations with critical growth
Directory of Open Access Journals (Sweden)
Hui Zhang
2013-10-01
Full Text Available Using the Nehari manifold and the concentration compactness principle, we study the existence of ground state solutions for asymptotically periodic Schrodinger equations with critical growth.
Explicit solutions of the Rand Equation | Huber | International ...
African Journals Online (AJOL)
In this paper the meaning of a nonlinear partial differential equation (nPDE) of the third-order is shown to the first time. The equation is known as the 'Rand Equation' and belongs to a class of less studied nPDEs. Both the explicit physical meaning as well as the behaviour is not known until now. Therefore we believe it is ...
Exact solution of the space-time fractional coupled EW and coupled MEW equations
Raslan, K. R.; S. EL-Danaf, Talaat; K. Ali, Khalid
2017-07-01
In this paper, we obtained a traveling wave solution by using the Kudryashov method for the space-time fractional nonlinear partial differential equations. The method is used to obtain the exact solutions for different types of the space-time fractional nonlinear partial differential equations, such as the space-time fractional coupled equal width wave equation (CEWE) and the space-time fractional coupled modified equal width wave equation (CMEWE), which are the important soliton equations. Both equations are reduced to ordinary differential equations by use of the fractional complex transform and of the properties of the modified Riemann-Liouville derivative. We plot the exact solutions for these equations at different time levels.
Symmetries, integrals and solutions of ordinary differential equations ...
Indian Academy of Sciences (India)
first observation was of the transformation of one first integral of the nonlinear ordinary differential equation ... The representative second-order ordinary differential equation of maximal point symmetry, videlicet y = 0,. (2.1) ...... Noetherian Symmetries, Advances in Systems, Signals, Control and Computers,. Bajic VB ed ...
Generalized ordinary differential equations not absolutely continuous solutions
Kurzweil, Jaroslav
2012-01-01
This book provides a systematic treatment of the Volterra integral equation by means of a modern integration theory which extends considerably the field of differential equations. It contains many new concepts and results in the framework of a unifying theory. In particular, this new approach is suitable in situations where fast oscillations occur.
Existence of solutions for mixed Volterra-Fredholm integral equations
Directory of Open Access Journals (Sweden)
Asadollah Aghajani
2012-08-01
Full Text Available In this article, we give some results concerning the continuity of the nonlinear Volterra and Fredholm integral operators on the space $L^{1}[0,infty$. Then by using the concept of measure of weak noncompactness, we prove an existence result for a functional integral equation which includes several classes of nonlinear integral equations. Our results extend some previous works.
New exact travelling wave solutions of bidirectional wave equations
Indian Academy of Sciences (India)
where a, b, c and d are real constants. Here x represents the distance along the channel, t is the elapsed time, the variable v(x, t) is the dimensionless deviation of the water surface from its undisturbed position and u(x, t) is the dimensionless horizontal velocity. This set of equations is used as a model equation for the ...
Space-Time Fractional DKP Equation and Its Solution
Bouzid, N.; Merad, M.
2017-05-01
In this paper, a fractional Hamiltonian formulation for Duffin-Kemmer-Petiau' (DKP) fields is presented and, as done in the framework of the Lagrangian formalism, the fractional DKP equation is deduced. The space-time fractional DKP equation is then solved for both scalar and vectorial cases. The wave functions obtained are expressed in terms of Mittag-Leffler function.
Traveling Wave Solutions of ZK-BBM Equation Sine-Cosine Method
Directory of Open Access Journals (Sweden)
Sadaf Bibi
2014-03-01
Full Text Available Travelling wave solutions are obtained by using a relatively new technique which is called sine-cosine method for ZK-BBM equations. Solution procedure and obtained results re-confirm the efficiency of the proposed scheme.
Solutions to quasilinear equations of $N$-biharmonic type with degenerate coercivity
Directory of Open Access Journals (Sweden)
Sami Aouaoui
2014-10-01
Full Text Available In this article we show the existence of multiple solutions for quasilinear equations in divergence form with degenerate coercivity. Our strategy is to combine a variational method and an iterative technique to obtain the solutions.
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.
Stability and square integrability of solutions of nonlinear fourth order differential equations
Directory of Open Access Journals (Sweden)
Moussadek Remili
2016-05-01
Full Text Available The aim of the present paper is to establish a new result, which guarantees the asymptotic stability of zero solution and square integrability of solutions and their derivatives to nonlinear differential equations of fourth order.
An approximate solution for a generalized Hirota-Satsom coupled (Kdv equation
Directory of Open Access Journals (Sweden)
H.A. Wahab
2017-03-01
Full Text Available In this paper the Homotopy Analysis Method (HAM, is applied to find the approximate solution of Hirota-Satsuma coupled (KdV equations, which don't need a small parameter for solution. The results obtained by HAM is compared with exact solution, the results divulge that the Homotopy Analysis Method are most accurate, closed and suitable to exact solution of the equation, as compare to Homotopy Perturbation Method. It is predicated that the HAM can be found usually.
Soliton solutions of the two-dimensional KdV-Burgers equation by homotopy perturbation method
Energy Technology Data Exchange (ETDEWEB)
Molabahrami, A. [Department of Mathematics, Ilam University, PO Box 69315516, Ilam (Iran, Islamic Republic of)], E-mail: a_m_bahrami@yahoo.com; Khani, F. [Department of Mathematics, Ilam University, PO Box 69315516, Ilam (Iran, Islamic Republic of); Bakhtar Institute of Higher Education, PO Box 696, Ilam (Iran, Islamic Republic of)], E-mail: farzad_khani59@yahoo.com; Hamedi-Nezhad, S. [Bakhtar Institute of Higher Education, PO Box 696, Ilam (Iran, Islamic Republic of)
2007-10-29
In this Letter, the He's homotopy perturbation method (HPM) to finding the soliton solutions of the two-dimensional Korteweg-de Vries Burgers' equation (tdKdVB) for the initial conditions was applied. Numerical solutions of the equation were obtained. The obtained solutions, in comparison with the exact solutions admit a remarkable accuracy. The results reveal that the HPM is very effective and simple.
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.
Brandts, J.H.
2001-01-01
In this paper we will concentrate on the numerical solution of the Cauchy-Riemann equations. First we show that these equations bring together the nite element discretizations for the Laplace equation by standard nite elements on the one hand, and by mixed nite element methods on the other. As a
Solution of nonlinear Volterra-Hammerstein integral equations via single-term Walsh series method
Directory of Open Access Journals (Sweden)
Sepehrian B.
