分数阶偏微分方程的小波算子矩阵解法

Solving Fractional Partial Differential Equations by Using the Wavelet Operational Matrix Method

推导并利用第二类Chebyshev小波的分数阶积分算子矩阵,给出了求解一类分数阶偏方程的数值方法,并证明了二元函数第二类Chebyshev小波展式的收敛性。研究结果表明,基于第二类Chebyshev小波算子矩阵的方法可将分数阶阶偏微分方程转化成Sylvester方程求解,减少方程的计算量。数值算例表明,随着参数m’的增大,数值解与精确解可以很好地吻合,证明了基于第二类Chebyshev小波算子矩阵方法数值求解分数阶偏微分方程的有效性和精确性。

In this paper, the second Chebyshev wavelet operational matrix of fractional integration is derived and used to solve a kind of fractional differential equations. Then the convergence of the two-dimensional second Chebyshev wavelet is proved. The initial equations are transformed into a Sylvester equation based on the proposed the second Chebyshev wavelet operational matrix method, which reduced the calculation time. Numerical examples are included to demonstrate that the numerical solutions are in very good agreement with exact solution when the value of m＇ is increasing. Therefore, the proposed second Chebyshev wavelet operational matrix method is accurate and effective.