On the Behavior of Solutions of a Class of Nonlinear Partial Differential Equations

The behavior of the steady-state (or the traveling wave) solutions for a class of nonlinear partial differential equations is studied. The nonlinearity in these equations is expressed by the presence of the convective term. It is shown that the steady-state (or the traveling wave) solution may explode at a finite value of the spatial (or the characteristic) variable. This holds whatever the order of the spatial derivative term in these equations. Furthermore, new special solutions of a set of equations in this class are also found.     

Conservation Laws and Exact Solutions for some Nonlinear Partial Differential Equations

An effective algorithmic method (Anco, S. C. and Bluman, G. (1996). Journal of Mathematical Physics 37, 2361; Anco, S. C. and Bluman, G. (1997). Physical Review Letters 78, 2869; Anco, S. C. and Bluman, G. (1998). European Journal of Applied Mathematics 9, 254; Anco, S. C. and Bluman, G. (2001). European Journal of Applied Mathematics 13, 547; Anco, S. C. and Bluman, G. (2002). European Journal of Applied Mathematics 13, 567 is used for finding the local conservation laws for some nonlinear partial differential equations. The method does not require the use or existence of a variational principle and reduces the calculation of conservation laws to solving a system of linear determining equations similar to that of finding symmetries. An explicit construction formula is derived which yields a conservation law for each solution of the determining system. Different methods to construct new exact solution classes for the same nonlinear partial differential equations are also presented, which are named hyperbolic function method and the Bäcklund transformations. On the other hand, other methods and transformations are developed to obtain exact solutions for some nonlinear partial differential equations.     

A Laplace Decomposition Method for Nonlinear Partial Differential Equations with Nonlinear Term of Any Order

A Laplace decomposition algorithm is adopted to investigate numerical solutions of a class of nonlinear partial differential equations with nonlinear term of any order, utt ＋ auxx ＋ bu ＋ cup ＋ du^2p-1 = 0, which contains some important equations of mathematical physics. Three distinct initial conditions are constructed and generalized numerical solutions are thereby obtained, including numerical hyperbolic function solutions and doubly periodic ones. Illustrative figures and comparisons between the numerical and exact solutions with different values of p are used to test the efficiency of the proposed method, which shows good results are azhieved.     

A Class of Traveling Wave Solutions to Some Nonlinear Partial Differential Equations

For the Noyes-Fields equations, two-dimensional hyperbolic equations of conversation laws, and theBurgers-KdV equation, a class of traveling wave solutions has been obtained by constructing appropriate functiontransformations. The main idea of solving the equations is that nonlinear partial differential equations are changed intosolving algebraic equations. This method has a wide-rangingpracticability.     

An Extension of Mapping Deformation Method and New Exact Solution for Three Coupled Nonlinear Partial Differential Equations

In this paper, we extend the mapping deformation method proposed by Lou. It is used to find new exacttravelling wave solutions of nonlinear partial differential equation or coupled nonlinear partial differential equations(PDEs). Based on the idea of the homogeneous balance method, we construct the general mapping relation betweenthe solutions of the PDEs and those of the cubic nonlinear Klein-Gordon (NKG) equation. By using this relation andthe abundant solutions of the cubic NKG equation, many explicit and exact travelling wave solutions of three systemsof coupled PDEs, which contain solitary wave solutions, trigonometric function solutions, Jacobian elliptic functionsolutions, and rational solutions, are obtained.     

New Solutions of Three Nonlinear Space- and Time-Fractional Partial Differential Equations in Mathematical Physics

Motivated by the widely used ansätz method and starting from the modified Riemann-Liouville derivative together with a fractional complex transformation that can be utilized to transform nonlinear fractional partial differential equations to nonlinear ordinary differential equations, new types of exact traveling wave solutions to three important nonlinear space- and time-fractional partial differential equations are obtained simultaneously in terms of solutions of a Riccati equation. The results are new and first reported in this paper.     

求解非线性偏微分方程的自适应小波精细积分法

以Burgers方程为例,提出了一种求解偏微分方程的自适应多层插值小波配置法,通过引入一种具有插值特性的拟Shannon小波并利用插值小波理论构造了多层自适应插值小波算子,从而在空间实现了偏微分方程的自适应离散.另外,精细时程积分方法和外推法的引入不但有助于提高求解速度和数值结果的精度,而且使时间积分步长的选取具有了自适应性.     

Relationship Between Soliton-like Solutions and Soliton Solutions to a Class of Nonlinear Partial Differential Equations

By using the generally projective Riccati equation method, a series of doubly periodic solutions (Jacobi elliptic function solution) for a class of nonlinear partial differential equations are obtained in a unified way. When the module m→1, these solutions exactly degenerate to the soliton solutions of the equations. Then we reveal the relationship between the soliton-like solutions obtained by other authors and these soliton solutions of the equations.     

Extended Mapping Transformation Method and Its Applications to Nonlinear Partial Differential Equation(s)

In this paper, we extend the mapping transformation method through introducing variable coefficients.By means of the extended mapping transformation method, many explicit and exact general solutions with arbitrary functions for some nonlinear partial differential equations, which contain solitary wave solutions, trigonometric function solutions, and rational solutions, are obtained.     

时变偏微分方程的贝叶斯稀疏识别方法

在数据驱动的建模中,通过测量或模拟得到时空数据,我们发现基于拉普拉斯先验的贝叶斯稀疏识别方法能有效地恢复时变偏微分方程的稀疏系数.本文将贝叶斯稀疏识别方法运用于各种时变偏微分方程模型(KdV方程、Burgers方程、Kuramoto-Sivashinsky方程、反应-扩散方程、非线性薛定谔方程和纳维-斯托克斯方程)的方...     

