分数阶微分方程的二维三尺度第3类Chebyshev小波法

Two-dimensional three-scale Chebyshev wavelets of the third kind for fractional differential equations

为了求解一类非线性分数阶微分方程,基于二维三尺度第3类Chebyshev小波,提出了的一个数值解法。首先,构造了标准正交的三尺度第3类Chebyshev小波,通过叉乘,得到了标准正交的二维三尺度第3类Chebyshev小波。其次,基于平移的第3类Chebyshev多项式,借助Laplace变换,推导出了三尺度第3类Chebyshev小波的Riemann-Liouville分数阶积分公式,并给出了二维三尺度第3类Chebyshev小波展开在L2范数意义下的一致收敛性分析和误差估计。最后,利用小波积分公式,结合Picard迭代和有效的配置法,将非线性分数阶微分方程离散为代数方程组问题求解。数值算例说明了该方法的有效性和高精度性。

Based on the two-dimensional three-scale Chebyshev wavelets of the third kind,a numerical method for solving a class of nonlinear fractional differential equations was proposed.Firstly,the orthonormal three-scale Chebyshev wavelets of the third kind were constructed,and two-dimensional three-scale Chebyshev wavelets of the third kind were obtained by cross multiplication of the one-dimensional ones.Secondly,based on the shifted Chebyshev polynomials of the third kind,Riemann Liouville fractional integral formulas of the present wavelets were derived by means of Laplace transform.The uniform convergence analysis of the expansion of the two-dimensional three-scale Chebyshev wavelets of the third kind and the error estimation in the sense of L 2-norm were given.Finally,by the integral formulas and Picard iteration,the nonlinear fractional differential equation was discretized into a system of algebraic equations by using the two-dimensional three-scale Chebyshev wavelets of the third kind and the effective collocation method.Numerical examples show the effectiveness and high accuracy of the method.