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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 535020, 7 pages
Note on Relation between Double Laplace Transform and Double Differential Transform
Mathematics Department, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
Received 12 May 2013; Accepted 8 June 2013
Academic Editor: Bessem Samet
Copyright © 2013 Hassan Eltayeb. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Double differential transform method has been employed to compute double Laplace transform. To illustrate the method, four examples of different forms have been prepared.
The concept of the DTM was first proposed by Zhou , who solved linear and nonlinear problems in electrical circuit problems. Chen and Ho  developed this method for partial differential equations and applied it to the system of differential equations. During the recent years, many authors have used this method for solving various types of equations. For example, this method has been used for differential algebraic equations , partial differential equations [4, 5], fractional differential equations , Volterra integral equations , and difference equations . The main goal of this paper is to extend the study of single Laplace transform by using differential transform (see ) and to compute double Laplace transform by means of double differential transform method. As we know, the standard derivation of Laplace transforms inherits an improper integration which may, in certain cases, not be analytically tractable. However, in contrast, the proposed straightforward approach merely requires easy differentiations and algebraic operations. Three examples are proposed.
The one-dimensional differential transform of the function is defined by the following formula: where and are the original and transform functions, respectively. The inverse differential transform of is specified as follows: Consider an analytical function of two variables; then this function can be represented as a series in using differential transform and inverse double differential transform From the definition of double Laplace transform, we can write where , and , is a complex value.
On using double direct and inverse differential transform with respect to and for both sides of the previous,
Lemma 1. Let and be a finite positive integers such that , and , , , ; then
Proof. By using mathematical induction, letting , , we have Then ; since we conclude that for , , Consequently Similarly Assume that, for , to , , it holds that and also Now, we are going to prove that By using the definition of polynomial, we have Thus
2. Relation between Double Laplace and Differential Transforms
In this section, we compute the double Laplace transform by means of double differential transform by proposing some examples as follows.
Example 2. Double Laplace transform of function ; consider In the next example, we apply double differential transform to compute double Laplace transform as follows.
Example 3. If we consider the function , then double Laplace transform is given by From the properties of double differential transform, we have On using definition of Kronecker delta function forces, we have From the previous equation, we have where and are constant coefficients of the polynomials generated by and , respectively. According to the lemma, the last summations of (23), , and are zeros, such that In the next example, we apply double integral transform as follows.
Example 4. If we consider the function , then double Laplace transform is given by By calculating the summation inside the bracket we have From the previous lemma, we know that, for , , , and , we have the following form: From the definition of infinite geometric series and the summation of the previous terms, we have
In the next example, we apply double differential transform to find double Laplace transform of the function as follows.
Example 5. The double Laplace transform of the function , as follows: By using series, we have By applying double differential transform, we have On using double inverse differential transform, we have According to the previous lemma, we have By using the definition of we have So that
Also, we can use the same idea to compute double Laplace transform for convolution function, single or double.
The author gratefully acknowledge, that this project was supported by King Saud University, Deanship of Scientific Research, College of Science Research Center.
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