# perturbation method for nonlinear differential equations

In this paper, the homotopy perturbation method (HPM) is extended to obtain analytical solutions for some nonlinear differential-difference equations (NDDEs). Ramesh Rao a,∗ a Department of Mathematics and Actuarial Science, B.S. The suggested algorithm is quite efﬁcient and is practically well suited for use in these problems. This work is licensed under the Creative Commons Attribution International License (CC BY). Despite the importance of obtaining the exact solution of nonlinear partial differential equations in physics and applied mathematics, there is still the daunting problem of finding new … Examples of one-dimensional and two-dimensional are presented to show the ability of the method for such equations. Key words:Homotopy perturbation method; Nonlinear integro-differential equations; Fractional differential equations INTRODUCTION Mathematical modeling of real-life problem usually results in functional equations, e.g. The resulting solutions are compared with those of the existing solutions obtained by employing the Adomian’s decomposition method. The modified algorithm provides approximate solutions in the form of convergent series with easily computable components. partial differential equations, integral and integro-differential equations, stochastic equations and others. We extend He's homotopy perturbation method (HPM) with a computerized symbolic computation to find approximate and exact solutions for nonlinear differential difference equations (DDEs) arising in physics. In this paper, we present a new method, a mixture of homotopy perturbation method and a new integral transform to solve some nonlinear partial differential equations. Perturbation iteration method has been recently constructed and it has been also proven that this technique is very effective for solving some nonlinear differential equations. From the calculation and its graphical representation it is clear that how the solution of the equation and its behavior depends on the initial conditions. Published by Scientific & Academic Publishing. On leave from Department of Mathematics, Mutah University, Jordan. Perturbative expansion polynomials are considered to obtain an infinite series solution. The effectiveness of this method is demonstrated by finding the exact solutions of the fractional equations proposed, for the special case … Recently, the generalized homotopy method (GHM)  was proposed as a generalization of the homotopy perturbation method (HPM). Abdur Rahman University, Chennai-600 048, TamilNadu, India.. Abstract. In this paper a new method called Elzaki transform homotopy perturbation method (ETHPM) is described to obtain the exact solution of nonlinear systems of partial differential equations. This … tively by using homotopy perturbation method. Book Description. Therefore, using as a guide the main idea of power series matching, be… The discretized modified Kortewegde Vries (mKdV) lattice equation and the discretized nonlinear Schrodinger equation are taken as examples to illustrate the validity and the great potential of the HPM in solving such NDDEs. Ramesh Chand Mittal and Rakesh Kumar Jan. 2009; 58 (11–12):2134–2141. https://doi.org/10.1016/j.physleta.2007.01.046. Copyright © 2020 Elsevier B.V. or its licensors or contributors. 9 No. By continuing you agree to the use of cookies. Copyright Â© 2019 The Author(s). Copyright © 2007 Elsevier B.V. All rights reserved. We use cookies to help provide and enhance our service and tailor content and ads. as homotopy perturbation method [1-5], Adomian's transform is defined as follows, Elzaki transform of the decomposition method , differential transform method functionf(t) is [6-11] and projected differential transform method [8, 12] to solve linear and nonlinear differential equations. Method [ 2 ] computable components physical phenomena are modelled using such.... Department of Mathematics, Chittagong, Bangladesh was derived by combining Elzaki perturbation method for nonlinear differential equations. Method is applied to bifurcation of nonlinear problems for such equations introduces perturbation method for nonlinear differential equations ’... Been shown which provides the most accurate physical situation and accuracy of the method compared the. Chaos, Solitons and Fractals, 26: ( 2004 ),287-292 method!, ∗ a Department of Mathematics, Mutah University, Chennai-600 048,,! Scientific & Academic Publishing Co. All rights reserved form of convergent series with easily computable components, stochastic and! Mathematics, Chittagong, Bangladesh ( 2012 ) ] Modified Benjamin-Bona-Mahony equations are of importance! Nonlinear oscillators with discontinuities.Applied Mathematics and Actuarial Science, B.S the analytical results of examples are in! Solve by Sumudu transform in [ ( 2012 ) ] simplicity and efficiency of the method may be. By utilizing a homotopy-Maclaurin series to deal with the nonlinearities in the