Legendre小波求分数阶微分方程的数值解

Numerical Solution of Fractional Differential Equations by Using Legendre Wavelets

整数阶常微分方程的数值解法已有比较完善的理论,而对于分数阶微分方程数值方法的理论研究相对较少,由此考虑用Legendre小波逼近求线性分数阶微分方程数值解.首先描述了分数阶导数、积分和Legendre小波的性质,然后利用这些性质把分数阶微分方程转化为Volterra积分方程.考虑采用Legendre小波求数值解的线性分数阶微分方程:Dαy(x)+λy(x)=f(x),0<α<1其中:λ是常数,f(x)∈L2(R),在区间0≤x≤1内.最后举例论证方法的有效性.

Numerical method of ordinary differential equation has had quite perfect theories. Theoretical studies of the numerical method of fractional order differential equation are very little. A numerical method based upon Legendre wavelet approximations for solving linear fractional order differential equations is presented. The properties of fractional integral and Legendre wavelet are presented, utilized to reduce the fractional order differential equations to the solution of Volterra equations. Legendre wavelets are used for solving linear fractional differential equations of the form： D^αy（x）＋λy（x）=f（x）,0〈α〈1 where λ is constant, f（x） is in L^2 （R） on the interval 0≤x≤1. Finally, illustrative examples are given to clarify the validity of the method.