分离变量法和函数逼近法在偏微分方程的解法分析

Analysis of the Methods of Separating Variables and Function Approxi⁃mation in Partial Differential Equations

以经典的Pennes传热方程为研究对象,通过有限区域傅里叶变换法和分离变量法将方程转化为常微分方程,然后利用分数阶导数的Laplace变换和逆变换求解进行了求解;结果表明,有限Fourier变换、Fourier-Laplace变换和分离变量法等方法能够求解传热方程;而函数逼近Adomian分解法可以求出方程的近似解。

Fractional differential equations have very important applications in many fields.In this paper,the classic Pennes heat transfer equation is used as the research object.The equation is transformed into ordinary differential equations through the limited area Fourier transform method and the separation variable method,and then the fractional derivative Laplace transform and inverse transform are used to solve the problem.The results showed that:methods such as finite Fourier transform,Fourier-Laplace transform and variable separation method can solve the heat transfer equation.The function approximation Adomian decomposition method can find the approximate solution of the equation.