应用多种方法求解非线性分数阶偏微分方程

A Variety of Methods to Solve Nonlinear Fractional Partial Differential Equations

为了发展非线性分数阶偏微分方程的求解技巧并丰富其解的形式,把若干非线性分数阶偏微分方程进行分数阶复变换,转化为整数阶常微分方程或偏微分方程。通过因式分解法求得分数阶CahnAllen方程的孤立波解;利用推广的(F/G)展开法求解了(2+1)维分数阶asymmetricNizhnikNovikovVeselov方程的完全分离变量形式的解,并得到了多Dromion孤子的结构激发;由重正规化方法分别求出在强、弱非线性下的分数阶KleinGordon方程的一级解析近似解,再采用线化和校正方法在无须特殊考虑非线性强度大小的情况下直接求得了该方程的一级近似解,并对两种近似方法所得结果进行比较。

In order to develop the solving techniques of nonlinear fractional partial differential equations and enrich the form of their solutions,some nonlinear fractional partial differential equations are transformed into integral order ordinary or partial differential equations.Then,the solitary wave solution of the fractional-order Cahn-Allen equation is obtained by the factorization method.The extended(F/G)-expansion method is used to solve the solution of the fully separated variable form of the(2+1)-dimensional fractional asymmetric-Nizhnik-Novikov-Veselov equation,and the structural excitation of the multi-Dromion soliton is obtained.Finally,the first-order analytical approximate solution of the fractional Klein-Gordon equation under strong and weak nonlinearity is obtained by the renormalization,and the linearization and correction method is adopted to get the first-order approximate solution of the equation without special consideration of the nonlinear intensity,and the results obtained by the two approximation methods are also compared.