Numerical solutions of nonlinear system of two-dimensional integral equations have been rarely investigated in the literature. In this study, we suggest a numerically practical algorithm to approximate the solutions of nonlinear system of two-dimensional Volterra–Fredholm and Volterra integral equations. This scheme is based on two-dimensional Legendre wavelet to reduce these nonlinear systems of integral equations to a system of nonlinear algebraic equations. The main characteristic of this approach is high accuracy and computational efficiency of performing which are the consequences of Legendre wavelet properties. The main benefit of this basic function is their ability to detect singularities and their efficiency in dealing with non-sufficiently smooth function in comparison with Legendre polynomials and they minimize the error. The convergence analysis and error bound of the proposed Legendre wavelet method is investigated. Numerical examples confirm that the Legendre wavelet collocation method is accurate, reliable for solving nonlinear system of two-dimensional integral equations.

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Legendre小波搭配方法对二维积分方程非线性系统解的影响

二维积分方程非线性系统的数值解在文献中鲜有研究。在这项研究中，我们提出了一种数值实用的算法来逼近二维 Volterra-Fredholm 和 Volterra 积分方程的非线性系统的解。该方案基于二维勒让德小波将这些非线性积分方程组化简为非线性代数方程组。这种方法的主要特点是执行的高精度和计算效率，这是勒让德小波特性的结果。与勒让德多项式相比，这个基本函数的主要好处是它们检测奇异点的能力和处理非充分平滑函数的效率，并且它们最大限度地减少了误差。研究了所提出的勒让德小波方法的收敛性分析和误差界限。数值算例证实了Legendre小波搭配法对于求解二维积分方程的非线性系统是准确、可靠的。