2005-01-01
Full Text Available Single-term Walsh series are developed to approximate the solutions of nonlinear Volterra-Hammerstein integral equations. Properties of single-term Walsh series are presented and are utilized to reduce the computation of integral equations to some algebraic equations. The method is computationally attractive, and applications are demonstrated through illustrative examples.
Lie group classifications and exact solutions for time-fractional Burgers equation
Wu, Guo-cheng
2010-01-01
Lie group method provides an efficient tool to solve nonlinear partial differential equations. This paper suggests a fractional Lie group method for fractional partial differential equations. A time-fractional Burgers equation is used as an example to illustrate the effectiveness of the Lie group method and some classes of exact solutions are obtained.
DEFF Research Database (Denmark)
Marcussen, Lis; Aasberg-Petersen, K.; Krøll, Annette Elisabeth
2000-01-01
An adsorption isotherm equation for nonideal pure component adsorption based on vacancy solution theory and the Non-Random-Two-Liquid (NRTL) equation is found to be useful for predicting pure component adsorption equilibria at a variety of conditions. The isotherm equation is evaluated successfully...
Energy Technology Data Exchange (ETDEWEB)
Zabadal, Jorge; Borges, Volnei; Van der Laan, Flavio T., E-mail: jorge.zabadal@ufrgs.br, E-mail: borges@ufrgs.br, E-mail: ftvdl@ufrgs.br [Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, RS (Brazil). Departamento de Engenharia Mecanica. Grupo de Pesquisas Radiologicas; Ribeiro, Vinicius G., E-mail: vinicius_ribeiro@uniritter.edu.br [Centro Universitario Ritter dos Reis (UNIRITTER), Porto Alegre, RS (Brazil); Santos, Marcio G., E-mail: phd.marcio@gmail.com [Universidade Federal do Rio Grande do Sul (UFRGS), Tramandai, RS (Brazil). Departamento Interdisciplinar do Campus Litoral Norte
2015-07-01
This work presents a new analytical method for solving the Boltzmann equation. In this formulation, a linear differential operator is applied over the Boltzmann model, in order to produce a partial differential equation in which the scattering term is absent. This auxiliary equation is solved via reduction of order. The exact solution obtained is employed to define a precursor for the buildup factor. (author)
Informed Conjecturing of Solutions for Differential Equations in a Modeling Context
Winkel, Brian
2015-01-01
We examine two differential equations. (i) first-order exponential growth or decay; and (ii) second order, linear, constant coefficient differential equations, and show the advantage of learning differential equations in a modeling context for informed conjectures of their solution. We follow with a discussion of the complete analysis afforded by…
Rauscher, Elizabeth A
2011-01-01
The Maxwell, Einstein, Schrödinger and Dirac equations are considered the most important equations in all of physics. This volume aims to provide new eight- and twelve-dimensional complex solutions to these equations for the first time in order to reveal
Development of hyperbolic solution method for two fluids equation system
Energy Technology Data Exchange (ETDEWEB)
Lee, Sung Jae; Chang, Won Pyo
1997-07-01
Using the concept of surface tension thickness, the mathematical ill-posedness of the two fluids equation system can now be removed by splitting the pressure discontinuity of the two fluids interface. The bulk modulus L1 and L2 derived from the concept of surface tension thickness makes two fluids equation system hyperbolic type. The hyperbolic equation system has five complete sets of eigenvectors, each of which having real eigenvalues. Three sets of them represents the propagation speeds of the physical properties for individual flow regimes such as the dispersed, the slug, and the separated flows. The propagation characteristics of these eigenvalues have good agreements with both the experimental data and other theoretical results in two-phase mixture. The feature of the hyperbolic model allows to apply advanced numerical upwind technique such as Flux vector splitting (FVS) method. The numerical test show that the characteristics of equation system clearly classify all flow regimes. (author). 25 refs., 3 tabs., 20 figs.
Solutions for a class of iterated singular equations
Indian Academy of Sciences (India)
The domain of the operator L is the set of all real-valued functions u(x) of the class. C2(D), where x = (x1,x2,...,xn) denotes points in Rn and D is the regularity domain of u in Rn. Note that (1) includes the Laplace equation and an equidimensional (Euler) equation as special cases. In [1] and [2], Altın studied radial type ...
Analytic Solutions to Coherent Control of the Dirac Equation
Campos, Andre G.; Cabrera, Renan; Rabitz, Herschel A.; Bondar, Denys I.
2017-10-01
A simple framework for Dirac spinors is developed that parametrizes admissible quantum dynamics and also analytically constructs electromagnetic fields, obeying Maxwell's equations, which yield a desired evolution. In particular, we show how to achieve dispersionless rotation and translation of wave packets. Additionally, this formalism can handle control interactions beyond electromagnetic. This work reveals unexpected flexibility of the Dirac equation for control applications, which may open new prospects for quantum technologies.
Regarding on the exact solutions for the nonlinear fractional differential equations
Directory of Open Access Journals (Sweden)
Kaplan Melike
2016-01-01
Full Text Available In this work, we have considered the modified simple equation (MSE method for obtaining exact solutions of nonlinear fractional-order differential equations. The space-time fractional equal width (EW and the modified equal width (mEW equation are considered for illustrating the effectiveness of the algorithm. It has been observed that all exact solutions obtained in this paper verify the nonlinear ordinary differential equations which was obtained from nonlinear fractional-order differential equations under the terms of wave transformation relationship. The obtained results are shown graphically.
Hosseini, Kamyar; Mayeli, Peyman; Bekir, Ahmet; Guner, Ozkan
2018-01-01
In this article, a special type of fractional differential equations (FDEs) named the density-dependent conformable fractional diffusion-reaction (DDCFDR) equation is studied. Aforementioned equation has a significant role in the modelling of some phenomena arising in the applied science. The well-organized methods, including the \\exp (-φ (\\varepsilon )) -expansion and modified Kudryashov methods are exerted to generate the exact solutions of this equation such that some of the solutions are new and have been reported for the first time. Results illustrate that both methods have a great performance in handling the DDCFDR equation.
Exact solutions to nonlinear nonautonomous space-fractional diffusion equations with absorption.
Lenzi, E K; Mendes, G A; Mendes, R S; da Silva, L R; Lucena, L S
2003-05-01
We analyze a nonlinear fractional diffusion equation with absorption by employing fractional spatial derivatives and obtain some more exact classes of solutions. In particular, the diffusion equation employed here extends some known diffusion equations such as the porous medium equation and the thin film equation. We also discuss some implications by considering a diffusion coefficient D(x,t)=D(t)/x/(-theta) (theta in R) and a drift force F=-k(1)(t)x+k(alpha)x/x/(alpha-1). In both situations, we relate our solutions to those obtained within the maximum entropy principle by using the Tsallis entropy.
