利用Chebyshev小波函数求解二维分数阶Poisson方程数值解

Applying Chebyshev Wavelets to Numerically Solve Two-Dimensional Fractional Poisson Equations

以Chebyshev小波函数为基函数对一类二维分数阶Poisson方程进行数值求解.首先引入Caputo分数阶导数和Riemann-Liouville分数阶积分算子定义,进而又给出Chebyshev小波函数的定义和Chebyshev小波积分算子矩阵的表示形式,利用该算子矩阵将原问题的解函数表示成Chebyshev小波的级数展开形式,进而结合边界条件将原问题进行转化,通过离散未知变量求得系数矩阵.文章结尾通过两个具体的测试问题对算法的有效性和可行性进行了验证.

In this paper,an effective numerical technique is presented to solve a class of two-dimensional fractional Poisson equations with Dirichlet boundary conditions via Cheby-shev wavelet.Firstly,the definitions of Caputo fractional differential operator and Riemann-Liouville fractional integral operator are introduced.Secondly,the Chebyshev wavelets and their fractional integral operational matrices are given.Thirdly,using the operational ma-trices,the original differential system is transformed into some products of vectors combing the boundary conditions.Lastly,solving the discrete system,the numerical solutions can be obtained.In the end of the paper,two test problems are proposed to verify the effectiveness andfeasibilityof thealgorithm.