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 Applied Mathematics , 2012, DOI: 10.5923/j.am.20120201.03 Abstract: In this paper, an iterated method is presented to determine the numerical solution of linear Volterra integral equations of the second kind (VIEs2). This method initially uses the solution of the direct method to obtain the more accurate solution. The convergence and error analysis of this method are given. Finally, numerical examples illustrate efficiency and accuracy of the proposed method. Also, the numerical results of this method are compared with the results of direct method, collocation method and iterated collocation method.
 Esmail Babolian;Zahra Masouri;Saeed Hatamzadeh-Varmazyar PIER B , 2008, DOI: 10.2528/PIERB08050505 Abstract: A new and effective direct method to determine the numerical solution of specific nonlinear Volterra-Fredholm integral and integro-differential equations is proposed. The method is based on vector forms of block-pulse functions (BPFs). By using BPFs and its operational matrix of integration, an integral or integro-differential equation can be transformed to a nonlinear system of algebraic equations. Some numerical examples are provided to illustrate accuracy and computational efficiency of the method. Finally, the error evaluation of this method is presented. The benefits of this method are low cost of setting up the equations without applying any projection method such as Galerkin, collocation, . . . . Also, the nonlinear system of algebraic equations is sparse.
 Zahra Masouri Advanced Computational Techniques in Electromagnetics , 2012, DOI: 10.5899/2012/acte-00108 Abstract: This paper presents a numerical expansion-iterative method for solving linear Volterra and Fredholm integral equations of the second kind. The method is based on vector forms of block-pulse functions and their operational matrix. By using this approach, solving the second kind integral equation reduces to solve a recurrence relation. The approximate solution is most easily produced iteratively via the recurrence relation. Therefore, computing the numerical solution does not need to directly solve any linear system of algebraic equations and to use any matrix inversion. Moreover, this approach does not use any projection method such as collocation, Galerkin, etc., for setting up the recurrence relation. To show convergence and stability of the method, some computable error bounds are obtained, and some test problems are provided to illustrate its accuracy and computational efficiency.
 Nasser Aghazadeh and Amir Ahmad Khajehnasiri Mathematical Sciences , 2013, DOI: 10.1186/2251-7456-7-3 Abstract: First, the two-dimensional block-pulse operational matrix of integration and differentiation has been presented. Then, by using this matrices, the nonlinear two-dimensional Volterra integro-differential equation has been reduced to an algebraic system. Some numerical examples are presented to illustrate the effectiveness and accuracy of the method.
 Advances in Numerical Analysis , 2012, DOI: 10.1155/2012/868279 Abstract: An efficient hybrid method is developed to approximate the solution of the high-order nonlinear Volterra-Fredholm integro-differential equations. The properties of hybrid functions consisting of block-pulse functions and Lagrange interpolating polynomials are first presented. These properties are then used to reduce the solution of the nonlinear Volterra-Fredholm integro-differential equations to the solution of algebraic equations whose solution is much more easier than the original one. The validity and applicability of the proposed method are demonstrated through illustrative examples. The method is simple, easy to implement and yields very accurate results. 1. Introduction Integral and integrodifferential equations have many applications in various fields of science and engineering such as biological models, industrial mathematics, control theory of financial mathematics, economics, electrostatics, fluid dynamics, heat and mass transfer, oscillation theory, queuing theory, and so forth . It is well known that it is extremely difficult to analytically solve nonlinear integrodifferential equations. Indeed, few of these equations can be solved explicitly. So it is required to devise an efficient approximation scheme for solving these equations. So far, several numerical methods are developed. The solution of the first order integrodifferential equations has been obtained by the numerical integration methods such as Euler-Chebyshev  and Runge-Kutta methods . Moreover, a differential transform method for solving integrodifferential equations was introduced in . Shidfar et al.  applied the homotopy analysis method for solving the nonlinear Volterra and Fredholm integrodifferential equations. As a concrete example, we can express the mathematical model of cell-to-cell spread of HIV-1 in tissue cultures considered by Mittler et al. . Yalcinbas and Sezer  proposed an approximation scheme based on Taylor polynomials for solving the high-order linear Volterra-Fredholm integrodifferential equations of the following form: Maleknejad and Mahmoudi  developed a numerical method by using Taylor polynomials to solve the following type of nonlinear Volterra-Fredholm integrodifferential equations: Darania and Ivaz  suggested an efficient analytical and numerical procedure for solving the most general form of nonlinear Volterra-Fredholm integrodifferential equations under the mixed conditions where and are constants and . Moreover, , , , , , and , , are functions that have suitable derivatives on the interval . These kinds of equations can be