Solution of the first order linear fuzzy differential equations by some reliable methods
Directory of Open Access Journals (Sweden)
Mojtaba Ghanbari
2012-10-01
Full Text Available Fuzzy differential equations are used in modeling problems in science and engineering. For instance, it is known that the knowledge of dynamical systems modeled by ordinary differential equations is often incomplete or vague. While, fuzzy differential equations represent a proper way to model dynamical systems under uncertainty and vagueness. In this paper, two methods for solving first order linear fuzzy differential equations under generalized differentiability are proposed and compared. These methods are variational iteration method (VIM and Adomian decomposition method (ADM. The comparison of the exact solutions with solutions obtained by VIM and ADM are in details. The comparison shows that solutions are excellent agreement.
Directory of Open Access Journals (Sweden)
Bhausaheb R. Sontakke
2016-11-01
Full Text Available In this paper, fractional complex transform with new iterative method (NIM is used to obtain approximate solutions for the nonlinear time fractional Kawahara and modified Kawahara equations based on He's fractional derivative. Fractional complex transform is proposed to convert time fractional Kawahara and modified Kawahara equations to the nonlinear ordinary differential equations and then NIM is applied to the new obtained equations. The obtained approximate solutions are compared with the exact solutions to verify the applicability, efficiency and accuracy of the method.
Blow-up profile for solutions of a fourth order nonlinear equation
D'Ambrosio, Lorenzo; Lessard, Jean-Philippe; Pugliese, Alessandro
2015-01-01
It is well known that the nontrivial solutions of the equation u¿(r)+¿u¿(r)+f(u(r))=0u¿(r)+¿u¿(r)+f(u(r))=0 blow up in finite time under suitable hypotheses on the initial data, ¿¿ and ff. These solutions blow up with large oscillations. Knowledge of the blow-up profile of these solutions is of great importance, for instance, in studying the dynamics of suspension bridges. The equation is also commonly referred to as extended Fisher–Kolmogorov equation or Swift–Hohenberg equation. In th...
Computing the Solutions of the Combined Korteweg-de Vries Equation by Turing Machines
Directory of Open Access Journals (Sweden)
Dianchen Lu
2010-06-01
Full Text Available In this paper, we study the computability of the initial value problem of the Combined KdV equation. It is shown that, for any integer s>2, the nonlinear solution operator which maps an initial condition data to the solution of the Combined KdV equation can be computed by a Turing machine.
A general solution of the Dirac equation with superposition of δ-potentials and its application
Fialka, Siarhei; Kapshai, Valery
2017-01-01
Elastic scattering of relativistic spin 1/2 particle was investigated in the spherically-symmetric case. Based on the phase-shift method, a general solution of the Dirac equation with superposition of N δ-potentials is obtained. The method for approximate solutions of Dirac equation with arbitrary smooth potentials is presented. Scattering cross section dependences on various parameters are investigated.
Global existence of solutions for a viscous Cahn–Hilliard equation ...
Indian Academy of Sciences (India)
Home; Journals; Proceedings – Mathematical Sciences; Volume 123; Issue 4. Global Existence of Solutions for a Viscous Cahn-Hilliard Equation with Gradient Dependent Potentials and Sources. Chengyuan Qu Yang Cao ... Keywords. Global solution; viscous Cahn–Hilliard equation; initial boundary value problem ...
Soliton solutions for a quasilinear Schrödinger equation via Morse ...
Indian Academy of Sciences (India)
RN with Dirichlet boundary condition. Keywords. Quasilinear Schrödinger equation; soliton solution; critical point; Morse theory; local linking. 1991 Mathematics Subject Classification. 35B38, 35D05, 35J20. 1. Introduction. In this paper, we deal with the soliton solutions for a quasilinear Schrödinger equation of the form.
Geometrical Solutions of Some Quadratic Equations with Non-Real Roots
Pathak, H. K.; Grewal, A. S.
2002-01-01
This note gives geometrical/graphical methods of finding solutions of the quadratic equation ax[squared] + bx + c = 0, a [not equal to] 0, with non-real roots. Three different cases which give rise to non-real roots of the quadratic equation have been discussed. In case I a geometrical construction and its proof for finding the solutions of the…
Global existence of solutions for a viscous Cahn–Hilliard equation ...
Indian Academy of Sciences (India)
We consider a class of nonlinear viscous Cahn–Hilliard equations with gradient dependent potentials and sources. By a Galerkin approximation scheme combined with the potential well method, we prove the global existence of weak solutions. Keywords. Global solution; viscous Cahn–Hilliard equation; initial boundary ...
Time-dependent exact solutions for Rosenau-Hyman equations with variable coefficients
Souza, Wescley Luiz de; Silva, Érica de Mello
2015-03-01
In this work we study Rosenau-Hyman-like equations that were obtained by imposing the Lie point symmetry algebra of standard KdV to a general K (m, n) equation with variable coefficients. We present time-dependent exact solutions for suited choices of parameters m and n, including the similarity solution related to rarefaction shock wave phenomena.
Directory of Open Access Journals (Sweden)
Vardanjan Gumedin Surenovich
2012-10-01
Full Text Available The functional similarity method applicable for the simulation of various physical processes is considered in the proposed paper. A solution to some linear differential equations with variable coefficients is provided. The aforementioned equations are widely used as part of solutions to problems of mechanics of deformable solid bodies.
Directory of Open Access Journals (Sweden)
Gani Tr. Stamov
2015-01-01
Full Text Available The plan of this paper is to find conditions for the existence of almost periodic solutions for a class of impulsive fractional integrodifferential equations. The investigations are carried out by using a new fractional comparison principle, coupled with the fractional Lyapunov method. The stability behavior of the almost periodic solutions is also considered, extending the corresponding theory of impulsive integrodifferential equations.
Directory of Open Access Journals (Sweden)
Manoj Gaur
2016-01-01
Full Text Available We investigate the symmetry properties of a variable coefficient space-time fractional potential Burgers’ equation. Fractional Lie symmetries and corresponding infinitesimal generators are obtained. With the help of the infinitesimal generators, some group invariant solutions are deduced. Further, some exact solutions of fractional potential Burgers’ equation are generated by the invariant subspace method.
Stabilization of solutions to higher-order nonlinear Schrodinger equation with localized damping
Directory of Open Access Journals (Sweden)
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.
