Comparison between homotopy perturbation method and optimal homotopy asymptotic method for the soliton solutions of Boussinesq–Burger equations

Gupta, A. K.; Ray, S. Saha

doi:10.1016/j.compfluid.2014.07.008

Cited by 66 publications

(43 citation statements)

References 13 publications

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“…It is very difficult to solve nonlinear problems and in general it is often more difficult to get an analytic approximation than a numerical one for a given nonlinear problem. To overcoming the shortcomings, many new analytical techniques have been successfully developed by diverse groups of mathematicians and physicists, such as, Perturbation Method [1], Homotopy Perturbation Method [2], Modified Homotopy Perturbation Method [3,4], Rational Homotopy Perturbation Method [5], He's Homotopy Perturbation Method [6], Modified He's homotopy Perturbation Method [7], Asymptotic Method [8][9][10][11], Optimal Iteration Perturbation Method [12], Generalization of Modified Differential Transforms Method [13][14][15][16], and so on. Several other authors used many powerful analytical methods in the field of approximate solutions especially for strongly nonlinear oscillators like Max-Min Approach Method [17,18], Algebraic Method [19], Parameter Expansion Method and Variational Iteration Method [20][21][22], Amplitude Frequency Formulation Method [23], Energy Balance Method [24,25], He's Energy Balance Method [26,27], Rational Energy Balance Method [28], Rational Harmonic Balance Method [29], Residue Harmonic Balance Method [30][31][32][33], Newton-harmonic Balancing Approach [34], and so on for solving NDEs.…”

Section: Introductionmentioning

confidence: 99%

A new reliable analytical solution for strongly nonlinear oscillator with cubic and harmonic restoring force

Hosen

Chowdhury

2015

Results in Physics

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a b s t r a c tIn the present paper, a complicated strongly nonlinear oscillator with cubic and harmonic restoring force, has been analysed and solved completely by harmonic balance method (HBM). Investigating analytically such kinds of oscillator is very difficult task and cumbersome. In this study, the offered technique gives desired results and to avoid numerical complexity. An excellent agreement was found between approximate and numerical solutions, which prove that HBM is very efficient and produces high accuracy results. It is remarkably important that, second-order approximate results are almost same with exact solutions. The advantage of this method is its simple procedure and applicable for many other oscillatory problems arising in science and engineering. IntroductionNonlinear oscillations are important fact in physical science, mechanical structures and other engineering problems. Naturally, all differential equations involving engineering and physical phenomena are nonlinear. The methods of solutions of linear differential equations are comparatively easy and well established. On the contrary, the techniques of solutions of nonlinear differential equations (NDEs) are less available and have no exact solution and, in general, linear approximations are frequently used. Nowadays, NDEs have been the subject of all-embracing studies in various branches of nonlinear science and engineering. A special class of analytical solutions named strongly nonlinear oscillator with cubic and harmonic restoring force has a lot of importance, because, most of the phenomena that arise in mathematical physics and engineering fields can be described by NDEs. Therefore, investigating strongly nonlinear oscillator with cubic and harmonic restoring force solutions is becoming increasingly attractive in nonlinear sciences. Moreover, obtaining exact solutions for nonlinear oscillatory problems has many difficulties. It is very difficult to solve nonlinear problems and in general it is often more difficult to get an analytic approximation than a numerical one for a given nonlinear problem. To overcoming the shortcomings, many new analytical techniques have been successfully developed by diverse groups of mathematicians and physicists, of Modified Differential Transforms Method [13][14][15][16], and so on. Several other authors used many powerful analytical methods in the field of approximate solutions especially for strongly nonlinear oscillators like Max-Min Approach Method [17,18], Algebraic Method [30-33], Newton-harmonic Balancing Approach [34], and so on for solving NDEs. The HBM is another technique for solving strongly nonlinear systems. Borges et al. [35] and Bobylev et al. [36] first provided overviews of HBM. Mickens [37-39] was first applied HBM in truly nonlinear oscillators. Due to his contribution he is known as father of HBM. Afterwards, Belendez et al. [40] and others researchers [41-43] has significantly improved the HBM.The HBM provides a general technique for calculating approximations to the periodic solutio...

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Section: Introductionmentioning

confidence: 99%

A new reliable analytical solution for strongly nonlinear oscillator with cubic and harmonic restoring force

Hosen

Chowdhury

2015

Results in Physics

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“…To demonstrate the basic ideas of optimal homotopy asymptotic method [11][12][13][14][15], consider the following general nonlinear differential equation…”

Section: Basic Idea Of Optimal Homotopy Asymptotic Methods (Oham)mentioning

confidence: 99%

A numerical investigation of time-fractional modified Fornberg–Whitham equation for analyzing the behavior of water waves

Ray

Gupta

2015

Applied Mathematics and Computation

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“…Various definitions and basic concept of fractional calculus are present in many books [1][2][3][4]. Therefore, several analytical and numerical methods were developed for solutions of fractional differential equations (both linear and nonlinear), among which Adomian's decomposition method [5][6][7], variation iteration method [8,9], homotopy perturbation method [10][11][12], homotopy analysis method [13][14][15], homotopy asymptotic method [16][17][18],differential transform method [19], and Galerkin method [20].…”

Section: Introductionmentioning

confidence: 99%

A Modified Analytical Approach for Fractional Discrete KdV Equations Arising in Particle Vibrations

Kumar

2017

Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci.

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In this paper, we present an effectiveness of the proposed analytical algorithm for nonlinear time fractional discrete KdV equations. The discrete KdV equation play an important role in modeling of complicated physical phenomena such as particle vibrations in lattices, current flow in electrical flow and pulse in biological chains. The proposed algorithm is amagalation of the homotopy analysis method, Laplace transform method and homotopy polynomials. First, we present an alternative framework of the proposed method which can be used simply and effectively to handle nonlinear problems arising in science and engineering. Then, mainly, we address the convergence study of nonlinear fractional differential equations with Laplace transform. Camparisons are made between the results of the proposed method and exact solutions. Illustrative examples are given to demonstrate the simplicity and efficiency of the proposed method. The new results reveal that the proposed method is explicit, efficient and easy to use.

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Comparison between homotopy perturbation method and optimal homotopy asymptotic method for the soliton solutions of Boussinesq–Burger equations

Cited by 66 publications

References 13 publications

A new reliable analytical solution for strongly nonlinear oscillator with cubic and harmonic restoring force

A new reliable analytical solution for strongly nonlinear oscillator with cubic and harmonic restoring force

A numerical investigation of time-fractional modified Fornberg–Whitham equation for analyzing the behavior of water waves

A Modified Analytical Approach for Fractional Discrete KdV Equations Arising in Particle Vibrations

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