scholarly journals Enhanced Multistage Differential Transform Method: Application to the Population Models

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Younghae Do ◽  
Bongsoo Jang
Keyword(s):  
Taylor Series ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
Differential Transform Method ◽  
Initial Condition ◽  
Initial Value ◽  
Transform Method ◽  
Large Domain ◽  
Runge Kutta Method ◽  
Differential Transform

We present an efficient computational algorithm, namely, the enhanced multistage differential transform method (E-MsDTM) for solving prey-predator systems. Since the differential transform method (DTM) is based on the Taylor series, it is difficult to obtain accurate approximate solutions in large domain. To overcome this difficulty, the multistage differential transform method (MsDTM) has been introduced and succeeded to have reliable approximate solutions for many problems. In MsDTM, it is the key to update an initial condition in each subdomain. The standard MsDTM utilizes the approximate solution directly to assign the new initial value. Because of local convergence of the Taylor series, the error is accumulated in a large domain. In E-MsDTM, we propose the new technique to update an initial condition by using integral operator. To demonstrate efficiency of the proposed method, several numerical tests are performed and compared with ones obtained by other numerical methods such as MsDTM, multistage variational iteration method (MVIM), and fourth-order Runge-Kutta method (RK4).

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 Modified Differential Transform Method for Solving the Model of Pollution for a System of Lakes

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Brahim Benhammouda ◽  
Hector Vazquez-Leal ◽  
Luis Hernandez-Martinez
Keyword(s):  
Approximate Solutions ◽  
Perturbation Parameter ◽  
Convergent Series ◽  
Differential Transform Method ◽  
Transform Method ◽  
Analytical Approximations ◽  
Runge Kutta Method ◽  
Power Series Solutions ◽  
Differential Transform ◽  
Fifth Order

This work presents the application of the differential transform method (DTM) to the model of pollution for a system of three lakes interconnected by channels. Three input models (periodic, exponentially decaying, and linear) are solved to show that DTM can provide analytical solutions of pollution model in convergent series form. In addition, we present the posttreatment of the power series solutions with the Laplace-Padé resummation method as a useful strategy to extend the domain of convergence of the approximate solutions. The Fehlberg fourth-fifth order Runge-Kutta method with degree four interpolant (RKF45) numerical solution of the lakes system problem is used as a reference to compare with the analytical approximations showing the high accuracy of the results. The main advantage of the proposed technique is that it is based on a few straightforward steps and does not generate secular terms or depend of a perturbation parameter.

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.

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

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.

scholarly journals Nonlinear Klein-Gordon and Schrödinger Equations by the Projected Differential Transform Method

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Younghae Do ◽  
Bongsoo Jang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Taylor Series ◽  
Nonlinear Partial Differential Equations ◽  
Differential Transform Method ◽  
Schrödinger Equations ◽  
Partial Differential ◽  
Transform Method ◽  
Differential Transform ◽  
Klein Gordon

The differential transform method (DTM) is based on the Taylor series for all variables, but it differs from the traditional Taylor series in calculating coefficients. Even if the DTM is an effective numerical method for solving many nonlinear partial differential equations, there are also some difficulties due to the complex nonlinearity. To overcome difficulties arising in DTM, we present the new modified version of DTM, namely, the projected differential transform method (PDTM), for solving nonlinear partial differential equations. The proposed method is applied to solve the various nonlinear Klein-Gordon and Schrödinger equations. Numerical approximations performed by the PDTM are presented and compared with the results obtained by other numerical methods. The results reveal that PDTM is a simple and effective numerical algorithm.

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