# COMPARISON OF ABOODH TRANSFORMATION AND DIFFERENTIAL TRANSFORMATION METHOD NUMERICALLY.

Byline: A.I.Ali and M.I.Bhatti

ABSTRACT

Differential Transformation gives semi analytical numerical solution. These methods are capable of reducing the calculation and works efficiently. In this paper we use Aboodh Transformation Method (ATM) and Differential Transformation Method (DTM) to get a Numerical solution of the system of linear ordinary differential equation. We compare the results to see which Transformation converges faster to true solution .we also presented the results with relative error.

Keywords: Taylor expansion, Fourier integral, Aboodh Transform Method, Differential Transform Method

INTRODUCTION

Differential Transform method was first proposed and applied to solve linear and nonlinear initial value problems in electric circuit analysis by X.Zhou [1]. Chen and Liu have applied this method to solve two boundary value problems [2].Jan, Chen and Liu apply the two dimensional DTM to solve partial differential equation [3]. Yu and Chen apply the DTM for the optimization of the rectangular fins with variable thermal parameters [4, 5]. The high order Taylor series method which required a lot of symbolic computations.

DTM is an iterative procedure for obtaining Taylor series solutions. The DTM is a numerical method based on a Taylor expansion .This method constructs an analytical solution in the form of a polynomial. This method is easy to handle and compute the work in less time even when apply to nonlinear or parameter varying systems. But, it is different from Taylor series method that requires computation of the high order derivatives. The DTM is an iterative procedure that is described by the transformed equation of original functions for solutions of differential equations. In recent year, many papers were devoted to the problem of approximate solution of system of differential equation. The implementation of the DTM [2, 4, 6, 7] amongst others has shown reliable results to solve ordinary differential equations, partial differential equations, Blasius equation, nonlinear fractional differential equations and delay differential equations.

Aboodh Transform is derived from the classical Fourier integral. Based on the mathematical simplicity of the Aboodh Transform and its fundamental properties, Aboodh Transform was introduced by Khalid Aboodh in 2013, to facilitate the process of solving ordinary and partial differential equations in the time domain. This transformation has deeper connection with the Laplace and Elzaki Transform. [8, 9].

2 Aboodh Transform Method (ATM)

The Aboodh Transform Method over a set of function define as

Equations

Application 1: Consider the System of Differential equation

Equations

Solution:-

Given system of ordinary differential equation is By Applying Aboodh transformation method

Equations

And after applying Inverse Aboodh Transform, we have

Equations

And after applying Inverse Aboodh Transform, we have

Equations

Using (2) in (1) we have

Table. 1 Values of x (t) and y (t) using Aboodh transformation

###t###x(t)###y(t)

###0.1###8.999661183103608###1.540537694897257

###0.2###10.770276550867314###-0.357428091595026

###0.3###13.664441871618232###-2.936142742064507

###0.4###18.210697503363541###-6.554464618612033

###0.5###25.199821595355118###-11.745458899298134

###0.6###35.813587322394937###-19.302294580813069

###0.7###51.816866832248216###-30.406367023237063

###0.8###75.844235411914170###-46.818415573632599

###0.9###111.827551629736960###-71.163620588652975

###1###165.633847305289920###-107.356902860431260

Application 2: Consider the system of differential equations

Equations

And after applying Inverse Aboodh Transform, we have

Equations

And after applying Inverse Aboodh Transform, we have

Equations

Table 2. Values of x (t) and y (t) using Aboodh transformation

t###x(t)###y(t)

0.1###2.766273059382557###2.700083341788971

0.2###4.321661578359802###5.638689189849844

0.3###7.286063579046839###10.862878814192005

0.4###12.794236271824374###20.275292870100440

0.5###22.918123549611138###37.338487219951865

0.6###41.436558684560900###68.358144646473022

0.7###75.237894764489695###124.821098358285770

0.8###136.875879161822160###227.655373394382740

0.9###249.226854756904460###414.992393855657040

1###453.975811052159030###756.310569426046300

Equations

Solution:-

Given system of ordinary differential equation is

Equations

By applying differential transformation on the initial conditions,

Equations

Table 3. For ( ) by Differential Transformation in application 1a

( ) by DTM

8.999661183103610

10.770276550867300

13.664441871618200

18.210697503363500

25.199821595355100

35.813587322395000

51.816866832248200

75.844235411914300

111.827551629737000

165.633847305290000

Table 3a.: Comparison Table for ( ) in application 1a , 1

COMPARISON TABLE FOR ( )

