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In this paper, we propose a novel approach for solving NONLINEAR interval Volterra INTEGRAL EQUATIONS (NIVIEs) based on the modifying Laplace decomposition method. We find the exact solutions of NIVIEs with less computation as compared with standard Laplace decomposition method, even there is no noise in the original problem. Finally, two illustrative examples have been solved to show the efficiency of the proposed method.

Using the mean-value theorem for INTEGRALs we tried to solved the NONLINEAR INTEGRAL EQUATIONS of Hammerstein type. The mean approach is to obtain an initial guess with unknown coefficients for unknown function y(x). The procedure of this method is so fast and don’t need high cpu and complicated programming. The advantages of this method is that we can applied for those INTEGRAL EQUATIONS which have not the unique solution too.

In this paper, a numerical method for solving NONLINEAR fractional INTEGRAL EQUATIONS (NFIE) is introduced. This method is based on the new basis functions (NFs) introduced in [M. Paripour and et al., Numerical solution of NONLINEAR Volterra Fredholm INTEGRAL EQUATIONS by using new basis functions, Communications in Numerical Analysis, (2013)]. Since the conventional operational matrices for fractional kernels are singular, the de nition of these matrices is modi ed. In order to increase the accuracy of approximating INTEGRALs, the operational matrices are exactly calculated and parametrically presented. Then, the solution procedure is proposed and applied on NFIE. Furthermore, the error analysis is performed and rate of convergence is obtained. In addition, various numerical examples are provided for a wide range of fractional orders and NONLINEARity of INTEGRAL EQUATIONS. Comparison of the results with the exact solutions and those reported in previous studies indicate the capability, salient accuracy, and superiority of the proposed method over similar ones.

A numerical method to solve NONLINEAR quadratic INTEGRAL EQUATIONS (QIE) is presented in this work. The method is based upon modification of hat functions (MHFs) and their operational matrices. By using this approach and the collocation points, solving the NONLINEAR QIE reduces to solve a NONLINEAR system of algebraic EQUATIONS. The proposed method does not need any integration for obtaining the constant coefficients. Hence, it can be applied in a simple and fast technique. Convergence analysis and associated theorems are considered. Some numerical examples illustrate the accuracy and computational efficiency of the proposed method.

In this paper, we provide a quadrature-based iterative approach to solve three-dimensional NONLINEAR fuzzy Volterra INTEGRAL EQUATIONS of the second kind. The error estimation as well as convergence analysis of the proposed method are also provided. Finally, numerical experiments validate the theoretical findings. The suggested method's advantages include   precision, accuracy and ease of use.

In this work, a new simple direct method to solve NONLINEAR Fredholm-Volterra INTEGRAL EQUATIONS is presented. By using Block-pulse (BP) functions, their operational matrices and Taylor expansion a NONLINEAR Fredholm-Volterra INTEGRAL equation converts to a NONLINEAR system. Some numerical examples illustrate accuracy and reliability of our solutions.

In this paper, we have introduced a computational method for a class of two-dimensional NONLINEAR Volterra INTEGRAL EQUATIONS, based on the expansion of the solution as a series of Haar functions. To achieve this aim it is necessary to define the INTEGRAL operator. The Banach fixed point theorem guarantees that under certain assumptions this operator has a unique fixed point, we have introduced an orthogonal projection and by interpolation property, we have achieved an operational matrix of integration. Also, by using the Banach fixed point theorem, we get an upper bound for the error of our method. Since our examples in this article are selected from different references, so should be the numerical results obtained here can be compared with other numerical methods.