Document Type : Research Paper

**Author**

Department of mathematics, Hashtgerd branch, Islamic Azad University, Hashtgerd, Iran.

**Abstract**

In the current study, a fast, accurate, and reliable numerical scheme for approximating second-kind nonlinear Fredholm, Volterra, and Fredholm-Volterra integral equations with a weakly singular kernel and invertible nonlinearity is presented. The computational approach is based upon function, especially the hybrid one. Hybrid functions give us the opportunity to achieve an appropriate solution by adjusting a suitable order for polynomials’ degrees and block-pulse functions. The basic idea of this method is based on using the invertibility of the nonlinear function as a benefit to reduce the total error and simplify the procedure. The scheme reduces these types of equations to nonlinear systems of algebraic equations. Convergence analysis of the method under the infinity norm is wellstudied. Numerical results indicate the superiority of the present method compared with another existing method in the literature

**Keywords**

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January 2023

Pages 183-196

**Receive Date:**28 August 2021**Revise Date:**13 February 2022**Accept Date:**30 March 2022**First Publish Date:**31 March 2022