Usage of the homotopy analysis method for solving the nonlinear and linear integral equations of the second kind (original) (raw)
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In this paper, we present an efficient modification of the homotopy analysis method (HAM) that will facilitate the calculations. We then conduct a comparative study between the new modification and the homotopy analysis method. This modification of the homotopy analysis method is applied to nonlinear integral equations and mixed Volterra-Fredholm integral equations, which yields a series solution with accelerated convergence. Numerical illustrations are investigated to show the features of the technique. The modified method accelerates the rapid convergence of the series solution and reduces the size of work.
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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.
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.
Asian Journal of Fuzzy and Applied Mathematics
In this paper, new powerful modification of homotopy analysis technique (NMHAM) was submitted to create an approximate solution of nonhomogeneous nonlinear ordinary and partial differential equations. The NMHAM is a combination of the new technique of homotopy analysis method(NHAM) [4] and the new technique of homotopy analysis method(nHAM) [7].Three illustrative examples are employed to illustrate the accuracy and computational proficiency of this approach. The outcomes uncover that the NMHAM is more accurate than the NHAM and nHAM.
APPLICATION OF HOMOTOPY ANALYSIS METHOD FOR SOLVING NONLINEAR PROBLEMS
In this project we introduced an analytic approximation method for nonlinear problem in general, namely the homotopy analysis method. The homotopy analysis method (HAM) is an analytic approximation method for highly nonlinear problems, proposed by the Liao in 1992.Unlike perturbation techniques; the HAM is independent of any small/large physical parameters at all: one can always transfer a nonlinear problem into an infinite number of linear sub problems by means of the HAM. Secondly, different from all of other analytic techniques, the HAM provides us a convenient way to guarantee the convergence of solution series so that it is valid even if nonlinearity becomes rather strong. Besides, based on the homotopy in topology, it provides us extremely large freedom to choose equation type of linear sub-problems, base function of solution, initial guess and so on, so that complicated nonlinear ODEs and PDEs can often be solved in a simple way. In this project, the homotopy analysis method is employed to solve non linear problems; the results reveal that the proposed method is effective.
Applied Mathematics and Computation, 2006
In this paper, a homotopy perturbation method is proposed to solve non-singular integral equations. Comparisons are made between AdomianÕs decomposition method and the proposed method. It is shown, AdomianÕs decomposition method is a homotopy, only. Finally, by using homotopy perturbation method, a new iterative scheme, like AdomianÕs decomposition method, is proposed for solving the non-singular integral equations of the first kind. The results reveal that the proposed method is very effective and simple.