scholarly journals Application of the Multistep Generalized Differential Transform Method to Solve a Time-Fractional Enzyme Kinetics

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Ahmed Alawneh
Keyword(s):  
Enzyme Kinetics ◽  
Fractional Derivatives ◽  
Differential Transform Method ◽  
Nonlinear Ordinary Differential Equations ◽  
Kutta Method ◽  
Transform Method ◽  
Enzyme Substrate ◽  
Runge Kutta Method ◽  
Differential Transform ◽  
Generalized Differential

The multistep differential transform method is first employed to solve a time-fractional enzyme kinetics. This enzyme-substrate reaction is formed by a system of nonlinear ordinary differential equations of fractional order. The fractional derivatives are described in the Caputo sense. A comparative study between the new algorithm and the classical Runge-Kutta method is presented in the case of integer-order derivatives. The results demonstrate reliability and efficiency of the algorithm developed.

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.

Comparison of Numerical Methods of the SEIR Epidemic Model of Fractional Order

2014 ◽  
Vol 69 (1-2) ◽  
pp. 81-89 ◽  
Author(s):  
Anwar Zeb ◽  
Madad Khan ◽  
Gul Zaman ◽  
Shaher Momani ◽  
Vedat Suat Ertürk
Keyword(s):  
Fractional Order ◽  
Epidemic Model ◽  
Nonnegative Solution ◽  
Incidence Rates ◽  
Kutta Method ◽  
Transform Method ◽  
Seir Model ◽  
Runge Kutta Method ◽  
Differential Transform ◽  
Generalized Differential

In this paper, we consider the SEIR (Susceptible-Exposed-Infected-Recovered) epidemic model by taking into account both standard and bilinear incidence rates of fractional order. First, the nonnegative solution of the SEIR model of fractional order is presented. Then, the multi-step generalized differential transform method (MSGDTM) is employed to compute an approximation to the solution of the model of fractional order. Finally, the obtained results are compared with those obtained by the fourth-order Runge-Kutta method and non-standard finite difference (NSFD) method in the integer case.

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 A semi-analytical solution of micro polar flow in a porous channel with mass injection by using differential transform method

2010 ◽  
Vol 15 (3) ◽  
pp. 341-350 ◽  
Author(s):  
M. M. Rashidi ◽  
S. A. Mohimanian Pour ◽  
N. Laraqi
Keyword(s):  
Approximate Solutions ◽  
Differential Transform Method ◽  
Porous Channel ◽  
Nonlinear Ordinary Differential Equations ◽  
Series Solutions ◽  
Transform Method ◽  
Runge Kutta Method ◽  
Mass Injection ◽  
Governing System ◽  
Differential Transform

In this letter, the differential transform method (DTM) was applied to the micro-polar flow in a porous channel with mass injection. Approximate solutions of the governing system of nonlinear ordinary differential equations were calculated in the form of DTM series with easily computable terms. The validity of the series solutions were verified by comparison with numerical results obtained using a fourth order Runge–Kutta method. The computed DTM velocity profiles are shown and the influence of Reynolds number on the velocity component in x-direction is discussed.

The Multi-Step Differential Transform Method and Its Application to Determine the Solutions of Non-Linear Oscillators

2012 ◽  
Vol 4 (04) ◽  
pp. 422-438 ◽  
Author(s):  
Vedat Suat Ertürk ◽  
Zaid M. Odibat ◽  
Shaher Momani
Keyword(s):  
Fourth Order ◽  
Differential Transform Method ◽  
Kutta Method ◽  
Transform Method ◽  
Promising Tool ◽  
Runge Kutta Method ◽  
Differential Transform ◽  
Non Linear ◽  
Runge Kutta

AbstractIn this paper, a reliable algorithm based on an adaptation of the standard differential transform method is presented, which is the multi-step differential transform method (MSDTM). The solutions of non-linear oscillators were obtained by MSDTM. Figurative comparisons between the MSDTM and the classical fourth-order Runge-Kutta method (RK4) reveal that the proposed technique is a promising tool to solve non-linear oscillators.

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.

Semi-Numerical, Semi-Analytical Approximations of the Rayleigh Equation for Gas-Filled Hyper-Spherical Bubble

2018 ◽  
Vol 16 (01) ◽  
pp. 1850094 ◽  
Author(s):  
Yupeng Qin ◽  
Zhen WANG ◽  
Li Zou ◽  
Mingfeng He
Keyword(s):  
Fourth Order ◽  
Differential Transform Method ◽  
Rayleigh Equation ◽  
Kutta Method ◽  
Transform Method ◽  
Spherical Bubble ◽  
Analytical Approximations ◽  
Runge Kutta Method ◽  
Differential Transform ◽  
Runge Kutta

A new semi-numerical, semi-analytical approach based on the differential transform method is proposed to solve the problems of a gas-filled hyper-spherical bubble governed by the Rayleigh equation. Semi-numerical, semi-analytical approximations are constructed for the Rayleigh equation in the form of piecewise functions. The proposed approach is compared with the standard fourth-order Runge–Kutta method and the standard differential transform method, respectively. The results reveal two main benefits of the new approach, one is that it possesses result with higher precision than the standard fourth-order Runge–Kutta method, the other is that it remains valid and accurate for longer time compared to the standard differential transform method. In addition, we also consider the Rayleigh equation in [Formula: see text] dimensions when the surface tension is not zero.

scholarly journals Generalized Differential Transform Method to Space-Time Fractional Telegraph Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-9 ◽  
Author(s):  
Mridula Garg ◽  
Pratibha Manohar ◽  
Shyam L. Kalla
Keyword(s):  
Closed Form ◽  
Fractional Derivatives ◽  
Telegraph Equation ◽  
Space Time ◽  
Differential Transform Method ◽  
Transform Method ◽  
Space And Time ◽  
Fractional Telegraph Equation ◽  
Differential Transform ◽  
Generalized Differential

We use generalized differential transform method (GDTM) to derive the solution of space-time fractional telegraph equation in closed form. The space and time fractional derivatives are considered in Caputo sense and the solution is obtained in terms of Mittag-Leffler functions.

Numerical Approximations to Solution of Biochemical Reaction Model

2011 ◽  
Vol 9 (1) ◽  
Author(s):  
Ahmet Yildirim ◽  
Ahmet Gökdogan ◽  
Mehmet Merdan
Keyword(s):  
Analytical Solution ◽  
Approximate Analytical Solution ◽  
Transformation Method ◽  
Reaction Model ◽  
Biochemical Reaction ◽  
Graphical Form ◽  
Kutta Method ◽  
Transform Method ◽  
Runge Kutta Method ◽  
Differential Transform

In this paper, approximate analytical solution of biochemical reaction model is used by the multi-step differential transform method (MsDTM) based on classical differential transformation method (DTM). Numerical results are compared to those obtained by the fourth-order Runge-Kutta method to illustrate the preciseness and effectiveness of the proposed method. Results are given explicit and graphical form.

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