非线性分数阶常微分方程的分段线性插值多项式方法

通过分段线性插值多项式方法构造了一类含有Hadamard有限部分积分的非线性常微分方程的数值离散格式.在时间方向上, 利用分段线性插值多项式方法对分数阶导数项进行近似, 并通过二阶向后差分格式来离散整数阶导数项.经过详细的证明, 得到了收敛精度为O(τmin{1+α,1+β})的误差估计结果.最后,通过数值算例和理论结果的对比直观地说明了理论分析的正确性.

A Piecewise Linear Interpolation Polynomial Method for Nonlinear Fractional Ordinary Differential Equations

A numerical scheme with the piecewise linear interpolation polynomial method was established to solve a class of nonlinear fractional ordinary differential equations including the Hadamard finite part integral. In the time direction, the fractional derivative was approximated with the piecewise linear interpolation polynomial method, and the integer order time derivative was discretized by means of the 2nd order backward difference scheme. Through detailed proof, the error estimates with an accuracy of OO(τmin{1+α,1+β}) were obtained. The comparison between the numerical results and the theoretical solution shows the correctness of the theoretical analysis.