Variable Separation and Derivative-Dependent Functional Separable Solutions to Generalized Nonlinear Wave Equations

Using the generalized conditional symmetry approach, we obtain a number of new generalized (1+1)-dimensional nonlinear wave equations that admit derivative-dependent functional separable solutions.

Extension of Variable Separable Solutions for Nonlinear Evolution Equations

We give the generalized definitions of variable separable solutions to nonlinear evolution equations, and characterize the relation between the functional separable solution and the derivative-dependent functional separable solution. The new definitions can unify various kinds of variable separable solutions appearing in references. As application, we classify the generalized nonlinear diffusion equations that admit special functional separable solutions and obtain some exact solutions to the resulting equations.

Variable Separation and Exact Separable Solutions for Equations of Type uxt=A（u,ux）uxx＋B（u,ux）

The generalized conditional symmetry is developed to study the variable separation for equations of type uxt = A（u,ux）uxx ＋ B（u, ux）. Complete classification of those equations which admit derivative-dependent functional separable solutions is obtained and some of their exact separable solutions are constructed.

Extended Group Foliation Method and Functional Separation of Variables to Nonlinear Wave Equations

Generalized functional separation of variables to nonlinear evolution equations is studied in terms of the extended group foliation method, which is based on the Lie point symmetry method. The approach is applied to nonlinear wave equations with variable speed and external force. A complete classification for the wave equation which admits functional separable solutions is presented. Some known results can be recovered by this approach.

Functional Separable Solutions to Nonlinear Diffusion Equations by Group Foliation Method

We consider the functional separation of variables to the nonlinear diffusion equation with source and convection term： ut = （A（x）D（u）ux）x ＋ B（x）Q（u）, Ax ≠ 0. The functional separation of variables to this equation is studied by using the group foliation method. A classification is carried out for the equations which admit the function separable solutions. As a consequence, some solutions to the resulting equations are obtained.

On the feasibility of variable separation method based on Hamiltonian system for plane magnetoelectroelastic solids

The eigenvalue problem of an infinite-dimensional Hamiltonian operator appearing in the isotropic plane magnetoelectroelastic solids is studied. First, all the eigenvalues and their eigenfunctions in a rectangular domain are solved directly. Then the completeness of the eigenfunction system is proved, which offers a theoretic guarantee of the feasibility of variable separation method based on a Hamiltonian system for isotropic plane magnetoelectroelastic solids. Finally, the general solution for the equation in the rectangular domain is obtained by using the symplectic Fourier expansion method.

Solving mKdV-sinh-Gordon equation by a modified variable separated ordinary differential equation method

By introducing a more general auxiliary ordinary differential equation （ODE）, a modified variable separated ODE method is developed for solving the mKdV-sinh-Gordon equation. As a result, many explicit and exact solutions including some new formal solutions are successfully picked up for the mKdV-sinh-Gordon equation by this approach.

Functional Variable Separation for Extended Nonlinear Elliptic Equations

This paper is devoted to the study of functional variable separation for extended nonlinear elliptic equations. By applying the functional variable separation approach to extended nonlinear elliptic equations via the generalized conditional symmetry, we obtain complete classification of those equations which admit functional separable solutions （FSSs） and construct some exact FSSs to the resulting equations.

The derivative-dependent functional variable separation for the evolution equations

This paper studies variable separation of the evolution equations via the generalized conditional symmetry. To illustrate, we classify the extended nonlinear wave equation utt = A（u, ux）uxx＋B（u, ux, ut） which admits the derivative- dependent functional separable solutions （DDFSSs）. We also extend the concept of the DDFSS to cover other variable separation approaches.

Analytical solution for the time-fractional heat conduction equation in spherical coordinate system by the method of variable separation

In this paper,using the fractional Fourier law,we obtain the fractional heat conduction equation with a time-fractional derivative in the spherical coordinate system.The method of variable separation is used to solve the timefractional heat conduction equation.The Caputo fractional derivative of the order 0 〈 α≤ 1 is used.The solution is presented in terms of the Mittag-Leffler functions.Numerical results are illustrated graphically for various values of fractional derivative.

Functional Separable Solutions of Nonlinear Heat Equations in Non-Newtonian Fluids

We study the functional separation of variables to the nonlinear heat equation： ut = （A（x）D（u）ux^n）x＋ B（x）Q（u）, Ax≠0. Such equation arises from non-Newtonian fluids. Its functional separation of variables is studied by using the group foliation method. A classification of the equation which admits the functional separable solutions is performed. As a consequence, some solutions to the resulting equations are obtained.

