Purpose

This paper aims to propose an efficient and convenient numerical algorithm for two-dimensional nonlinear Volterra-Fredholm integral equations and fractional integro-differential equations (of Hammerstein and mixed types).

Design/methodology/approach

The main idea of the presented algorithm is to combine Bernoulli polynomials approximation with Caputo fractional derivative and numerical integral transformation to reduce the studied two-dimensional nonlinear Volterra-Fredholm integral equations and fractional integro-differential equations to easily solved algebraic equations.

Findings

Without considering the integral operational matrix, this algorithm will adopt straightforward discrete data integral transformation, which can do good work to less computation and high precision. Besides, combining the convenient fractional differential operator of Bernoulli basis polynomials with the least-squares method, numerical solutions of the studied equations can be obtained quickly. Illustrative examples are given to show that the proposed technique has better precision than other numerical methods.

Originality/value

The proposed algorithm is efficient for the considered two-dimensional nonlinear Volterra-Fredholm integral equations and fractional integro-differential equations. As its convenience, the computation of numerical solutions is time-saving and more accurate.

中文翻译：

---

二维非线性 Volterra-Fredholm 积分方程和分数阶积分微分方程（Hammerstein 和混合类型）的数值算法

目的

本文旨在为二维非线性 Volterra-Fredholm 积分方程和分数阶积分-微分方程（Hammerstein 和混合类型）提出一种高效便捷的数值算法。

设计/方法/方法

该算法的主要思想是将伯努利多项式逼近与Caputo分数阶导数和数值积分变换相结合，将所研究的二维非线性Volterra-Fredholm积分方程和分数阶积分-微分方程简化为易于求解的代数方程。

发现

该算法不考虑积分运算矩阵，采用直接的离散数据积分变换，计算量小，精度高。此外，将方便的伯努利基多项式分数阶微分算子与最小二乘法相结合，可以快速得到所研究方程的数值解。给出了说明性的例子，以表明所提出的技术比其他数值方法具有更好的精度。

原创性/价值

所提出的算法对于所考虑的二维非线性Volterra-Fredholm积分方程和分数阶积分-微分方程是有效的。由于其方便，数值解的计算既省时又准确。