Jacobi多项式解变分数阶非线性微积分方程

Numerical solution of variable fractional order nonlinear differentialintegral equation by Jacobi polynomial

为求解一类变分数阶非线性微积分方程,提出了一种求解该类方程数值解的方法.该方法主要利用移位的Jacobi多项式将方程中的函数逼近,再结合Captuo类型的变分数阶微积分定义,推导出移位Jacobi多项式的微积分算子矩阵,将最初的方程转化为矩阵相乘的形式,然后通过离散变量,将原方程转化为一系列非线性方程组.通过解该非线性方程组得到移位Jacobi多项式的系数,进而可得原方程的数值解.最后,通过数值算例的精确解和数值解的绝对误差验证了该方法的高精度性和有效性.

In order to solve a class of variable fractional order nonlinear differential-integral equations, the numerical solution is proposed. Function approximation based on Shifted Jacobi polynomials, combining Caputo-type variable order fractional derivate definition, which are used to get the operational matrixes of Shifted Jacobi polynomials, are the main characteristic behind this method. With the operational matrix, the original equation is translated into the products of several dependent matrixes, which can be regarded as a system of nonlinear equations after dispersing the variable. By solving the nonlinear system of algebraic equations, the coefficients of Shifted Jacobi polynomials are got, then the numerical solntions of the original equation are acquired. Finally, some numerical examples illustrate the accuracy and effectiveness of the method.