The projective Riccati equation expansion method and variable separation solutions for the nonlinear physical differential equation in physics

Using the projective Riccati equation expansion （PREE） method, new families of variable separation solutions （including solitary wave solutions, periodic wave solutions and rational function solutions） with arbitrary functions for two nonlinear physical models are obtained. Based on one of the variable separation solutions and by choosing appropriate functions, new types of interactions between the multi-valued and single-valued solitons, such as a peakon-like semi-foldon and a peakon, a compacton-like semi-foldon and a compacton, are investigated.

Variable Separation for(1+1)-Dimensional Nonlinear Evolution Equations with Mixed Partial Derivatives

We present basic theory of variable separation for (1 + 1)-dimensional nonlinear evolution equations withmixed partial derivatives.As an application,we classify equations u_(xt)=A(u,u_x)u_(xxx)+B(u,u_x) that admits derivative-dependent functional separable solutions (DDFSSs) and illustrate how to construct those DDFSSs with some examples.

Multi-linear Variable Separation Approach to Solve a (2+1)-DimensionalGeneralization of Nonlinear Schroedinger System

By using a Baecklund transformation and the multi-linear variable separationapproach, we find a new general solution of a (2+1)-dimensional generalization of the nonlinearSchroedinger system. The new 'universal' formula is defined, and then, rich coherent structures canbe found by selecting corresponding functions appropriately.

Solving (2+1)-dimensional sine-Poisson equation by a modified variable separated ordinary differential equation method

By introducing a more general auxiliary ordinary differential equation （ODE）, a modified variable separated ordinary differential equation method is presented for solving the （2 ＋ 1）-dimensional sine-Poisson equation. As a result, many explicit and exact solutions of the （2 ＋ 1）-dimensional sine-Poisson equation are derived in a simple manner by this technique.

Separation of Variable Treatment for Solving Time—Dependent Potentials

We use the separation of variable treatment to treat some time-dependent systems, and point out that the condition of separability is the same as the condition of existence of invariant, and the separation of variable treatment is interrelated with the quantum-invariant method and the propagator method. We directly use the separation of variable treatment to obtain the wavefunctions of the time-dependent Coulomb potential and the time-dependent Hulthén potential.

Variable Separated Solutions and Four-Dromion Excitations for (2+1)-Dimensional Nizhnik-Novikov-Veselov Equation

Using the mapping approach via the projective Riccati equations, several types of variable separated solutions of the （2＋1）-dimensional Nizhnik-Novikov-Veselov equation are obtained, including multiple-soliton solutions, periodic-soliton solutions, and Weierstrass function solutions. Based on a periodic-soliton solution, a new type of localized excitation, i.e., the four-dromion soliton, is constructed and some evolutional properties of this localized structure are briefly discussed.

Variable Separation Approach to Solve （2 ＋ 1）-Dimensional Generalized Burgers System： Solitary Wave and Jacobi Periodic Wave Excitations

By means of the standard truncated Painlevé expansion and a variable separation approach, a general variable separation solution of the generalized Burgers system is derived. In addition to the usual localized coherent soliton excitations like dromions, lumps, rings, breathers, instantons, oscillating soliton excitations, peakons, foldons, and previously revealed chaotic and fractal localized solutions, some new types of excitations — compacton and Jacobi periodic wave solutions are obtained by introducing appropriate lower dimensional piecewise smooth functions and Jacobi elliptic functions.

A SPECIAL METHOD OF FOURIER SERIES WHICH IS EQUAL TO THE METHOD OF SEPARATION OF VARIABLES ON BOUNDARY VALUE PROBLEM

By the separation of singularity, a special Fourier series solution of the boundary value problem for plane is obtained, which can satisfy all boundary conditions and converges rapidly. II is proved that the solution is equal to the result of separation of variables. As a result, the non-linear characteristic equations resulting from the method of separation of variables are transformed into polynomial equations that can provide a foundation for approximate computation and asymptotic analysis.

Some discussions about method for solving the variable separating nonlinear models

Through analysing the exact solution of some nonlinear models, the role of the variable separating method in solving nonlinear equations is discussed. We find that rich solution structures of some special fields of these equations come from the nonzero seed solution. However, these nonzero seed solutions is likely to result in the divergent phenomena for the other field component of the same equation. The convergence and the signification of all field components should be discussed when someone solves the nonlinear equation using the variable separating method.