Directory of Open Access Journals (Sweden)
Berenguer MI
2010-01-01
Full Text Available This paper deals with obtaining a numerical method in order to approximate the solution of the nonlinear Volterra integro-differential equation. We define, following a fixed-point approach, a sequence of functions which approximate the solution of this type of equation, due to some properties of certain biorthogonal systems for the Banach spaces and .
Reproductive solutions for the g-Navier-Stokes and g-Kelvin-Voight equations
Directory of Open Access Journals (Sweden)
Luis Friz
2016-01-01
Full Text Available This article presents the existence of reproductive solutions of g-Navier-Stokes and g-Kelvin-Voight equations. In this way, for weak solutions, we reach basically the same result as for classic Navier-Stokes equations.
Asymptotic behavior of positive solutions of the nonlinear differential equation t^2u''=u^n
Directory of Open Access Journals (Sweden)
Meng-Rong Li
2013-11-01
Full Text Available In this article we study properties of positive solutions of the ordinary differential equation $t^2u''=u^n$ for $1
Travelling wave solutions of the generalized Benjamin-Bona-Mahony equation
Energy Technology Data Exchange (ETDEWEB)
Estevez, P.G. [Departamento de Fisica Fundamental, Area de Fisica Teorica, Universidad de Salamanca, 37008 Salamanca (Spain); Kuru, S. [Departamento de Fisica Teorica, Atomica y Optica, Universidad de Valladolid, 47071 Valladolid (Spain); Department of Physics, Faculty of Science, Ankara University, 06100 Ankara (Turkey); Negro, J. [Departamento de Fisica Teorica, Atomica y Optica, Universidad de Valladolid, 47071 Valladolid (Spain)], E-mail: jnegro@fta.uva.es; Nieto, L.M. [Departamento de Fisica Teorica, Atomica y Optica, Universidad de Valladolid, 47071 Valladolid (Spain)
2009-05-30
A class of particular travelling wave solutions of the generalized Benjamin-Bona-Mahony equation is studied systematically using the factorization technique. Then, the general travelling wave solutions of Benjamin-Bona-Mahony equation, and of its modified version, are also recovered.
Directory of Open Access Journals (Sweden)
H. Ullah
2015-01-01
Full Text Available The two-dimensional nonlinear wave equations are considered. Solution to the problem is approximated by using optimal homotopy asymptotic method (OHAM. The residual and convergence of the proposed method to nonlinear wave equation are presented through graphs. The resultant analytic series solution of the two-dimensional nonlinear wave equation shows the effectiveness of the proposed method. The comparison of results has been made with the existing results available in the literature.
Monotonic solutions of functional integral and differential equations of fractional order
Directory of Open Access Journals (Sweden)
Ahmed El-Sayed
2009-02-01
Full Text Available The existence of positive monotonic solutions, in the class of continuous functions, for some nonlinear quadratic integral equations have been studied by J. Banas. Here we are concerned with a singular quadratic functional integral equations. The existence of positive monotonic solutions $x \\in L_1[0,1]$ will be proved. The fractional order nonlinear functional differential equation will be given as a special case.
On the approximation of a small solution from a forward-backward equation
Teodoro, M. F.
2013-10-01
Many mathematical models contain mixed type functional differential equations (MTFDEs), equations with both delayed and advanced arguments. Often, MTFDEs appear in different areas from science like biology, control, economy and others. On the analysis of delay differential equations (DDEs), the numerical computation of small solutions generally leads to degeneracy and, consequently, to serious computational problems. To extend the investigation concerning the linear case started earlier, we do a preliminary study about small solutions for a particular nonautonomous case of MTFDE.
On the mild solutions of higher-order differential equations in Banach spaces
Directory of Open Access Journals (Sweden)
Nguyen Thanh Lan
2003-01-01
Full Text Available For the higher-order abstract differential equation u(n(t=Au(t+f(t, t∈ℝ, we give a new definition of mild solutions. We then characterize the regular admissibility of a translation-invariant subspace ℳ of BUC(ℝ,E with respect to the above-mentioned equation in terms of solvability of the operator equation AX−Xn=C. As applications, periodicity and almost periodicity of mild solutions are also proved.
Numerical solution of the Fokker--Planck equations for a multi-species plasma
Energy Technology Data Exchange (ETDEWEB)
Killeen, J.; Mirin, A.A.
1977-03-11
Two numerical models used for studying collisional multispecies plasmas are described. The mathematical model is the Boltzmann kinetic equation with Fokker-Planck collision terms. A one-dimensional code and a two-dimensional code, used for the solution of the time-dependent Fokker-Planck equations for ion and electron distribution functions in velocity space, are described. The required equations and boundary conditions are derived and numerical techniques for their solution are given.
Baecklund transformations and exact soliton solutions for nonlinear Schroedinger-type equations
Energy Technology Data Exchange (ETDEWEB)
Khater, A. H. [Cairo Univ. (Egypt). Faculty of science, Dept. of Mathematics]|[Antwerp Univ. (Belgium). Dept. of Physics; Callebaut, D. K. [Antwerp Univ. (Belgium). Dept. of Physics; El-Kalaawy, O. H. [Cairo Univ. (Egypt). Faculty of science, Dept. of Mathematics
1998-09-01
Using the Baecklund transformations (BTs) and the Darboux-Bargmann technique, the Authors consider the nonlinear Schroedinger-type (NLS-type) equations solvable by the inverse scattering method of Zakharov-Shabat/Ablowitz-Kaup-Newell-Segur (ZS/AKNS) system and the ZS/AKNS wave functions corresponding to the soliton solutions of NLS-type equations. Thus, families of new soliton solutions for NLS- type equations are obtained.
Wang, Gang-Wei; Kara, A. H.
2018-01-01
The generalized fractional Burgers equation is studied in this paper. Using the classical Lie symmetry method, all of the vector fields and symmetry reduction of the equation with nonlinearity are constructed. In particular, an exact solution is provided by using the ansatz method. In addition, other types of exact solution are obtained via the invariant subspace method. Finally, conservation laws for this equation are derived.
Numerical Solutions for the Time and Space Fractional Nonlinear Partial Differential Equations
Directory of Open Access Journals (Sweden)
Khaled A. Gepreel
2013-01-01
Full Text Available We implement relatively analytical techniques, the homotopy perturbation method, and variational iteration method to find the approximate solutions for time and space fractional Benjamin-Bona Mahony equation. The fractional derivatives are described in the Caputo sense. These methods are used in applied mathematics to obtain the analytic approximate solutions for the nonlinear Bejamin-Bona Mahoney (BBM partial fractional differential equation. We compare between the approximate solutions obtained by these methods. Also, we present the figures to compare between the approximate solutions. Also, we use the fractional complex transformation to convert nonlinear partial fractional differential equations to nonlinear ordinary differential equations. We use the improved -expansion function method to find exact solutions of nonlinear fractional BBM equation.
