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 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
 A.A. El -Bary Journal of Applied Sciences , 2004, Abstract: This study considers an integral equation of Fredholm-Volterra type, where the Fredholm integral term is measured with respect to the position, while Volterra integral term is measured with respect to the time. Also, we obtained the solution of Fredholm-Volterra integral equation in series form.
 Le Matematiche , 2000, Abstract: A series method is used to separate the variables of position and time for the Fredholm-Volterra integral equation of the first kind and the solution of the system in L_2 [0,1] × C[0,T], 0 ≤ t ≤ T < ∞ is obtained, the Fredholm integral equation is discussed using Krein's method. The kernel is written in a Legendre polynomial form. Some important relations are also, established and discussed.
 Mathematics , 2009, Abstract: The purpose of this paper is to give a convergence analysis of the iterative scheme: \bee u_n^\dl=qu_{n-1}^\dl+(1-q)T_{a_n}^{-1}K^*f_\dl,\quad u_0^\dl=0,\eee where $T:=K^*K,\quad T_a:=T+aI,\quad q\in(0,1),\quad a_n:=\alpha_0q^n, \alpha_0>0,$ with finite-dimensional approximations of $T$ and $K^*$ for solving stably Fredholm integral equations of the first kind with noisy data.
 Proyecciones (Antofagasta) , 2012, DOI: 10.4067/S0716-09172012000100007 Abstract: in previous studies, the effectiveness of the half-sweep geometric mean (hsgm) iterative method has been shown in solving first and second kind linear fredholm integral equations using repeated trapezoidal (rt) discretization scheme. in this work, we investigate the efficiency of the hsgm method to solve dense linear system generated from the discretization of the second kind linear fredholm integral equations by using repeated simpson's ^ (rs1) scheme. the formulation and implementation ofthe proposed method are also presented. in addition, several numerical simulations and computational complexity analysis were also included to verify the efficiency of the proposed method.
 Proyecciones (Antofagasta) , 2012, Abstract: In previous studies, the effectiveness of the Half-Sweep Geometric Mean (HSGM) iterative method has been shown in solving first and second kind linear Fredholm integral equations using repeated trapezoidal (RT) discretization scheme. In this work, we investigate the efficiency of the HSGM method to solve dense linear system generated from the discretization of the second kind linear Fredholm integral equations by using repeated Simpson's ^ (RS1) scheme. The formulation and implementation ofthe proposed method are also presented. In addition, several numerical simulations and computational complexity analysis were also included to verify the efficiency of the proposed method.
 A. Shahsavaran American Journal of Computational Mathematics (AJCM) , 2011, DOI: 10.4236/ajcm.2011.12014 Abstract: In this work, we present a computational method for solving nonlinear Fredholm-Volterra integral equations of the second kind which is based on replacement of the unknown function by truncated series of well known Block-Pulse functions (BPfs) expansion. Error analysis is worked out that shows efficiency of the method. Finally, we also give some numerical examples.
 Journal of Applied Mathematics , 2012, DOI: 10.1155/2012/370894 Abstract: With the aid of fixed-point theorem (an equivalent version for the linear case) and biorthogonal systems in adequate Banach spaces, the problem of approximating the solution of a linear Fredholm-Volterra integro-differential equation is turned into a numerical algorithm, so that it can be solved numerically.
 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.
 shadan sadigh behzadi Communications in Numerical Analysis , 2012, DOI: 10.5899/2012/cna-00108 Abstract: In this present paper, we solve a two-dimensional nonlinear Volterra-Fredholm integro-differential equation by using the following powerful, efficient but simple methods: (i) Modified Adomian decomposition method (MADM), (ii) Variational iteration method (VIM), (iii) Homotopy analysis method (HAM) and (iv) Modified homotopy perturbation method (MHPM). The uniqueness of the solution and the convergence of the proposed methods are proved in detail. Numerical examples are studied to demonstrate the accuracy of the presented methods.
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