Solving the Linear Time-Fractional Wave Equation by Generalized Differential Transform Method

In this paper, the generalized differential transform method is implemented for solving time-fractional wave equations in fluid mechanics. This method is based on the two-dimensional differential transform method (DTM) and generalized Taylor’s formula. The results reveal the method is feasible and convenient for handling approximate solutions of time-fractional partial differential equations.

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Analytical Study of Fractional-Order Multiple Chaotic FitzHugh-Nagumo Neurons Model Using Multistep Generalized Differential Transform Method

Abstract and Applied Analysis ◽

10.1155/2014/276279 ◽

2014 ◽

Vol 2014 ◽

pp. 1-10 ◽

Cited By ~ 13

Author(s):

Shaher Momani ◽

Asad Freihat ◽

Mohammed AL-Smadi

Keyword(s):

Fractional Order ◽

Fractional Derivatives ◽

Analytical Study ◽

Approximate Solutions ◽

Differential Transform Method ◽

Kutta Method ◽

Transform Method ◽

Runge Kutta Method ◽

Differential Transform ◽

Generalized Differential

The multistep generalized differential transform method is applied to solve the fractional-order multiple chaotic FitzHugh-Nagumo (FHN) neurons model. The algorithm is illustrated by studying the dynamics of three coupled chaotic FHN neurons equations with different gap junctions under external electrical stimulation. The fractional derivatives are described in the Caputo sense. Furthermore, we present figurative comparisons between the proposed scheme and the classical fourth-order Runge-Kutta method to demonstrate the accuracy and applicability of this method. The graphical results reveal that only few terms are required to deduce the approximate solutions which are found to be accurate and efficient.

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Analytical Approach to Time-Fractional Partial Differential Equations in Fluid Mechanics

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.347-353.463 ◽

2011 ◽

Vol 347-353 ◽

pp. 463-466

Author(s):

Xue Hui Chen ◽

Liang Wei ◽

Lian Cun Zheng ◽

Xin Xin Zhang

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Fluid Mechanics ◽

Approximate Solutions ◽

Differential Transform Method ◽

Fractional Partial Differential Equations ◽

Partial Differential ◽

Transform Method ◽

Differential Transform ◽

Generalized Differential

The generalized differential transform method is implemented for solving time-fractional partial differential equations in fluid mechanics. This method is based on the two-dimensional differential transform method (DTM) and generalized Taylor’s formula. Results obtained by using the scheme presented here agree well with the numerical results presented elsewhere. The results reveal the method is feasible and convenient for handling approximate solutions of time-fractional partial differential equations.

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Adaptation of Differential Transform Method for the Numeric-Analytic Solution of Fractional-Order Rössler Chaotic and Hyperchaotic Systems

Abstract and Applied Analysis ◽

10.1155/2012/934219 ◽

2012 ◽

Vol 2012 ◽

pp. 1-13 ◽

Cited By ~ 8

Author(s):

Asad Freihat ◽

Shaher Momani

Keyword(s):

Fractional Order ◽

Nonlinear Problems ◽

Approximate Solutions ◽

Differential Transform Method ◽

Time Step ◽

Transform Method ◽

Runge Kutta Method ◽

Differential Transform ◽

Generalized Differential ◽

Hyperchaotic Systems

A new reliable algorithm based on an adaptation of the standard generalized differential transform method (GDTM) is presented. The GDTM is treated as an algorithm in a sequence of intervals (i.e., time step) for finding accurate approximate solutions of fractional-order Rössler chaotic and hyperchaotic systems. A comparative study between the new algorithm and the classical Runge-Kutta method is presented in the case of integer-order derivatives. The algorithm described in this paper is expected to be further employed to solve similar nonlinear problems in fractional calculus.

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On New Solutions of Time-Fractional Wave Equations Arising in Shallow Water Wave Propagation

Mathematics ◽

10.3390/math7080722 ◽

2019 ◽

Vol 7 (8) ◽

pp. 722 ◽

Cited By ~ 16

Author(s):

Rajarama Mohan Jena ◽

Snehashish Chakraverty ◽

Dumitru Baleanu

Keyword(s):

Wave Equations ◽

Perturbation Technique ◽

Adomian Decomposition Method ◽

Primary Objective ◽

Differential Transform Method ◽

Transform Method ◽

Physical Problems ◽

Adomian Decomposition ◽

Differential Transform ◽

Fractional Wave Equations

The primary objective of this manuscript is to obtain the approximate analytical solution of Camassa–Holm (CH), modified Camassa–Holm (mCH), and Degasperis–Procesi (DP) equations with time-fractional derivatives labeled in the Caputo sense with the help of an iterative approach called fractional reduced differential transform method (FRDTM). The main benefits of using this technique are that linearization is not required for this method and therefore it reduces complex numerical computations significantly compared to the other existing methods such as the perturbation technique, differential transform method (DTM), and Adomian decomposition method (ADM). Small size computations over other techniques are the main advantages of the proposed method. Obtained results are compared with the solutions carried out by other technique which demonstrates that the proposed method is easy to implement and takes small size computation compared to other numerical techniques while dealing with complex physical problems of fractional order arising in science and engineering.

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