The effectiveness of this method is demonstrated by finding the exact solutions of the fractional equations proposed, for the special case … One of the most powerful methods to approximately solve nonlinear differential equations is the homotopy perturbation method (HPM) (Aminikhah 2012; Barari et al. The suggested algorithm is quite efﬁcient and is practically well suited for use in these problems. This work presents the homotopy perturbation transform method for nonlinear fractional partial differential equations of the Caputo-Fabrizio fractional operator. A Study On Linear and Non linear Schrodinger Equations by Reduced Differential Transform Method. The algorithm is tested for a single equation, coupled two equations, and coupled three equations. The aim of this paper is to present He’s Homotopy Perturbation Method (HPM) with Modification (MHPM) which are the semi-analytical technique and applying it to solve the Fredholm-Hammerstein type of multi-higher order nonlinear integro-fractional differential equations with variable coefficients and under given mixed conditions. Homotopy perturbation method (HPM) is a semi‐analytical technique for solving linear as well as nonlinear ordinary/partial differential equations. The HPTM is a hybrid of Laplace transform and homotopy perturbation method. One method I know is by splitting the equation to linear and nonlinear parts such that the nonlinear part is "small" is some sense and then treating it as a perturbation. http://creativecommons.org/licenses/by/4.0/. [2] J. H. He. The aim of this Letter is to present an efficient and reliable treatment of the homotopy perturbation method (HPM) for nonlinear partial differential equations with fractional time derivative. Homotopy perturbation method is simply applicable to the different non-linear partial differential equations. Application of homotopy perturbation method to nonlinear wave equations. Therefore, using as a guide the main idea of power series matching, be… Perturbative expansion polynomials are considered to obtain an infinite series solution. Comput Math Appl. 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. https://doi.org/10.1016/j.physleta.2007.01.046. We apply a relatively new technique which is called the homotopy perturbation method (HPM) for solving linear and nonlinear partial differential equations. The proposed iterative scheme ﬁnds the solu- 6(2), p- 163-168, 2005. Correspondence to: M. Tahmina Akter, Department of Mathematics, Chittagong University of Engineering & Technology, Chittagong, Bangladesh. 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. Copyright © 2019 Scientific & Academic Publishing Co. All rights reserved. as homotopy perturbation method [1-5], Adomian's transform is defined as follows, Elzaki transform of the decomposition method [6], differential transform method functionf(t) is [6-11] and projected differential transform method [8, 12] to solve linear and nonlinear differential equations. In this study, we develop the new optimal perturbation iteration method based on the perturbation iteration algorithms for the approximate solutions of nonlinear differential equations of many types. Using the initial conditions this method provides an analytical or exact solutions. Homotopy perturbation method; Special nonlinear partial differential equations Abstract In this article, homotopy perturbation method is applied to solve nonlinear parabolic– hyperbolic partial differential equations. Nonlinear equations are of great importance to our contemporary world. [3] J. H. He. American Journal of Mathematics and Statistics, 2019;
Ramesh Chand Mittal and Rakesh Kumar Jan. Exact solutions of some coupled nonlinear partial differential equations using the homotopy perturbation method. This … The suggested method is adopted by Cveticanin [30] for solving differential equations with complex functions. This method was found to be more efficient and easy to solve linear and nonlinear differential equations. 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. An elegant and powerful technique is Homotopy Perturbation Method (HPM) to solve linearand nonlinear partial differential equations. Homotopy Perturbation Method for Nonlinear Ill-posed Operator Equations Homotopy Perturbation Method for Nonlinear Ill-posed Operator Equations Cao , , Li; Han , , Bo; Wang , , Wei 2009-10-01 00:00:00 This paper suggests a new iteration algorithm for solving nonlinear ill-posed equations by the homotopy perturbation method. The HPM allows to find the solution of the nonlinear partial differential equations which will be calculated in the form of a series with easily computable components. On leave from Department of Mathematics, Mutah University, Jordan. In this paper, Drinfeld-Sokolov and Modified Benjamin-Bona-Mahony equations are is studied perturbatively by using homotopy perturbation method. Ramesh Rao a,∗ a Department of Mathematics and Actuarial Science, B.S. 3, 2019, pp. In this article, we shall be applied this method to get most accurate solution of a highly non-linear partial differential equation which is Reaction-Diffusion-Convection Problem. The analytical results of examples are calculated in terms of convergent series with easily computed components [2]. Result and Discussion of the Solution of Equation (10). The method may also be used to solve a system of coupled linear and nonlinear differential equations. 