The collocation methods based on cubic B-spline, are developed to approximate solution of the second and first kind FREDHOLM INTEGRAL EQUATIONS. First we collocate the solution by B-spline and the Newton-Cotes formula is used to approximate INTEGRAL. Convergence analysis has been investigated and proved that the quadratur rule is fourth order convergent. The presented methods are tested to the problem, and the absolute error in the solution are compared with existing methods [1, 2, 7, 10] to verify the accuracy and convergent nature of proposed methods.

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In this study, a new application of Taylor expansion is considered to estimate the solution of Volterra-FREDHOLM INTEGRAL EQUATIONS (VFIEs) and systems of Volterra-FREDHOLM INTEGRAL EQUATIONS (SVFIEs). Our proposed method is based upon utilizing the nth-order Taylor polynomial of unknown function at an arbitrary point and employing integration method to convert VFIEs into a system of linear EQUATIONS with respect to unknown function and its derivatives. An approximate solution can be easily determined by solving the obtained system. Furthermore, this method leads always to the exact solution if the exact solution is a polynomial function of degree up to n. Also, an error analysis is given. In addition, some problems are provided to demonstrate the validity and applicability of the proposed method.

This paper suggests a simple analytical method for Volterra-FREDHOLM INTEGRAL EQUATIONS, the solution process is similar to that by variational-based analytical method, e. g., Ritz method, however, the method requires no establishment of the variational principle for the discussed problem, making the method much attractive for practical applications. The examples show the method is straightforward and effective, and the method can also be extended to other nonlinear problems.

In this paper Homotopy Perturbation Method (HPM) to solve fuzzy FREDHOLM INTEGRAL EQUATIONS is proposed. The method is discussed in details and it is illustrated by solving some numerical examples.

The method used in this research consists of a hybrid of the Block-Pulse functions and third-kind Chebyshev polynomials for solving systems of FREDHOLM INTEGRAL differential EQUATIONS. Through the use of an operational matrix representing the derivation, the problem is represented by a system of algebraic EQUATIONS. Some examples are provided to illustrate the simplicity and effectiveness of the utilized method. In addition, results of the presented method have been compared with those obtained from the Tau method and variational iteration method that reveal the proposed scheme to be more applicable