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 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 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.

scholarly journals Modified Reduced Differential Transform Method for Partial Differential-Algebraic Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Brahim Benhammouda ◽  
Hector Vazquez-Leal ◽  
Arturo Sarmiento-Reyes
Keyword(s):  
Perturbation Parameter ◽  
Convergent Series ◽  
Differential Algebraic Equations ◽  
Differential Transform Method ◽  
Algebraic Equations ◽  
Partial Differential ◽  
Transform Method ◽  
Partial Differential Algebraic Equations ◽  
Differential Transform ◽  
Reduced Differential Transform Method

This work presents the application of the reduced differential transform method (RDTM) to find solutions of partial differential-algebraic equations (PDAEs). Two systems of index-two and index-three are solved to show that RDTM can provide analytical solutions for PDAEs in convergent series form. In addition, we present the posttreatment of the power series solutions with the Laplace-Padé resummation method as a useful technique to find exact solutions. 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 on a perturbation parameter.

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 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.

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.

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.

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