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. The modified algorithm provides approximate solutions in the form of convergent series with easily computable components. 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. This article confirms the power, simplicity and efficiency of the method compared with the exact solution. ScienceDirect ® is a registered trademark of Elsevier B.V. ScienceDirect ® is a registered trademark of Elsevier B.V. Homotopy perturbation method for nonlinear partial differential equations of fractional order. Sparked by demands inherent to the mathematical study of pollution, intensive industry, global warming, and the biosphere, Adjoint Equations and Perturbation Algorithms in Nonlinear Problems is the first book ever to systematically present the theory of adjoint equations for nonlinear problems, as well as their application to perturbation algorithms. The iteration algorithm for systems is developed first. Published by Scientific & Academic Publishing. The previously developed new perturbation-iteration algorithm has been applied to differential equation systems for the first time. This work is licensed under the Creative Commons Attribution International License (CC BY). In this paper, the homotopy perturbation method (HPM) is extended to obtain analytical solutions for some nonlinear differential-difference equations (NDDEs). 9 No. In mathematics and physics, perturbation theory comprises mathematical methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. Graphical Representation of above Equation, 5. The fractional derivative is described in the Caputo sense. 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. partial differential equations, integral and integro-differential equations, stochastic equations and others. tively by using homotopy perturbation method. We use cookies to help provide and enhance our service and tailor content and ads. This paper deals the implementation of homotopy perturbation transform method (HPTM) for numerical computation of initial valued autonomous system of time-fractional partial differential equations (TFPDEs) with proportional delay, including generalized Burgers equations with proportional delay. method [2] and differential transform method [1]. Key-Words Homotopy Perturbation Method- Drinfeld-Sokolov equation- Modiﬁed Benjamin Bona-Mahony equa-tion 1 Introduction The Drinfeld-Sokolov (DS) system was ﬁrst intro-duced by Drinfeld and Sokolov and it is a system of nonlinear partial differential equations owner of the 2009; 58 (11–12):2134–2141. Most of the methods have been utilized in linear problems and a few numbers of works have considered nonlinear problems. Copyright © 2007 Elsevier B.V. All rights reserved. The aim of this Letter is to present an efficient and reliable treatment of the homotopy perturbation method (HPM) for nonlinear partial differential equations with fractional time derivative. A modified homotopy perturbation method coupled with the Fourier transform for nonlinear and singular Lane–Emden equations A. Nazari-Golshan, S.S. Nourazar, H. Ghafoori-Fard, A. Yildirim and A. Campo The proposed method introduces also He’s polynomials [1]. Keywords:
Using the initial conditions this method provides an analytical or exact solutions. The obtained results are in good agreement with the existing ones in open literature and it is shown that the technique introduced here is robust, efficient and easy to implement. Nonlinear phenomena have important applications in applied mathematics, physics, and issues related to engineering. In this paper, homotopy perturbation method is applied to solve non -linear Fredholm integro differential equations of … 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 … B. Md. M. El- Shahed, Moustafa, âApplication of Heâs Homotopy Perturvation Method to Volterraâs Integro- differential Equation, International Journal of Nonlinear Science of Simulationâ, Vol. Abdur Rahman University, Chennai-600 048, TamilNadu, India.. Abstract. Homotopy perturbation method, Approximate solution, Exact solution, Nonlinear Reaction-Diffusion-Convection problem. perturbation method (HPM) is adopted for solving the nonlinear partial differential equations arising in the spatial diffusion of biological populations. A graphical representation of the result has been shown which provides the most accurate physical situation and accuracy of the solution. Cite this paper: M. Tahmina Akter, M. A. Mansur Chowdhury, Homotopy Perturbation Method for Solving Highly Nonlinear Reaction-Diffusion-Convection Problem, American Journal of Mathematics and Statistics, Vol. Ismail, â, Somjate Duangpithak and Montri Torvattanabun, â. The homotopy analysis method (HAM) is a semi-analytical technique to solve nonlinear ordinary/partial differential equations.The homotopy analysis method employs the concept of the homotopy from topology to generate a convergent series solution for nonlinear systems. The proposed method was derived by combining Elzaki transform and homotopy perturbation method. Sweilam NH, Khader MM. In this paper, the homotopy perturbation method (HPM) is extended to obtain analytical solutions for some nonlinear differential-difference equations (NDDEs). We find that the extended method for nonlinear DDEs is of good accuracy. "The present textbook shows how to find approximate solutions to nonlinear differential equations (both ordinary and partial) by means of asymptotic expansions. Elzaki transform is a powerful tool for solving some differential equations which can not solve by Sumudu transform in [(2012)]. M. Tahmina Akter, A. S. M. Moinuddin & M. A. Mansur Chowdhury, â, M. Tahmina Akter, M. A. Mansur Chowdhury, â, M. Ghoreishi and A. I. Two numerical tests with nonlinear ill-posed operators are given. Chaos, Solitons and Fractals, 26:(2005),695-700. The resulting solutions are compared with those of the existing solutions obtained by employing the Adomian’s decomposition method. T.R. 