两类基于Riemann-Liouville分数阶导数的非线性偏微分方程的对称分析

Symmetry Analysis of Two Kinds of Nonlinear Partial Differential Equations Based on Riemann-Liouville Fractional Derivatives

针对热传导类和扩散类这两类Riemann-Liouville分数阶微分方程,采用了Lie对称方法,研究了这两类分数阶微分方程所允许的Lie代数。给出两类方程拥有的对称,运用部分Lie对称变换把对应的偏微分方程化为新变量下的分数阶常微分方程,表明Lie对称方法适用于此类方程,可以使方程实现约化,进而更易求解,使得热传导类和扩散类Riemann-Liouville分数阶微分方程可以更加广泛地应用于对事物现象的描述。

For two kinds of Riemann-Liouville fractional differential equations of heat conduction and diffu-sion, the Lie algebras allowed for these two kinds of fractional differential equations are studied by using Lie symmetry method. The symmetry of the two kinds of equations is given, and the corre-sponding partial lie symmetry transformation is used to transform the corresponding partial dif-ferential equations into fractional ordinary differential equations with new variables. It shows that the Lie symmetry method is suitable for such equations, which can reduce the equations and make them easier to solve. The Riemann-Liouville fractional differential equations of heat conduction and diffusion can be more widely used to describe the phenomena of things.