Abstract Numerical schemes for nonlinear weakly singular Volterra and Fredholm integral equations of the first kind are rarely investigated in the literature. In this paper, we present numerical solutions for these types of equations including Abel equations by hybrid block-pulse functions and Legendre polynomials. Hybrid functions give us the opportunity to attend a highly accurate solution by adjusting the orders of block-pulse functions and Legendre polynomials. The main idea of the scheme is based on using the precise forms of the known functions in the approximation procedure which yields the simplicity, reliability and high accuracy of the method. This simple scheme converts these types of equations into a linear system of equations. The focus of this paper is to investigate the convergence analysis and to show high convergence rate of the scheme. Numerical examples confirm the efficiency and superiority of the present approach in comparison with those already available in the literature.

中文翻译：

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一些非线性类阿贝尔积分方程使用混合展开的近似解

摘要 第一类非线性弱奇异Volterra 和Fredholm 积分方程的数值格式在文献中鲜有研究。在本文中，我们通过混合块脉冲函数和勒让德多项式给出了这些类型方程的数值解，包括阿贝尔方程。混合函数让我们有机会通过调整块脉冲函数和勒让德多项式的阶来获得高度准确的解决方案。该方案的主要思想是基于在近似过程中使用已知函数的精确形式，从而产生该方法的简单性、可靠性和高精度。这个简单的方案将这些类型的方程转换为线性方程组。本文的重点是研究收敛性分析并显示该方案的高收敛速度。与文献中已有的方法相比，数值例子证实了本方法的效率和优越性。