 Abdallah A. Badr Mathematical Problems in Engineering , 2010, DOI: 10.1155/2010/537909 Abstract: We consider a nonlinear Volterra-Fredholm integral equation (NVFIE) of the second kind. The Volterra kernel is time dependent, and the Fredholm kernel is position dependent. Existence and uniqueness of the solution to this equation, under certain conditions, are discussed. The block-by-block method is introduced to solve such equations numerically. Some numerical examples are given to illustrate our results. 1. Introduction Different methods are used to solve integral equations which are investigated from many physical applications such as the mixed problems in the theory of elasticity. Popov  applied the orthogonal polynomials method to solve the mixed problem in the mechanics of continuous media. Badr  applied Toeplitz matrix method to solve a NVFIE. Abdou et al.  discussed the solution of Harmmerstein-Volterra integral equation of the second kind. In , Haci obtained, numerically, the solution of a system of Harmmerstein integral equations in the space . The equivalence between Volterra integral equation with degenerate kernel and a linear system of differential equations is mentioned by Cochran . Although there are some works on Hermite-type collocation method for the second-kind VIEs with smooth kernels, not too many studies have dealt with weakly singular kernel. For example, Papatheodorou and Jesanis  used the collocation method and obtained the solution of Volterra integrodifferential equation with weakly singular kernels. More information about different analytical and numerical solutions of Volterra equations can be found in Davis , Linz , Volterra , and Wolkenfelt . In this paper, we consider the following NVFIE: The existence of a unique solution for the above equation, under certain conditions, is granted using fixed point theorem, where is the Fredholm kernel and is the Volterra kernel. is called the free term, and the unknown function, , is called the potential function in the applied mathematics, and it will be determined. Both two functions and are assumed in the same space. The parameter has many physical meanings. A numerical method is applied to this equation, and it is reduced it to a system of Volterra integral equations of the second kind. Finally, the block-by-block method is used to obtain the numerical solution of this system. Some examples are stated to illustrate the results. 2. Existence and Uniqueness of Solution To guarantee the existence and uniqueness of solution to (1.1), we write (1.1) in the integral operator form where Also, we assume the following conditions: (i) and satisfies, in
 Mathematical Sciences Quarterly Journal , 2010, Abstract: In this paper, the properties of the hybrid functions which consist of block-pulse functions plus Legendre polynomials are presented. Then, integral equations are converted into an algebraic system by hybrid of general block-pulse functions and the Legendre polynomials. In continue, approximate solutions of integral equations are derived, finally the numerical examples are included to demonstrate the validity and applicability of the algorithm.
 Mathematics , 2015, Abstract: Numerical solution of one-dimensional stochastic integral equations because of the randomness has its own problems, i.e. some of them no have analytically solution or finding their analytic solution is very difficult. This problem for two-dimensional equations is twofold. Thus, finding an efficient way to approximate solutions of these equations is an essential requirement. To begin this important issue in this paper, we will give an efficient method based on Haar wavelet to approximate a solution for the two-dimensional linear stochastic Volterra integral equation. We also give an example to demonstrate the accuracy of the method.
 Jean Marc Owo Mathematics , 2009, Abstract: In this paper, we study backward doubly stochastic integral equations of the Volterra type (BDSIEVs in short). Under uniform Lipschitz assumptions, we establish an existence and uniqueness result.
 American Journal of Computational and Applied Mathematics , 2012, DOI: 10.5923/j.ajcam.20120201.10 Abstract: In this paper, the piecewise constant Block-Pulse functions and their operational matrices of integration have directly been used to solve a two-dimensional Fredholm-Volterra integral equation of second kind. This method presents a computational technique through converting this integral equation into a system of linear equations which can be easily solved by the known methods. Also the error analysis of this method will be considered. The efficiency and accuracy of the proposed method are illustrated by some examples.
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