On the application of Liao’s method for solving linear systems (original) (raw)
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Research Article Convergent Homotopy Analysis Method for Solving Linear Systems
2016
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. By using homotopy analysis method (HAM), we introduce an iterative method for solving linear systems.This method (HAM) can be used to accelerate the convergence of the basic iterative methods. We also show that by applying HAM to a divergent iterative scheme, it is possible to construct a convergent homotopy-series solution when the iteration matrix G of the iterative scheme has particular properties such as being symmetric, having real eigenvalues. Numerical experiments are given to show the efficiency of the new method. 1.
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.
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.
On the Application of Homotopy Perturbation Method for Solving Systems of Linear Equations
International Scholarly Research Notices, 2014
The application of homotopy perturbation method (HPM) for solving systems of linear equations is further discussed and focused on a method for choosing an auxiliary matrix to improve the rate of convergence. Moreover, solving of convection-diffusion equations has been developed by HPM and the convergence properties of the proposed method have been analyzed in detail; the obtained results are compared with some other methods in the frame of HPM. Numerical experiment shows a good improvement on the convergence rate and the efficiency of this method.
An improved adaptation of homotopy analysis method
Mathematical Sciences, 2017
An improved adaptation of the well-known homotopy analysis method (HAM) is proposed to approximate the solutions of strongly nonlinear differential problems in terms of a rapidly convergent series. The proposed method involves simpler integrals and less computations than the standard HAM. The method is illustrated using different numerical examples. The comparative analysis confirms the applicability and efficiency of the proposed technique.
Homotopy analysis method: A new analytical technique for nonlinear problems
Communications in Nonlinear Science and Numerical Simulation, 1997
In this paper, the basic ideas of a new kind of analytical technique, namely the Homotopy Analysis Method (HAM), are briefly described. Different from perturbation techniques, the HAM does not depend on whether or not there exist small parameters in nonlinear equations under consideration. Therefore, it provides us with a powerful tool to analyse strongly nonlinear problems. A simple but typical example is used to illustrate the validity and the great potential of the HAM. Moreover, a pure mathematical theorem, namely the General Taylor Theorem, is given in appendix, which provides us with some rational knowledge for the validity of this new analytical technique.
HOMOTOPY ANALYSIS METHOD: A NEW ANALYTIC METHOD FOR NONLINEAR PROBLEMS
In this paper, the basic ideas of a new (malytic techniq,te, ntmwly the llomotopy Analysis:Method (HAM). are described. Diff'erent from perturl~ttioJt methods, the ralidit.r of the tlAM is huh'l)ende, t o, whether or not there exist sin,l/pammwters h~ considered notdinear equations. Therefore, it provides tls with a poweJ]'zd alttdytic tool Jbr strongly nonlhtear problems. A typical no,lhwar problem is used as an example to ver(])' the validity and the great potential of the HAM.
Newton-homotopy analysis method for nonlinear equations
Applied Mathematics and Computation, 2007
In this paper, we present an efficient numerical algorithm for solving nonlinear algebraic equations based on Newton-Raphson method and homotopy analysis method. Also, we compare homotopy analysis method with Adomian's decomposition method and homotopy perturbation method. Some numerical illustrations are given to show the efficiency of algorithm.
New iterative methods for solving nonlinear equation by using homotopy perturbation method
Applied Mathematics and Computation, 2012
In this paper, we suggest and analyze a new class of iterative methods for solving nonlinear equations by using the homotopy perturbation method. Convergence of their method is also considered. Here we also discuss the efficiency index and computational order of convergence of new methods. Several numerical examples are given to illustrate the efficiency and performance of these new methods. These new iterative methods may be viewed as an extension and generalization of the existing methods for solving nonlinear equations.
Numerically solving non-linear problems by the homotopy analysis method
Computational Mechanics, 1997
In this paper, the Homotopy Analysis Method (HAM) proposed by Liao (is greatly improved by introducing a nonzero variable " h. Based on the HAM, a new numerical approach for strongly non-linear problems is proposed and applied to solve, as an example, a non-linear heat transfer problem, i.e. microwave heating of an unit plate, so as to verify its validity and great potential. Our numerical experiments show that, by the proposed approach, iteration is not absolutely necessary for solving non-linear problems. This fact may deepen our understanding about numerical techniques for non-linear problems and widen our ®eld of vision. Moreover, the basic ideas proposed in this paper may afford us a great possibility to greatly improve our current numerical techniques.