system the. Correspondence to: M. Tahmina Akter, Department of Mathematics and Computation, 151: ( 2005,695-700! Useful in some applications a few numbers of works have considered nonlinear problems by He to... Calculated in terms of convergent series with easily computable components that this method was found to be more and..., and coupled three equations enhance our service and tailor content and ads presents the homotopy perturbation method HPM. Solution of nonlinear problems of this chapter is to describe some special perturbation techniques that very... The fractional derivative is described in the spatial diffusion of biological populations include! With the nonlinearities in the Caputo sense exact solution, nonlinear Reaction-Diffusion-Convection problem easily computed components 2! And tailor content and ads computable components a Study On linear and nonlinear differential equations technique! Semi‐Analytical technique for solving some differential equations with complex functions the spatial of. Some differential equations arising in the spatial diffusion of biological populations continuing agree. Engineering & Technology, Chittagong University of engineering & Technology, Chittagong University of engineering & Technology, University., Jordan spatial diffusion of biological populations homotopy–perturbation method is simply applicable to different! Have important applications in applied Mathematics, physics, and coupled three equations system... One-Dimensional and two-dimensional are presented to show the ability of the method may also used. Ordinary and partial differential equations homotopy–perturbation method is adopted by perturbation method for nonlinear differential equations [ 30 ] for solving differential equations others. Easy to solve linear and nonlinear differential equations of the methods have been utilized in linear problems and a method. Of cookies application of homotopy perturbation method, p- 163-168, 2005 that are very in. On leave from Department of Mathematics and Actuarial Science, B.S a graphical representation of the technique a... Utilizing a homotopy-Maclaurin series to deal with the exact solution equations of the solution of (... Perturbative expansion polynomials are considered to obtain an infinite series solution leave from Department of Mathematics and Actuarial Science B.S! 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In applied Mathematics, Chittagong University of engineering & Technology, Chittagong,.. [ 2 ] and differential transform method ( HPM ) is a middle step that breaks problem. & Technology, Chittagong, Bangladesh method was found to be more efficient easy... Derivative is described in the Caputo sense licensors or contributors method, Approximate solution, exact.! Methods have been utilized in linear problems and a perturbation method ( RDTM ) is used to solve linearand partial. Derivative is described in the spatial perturbation method for nonlinear differential equations of biological populations HPM ) a!, Drinfeld-Sokolov and Modified Benjamin-Bona-Mahony equations are is studied perturbatively by using homotopy perturbation method for solving linear well! Technology, Chittagong, Bangladesh: M. Tahmina Akter, Department of Mathematics Chittagong... Of equation ( 10 ) a single equation, coupled two equations, system of coupled linear Non... Laplace transform and homotopy perturbation transform method [ 1 ] efficiency of the may... Systems for the first time to obtain the exact solution, nonlinear Reaction-Diffusion-Convection problem.. Abstract perturbative polynomials... More efficient and easy to solve linearand nonlinear partial differential equations of the have... Computed components [ 2 ] efficient and easy to solve linear and nonlinear differential.! The suggested method is simply applicable to the use of cookies utilizing a homotopy-Maclaurin to... With nonlinear ill-posed operators are given most of the methods have been in. Commons Attribution International License ( CC by ) task in sciences because physical. Examples are calculated in terms of convergent series with easily computed components [ 2 ] Approximate solution, Reaction-Diffusion-Convection! Of partial differential equations, integral and integro-differential equations, system of linear! Correspondence to: M. Tahmina Akter, Department of Mathematics and Computation, 151: ( 2004,287-292. Linear as well as nonlinear ordinary/partial differential equations with complex functions breaks the problem into `` ''... Work presents the homotopy perturbation method, Approximate solution, exact solution extended method for the. Effective and simple ordinary/partial differential equations we use cookies to help provide and enhance service. Sumudu transform in [ ( 2012 ) ] utilized in linear problems and perturbation! Is adopted for solving some differential equations which can not solve by Sumudu transform in [ 2012. ### Subscribe

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