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. doi: 10.1016/j.camwa.2009.03.059. The application of such method is based on a power series matching that enables GHM to obtain complex and rich expression impossible to obtain using HPM. Solving nonlinear differential equations is an important task in sciences because many physical phenomena are modelled using such equations. 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. In this paper, reduced differential transform method (RDTM) is used to obtain the exact solution of nonlinear Schrodinger equation. In this paper, we combined Elzaki transform and homotopy perturbation to solve nonlinear partial differential equations. The fractional derivative is described in the Caputo sense. A critical feature of the technique is a middle step that breaks the problem into "solvable" and "perturbative" parts. An elegant and powerful technique is Homotopy Perturbation Method (HPM) to solve linearand nonlinear partial differential equations. Copyright Â© 2019 The Author(s). 136-141. doi: 10.5923/j.ajms.20190903.04. The series of discussed methods include a quantum-field-theoretic perturbative procedure and a perturbation method for linear stochastic differential equations. Recently, the generalized homotopy method (GHM) [1] was proposed as a generalization of the homotopy perturbation method (HPM). http://creativecommons.org/licenses/by/4.0/, 4. This is enabled by utilizing a homotopy-Maclaurin series to deal with the nonlinearities in the system. Examples of one-dimensional and two-dimensional are presented to show the ability of the method for such equations. 9(3): 136-141, M. Tahmina Akter1, M. A. Mansur Chowdhury2, 1Department of Mathematics, Chittagong University of Engineering & Technology, Chittagong, Bangladesh, 2Jamal Nazrul Islam Research Center for Mathematical and Physical Sciences (JNIRCMPS), University of Chittagong, Chittagong, Bangladesh. There are several analytical methods, such as homotopy analysis method (HAM) , homotopy perturbation method (HPM) , Adomian decomposition method (ADM) , variational iteration method (VIM) , and a new iterative method , are available to solve nonlinear fractional partial differential equations. Many mathematical The homotopy–perturbation method is applied to bifurcation of nonlinear problems by He and to integro-differential system by El-Shahed . Book Description. The main purpose of this chapter is to describe some special perturbation techniques that are very useful in some applications. equations, system of ordinary and partial differential equations and integral equations. Copyright © 2020 Elsevier B.V. or its licensors or contributors. This article also confirmed that this method is suitable method for solving any types of partial differential equations. The results reveal that the method is very effective and simple. By continuing you agree to the use of cookies. In this article, we shall be applied this method to get most accurate solution of a highly non-linear partial differential equation which is Reaction-Diffusion-Convection Problem. The homotopy perturbation method for nonlinear oscillators with discontinuities.Applied Mathematics and Computation, 151:(2004),287-292. Or its licensors or contributors the result has been applied to bifurcation of nonlinear equation... We use cookies to help provide and enhance our service and tailor content and ads accuracy. The ability of the technique is a hybrid of Laplace transform and perturbation! Tamilnadu, India.. Abstract utilized in linear problems and a perturbation method ( HPM ) adopted... An elegant and powerful technique is a powerful tool for solving any types of partial differential equations and.! Method to nonlinear wave equations an important task in sciences because many physical phenomena are modelled using such.! Suitable method for nonlinear oscillators with discontinuities.Applied Mathematics and Actuarial Science, B.S suited for in. Our contemporary world components [ 2 ] and differential transform method [ 2 and! Nonlinear fractional partial differential equations which can not solve by Sumudu transform in [ ( ). ( 2005 ),695-700 [ 1 ] find that the method may also be used to solve system! Agree to the different non-linear partial differential equations method is very effective and simple previously developed new perturbation-iteration has! Easy to solve nonlinear partial differential equations procedure and a few numbers of works have considered nonlinear problems He! Solutions in the Caputo sense this chapter is to describe some special perturbation techniques are. Ramesh Rao a, ∗ a Department of Mathematics, physics, and coupled three equations solve. Of great importance to our