Application of the modified differential transform method to fractional oscillators
TL;DR: It is shown that the results reveal that the modified differential transform method in many instances gives better results than the fourth‐order Runge‐Kutta method.
Abstract: Purpose – The purpose of this paper is to find a semi‐analytic solution to the fractional oscillator equations. In this paper, the authors apply the modified differential transform method to find approximate analytical solutions to fractional oscillators.Design/methodology/approach – The modified differential transform method is used to obtain the solutions of the systems. This approach rests on the recently developed modification of the differential transform method. Some examples are given to illustrate the ability and reliability of the modified differential transform method for solving fractional oscillators.Findings – The main conclusion is that the proposed method is a good way for solving such problems. The results are compared with those obtained by the fourth‐order Runge‐Kutta method. It is shown that the results reveal that the modified differential transform method in many instances gives better results.Originality/value – The paper demostrates that a hybrid method of differential transform met...
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TL;DR: It is concluded that, for the sake of clarity, when the D TM is applied to ODEs, it should be mentioned that the DTM exactly coincides with the traditional Taylor method, contrary to what is currently written.
61 citations
TL;DR: A theory of equivalent systems with respect to three classes of fractional oscillators is presented and it is suggested that this may facilitate the application of the theory of fractionAL oscillators to practice.
Abstract: This article addresses three classes of fractional oscillators named Class I, II and III. It is known that the solutions to fractional oscillators of Class I type are represented by the Mittag-Leffler functions. However, closed form solutions to fractional oscillators in Classes II and III are unknown. In this article, we present a theory of equivalent systems with respect to three classes of fractional oscillators. In methodology, we first transform fractional oscillators with constant coefficients to be linear 2-order oscillators with variable coefficients (variable mass and damping). Then, we derive the closed form solutions to three classes of fractional oscillators using elementary functions. The present theory of equivalent oscillators consists of the main highlights as follows. (1) Proposing three equivalent 2-order oscillation equations corresponding to three classes of fractional oscillators; (2) Presenting the closed form expressions of equivalent mass, equivalent damping, equivalent natural frequencies, equivalent damping ratio for each class of fractional oscillators; (3) Putting forward the closed form formulas of responses (free, impulse, unit step, frequency, sinusoidal) to each class of fractional oscillators; (4) Revealing the power laws of equivalent mass and equivalent damping for each class of fractional oscillators in terms of oscillation frequency; (5) Giving analytic expressions of the logarithmic decrements of three classes of fractional oscillators; (6) Representing the closed form representations of some of the generalized Mittag-Leffler functions with elementary functions. The present results suggest a novel theory of fractional oscillators. This may facilitate the application of the theory of fractional oscillators to practice.
54 citations
Cites background from "Application of the modified differe..."
...[61] discussed nor those in the sense of subharmonic oscillations as stated by Den Hartog ([3], Sections 8–10, Chapter 4), Ikeda [62], Fudan Univ....
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TL;DR: In this article, the analytical solution of non-homogeneous fractional second-order RLC circuit is presented in terms of convergent series, where the authors use two different methods, one is modern and the other is traditional, namely generalized differential transform Method (GDTM) and Laplace transform method (LTM) to obtain the analytical solutions.
Abstract: Abstract Systematic construction of fractional ordinary differential equations [FODEs] has gained much attention nowadays research because dimensional homogeneity plays a major role in mathematical modeling. In order to keep up the dimension of the physical quantities, we need some auxiliary parameters. When we utilize auxiliary parameters, the FODE turns out to be more intricate. One of such kind of model is non-homogeneous fractional second order RLC circuit. To solve this kind of complicated FODEs, we need proficient modern analytical method. In this paper, we use two different methods, one is modern and the other is traditional, namely generalized differential transform Method (GDTM) and Laplace transform method (LTM) to obtain the analytical solution of non-homogeneous fractional second order RLC circuit. We present the solution in terms of convergent series. Though GDTM and LTM are capable to produce the exact solution of fractional RLC circuit, great strength of GDTM over LTM is that differential transform of initial conditions occupy the coefficients of first two terms in series solution so that we arrive exact solution with few iterations and also, it does not allow the noise terms while computing the coefficients. Due to this, GDTM takes less time to converge than LTM and it has been demonstrated. Furthermost, we discuss the characteristics of non-homogeneous fractional second order RLC circuit through numerical illustrations.
7 citations
TL;DR: In this paper, a new method incorporated by the fractional oscillator type equations is proposed to describe several phenomenon in mathematical physics, engineering and biology, and the method is applied to the problem of finite oscillator types.
Abstract: Fractional oscillator type equations are well-known model equations to describe several phenomenon in mathematical physics, engineering and biology. In this paper, a new method incorporated by the ...
5 citations
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TL;DR: Three-dimensional differential transform method has been introduced and fundamental theorems have been defined for the first time and exact solutions of linear and non-linear systems of partial differential equations have been investigated.
383 citations
TL;DR: Using two-dimensional differential transform to solve Partial Differential Equations (PDE) is proposed in this study and three PDE problems with constant and variable coefficients are solved.
334 citations
TL;DR: It is demonstrated that the differential transform is a feasible tool for obtaining the analytic form solutions of linear and nonlinear partial differential equation.
269 citations
TL;DR: The commonly encountered linear and nonlinear integro-differential equations that appear in literature are solved as an illustration for the efficiency of the differential transform method.
240 citations
"Application of the modified differe..." refers methods in this paper
...The method is well addressed in Arikoglu and Ozkol (2005)...
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