A numerical method for solving Volterra and Fredholm integral equations using homotopy analysis method (original) (raw)
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Numerically Solving Volterra and Fredholm Integral Equations
2012
Homotopy Analysis Method (HAM) and Homotopy Perturbation Method (HPM) is known to be two powerful tools for solving many functional equations such as ordinary and partial differential and integral equations. In this paper (HAM) is applied to solve linear Fredholm and Volterra first and second kind integral equations, the deformation equations are solved analytically by using MATLAB integration functions. Numerical techniques for solving deformation equation are also applied using interpolation methods and Gaussian integration. Auxiliary parameter h is also used experimentally to control convergence of partial sum of series solution. Several examples are tested by these methods and numerical results are compared with exact solution or existing numerical results to demonstrate the efficiency of the methods. These methods can be generalized to non-linear Volterra and Fredholm integral equation. The analytical and numerical results show the perfo rmance and reliability of presented method.
Numerically Solving Volterra and Fredholm Integral Equations 1
2013
Abstract: Homotopy Analysis Method (HAM) and Homotopy Perturbation Method (HPM) is known to be two powerful tools for solving many functional equations such as ordinary and partial differential and integral equations. In this paper (HAM) is applied to solve linear Fredholm and Volterra first and second kind integral equations, the deformation equations are solved analytically by using MATLAB integration functions. Numerical techniques for solving deformation equation are also applied using interpolation methods and Gaussian integration. Auxiliary parameter h is also used experimentally to control convergence of partial sum of series solution. Several examples are tested by these methods and numerical results are compared with exact solution or existing numerical results to demonstrate the efficiency of the methods. These methods can be generalized to non-linear Volterra and Fredholm integral equation. The analytical and numerical results show the performance and reliability of prese...
Applied Mathematics and Computation, 2007
Homotopy perturbation method is applied to the numerical solution for solving system of Fredholm integral equations. Comparison of the result obtained by the present method with that obtained by Taylor-series expansion method [K. Maleknejad, N. Aghazade, M. Rabbani, Numerical solution of second kind Fredholm integral equations system by using a Taylor-series expansion method, Appl. Math. Comput. 175 (2006) 1229-1234 reveals that the present method is very effective and convenient.
Applied Mathematics Letters, 2013
In this work the solution of the Volterra-Fredholm integral equations of the second kind is presented. The proposed method is based on the homotopy perturbation method, which consists in constructing the series whose sum is the solution of the problem considered. The problem of the convergence of the series constructed is formulated and a proof of the formulation is given in the work. Additionally, the estimation of the approximate solution errors obtained by taking the partial sums of the series is elaborated on.
Homotopy perturbation method for the mixed Volterra–Fredholm integral equations
Chaos Solitons & Fractals, 2009
This article presents a numerical method for solving nonlinear mixed Volterra-Fredholm integral equations. The method combined with the noise terms phenomena may provide the exact solution by using two iterations only. Two numerical illustrations are given to show the pertinent features of the technique. The results reveal that the proposed method is very effective and simple.
Note on new homotopy perturbation method for solving non-linear integral equations
2016
In this paper, exact solution for the second kind of nonlinear integral equations are presented. An application of modified new homotopy perturbation method is applied to solve the second kind of non-linear integral equations such that Voltrra and Fredholm integral equations. The results reveal that the modified new homotopy perturbation method is very effective and simple and gives the exact solution. Also the comparison of the results of applying this method with those of applying the homotopy perturbation method reveals the effectiveness and convenience of the new technique.
European Journal of Pure and Applied Mathematics
This paper presents the application of the Homotopy Analysis Method (HAM) for solving nonlinear system of Volterra integral equations used to obtain a reasonably approximate solution. The HAM contains the auxiliary parameter h which provides a simple way to adjust and control the convergence region of the solution series. The results show that the HAM is a very effective method as well. The results were compared with the solutions obtained by developing a homotopy analysis method using the genetic algorithm (HAM-GA), considering the residual error function as a fitness function of the genetic algorithm.