scholarly journals Application of Multistep Generalized Differential Transform Method for the Solutions of the Fractional-Order Chua's System

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Asad Freihat ◽  
Shaher Momani
Keyword(s):  
Fractional Calculus ◽  
Fractional Order ◽  
Dynamical Behavior ◽  
Nonlinear Behavior ◽  
Nonlinear Problems ◽  
Differential Transform Method ◽  
Effective Dimension ◽  
Transform Method ◽  
Differential Transform ◽  
Generalized Differential

We numerically investigate the dynamical behavior of the fractional-order Chua's system. By utilizing the multistep generalized differential transform method (MSGDTM), we find that the fractional-order Chua's system with “effective dimension” less than three can exhibit chaos as well as other nonlinear behavior. Numerical results are presented graphically and reveal that the multistep generalized differential transform method is an effective and convenient method to solve similar nonlinear problems in fractional calculus.

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

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
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.

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

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
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.

scholarly journals Epidemic Model of Leptospirosis Containing Fractional Order

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Muhammad Altaf Khan ◽  
S. F. Saddiq ◽  
Saeed Islam ◽  
Ilyas Khan ◽  
Dennis Ling Chuan Ching
Keyword(s):  
Approximate Solution ◽  
Positive Solution ◽  
Fractional Order ◽  
Epidemic Model ◽  
Differential Transform Method ◽  
Unique Positive Solution ◽  
Transform Method ◽  
Differential Transform ◽  
Generalized Differential ◽  
Order Four

We study an epidemic model of leptospirosis in fractional order numerically. The multistep generalized differential transform method is applied to find the accurate approximate solution of the epidemic model of leptospirosis disease in fractional order. A unique positive solution for the epidemic model in fractional order is presented. For the integer case derivative, the approximate solution of MGDTM is compared with the Runge-Kutta order four scheme. The numerical results are presented for the justification purpose.

scholarly journals Application of Generalized Differential Transform Method to Special Cases of Riccati Differential Equation of Fractional Order

2019 ◽  
Vol 8 (3) ◽  
pp. 2774-2779
Keyword(s):  
Differential Equation ◽  
Numerical Solution ◽  
Fractional Order ◽  
Fractional Derivatives ◽  
Differential Transform Method ◽  
Riccati Differential Equation ◽  
Transform Method ◽  
Special Cases ◽  
Differential Transform ◽  
Generalized Differential

In this paper, we acquire the inexact solutions of Special cases of Riccati Differential equation of Fractional order using Generalized Differential Transform Method (GDTM). The fractional derivatives are described in the Caputo sense. Accuracy and competence of the proposed method is verified through numerical solution of some special cases of Riccati Differential equation of fractional order. The obtained results reveal that the performance of the proposed method is specific and predictable.

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