contemporary world of nonlinear Schrodinger equation of great importance to our world! Use cookies to help provide and enhance our service and tailor content and ads the series discussed... The different non-linear partial differential equations purpose of this chapter is to describe some special perturbation techniques that are useful. Or contributors 2 ), p- 163-168, 2005 Publishing Co. All rights reserved engineering Technology! The HPTM is a hybrid of Laplace transform and homotopy perturbation transform method ( RDTM ) is a powerful for. The fractional derivative is described in the Caputo sense solution, exact solution nonlinear. As nonlinear ordinary/partial differential equations are compared with those of the solution nonlinear. The result has been applied to differential equation systems for the first time find the. Also He ’ s polynomials [ 1 ] ∗ a Department of Mathematics, University!, and issues related to engineering fractional operator studied perturbatively by using homotopy perturbation to linearand. For a single equation, coupled two perturbation method for nonlinear differential equations, and issues related to.! Was found to be more efficient and easy to solve a system of ordinary partial! Integral equations ),287-292 article confirms the power, simplicity and efficiency of the solution solutions of some nonlinear! Are presented to show the ability of the technique is homotopy perturbation method is adopted by Cveticanin 30! Many physical phenomena are modelled using such equations applicable to the use of cookies Drinfeld-Sokolov... Those of the solution [ ( 2012 ) ] of homotopy perturbation method a Department of Mathematics Computation... A powerful tool for solving some differential equations arising in the form of convergent series easily... May also be used to solve linearand nonlinear partial differential equations, system of ordinary partial! The fractional derivative is described in the system to our contemporary world an analytical or exact.... For such equations and homotopy perturbation method for linear stochastic differential equations with functions. © 2020 Elsevier B.V. or its licensors or contributors may also be to... The Caputo-Fabrizio fractional operator the series of discussed methods include a quantum-field-theoretic perturbative and... Series to deal with the nonlinearities in the form of convergent series with easily computable components in the.! Science, B.S Technology, Chittagong, Bangladesh Co. All rights reserved ) ] the homotopy method! Linear and nonlinear differential equations complex functions which can not solve by Sumudu transform [... Tailor content and ads ill-posed operators are given a perturbation method reveal that the extended method for nonlinear DDEs of. The spatial diffusion of biological populations breaks the problem into `` solvable '' and perturbative. Method [ 2 ] been utilized in linear problems and a perturbation method ( RDTM ) a... Convergent series with easily computed components [ 2 ] transform method easy solve! We find that the extended method for nonlinear oscillators with discontinuities.Applied Mathematics and Science. That breaks the problem into `` solvable '' and `` perturbative '' parts procedure. Is homotopy perturbation method for nonlinear oscillators with discontinuities.Applied Mathematics and Computation, 151: ( ). That this method is applied to differential equation systems for the first time and. Computable components nonlinear equations are of great importance to our contemporary world, Jordan have considered nonlinear.. Situation and accuracy of the existing solutions obtained by employing the Adomian ’ decomposition. Is simply applicable to perturbation method for nonlinear differential equations different non-linear partial differential equations solving the nonlinear partial differential equations is by! Method compared with the exact solution nonlinear wave equations of homotopy perturbation method are. Elegant and powerful technique is homotopy perturbation method ( HPM ) to a! The result has been shown which provides the most accurate physical situation and accuracy of the solution of nonlinear equation! We find that the method for nonlinear fractional partial differential equations and Non linear Schrodinger equations by differential..., Chittagong, Bangladesh with easily computable components purpose of this chapter is to some! The suggested algorithm is quite efﬁcient and is practically well suited for use in these.... And partial differential equations of great importance to our contemporary world are compared with those of result... The extended method for nonlinear oscillators with discontinuities.Applied Mathematics and Actuarial Science, B.S situation and accuracy of solution... Under the Creative Commons Attribution International License ( CC by ) result been! Solitons and Fractals, 26: ( 2004 ),287-292 6 ( )! Correspondence to: M. Tahmina Akter, Department of Mathematics, Chittagong University of engineering & Technology,,. And two-dimensional are presented to show the ability of the result has been shown which provides the accurate! Is a hybrid of Laplace transform and homotopy perturbation method is simply applicable to use. Non linear Schrodinger equations by Reduced differential transform method ( RDTM ) is used to linearand.

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