Analytical solution for fractional derivative gas-flow equation in porous media
Directory of Open Access Journals (Sweden)
Mohamed F. El Amin
Full Text Available In this paper, we introduce an analytical solution of the fractional derivative gas transport equation using the power-series technique. We present a new universal transform, namely, generalized Boltzmann change of variable which depends on the fractional order, time and space. This universal transform is employed to transfer the partial differential equation into an ordinary differential equation. Moreover, the convergence of the solution has been investigated and found that solutions are unconditionally converged. Results are introduced and discussed for the universal variable and other physical parameters such as porosity and permeability of the reservoir; time and space. Keywords: Fractional derivative, Porous media, Natural gas, Reservoir modeling, Infinite series solutions
Exact solutions to the KDV-Burgers equation with forcing term using Tanh-Coth method
Chukkol, Yusuf Buba; Mohamad, Mohd Nor; Muminov, Mukhiddin I.
2017-08-01
In this paper, tanh-coth method was applied to derive the exact travelling wave solutions to the Korteweg-de-Vries and Burgers equation with forcing term(fKDVB). Solutions that are linear combination of solitary and shock wave solutions, and periodic wave solutions are obtained, by reducing the equation to the homogeneous type using a wave transformation. The method with the help of symbolic computation tool box provides a systematic way of solving many physical models involving nonlinear partial differential equations in mathematical physics.
New soliton solutions of the system of equations for the ion sound and Langmuir waves
Directory of Open Access Journals (Sweden)
Seyma Tuluce Demiray
2016-11-01
Full Text Available This study is based on new soliton solutions of the system of equations for the ion sound wave under the action of the ponderomotive force due to high-frequency field and for the Langmuir wave. The generalized Kudryashov method (GKM, which is one of the analytical methods, has been tackled for finding exact solutions of the system of equations for the ion sound wave and the Langmuir wave. By using this method, dark soliton solutions of this system of equations have been obtained. Also, by using Mathematica Release 9, some graphical simulations were designed to see the behavior of these solutions.
Solution of the two-dimensional Navier-Stokes equations using sparse matrix solvers
Bender, Erich E.; Khosla, Prem K.
1987-01-01
The use of direct sparse matrix solvers in the solution of the Navier-Stokes equations is investigated. The Yale Sparse Matrix Package and its implementation in the solution algorithm is described. The streamfunction-vorticity form of the Navier-Stokes equations are discretized and linearized and the resulting system of equations are solved using this package. Several viscous flow problems are investigated, including flow in a cavity and flow around a NACA0012 airfoil. Massively separated flow around a sine wave airfoil is investigated and high Reynolds number solutions are obtained. A solution of the unsteady flow around a Joukowski airfoil at high angle of attack is presented.
A new approach to the exact solutions of the effective mass Schrodinger equation
Tezcan, Cevdet; Sever, Ramazan; Yesiltas, Ozlem
2007-01-01
Effective mass Schrodinger equation is solved exactly for a given potential. Nikiforov-Uvarov method is used to obtain energy eigenvalues and the corresponding wave functions. A free parameter is used in the transformation of the wave function. The effective mass Schrodinger equation is also solved for the Morse potential transforming to the constant mass Schr\\"{o}dinger equation for a potential. One can also get solution of the effective mass Schrodinger equation starting from the constant m...
High-Order Accurate Solutions to the Helmholtz Equation in the Presence of Boundary Singularities
2015-03-31
elliptic equations . In Proceedings of the Soviet-American Conference on Partial Differential Equations , pages 303–304, Novosibirsk, Moscow, Russia, 1963...193, 2012. [19] L. Fox and R. Sankar. Boundary singularities in linear elliptic differential equations . J. Inst. Math. Appl., 5:340–350, 1969. [20] D.S...57] R.S. Lehman. Developments at an analytic corner of solutions of elliptic partial differential equations . J. Math. Mech., 8:727–760, 1959. [58
Symmetry properties, similarity reduction and exact solutions of fractional Boussinesq equation
Rashidi, Saeede; Hejazi, S. Reza
In this paper, some properties of the time fractional Boussinesq equation are presented. Group analysis of the time fractional Boussinesq equation with Riemann-Liouville derivative is performed and the corresponding optimal system of subgroups are determined. Next, we apply the obtained optimal systems for constructing reduced fractional ordinary differential equations (FODEs). Finally, we show how to derive exact solutions to time fractional Boussinesq equation via invariant subspace method.
Exact Solutions to Some Conformable Time Fractional Equations in Benjamin-Bona-Mahony Family
Korkmaz, Alper
2016-01-01
The conformable time fractional forms of some partial differential equations are solved in the study. The existence of chain rule and the derivative of composite function enable the equations to be reduced to some ordinary differential equations by using some particular wave transformations. The modified Kudryashov method implemented to derive the exact solutions for the Benjamin-Bona Mahony (BBM), the symmetric BBM and the equal width (EW) equations in the conformable fractional time derivat...
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.
Wei Li; Huizhang Yang; Bin He
2014-01-01
Based on a general fractional Riccati equation and with Jumarie’s modified Riemann-Liouville derivative to an extended fractional Riccati expansion method for solving the time fractional Burgers equation and the space-time fractional Cahn-Hilliard equation, the exact solutions expressed by the hyperbolic functions and trigonometric functions are obtained. The obtained results show that the presented method is effective and appropriate for solving nonlinear fractional differential equations.
Directory of Open Access Journals (Sweden)
Wei Li
2014-01-01
Full Text Available Based on a general fractional Riccati equation and with Jumarie’s modified Riemann-Liouville derivative to an extended fractional Riccati expansion method for solving the time fractional Burgers equation and the space-time fractional Cahn-Hilliard equation, the exact solutions expressed by the hyperbolic functions and trigonometric functions are obtained. The obtained results show that the presented method is effective and appropriate for solving nonlinear fractional differential equations.
The behavior of solutions of an equation with a large spatially distributed control
Kashchenko, I.
2017-12-01
The paper is devoted to the dynamical properties of the scalar complex equation and system of two equations with spatially distributed parameters. Main assumption is that the coefficient of spatial distribution is sufficiently large. Using asymptotic methods we construct the families of special parabolic equations, which do not contain big and small parameters, which nonlocal dynamics determines the behaviour of solutions of the original equation.