DTM###Aboodh###Error

###8.99966118310###0.0000000000

###3608###00000

###10.7702765508###0.0000000000

###67314###00000

###13.6644418716###0.0000000000

###18232###00000

###18.2106975033###0.0000000000

###63541###00000

###25.1998215953###0.0000000000

###55118###00000

###35.8135873223###0.0000000000

###94937###00099

###51.8168668322###0.0000000000

###48216###00000

###75.8442354119###0.0000000000

###14170###00199

###111.827551629###0.0000000000

###736960###00995

###165.633847305###0.0000000000

###289920###00995

Table 4. For ( ) by Differential Transformation in Applcation 1a

###0.1###1.540537694897260

###0.2###-0.357428091595026

###0.3###-2.936142742064510

###0.4###-6.554464618612040

###0.5###-11.745458899298100

###0.6###-19.302294580813100

###0.7###-30.406367023237100

###0.8###-46.818415573632600

###0.9###-71.163620588653100

###1###-107.356902860432000

Table 4a.COMPARISON TABLE FOR ( ) in Application 1a, 1

###DTM###Aboodh###Error

###0.1###1.540537694897260###1.540537694897250

###0.2###-0.357428091595026###-0.357428091595026

###0.3###-2.936142742064510###-2.936142742064500

###0.4###-6.554464618612040###-6.554464618612030

###0.5###-11.745458899298100###-11.745458899298100

###0.6###-19.302294580813100###-19.302294580813000

###0.7###-30.406367023237100###-30.406367023237000

###0.8###-46.818415573632600###-46.818415573632500

###0.9###-71.163620588653100###-71.163620588652900

###1###-107.356902860432000###-107.356902860431000

Application No. 2a Consider the system of differential equations

Equations

By applying differential transformation method

Equations

By applying differential transformation on the initial conditions,

Table 5. For ( ) by differential Transformation in Application 2a

###t###x(t) by DTM

###0.1###2.766273059382560

###0.2###4.321661578359800

###0.3###7.286063579046840

###0.4###12.794236271824400

###0.5###22.918123549611200

###0.6###41.436558684561000

###0.7###75.237894764489800

###0.8###136.875879161822000

###0.9###249.226854756905000

###1###453.975811052161000

Table 5a. COMPARISON TABLE FOR ( ) in Application 2a , 2

###DTM###Aboodh###Error

###2.766273059382560###2.766273059382557

###4.321661578359800###4.321661578359802

###7.286063579046840###7.286063579046839

###12.794236271824400###12.794236271824374

###22.918123549611200###22.918123549611138

###41.436558684561000###41.436558684560900

###75.237894764489800###75.237894764489695

###136.875879161822000###136.875879161822160

###249.226854756905000###249.226854756904460

###453.975811052161000###453.975811052159030

Table 6. For ( ) by Differential Transformation in application 2a

###t###( )by DTM

###0.1###2.700083341788970

###0.2###5.638689189849850

###0.3###10.862878814192000

###0.4###20.275292870100500

###0.5###37.338487219951900

###0.6###68.358144646473200

###0.7###124.821098358286000

###0.8###227.655373394384000

###0.9###414.992393855659000

###1###756.310569426050000

Table 6a. COMPARISON TABLE FOR ( ) in Application 2a , 2

###DTM###Aboodh###Error

###2.700083341788970

###5.638689189849850

###10.862878814192000

###20.275292870100500

###37.338487219951900

###68.358144646473200

###124.821098358286000

###227.655373394384000

###414.992393855659000

###756.310569426050000

CONCLUSION

In this paper, solved system of differential equations by two different transformations, one is Aboodh Integral Transformations and the other one is Differential transformation; and solutions obtained by these two transformations are compared. Solution obtained by DTM is series solutions; on the other hand Aboodh integral Transformation gives exact solution. In this paper, it is found that solutions by DTM converge rapidly. Results are compared in tabular form as well as with graph. It is found that results are approximately near to the exact solution. Hence DTM is a reliable, effective tool for the solution of system of differential equations.

REFERENCES

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[3] M.J. Jang, C.L. Chen Analysis of the response of a strongly nonlinear damped system using a differential transformation technique", Appl. Math. Comput. 88,137-151(1997).

[4] L.T. Yu, C.K. Chen The solution of the Blasius equation by the differential transformation method", Math. Comput. Modelling 28(1), 101-111 (1998).

[5] L.T. Yu, C.K. Chen Application of Taylor transformation to optimize rectangular fins with variable thermal parameters", Appl. Math. Model 22, 11-21(1998).

[6] F.Ayas Solutions of the system of differential equations by differential transform method", Appl. Math. Comput. 147,547-567(2004).

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[8] K. S. Aboodh, The New Integral Transform Aboodh Transform" Global Journal of pure and Applied Mathematics, 9(1), 35-43(2013).

[9] K. S. Aboodh, Application of New Transform Aboodh transform" to Partial Differential Equations, Global Journal of pure and Applied Math, 10(2),249- 254(2014).