A new algorithm for calculating one-dimensional differential transform of nonlinear functions

doi:10.1016/j.amc.2007.05.026

Applied Mathematics and Computation

Volume 195, Issue 2, 1 February 2008, Pages 799-808

https://doi.org/10.1016/j.amc.2007.05.026 Get rights and content

Abstract

A new technique for calculating the one-dimensional differential transform of nonlinear functions is developed in this paper. This new technique avoids the difficulties and massive computational work that usually arise from the standard method. The algorithm will be illustrated by studying suitable forms of nonlinearity. Several nonlinear ordinary differential equations, including Troesch’s and Bratu-type problems, are then solved to demonstrate the reliability and efficiency of the proposed scheme. The present algorithm offers a computationally easier approach to compute the transformed function for all forms of nonlinearity. This gives the technique much wider applicability.

Introduction

The one-dimensional differential transform method [1] has been successfully applied to a wide class of nonlinear ordinary differential equations (ODEs) arising in many areas of science and engineering such as viscous flow [2], predictive control [3], vibration [4], [5], [6] and steady heat conduction [7] problems. The main advantage of this method is that it can be applied directly to nonlinear ODEs without requiring linearization, discretization or perturbation. Another important advantage is that this method is capable of greatly reducing the size of computational work while still accurately providing the series solution with fast convergence rate.

Although this newly emerged method has been proved to be an efficient tool for handling nonlinear problems, the nonlinear function f(y(x)) used in these studies is restricted to the types of nonlinear polynomials and derivatives. For other type of nonlinearity such as exponential function, the standard way to calculate its transformed function introduced by Zhou [1] is to expand this nonlinear function in an infinite power series, given by: $f (y) = \sum_{i = 0}^{\infty} a_{i} y^{i} .$ Take the differential transform of this equation to obtain: $F (k) = \sum_{i = 0}^{\infty} a_{i} W_{i} (k),$ where W_i(k) is the differential transform of yⁱ and expressed as: $Y (0) W_{i} (k) = \{\begin{matrix} [Y (0)]^{1 + i}, & k = 0, \\ \sum_{m = 1}^{k} \frac{(1 + i) m - k}{k} Y (m) W_{i} (k - m), & k ⩾ 1 . \end{matrix}$ The problem with this approach is that the computational difficulties will arise in determining the differential transform of nonlinear function while working with this infinite series. Therefore, it is essential to develop an alternative technique that can be used to calculate the differential transform of nonlinear function in an easy way.

In this paper, a simple and reliable algorithm used to calculate the differential transform of nonlinear function is introduced. The developed technique depends only on the fundamental operation properties of differential transform and calculus. This new technique significantly reduce the volume of computational work. Several strongly nonlinear ODEs are then solved using the present algorithm. The calculated results are exactly the same as those obtained by other analytical or approximate methods and demonstrate the reliability and efficiency of the technique.

Section snippets

Differential transform

The basic definitions and fundamental Theorem 1, Theorem 2, Theorem 3, Theorem 4, Theorem 5, Theorem 6 of one-dimensional differential transform are defined and proved in [1] and will be stated in brief in this section.

Differential transform of function y(x) is defined as follows: $Y (k) = \frac{1}{k!} {[\frac{d^{k} y (x)}{d x^{k}}]}_{x = 0},$ where y(x) is the original function and Y(k) is the transformed function, which is also called the T-function. In this paper, the lowercase and uppercase letters represent the original and

Analysis of the method

In this section, we will introduce a reliable and efficient algorithm to calculate the differential transform of nonlinear functions. As will be seen later, the proposed technique is based only on the differentiation and fundamental Theorem 1, Theorem 2, Theorem 3, Theorem 4, Theorem 5. Further, the transformed function is determined easily through a recursive relation that only involves the finite series. Here, a and b are constants.

Case 1. Exponential nonlinearity: f(y) = e^ay.

From the

Application

Example 1

Consider the nonlinear initial-value problem: $\begin{matrix} y^{″} (x) = 2 y + 4 y \ln y, y > 0, \\ y (0) = 1, y^{'} (0) = 0 . \end{matrix}$

Applying the differential transform to ODE and using the initial condition y(0) = 1, y′(0) = 0, we obtain: $(k + 1) (k + 2) Y (k + 2) = 2 Y (k) + 4 \sum_{m = 0}^{k} Y (m) F (k - m),$ $Y (0) = 1, Y (1) = 0,$ where F(k) is the T-function of ln y and given by Eq. (19): $F (k) = \{\begin{matrix} \ln (Y (0)), & k = 0, \\ \frac{Y (1)}{Y (0)}, & k = 1, \\ \frac{Y (k)}{Y (0)} - \sum_{m = 0}^{k - 2} \frac{m + 1}{kY (0)} F (m + 1) Y (k - 1 - m), & k ⩾ 2 . \end{matrix}$ Substituting Eq. (36) and k = 0 into Eqs. (37), (35) to get: $F (0) = 0, Y (2) = 1 .$ Substituting Eqs. (36), (38) and k = 1 into Eqs. (37), (35),

Conclusions

A simple and reliable algorithm based only on the fundamental operation properties of differential transform and calculus for calculating the one-dimensional differential transform of nonlinear functions has been developed. This new technique avoids the difficulties and massive computational work that usually arise from the standard method. The proposed algorithm is illustrated by solving several nonlinear ordinary differential equations including Troesch’s and Bratu-type problems. The

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A new algorithm for calculating one-dimensional differential transform of nonlinear functions

Abstract

Introduction

Section snippets

Differential transform

Analysis of the method

Application

Conclusions

Appl. Math. Model.

Appl. Math. Comput.

Appl. Math. Comput.

Appl. Math. Comput.

Appl. Math. Comput.