In this article we investigate the two-term Abel’s Integral equations. We will do this in two different ways and show that such equation is reducible to an integro-differential equation of Volterra type.

A new six order method developed for the approximation Fredholm Integral equation of the second kind. This method is based on the quintic spline functions (QSF). In our approach, we , rst formulate the Quintic polynomial spline then the solution of Integral equation approximated by this spline. But we need to develop the end conditions which can be associated with the quntic spline. To avoid the reduction accuracy, we formulate the end condition in such a way to obtain the band matrix and also to obtain the same order of accuracy. The convergence of the method is discussed by using matrix algebra. Finally, four test problems have been used for numerical illustration to demonstrate the practical ability of the new method.

In this paper, using orthogonally of Tchebychev polynomials, we present an orthonormal wavelet basis for L2[0, 1]. We use this basis for solving Neumann problems with Galerkin method. The property of this basis is that a variety of Integral operators is represented in this basis as sparse matrices, to high precision. Some examples are solved to illustrate the efficiency and accuracy of this method.

In this paper, we compute the approximate numerical solution for the Volterra-Fredholm Integral equation (VFIE) by using the shifted Jacobi collocation (SJC) method which depends on the operational matrices. Some properties of the shifted Jacobi polynomials are introduced. These properties allow us to transform the VolterraFredholm Integral equation into a system of algebraic equations in a nice form with the expansion coefficients of the solution. Also, the convergence and error analysis are studied extensively. Finally, some examples which verify the efficiency of the given method are supplied and compared with other methods.

This paper presents an efficient numerical method to solve two versions of the Duffing equation by the hybrid functions based on the combination of Block-pulse functions and Legendre polynomials. This method reduces the solution of the considered problem to the solution of a system of algebraic equations. Moreover, the convergence of the method is studied. Some examples are given to demonstrate the applicability and effectiveness of the proposed method. Also, the obtained results are compared with some other results.

In this paper, we propose a novel numerical method for solving Hallen’s Integral equation, based on the sinc collocation approximation. The key innovation of our approach lies in the incorporation of weight functions into the traditional sinc-expansion framework. By leveraging the properties of sinc collocation, we transform Hallen’s Integral equation into a system of algebraic equations, which can be solved efficiently.  Our method involves discretizing the singular kernel of Hallen’s Integral equation and then applying the sinc approximation. Additionally, we provide a detailed analysis of the convergence and error estimation of the proposed method. Numerical results are presented for three distinct values of $\lambda$ and $l$, as well as for three different weight functions: $w(t)=1+\sin(\pi t)$, $w(t)=1+\cos(\frac{\pi t}{2})$ and $w(t)=1+t$.