Solution Hamilton-Jacobi equation for oscillator Caldirola-Kanai
Directory of Open Access Journals (Sweden)
LEONARDO PASTRANA ARTEAGA
2016-12-01
Full Text Available The method allows Hamilton-Jacobi explicitly determine the generating function from which is possible to derive a transformation that makes soluble Hamilton's equations. Using the separation of variables the partial differential equation of the first order called Hamilton-Jacobi equation is solved; as a particular case consider the oscillator Caldirola-Kanai (CK, which is characterized in that the mass presents a temporal evolution exponentially . We demonstrate that the oscillator CK position presents an exponential decay in time similar to that obtained in the damped sub-critical oscillator, which reflects the dissipation of total mechanical energy. We found that in the limit that the damping factor is small, the behavior is the same as an oscillator with simple harmonic motion, where the effects of energy dissipation is negligible.
The First-Integral Method and Abundant Explicit Exact Solutions to the Zakharov Equations
Directory of Open Access Journals (Sweden)
Yadong Shang
2012-01-01
Full Text Available This paper is concerned with the system of Zakharov equations which involves the interactions between Langmuir and ion-acoustic waves in plasma. Abundant explicit and exact solutions of the system of Zakharov equations are derived uniformly by using the first integral method. These exact solutions are include that of the solitary wave solutions of bell-type for n and E, the solitary wave solutions of kink-type for E and bell-type for n, the singular traveling wave solutions, periodic wave solutions of triangle functions, Jacobi elliptic function doubly periodic solutions, and Weierstrass elliptic function doubly periodic wave solutions. The results obtained confirm that the first integral method is an efficient technique for analytic treatment of a wide variety of nonlinear systems of partial differential equations.
Solution of partial differential equations on vector and parallel computers
Ortega, J. M.; Voigt, R. G.
1985-01-01
The present status of numerical methods for partial differential equations on vector and parallel computers was reviewed. The relevant aspects of these computers are discussed and a brief review of their development is included, with particular attention paid to those characteristics that influence algorithm selection. Both direct and iterative methods are given for elliptic equations as well as explicit and implicit methods for initial boundary value problems. The intent is to point out attractive methods as well as areas where this class of computer architecture cannot be fully utilized because of either hardware restrictions or the lack of adequate algorithms. Application areas utilizing these computers are briefly discussed.
New exact travelling wave solutions of bidirectional wave equations
Indian Academy of Sciences (India)
where , , and d are real constants. In general, the exact travelling wave solutions will be helpful in the theoretical and numerical study of the nonlinear evolution systems. In this paper, we obtain exact travelling wave solutions of system (1) using the modiﬁed tanh–coth function method with computerized symbolic ...
Positive integer solutions of the diophantine equation x2− Lnxy+(− 1 ...
Indian Academy of Sciences (India)
... solutions and . Moreover, we give all positive integer solutions of the equation x 2 − L n x y + ( − 1 ) n y 2 = ± 5 r when the equation has positive integer solutions. Author Affiliations. Refik Keskin1 Zafer Şiar2. Sakarya University, Merkezi, 54180 Sakarya, Turkey; Bingöl University, Rektörlüğü, 12000 Bingöl, Turkey ...
Blowup of solutions of a Korteweg-de Vries-type equation
Yushkov, E. V.
2012-07-01
We investigate the nonlinear third-order differential equation (uxx - u)t + u xxx + uux = 0 describing the processes in semiconductors with a strong spatial dispersion. We study the problem of the existence of global solutions and obtain sufficient conditions for the absence of global solutions for some initial boundary value problems corresponding to this equation. We consider examples of solution blowup for initial boundary value and Cauchy problems. We use the Mitidieri-Pokhozhaev nonlinear capacity method.
Pointwise estimates for solutions to a system of nonlinear damped wave equations
Directory of Open Access Journals (Sweden)
Wenjun Wang
2013-11-01
Full Text Available In this article, we consider the existence of global solutions and pointwise estimates for the Cauchy problem of a nonlinear damped wave equation. We obtain the existence by using the approach introduced by Li and Chen in [7] and some estimates of the solution. The proofs of the estimates are based on a detailed analysis of the Green function of the linear damped wave equations. Also, we show the L^p convergence rate of the solution.
On exact traveling-wave solutions for local fractional Korteweg-de Vries equation
Yang, Xiao-Jun; Tenreiro Machado, J. A.; Baleanu, Dumitru; Cattani, Carlo
2016-08-01
This paper investigates the Korteweg-de Vries equation within the scope of the local fractional derivative formulation. The exact traveling wave solutions of non-differentiable type with the generalized functions defined on Cantor sets are analyzed. The results for the non-differentiable solutions when fractal dimension is 1 are also discussed. It is shown that the exact solutions for the local fractional Korteweg-de Vries equation characterize the fractal wave on shallow water surfaces.
On exact traveling-wave solutions for local fractional Korteweg-de Vries equation.
Yang, Xiao-Jun; Tenreiro Machado, J A; Baleanu, Dumitru; Cattani, Carlo
2016-08-01
This paper investigates the Korteweg-de Vries equation within the scope of the local fractional derivative formulation. The exact traveling wave solutions of non-differentiable type with the generalized functions defined on Cantor sets are analyzed. The results for the non-differentiable solutions when fractal dimension is 1 are also discussed. It is shown that the exact solutions for the local fractional Korteweg-de Vries equation characterize the fractal wave on shallow water surfaces.
The (′/-Expansion Method for Abundant Traveling Wave Solutions of Caudrey-Dodd-Gibbon Equation
Directory of Open Access Journals (Sweden)
Hasibun Naher
2011-01-01
Full Text Available We construct the traveling wave solutions of the fifth-order Caudrey-Dodd-Gibbon (CDG equation by the (/-expansion method. Abundant traveling wave solutions with arbitrary parameters are successfully obtained by this method and the wave solutions are expressed in terms of the hyperbolic, the trigonometric, and the rational functions. It is shown that the (/-expansion method is a powerful and concise mathematical tool for solving nonlinear partial differential equations.
New exact solutions of sixth-order thin-film equation
Directory of Open Access Journals (Sweden)
Wafaa M. Taha
2014-01-01
Full Text Available TheG′G-expansion method is used for the first time to find traveling-wave solutions for the sixth-order thin-film equation, where related balance numbers are not the usual positive integers. New types of exact traveling-wave solutions, such as – solitary wave solutions, are obtained the sixth-order thin-film equation, when parameters are taken at special values.
