非线性分数阶微分方程数值解的三尺度第3类Chebyshev小波配点法

基于三尺度第3类Chebyshev小波，提出了一类非线性分数阶微分方程数值解的一个小波配点法。首先，构造了三尺度第3类Chebyshev小波函数，证明了该小波函数的标准正交性，并给出了小波函数展开的L2范数意义下的一致收敛性分析和误差估计。其次，基于平移第3类Chebyshev多项式，借助Laplace变换推导出了三尺度第3类Chebyshev小波函数在Riemann-Liouville分数阶意义下的积分公式。最后，结合Picard迭代，利用三尺度第3类Chebyshev小波配点法，将非线性分数阶微分方程的初值问题及边值问题离散为代数方程组求解。数值算例说明了该方法的有效性和高精度性。

THE THREE-SCALE THIRD KIND OF CHEBYSHEV WAVELET COLLOCATION METHOD FOR NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS

Based on the three-scale third kind of Chebyshev wavelet, a wavelet collocation method for numerical solutions of a class of nonlinear fractional differential equations was proposed. Firstly, the three-scale third kind of Chebyshev wavelet functions were constructed, the orthogonality of the wavelet functions was proved, and the uniform convergence and error estimation in the sense of L2 norm of wavelet functions expansion were given. Secondly, based on the translational Chebyshev polynomials of the third kind, the integral formula of the three-scale third kind of Chebyshev wavelet in the sense of Riemann-Liouville fractional order was derived by means of Laplace transform. Finally, combined with Picard iteration, the initial value problem and boundary value problem of nonlinear fractional differential equation were discretized into algebraic equations by using the three-scale third kind of Chebyshev wavelet collocation method. Numerical examples showed the effectiveness and high accuracy of the method.