This paper is concerned with the study of wavelet approximation scheme based on Legendre and Chebyshev wavelets for finding the approximate solutions of distributed order linear differential equations. The operational matrix for distributed order fractional differential operator is derived for Legendre and Chebyshev wavelets basis. Furthermore, the obtained operational matrix along with Gauss Legendre quadrature formula and standard Tau method are utilized to reduce the distributed order linear differential equations into the system of linear algebraic equations. For the better understanding of the method, numerical algorithms are also provided for the considered problems. In order to verify the desired accuracy of the proposed method, five test examples are included and numerical experiments confirm the theoretical results and illustrate applicability and efficiency of the proposed method. Also, the convergence analysis, error bounds, error estimate and numerical stability of presented method via Legendre and Chebyshev wavelets are investigated. Moreover, comparison of the numerical results obtained by the proposed approach is provided with the results of existing method.

本文针对基于Legendre和Chebyshev小波的小波逼近方案的研究，以寻找分布式线性微分方程的近似解。以Legendre和Chebyshev小波为基础，推导了分布式阶数分数阶微分算子的运算矩阵。此外，利用所获得的运算矩阵以及高斯勒让德正交公式和标准Tau方法，将分布阶线性微分方程简化为线性代数方程组。为了更好地理解该方法，还为所考虑的问题提供了数值算法。为了验证所提出方法的所需准确性，包括五个测试示例，数值实验证实了理论结果，并说明了该方法的适用性和有效性。此外，还研究了通过Legendre和Chebyshev小波的收敛性分析，误差界，误差估计和数值稳定性。此外，将所提出的方法获得的数值结果与现有方法的结果进行了比较。