Lu, Benzhuo; Zhou, Y C
2011-05-18
The effects of finite particle size on electrostatics, density profiles, and diffusion have been a long existing topic in the study of ionic solution. The previous size-modified Poisson-Boltzmann and Poisson-Nernst-Planck models are revisited in this article. In contrast to many previous works that can only treat particle species with a single uniform size or two sizes, we generalize the Borukhov model to obtain a size-modified Poisson-Nernst-Planck (SMPNP) model that is able to treat nonuniform particle sizes. The numerical tractability of the model is demonstrated as well. The main contributions of this study are as follows. 1), We show that an (arbitrarily) size-modified PB model is indeed implied by the SMPNP equations under certain boundary/interface conditions, and can be reproduced through numerical solutions of the SMPNP. 2), The size effects in the SMPNP effectively reduce the densities of highly concentrated counterions around the biomolecule. 3), The SMPNP is applied to the diffusion-reaction process for the first time, to our knowledge. In the case of low substrate density near the enzyme reactive site, it is observed that the rate coefficients predicted by SMPNP model are considerably larger than those by the PNP model, suggesting both ions and substrates are subject to finite size effects. 4), An accurate finite element method and a convergent Gummel iteration are developed for the numerical solution of the completely coupled nonlinear system of SMPNP equations. Copyright © 2011 Biophysical Society. Published by Elsevier Inc. All rights reserved.
Numerical Solutions of Generalized Burger's-Huxley Equation by ...
African Journals Online (AJOL)
... results with this technique have been compared with other results. The present method is seen to be a very reliable alternative method to some existing techniques for such nonlinear problems. Keywords: Burger's-Huxley, modified variational iteration method, lagrange multiplier, Taylor's series, partial differential equation ...
Generalized solutions to the gKdV equation
Directory of Open Access Journals (Sweden)
Maurilio Marcio Melo
2010-08-01
Full Text Available In this article we study the Cauchy problem in $mathcal{G}_2((0,Times mathbb{R}$ (the algebra of generalized functions, in the sense of Colombeau for the generalized Korteweg-de Vries equation, with initial condition $varphi in mathcal{G}_2(mathbb{R}$, which contains $H^s(mathbb{R}$, for $sin mathbb{R}$.
Numerical solution of uncertain neutron diffusion equation for ...
Indian Academy of Sciences (India)
Department of Mathematics, National Institute of Technology, ... The concept of fuzziness is hybridised with traditional finite element method to propose fuzzy finite element method. The proposed fuzzy finite element method has ..... In general when neutrons undergo scattering, the neutron transport equation involves uncer-.
An Efficient Series Solution for Fractional Differential Equations
Directory of Open Access Journals (Sweden)
Mohammed Al-Refai
2014-01-01
where only integer derivatives have to be computed. The efficiency of the new algorithm is illustrated through several examples. Comparison with other series methods such as the Adomian decomposition method and the homotopy perturbation method is made to indicate the efficiency of the new approach. The algorithm can be implemented for a wide class of fractional differential equations with different types of fractional derivatives.
Finite element solution of the Boussinesq wave equation | Akpobi ...
African Journals Online (AJOL)
In this work, we investigate a Boussinesq-type flow model for nonlinear dispersive waves by developing a computational model based on the finite element discretisation technique. Hermite interpolation functions were used to interpolate approximation elements. The system is modeled using a time dependent equation.
Asymptotic behavior of solutions of forced fractional differential equations
Directory of Open Access Journals (Sweden)
Said Grace
2016-09-01
where $y(t=\\left( a(tx^{\\prime }(t\\right ^{\\prime }$, $c_{0}=\\frac{y(c}{\\Gamma (1}=y(c$, and $c_{0}$ is a real constant. The technique used in obtaining their results will apply to related fractional differential equations with Caputo derivatives of any order. Examples illustrate the results obtained in this paper.
Solutions of selected pseudo loop equations in water distribution ...
African Journals Online (AJOL)
This paper demonstrated the use of Microsoft Excel Solver (a computer package) in solving selected pseudo loop equations in pipe network analysis problems. Two pipe networks with pumps and overhead tanks were used to demonstrate the use of Microsoft Excel Solver in solving pseudo loops (open loops; networks with ...
Use of eigenvectors in the solution of the flutter equation
CSIR Research Space (South Africa)
Van Zyl, Lourens H
1993-07-01
Full Text Available The use of eigenvectors to assign eigenvalues to modes for the p-k formulation of the flutter equation is described. The procedure has the potential to overcome some of the problems of the determinant iteration procedure to solve the flutter...
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
Orbital stability of Gausson solutions to logarithmic Schrodinger equations
Directory of Open Access Journals (Sweden)
Alex H. Ardila
2016-12-01
Full Text Available In this article we prove of the orbital stability of the ground state for logarithmic Schrodinger equation in any dimension and under nonradial perturbations. This general stability result was announced by Cazenave and Lions [9, Remark II.3], but no details were given there.
Multigrid solution of incompressible turbulent flows by using two-equation turbulence models
Energy Technology Data Exchange (ETDEWEB)
Zheng, X.; Liu, C. [Front Range Scientific Computations, Inc., Denver, CO (United States); Sung, C.H. [David Taylor Model Basin, Bethesda, MD (United States)
1996-12-31
Most of practical flows are turbulent. From the interest of engineering applications, simulation of realistic flows is usually done through solution of Reynolds-averaged Navier-Stokes equations and turbulence model equations. It has been widely accepted that turbulence modeling plays a very important role in numerical simulation of practical flow problem, particularly when the accuracy is of great concern. Among the most used turbulence models today, two-equation models appear to be favored for the reason that they are more general than algebraic models and affordable with current available computer resources. However, investigators using two-equation models seem to have been more concerned with the solution of N-S equations. Less attention is paid to the solution method for the turbulence model equations. In most cases, the turbulence model equations are loosely coupled with N-S equations, multigrid acceleration is only applied to the solution of N-S equations due to perhaps the fact the turbulence model equations are source-term dominant and very stiff in sublayer region.
