Research Article Convergent Homotopy Analysis Method for Solving Linear Systems (original) (raw)
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One of the efficient and powerful schemes to solve linear and nonlinear equations is homotopy analysis method (HAM). In this work, we obtain the approximate solution of a system of partial differential equations (PDEs) by means of HAM. For this purpose, we develop the concept of HAM for a system of PDEs as a matrix form. Then, we prove the convergence theorem and apply the proposed method to find the approximate solution of some systems of PDEs. Also, we show the region of convergence by plotting the H-surface.
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Ain Shams Eng J (Elsevier), 2012
in this paper, an analytical attitude is proposed for solving linear systems by Homotopy Analysis Method (HAM). On the basis of HAM we design new iterative methods. The convergence properties of the proposed method have been analyzed. Numerical examples show that our method is effective and simple for applications."
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.
BMC Research Notes, 2022
Objectives: This paper proposes three iterative methods of order three, six and seven respectively for solving nonlinear equations using the modified homotopy perturbation technique coupled with system of equations. This paper also discusses the analysis of convergence of the proposed iterative methods. Results: Several numerical examples are presented to illustrate and validation of the proposed methods. Implementation of the proposed methods in Maple is discussed with sample computations.