A Laplace Decomposition Method for Nonlinear Partial Diferential Equations with Nonlinear Term of Any Order

A Laplace decomposition algorithm is adopted to investigate numerical solutions of a class of nonlinear partial diferential equations with nonlinear term of any order,utt+auxx+bu+cup+du2p 1=0,which contains some important equations of mathematical physics.Three distinct initial conditions are constructed and generalized numerical solutions are thereby obtained,including numerical hyperbolic function solutions and doubly periodic ones.Illustrative figures and comparisons between the numerical and exact solutions with diferent values of p are used to test the efciency of the proposed method,which shows good results are achieved.     

从噪声数据学习偏微分方程的复合神经网络

提出一种名为NN-PDE(neural network-partial differential equations)的复合神经网络方法, 用于噪声数据预处理和学习偏微分方程。NN-PDE用一套神经网络负责数据预处理, 另一套网络耦合备选的方程信息, 进而学习潜在的控制方程。两套网络复合为一套网络, 可更加高效地处理噪声数据, 有效减小噪声的影响。使用NN-PDE学习多种物理方程(如Burgers方程、Korteweg-de Vries方程、Kuramoto-Sivashinsky方程和Navier-Stokes方程)的噪声数据, 均可获得准确的控制方程。     

A Note on Nonclassical Symmetries of a Class of Nonlinear Partial Differential Equations and Compatibility

The nonclassical symmetries of a class of nonlinear partial differential equations obtained by the compatibility method is investigated. We show thenonclassical symmetries obtained in [J. Math. Anal. Appl. 289 (2004) 55, J. Math. Anal. Appl. 311 (2005) 479] are not all the nonclassical symmetries. Based on a new assume on the form of invariant surface condition, all the nonclassical symmetries for a class of nonlinear partial differential equations can be obtained through the compatibility method. The nonlinear Klein-Gordonequation and the Cahn-Hilliard equations all serve as examples showing the compatibility method leads quickly and easily to the determining equations for their all nonclassical symmetries for two equations.     

A Note on Nonclassical Symmetries of a Class of Nonlinear Partial Differential Equations and Compatibility

The nonclassical symmetries of a class of nonlinear partial differential equations obtained by the compatibility method is investigated. We show the nonclassicaJ symmetries obtained in [J. Math. Anal. Appl. 289 （2004） 55, J. Math. Anal. Appl. 311 （2005） 479] are not all the nonclassical symmetries. Based on a new assume on the form of invariant surface condition, all the nonclassical symmetries for a class of nonlinear partial differential equations can be obtained through the compatibility method. The nonlinear Klein-Gordon equation and the Cahn-Hilliard equations all serve as examples showing the compatibility method leads quickly and easily to the determining equations for their all nonclassical symmetries for two equations.     

New Exact Solutions for a Class of Nonlinear Coupled Differential Equations

More new exact solutions for a class of nonlinear coupled differential equations are obtained by using a direct and efficient hyperbola function transform method based on the idea of the extended homogeneous balance method.     

Nonlinear Neutral Delay Differential Equations of Fourth-Order: Oscillation of Solutions

The objective of this paper is to study oscillation of fourth-order neutral differential equation. By using Riccati substitution and comparison technique, new oscillation conditions are obtained which insure that all solutions of the studied equation are oscillatory. Our results complement some known results for neutral differential equations. An illustrative example is included.     

Differential Equations Invariant Under Conditional Symmetries

Nonlinear PDE’s having given conditional symmetries are constructed. They are obtained starting from the invariants of the conditional symmetry generator and imposing the extra condition given by the characteristic of the symmetry. Series of examples starting from the Boussinesq and including non-autonomous Korteweg–de Vries like equations are given to show and clarify the methodology introduced.     

Nonclassical Symmetries for Nonlinear Partial Differential Equations via Compatibility

The determining equations for the nonclassical symmetry reductions of nonlinear partial differential equations with arbitrary order can be obtained by requiring the compatibility between the original equations and the invariant surface conditions. The (2+1)-dimensional shallow water wave equation,
Boussinesq equation, and the dispersive wave equations in shallow water serve as examples illustrating how compatibility leads quickly and easily to the determining equations for their nonclassical symmetries.     

A New Fractional Projective Riccati Equation Method for Solving Fractional Partial Differential Equations

In this paper, a new fractional projective Riccati equation method is proposed to establish exact solutions for fractional partial differential equations in the sense of modified Riemann—Liouville derivative. This method can be seen as the fractional version of the known projective Riccati equation method. For illustrating the validity of this method, we apply this method to solve the space—time fractional Whitham—Broer—Kaup (WBK) equations and the nonlinear fractional Sharma—Tasso—Olever (STO) equation, and as a result, some new exact solutions for them are obtained.     

Consistent Riccati Expansion Method and Its Applications to Nonlinear Fractional Partial Differential Equations

In this paper, a consistent Riccati expansion method is developed to solve nonlinear fractional partial differential equations involving Jumarie's modified Riemann–Liouville derivative. The efficiency and power of this approach are demonstrated by applying it successfully to some important fractional differential equations, namely, the time fractional Burgers, fractional Sawada–Kotera, and fractional coupled mKdV equation. A variety of new exact solutions to these equations under study are constructed.