Approximate analytical solutions of systems of PDEs by homotopy analysis method (original) (raw)
Journal of Applied Mathematics, 2013
A modifiedq-homotopy analysis method (mq-HAM) was proposed for solvingnth-order nonlinear differential equations. This method improves the convergence of the series solution in thenHAM which was proposed in (see Hassan and El-Tawil 2011, 2012). The proposed method provides an approximate solution by rewriting thenth-order nonlinear differential equation in the form ofnfirst-order differential equations. The solution of thesendifferential equations is obtained as a power series solution. This scheme is tested on two nonlinear exactly solvable differential equations. The results demonstrate the reliability and efficiency of the algorithm developed.
Homotopy Analysis Method for Solving Non-linear Various Problem of Partial Differential Equations
2014
In this paper, solve several important equations such as korteweg-devries (kdv) problem, Boussinesq equation of non-homogeneous problem and non-homogeneous system Hirota-Satsuma problem of partial differential equation by Homotopy analysis method (HAM). Studied comparison exact solution with numerical results , this method have shown that is very effective and convenient and gives numerical solutions in the form of convergent series with easily computable components for solving non-linear various problem of partial differential equation .
Journal of Linear and Topological Algebra, 2015
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.
Homotopy Analysis Method for Solving System of Non-Linear Partial Differential Equations
International Journal of Emerging Multidisciplinaries: Mathematics
This paper applies Homotopy Analysis Method (HAM) to obtain analytical solutions of system of non-linear partial differential equations. Numerical results clearly reflect complete compatibility of the proposed algorithm and discussed problems. Moreover, the validity of the present solution and suggested scheme is presented and the limiting case of presented findings is in excellent agreement with the available literature. The computed solution of the physical variables against the influential parameters is presented through graphs. Several examples are presented to show the efficiency and simplicity of the method.
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.
The exact solutions of nonlinear problems by Homotopy Analysis Method (HAM
The present paper presents the comparison of analytical techniques. We establish the existence of the phenomena of the noise terms in the perturbation series solution and find the exact solution of the nonlinear problems. If the noise terms exist, the Homotopy Analysis method gives the same series solution as in Adomian Decomposition Method as well as homotopy Perturbation Method (Wahab et al, 2015) and we get the exact solution using the initial guess in Homotopy Analysis Method using the results obtained by Adomian Decomposition Method.
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.
Communications in Nonlinear Science and Numerical Simulation, 2009
In this paper, the homotopy analysis method (HAM) proposed by Liao in 1992 and the homotopy perturbation method (HPM) proposed by He in 1998 are compared through an evolution equation used as the second example in a recent paper by . It is found that the HPM is a special case of the HAM when = −1. However, the HPM solution is divergent for all x and t except t = 0. It is also found that the solution given by the variational iteration method (VIM) is divergent too. On the other hand, using the HAM, one obtains convergent series solutions which agree well with the exact solution. This example illustrates that it is very important to investigate the convergence of approximation series. Otherwise, one might get useless results.
IOSR Journal of Mathematics, 2013
This research paper deals with the systems of partial differential equations by using the New Variational Homotopy Perturbation Method. The New Method does not require discritization, linearization or any restrictive assumption of any form in providing analytical or approximate solutions to linear and nonlinear equation. Theses virtues make it to be reliable and its efficiency is demonstrated with numerical e x ample s.
The Combined Laplace-homotopy Analysis Method for Partial Differential Equations
Journal of Mathematics and Computer Science
In this paper, the Laplace transform homotopy analysis method (LHAM) is employed to obtain approximate analytical solutions of the linear and nonlinear differential equations. This method is a combined form of the Laplace transform method and the homotopy analysis method. The proposed scheme finds the solutions without any discretization or restrictive assumptions and is free from round-off errors and therefore, reduces the numerical computations to a great extent. Some illustrative examples are presented and the numerical results show that the solutions of the LHAM are in good agreement with those obtained by exact solution.
Solution of coupled system of nonlinear differential equations using homotopy analysis method
Nonlinear Dynamics, 2008
In this article, the homotopy analysis method has been applied to solve a coupled nonlinear diffusion-reaction equations. The validity of this method has been successful by applying it for these nonlinear equations. The results obtained by this method have a good agreement with one obtained by other methods. This work illustrates the validity of the homotopy analysis method for the nonlinear differential equations. The basic ideas of this approach can be widely employed to solve other strongly nonlinear problems. Keywords Homotopy analysis method • Numerical solution • Coupled system of nonlinear equations 1 Introduction Nonlinear evaluation equations arise in many areas of science and technology especially in mechanics, solid state physics, plasma physics, and chemical physics. There has been a large amount of literature regarding
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.
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 and Pade Methods for Solving Two Nonlinear Equations
Journal of Mathematical Extension, 2011
In this paper, we are giving analytic approximate so- lutions to nonlinear PDEs using the Homotopy Analysis Method (HAM) and Homotopy Pade Method(HPadeM). The HAM con- tains the auxiliary parameter h, which provides us with a simple way to adjust and control the convergence regions of solution se- ries. It is illustrated that HPadeM accelerates the convergence of the related series. The results reveal these methods are remarkably eective. AMS Subject Classification: 65M99; 65N99. Keywords and Phrases: Drinfeld-Sokolov system; Drinfeld- Sokolov-Wilson equation; homotopy analysis method; homotopy pade method.
A Study of General First-order Partial Differential Equations Using Homotopy Perturbation Method
International Journal of Mathematics Trends and Technology
In this work, we have studied a general class of linear first-order partial differential equations which is used as mathematical models in many physically significant fields and applied science. The homotopy perturbation method (HPM) has been used for solving generalized linear first-order partial differential equation. Also, we have tested the HPM on the solving of different implementations which show the efficiency and accuracy of the method. The approximated solutions are agree well with analytical solutions for the tested problems Moreover, the approximated solutions proved that the proposed method to be efficient and high accurate.
Study of convergence of homotopy perturbation method for systems of partial differential equations
Computers & Mathematics with Applications, 2009
The aim of this paper is convergence study of homotopy perturbation method for systems of nonlinear partial differential equations. The sufficient condition for convergence of the method is addressed. Since mathematical modeling of numerous scientific and engineering experiments lead to Brusselator and Burgers' system of equations, it is worth trying new methods to solve these systems. We construct a new efficient recurrent relation to solve nonlinear Burgers' and Brusselator systems of equations. Comparison of the results obtained by homotopy perturbation method with those of Adomian's decomposition method and dual-reciprocity boundary element method leads to significant consequences. Two standard problems are used to validate the method.