A reliable aftertreatment for improving the differential transformation method and its application to nonlinear oscillators with fractional nonlinearities

Ebaid, Abdelhalim

doi:10.1016/j.cnsns.2010.03.012

Cited by 55 publications

(43 citation statements)

References 20 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, 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.…”

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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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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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“…However, this kind of solutions have narrow domains of convergence. Hence, Laplace-Padé [23][24][25][26] resummation method is used to extend the domain of convergence of the solutions or to obtain exact solutions. We describe the LPRDTM which is combination of RDTM and Laplace-Padé resummation method as follows.…”

Section: Laplace-padé Reduced Differential Transform Methods (Lprdtm)mentioning

confidence: 99%

A new technique of Laplace Pade reduced differential transform method for (1 3) dimensional wave equations

Keskin¹,

Acan²

2017

ntmsci

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Abstract:The aim of this paper is to give a good strategy for solving some linear and non-linear partial differential equations in mechanics, physics, engineering and various other technical fields by Modified Reduced Differential Transform Method. In this article we use the method named with Laplace-Padé Reduced Differential Transform Method. This method is obtained by combining LaplacePadé resummation method, which is a useful technique to find exact solutions, and the Reduced Differential Transform Method. We apply the method to the wave equations and give some examples to see its effectiveness and usefulness. The results and the findings showed that this method leads us to exact solutions with a few iterations or the approximate solutions with small errors.

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“…Sometimes, even a combination of these methods are required in order to satisfactorily extend the domain of validity of the series solution. In this article, we follow the idea of [4] - [19] to employ a combination of the Laplace transform approach and Padé approximant in order to obtain exact solution or significantly improved solution to the problem. This approach is described below.…”

Section: Laplace-padé Post Processingmentioning

confidence: 99%

Improved Parker-Sochacki Approach for Closed Form Solution of Enzyme Catalyzed Reaction Model

Ogundare

Akindeinde

Adewumi

et al. 2017

J. of Mod. Meth. in Numer. Math.

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In this article, the Improved Parker-Sochacki Method (IPSM) is applied to solving nonlinear MichaelisMenten enzyme catalyzed reaction model. The global form of the solution for the concentrations of the substrate, enzyme and the enyzme-free product are obtained. Employing the Laplace-Pade resummation as a post processing technique on the computed series solution, the domain of convergence of the solution is greatly extended. The solution is therefore devoid of limited convergence interval that is typical of series solution of nonlinear differential equations. The proposed method showed a significant improvement over the conventional Parker-Sochacki Method (PSM). Furthermore, comparison of the results with numerically computed solutions via Runge-Kutta method elucidated the simplicity and accuracy of the proposed method.

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A reliable aftertreatment for improving the differential transformation method and its application to nonlinear oscillators with fractional nonlinearities

Cited by 55 publications

References 20 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 new technique of Laplace Pade reduced differential transform method for (1 3) dimensional wave equations

Improved Parker-Sochacki Approach for Closed Form Solution of Enzyme Catalyzed Reaction Model

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