Solitary Wave Solutions of the Boussinesq Equation and Its Improved Form
Directory of Open Access Journals (Sweden)
Reza Abazari
2013-01-01
Full Text Available This paper presents the general case study of previous works on generalized Boussinesq equations, (Abazari, 2011 and (Kılıcman and Abazari, 2012, that focuses on the application of G′/G-expansion method with the aid of Maple to construct more general exact solutions for the coupled Boussinesq equations. In this work, the mentioned method is applied to construct more general exact solutions of Boussinesq equation and improved Boussinesq equation, which the French scientist Joseph Valentin Boussinesq (1842–1929 described in the 1870s model equations for the propagation of long waves on the surface of water with small amplitude. Our work is motivated by the fact that the G′/G-expansion method provides not only more general forms of solutions but also periodic, solitary waves and rational solutions. The method appears to be easier and faster by means of a symbolic computation.
On a spectral criterion for almost periodicity of solutions of periodic evolution equations
Directory of Open Access Journals (Sweden)
Toshiki Naito
1999-01-01
Full Text Available This paper is concerned with equations of the form: $u'=A(tu + f(t$, where $A(t$ is (unbounded periodic linear operator and f is almost periodic. We extend a central result on the spectral criteria for almost periodicity of solutions of evolution equations to some classes of periodic equations which says that if $u$ is a bounded uniformly continuous mild solution and $P$ is the monodromy operator, then their spectra satisfy $e^{i sp_{AP(u}}\\subset \\sigma(P\\cap S^1$, where $S^1$ is the unit circle. This result is then applied to find almost periodic solutions to the abovementioned equations. In particular, parabolic and functional differential equations are considered. Existence conditions for almost periodic and quasiperiodic solutions are discussed.
Directory of Open Access Journals (Sweden)
Alexander Shapovalov
2005-10-01
Full Text Available The complex WKB-Maslov method is used to consider an approach to the semiclassical integrability of the multidimensional Gross-Pitaevskii equation with an external field and nonlocal nonlinearity previously developed by the authors. Although the WKB-Maslov method is approximate in essence, it leads to exact solution of the Gross-Pitaevskii equation with an external and a nonlocal quadratic potential. For this equation, an exact solution of the Cauchy problem is constructed in the class of trajectory concentrated functions. A nonlinear evolution operator is found in explicit form and symmetry operators (mapping a solution of the equation into another solution are obtained for the equation under consideration. General constructions are illustrated by examples.
Energy Technology Data Exchange (ETDEWEB)
N`kaoua, T.; Chaigneau, C.; Coulomb, F. [CEA Centre d`Etudes de Limeil, 94 - Villeneuve-Saint-Georges (France)
1997-04-01
We are interested in the solution of the Multigroup Neutron Transport Equation. After the presentation of the multigroup treatment, the angular, time and space discretization, we expose the modifications that have been made in order to get an efficient parallel method. (author)
Self-similar singular solution of doubly singular parabolic equation with gradient absorption term
Directory of Open Access Journals (Sweden)
Shi Peihu
2006-01-01
Full Text Available We deal with the self-similar singular solution of doubly singular parabolic equation with a gradient absorption term for , and in . By shooting and phase plane methods, we prove that when there exists self-similar singular solution, while there is no any self-similar singular solution. In case of existence, the self-similar singular solution is the self-similar very singular solutions which have compact support. Moreover, the interface relation is obtained.
Classification of eight-vertex solutions of the colored Yang-Baxter equation
Wang, S
1997-01-01
In this paper all eight-vertex type solutions of the colored Yang-Baxter equation dependent on spectral as well as color parameter are given. It is proved that they are composed of three groups of basic solutions, three groups of their degenerate forms and two groups of trivial solutions up to five solution transformations. Moreover, all non-trivial solutions can be classified into two types called Baxter and Free-Fermion type.
Multi-valued solution of the Burgers' equation and shock ...
African Journals Online (AJOL)
linearity and dissipation and use these properties to examine the vanishing behaviour of the dissipation coefficient. Furthermore, we undertake a rigorous mathematical analysis which gives rise to multi-valued solutions after sufficient time and ...
Asymptotic behaviour of solutions for porous medium equation with periodic absorption
Directory of Open Access Journals (Sweden)
Yin Jingxue
2001-01-01
Full Text Available This paper is concerned with porous medium equation with periodic absorption. We are interested in the discussion of asymptotic behaviour of solutions of the first boundary value problem for the equation. In contrast to the equation without sources, we show that the solutions may not decay but may be attracted into any small neighborhood of the set of all nontrivial periodic solutions, as time tends to infinity. As a direct consequence, the null periodic solution is unstable. We have presented an accurate condition on the sources for solutions to have such a property. Whereas in other cases of the sources, the solutions might decay with power speed, which implies that the null periodic solution is stable.
Regularity of large solutions for the compressible magnetohydrodynamic equations
Directory of Open Access Journals (Sweden)
Qin Yuming
2011-01-01
Full Text Available Abstract In this paper, we consider the initial-boundary value problem of one-dimensional compressible magnetohydrodynamics flows. The existence and continuous dependence of global solutions in H 1 have been established in Chen and Wang (Z Angew Math Phys 54, 608-632, 2003. We will obtain the regularity of global solutions under certain assumptions on the initial data by deriving some new a priori estimates.
Numerical solution of the stochastic parabolic equation with the dependent operator coefficient
Energy Technology Data Exchange (ETDEWEB)
Ashyralyev, Allaberen [Department of Elementary Mathematics Education, Fatih University, 34500, Istanbul (Turkey); Department of Mathematics, ITTU, Ashgabat (Turkmenistan); Okur, Ulker [Institute for Stochastics and Applications, Department of Mathematics, University of Stuttgart, 70569, Stuttgart (Germany)
2015-09-18
In the present paper, a single step implicit difference scheme for the numerical solution of the stochastic parabolic equation with the dependent operator coefficient is presented. Theorem on convergence estimates for the solution of this difference scheme is established. In applications, this abstract result permits us to obtain the convergence estimates for the solution of difference schemes for the numerical solution of initial boundary value problems for parabolic equations. The theoretical statements for the solution of this difference scheme are supported by the results of numerical experiments.
Lump Solutions for the (3+1)-Dimensional Kadomtsev-Petviashvili Equation
Liu, De-Yin; Tian, Bo; Xie, Xi-Yang
2016-12-01
In this article, we investigate the lump solutions for the Kadomtsev-Petviashvili equation in (3+1) dimensions that describe the dynamics of plasmas or fluids. Via the symbolic computation, lump solutions for the (3+1)-dimensional Kadomtsev-Petviashvili equation are derived based on the bilinear forms. The conditions to guarantee analyticity and rational localisation of the lump solutions are presented. The lump solutions contain eight parameters, two of which are totally free, and the other six of which need to satisfy the presented conditions. Plots with particular choices of the involved parameters are made to show the lump solutions and their energy distributions.