Approximate periodic solutions for the non-linear relativistic harmonic oscillator via differential transformation method

Ebaid, Abdelhalim

doi:10.1016/j.cnsns.2009.07.003

Cited by 35 publications

(29 citation statements)

References 25 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%

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confidence: 99%

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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%

mentioning

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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“…In this first example, we consider the particular case of (14) such that β = 0.1; this case was studied in [16] via differential transformation method (DTM) and also in [9] through the homotopy perturbation method (HPM). Good approximations were obtained in both works in comparison with the first known approximation solution of (14) obtained in [19] by the harmonic balance method (HBM).…”

Section: Application To the Relativistic Harmonic Oscillatormentioning

confidence: 99%

“…All the numerical work was accomplished with the Mathematica software package. Example 2 In this second example, we consider the particular case of (14) such that β = 0.2; this case was studied in [16] via DTM and also in [9] using HPM. Once again, in both works, good approximations were found in comparison with the first obtained in (14) and the one obtained in [19] by HBM.…”

Section: Application To the Relativistic Harmonic Oscillatormentioning

confidence: 99%

Solution for the Nonlinear Relativistic Harmonic Oscillator via Laplace-Adomian Decomposition Method

González-Gaxiola

Santiago

Chávez

2016

Int. J. Appl. Comput. Math

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Far as we know there are not exact solutions to the equation of motion for a relativistic harmonic oscillator. In this paper, the relativistic harmonic oscillator equation which is a nonlinear ordinary differential equation is studied by means of a combined use of the Adomian Decomposition Method and the Laplace Transform (LADM). The results that we have obtained, a series of powers of functions, have never been reported and show a very good match when compared with other approximate solutions, obtained by different methods. The method here proposed works with high degree of accuracy and because it requires less computational effort, it is very convenient to solve this kind of nonlinear differential equations.

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“…Due of this difficulty, most of the researchers in the applied and the engineering sciences directly resort to the numerical methods [1] or the numerical codes to solve their physical models. However, many other authors investigated their physical models by using one of the semi-analytical methods such as Adomian decomposition method (ADM) [2][3][4][5][6][7], differential transformation/Taylor method (DTM) [8][9], homotopy perturbation method (HPM) [10][11][12][13], and homotopy analysis method (HAM) [14]. Unfortunately, the accuracy of the numerical solutions derived from these semi-analytical methods cannot be checked without addressing the convergence issue.…”

Section: Introductionmentioning

confidence: 99%

Effect of a Convective Boundary Condition on Boundary Layer Slip Flow and Heat Transfer Over a Stretching Sheet in View of the Exact Solution

Aljoufi¹,

Ebaid²

2016

Journal of Theoretical and Applied Mechanics

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The exact solutions of a nonlinear differential equations system, describing the boundary layer flow over a stretching sheet with a convective boundary condition and a slip effect have been obtained in this paper. This problem has been numerically solved by using the shooting method in literature. The aim of the current paper is to check the accuracy of these published numerical results. This goal has been achieved via first obtaining the exact solutions of the governing nonlinear differential equations and then, by comparing them with the approximate numerical results reported in literature. The effects of the physical parameters on the flow field and the temperature distribution have been re-investigated through the new exact solutions. The main advantage of the current paper is the simple computational approach that has been introduced to analyze exactly the present physical problem. This simple analytical approach can be further applied to investigate similar problems. Although no remarkable differences have been detected between the current figures and those obtained in literature, the authors believe that if some numerical calculations were available for the fluid velocity and the temperature in literature then the convergence criteria and the accuracy of the shooting method used in Ref. [15] can be validated in view of the current exact expressions.

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Approximate periodic solutions for the non-linear relativistic harmonic oscillator via differential transformation method

Cited by 35 publications

References 25 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

Solution for the Nonlinear Relativistic Harmonic Oscillator via Laplace-Adomian Decomposition Method

Effect of a Convective Boundary Condition on Boundary Layer Slip Flow and Heat Transfer Over a Stretching Sheet in View of the Exact Solution

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