Purpose

This paper aims to present a new method for the approximate solution of two-dimensional nonlinear Volterra–Fredholm partial integro-differential equations with boundary conditions using two-dimensional Chebyshev wavelets.

Design/methodology/approach

For this purpose, an operational matrix of product and integration of the cross-product and differentiation are introduced that essentially of Chebyshev wavelets. The use of these operational matrices simplifies considerably the structure of the computation used for a set of the algebraic system has been obtained by using the collocation points and solved.

Findings

Theorem for convergence analysis and some illustrative examples of using the presented method to show the validity, efficiency, high accuracy and applicability of the proposed technique. Some figures are plotted to demonstrate the error analysis of the proposed scheme.

Originality/value

This paper uses operational matrices of two-dimensional Chebyshev wavelets and helps to obtain high accuracy of the method.

中文翻译：

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二维Chebyshev小波的运算矩阵及其在非线性偏积分微分方程中的应用。

目的

本文旨在提出一种使用二维切比雪夫小波近似求解带边界条件的二维非线性Volterra-Fredholm偏微分积分方程的新方法。

设计/方法/方法

为此目的，引入了乘积运算，叉积积分和微分积分，基本上是切比雪夫小波的矩阵。这些运算矩阵的使用大大简化了通过使用搭配点获得并求解的一组代数系统所使用的计算结构。

发现

收敛性分析定理和使用本文提出的方法的一些说明性例子说明了该技术的有效性，效率，高精度和适用性。绘制了一些数字以证明所提出方案的误差分析。

创意/价值

本文使用二维切比雪夫小波的运算矩阵，有助于获得该方法的高精度。