The main aim of this paper is to use two-dimensional Bernoulli wavelet for obtaining the numerical solution of a nonlinear two-dimensional fractional partial Volterra integral equation. To do this, first, we construct the operational matrix of fractional integration as well as derivative of two-dimensional Bernoulli wavelet. Then, by applying the operational matrices and collocation method, we reduce the considered problem to a system of algebraic equations. Moreover, by preparing some theorems, we investigate the convergence analysis of the method. Finally, to show the accuracy and the applicability of the proposed method, we give some numerical examples.

中文翻译：

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用伯努利小波求解分数维二维非线性部分Volterra积分方程

本文的主要目的是使用二维伯努利小波来获得非线性二维分数阶Volterra积分方程的数值解。为此，首先，我们构造分数积分的运算矩阵以及二维Bernoulli小波的导数。然后，通过应用运算矩阵和搭配方法，我们将考虑的问题简化为一个代数方程组。此外，通过准备一些定理，我们研究了该方法的收敛性分析。最后，为了说明所提方法的准确性和适用性，我们给出了一些数值例子。