Solution of Non linear Heat Equation Using Differential Transform Method and Taylor Series Method

Available online at: http://www.ijmtst.com/vol3issue10.html

International Journal for Modern Trends in Science and Technology ISSN: 2455-3778 :: Volume: 03, Issue No: 10, October 2017

Solution of Non linear Heat Equation Using Differential Transform Method and Taylor Series Method R.Selvi1 | T.Ramesh2 1PG

Scholar, Department of Mathematics, Dr.SNS Rajalakshmi College of Arts and Science, Coimbatore, Tamilnadu, India.

Professor, Department of Mathematics, Dr.SNS Rajalakshmi College of Arts and Science, Coimbatore, Tamilnadu, India.

2Assistant

To Cite this Article R.Selvi and T.Ramesh, “Solution of Non linear Heat Equation Using Differential Transform Method and Taylor Series Method�, International Journal for Modern Trends in Science and Technology, Vol. 03, Issue 10, October 2017, pp:01-03.

ABSTRACT This paper the application of differential transform method (DTM) and Taylor series method (TSM) to find the exact and approximate solutions of the heat equation with nonlinearity. Although the both of methods provide the solution in an infinite series, the DTM provides a fast convergent series of easily computable components and eliminates heavy computational work needed by TSM. KEYWORDS: Differential Transform Method, Nonlinear Heat Equation, Taylor Series Method.

Copyright Š 2017 International Journal for Modern Trends in Science and Technology All rights reserved.

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I. INTRODUCTION The DTM is a semi analytical-numerical method that depends on Taylor series. It was introduced by zhou in 1986 for solving the linear and nonlinear initial value problem that appear in electrical circuits. It was found that, unlike other series solution methods, the DTM is easy to program in engineering problems, and provides solution terms without linearization and discretization. In [2] ,the advantage of the ADM has been expressed. The most recent application of the DTM and TSM lies in solving the nonlinear heat equation.

The DTM for the heat equation Consider the following problem with solutions Ut(x,t)=uxx+um‌ (1) U(x,0)=f(x) ‌ (2) If the function u(x,t) is analytic and differential continuously with respect to time t and space x in the domain of interest , then let

đ?‘˘đ?‘˜ đ?‘Ľ = đ?‘˜ !

đ?œ•đ?‘˜ đ?œ•đ?‘Ą đ?‘˜

�(�, �) ‌ (3)

Where the t-dimensional spectrum function uk(x) is transformed of uk(x) is defined as đ?‘˘ đ?‘Ľ, đ?‘Ą = ∞đ?‘˜=0 đ?‘˘đ?‘˜ đ?‘Ľ đ?‘Ą đ?‘˜ ‌ (4) From (1) and (2) the function u(x,t) can be described as 1

đ?œ•đ?‘˜

đ?‘˘ đ?‘Ľ, đ?‘Ą = ∞đ?‘˜=0 đ?‘˘(đ?‘Ľ, đ?‘Ą) đ?‘Ą đ?‘˜ ‌ (5) đ?‘˜ ! đ?œ•đ?‘Ą đ?‘˜ The differential transform method to obtain the solution of equation (1) and (2) by taking the differential transform on both sides we have, đ?œ•đ?‘˜

đ?‘˜ + 1 đ?‘˘đ?‘˜+1 đ?‘Ľ = đ?‘˜ đ?‘˘đ?‘˜ đ?‘Ľ + đ?‘“đ?‘˜ (đ?‘Ľ) ‌ (6) đ?œ•đ?‘Ą U0(x) = f(x) Where Fk(x) are, F0(x) = đ?‘˘0đ?‘š (đ?‘Ľ) đ??š1 đ?‘Ľ = đ?‘šđ?‘˘0đ?‘š −1 (đ?‘Ľ)đ?‘˘1 (đ?‘Ľ) 1 đ??š2 đ?‘Ľ = 2 đ?‘š đ?‘š − 1 đ?‘˘0đ?‘š −2 đ?‘Ľ đ?‘˘12 đ?‘Ľ + đ?‘šđ?‘˘0đ?‘š −1 (đ?‘Ľ)đ?‘˘2 (đ?‘Ľ) 1 đ?‘š đ?‘šâˆ’1 6 đ?‘Ľ đ?‘˘13 đ?‘Ľ + đ?‘š đ?‘š − 1 đ?‘˘0đ?‘š −2 đ?‘Ľ đ?‘˘2 đ?‘Ľ đ?‘˘1 đ?‘Ľ + đ?‘šđ?‘˘0đ?‘šâˆ’1 (đ?‘Ľ)đ?‘˘3 (đ?‘Ľ) đ??š3 đ?‘Ľ =

đ?‘š − 2 đ?‘˘0đ?‘šâˆ’3

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R.Selvi and T.Ramesh : Solution of Non linear Heat Equation Using Differential Transform Method and Taylor Series Method From (4) we have, đ?‘˘ đ?‘Ľ, đ?‘Ą = đ?‘˘0 đ?‘Ľ + đ?‘˘1 đ?‘Ľ đ?‘Ą + đ?‘˘2 đ?‘Ľ đ?‘Ą 2 + đ?‘˘3 đ?‘Ľ đ?‘Ą 3 + â‹Ż + đ?‘˘đ?‘› đ?‘Ľ đ?‘Ą đ?‘› + â‹Ż (7) đ?‘˘đ?‘› đ?‘Ľ, đ?‘Ą = đ?‘›đ?‘˜=0 đ?‘˘đ?‘˜ (đ?‘Ľ) đ?‘Ą đ?‘˜ ‌ (8) Where n is order of approximation solution Therefore, the exact solution of the equation (1) is given by, đ?‘˘ đ?‘Ľ, đ?‘Ą = limđ?‘›â†’∞ đ?‘˘đ?‘› (đ?‘Ľ, đ?‘Ą)‌ (9)

� �, � =

Solution with TSM U(x,t)=f(x)

The TSM for the heat equation

U'(x0)=

A Taylor series is a series expansion of a function based on the values of the values of the function and derivatives at one point one form for a taylor series expansion is, F(x)=f(x0)+f’(x0)(x-x0)+f�(x0)/2(x-x0)2+‌‌‌.

(10) When x0=0 it is also called Machaurin series for instance cos đ?‘Ľ = cos 0 − sin 0 đ?‘Ľ − cos 0

đ?‘Ľ2 2!

+â‹Ż

đ?œ•đ?‘“ đ?œ•đ?‘Ľ

đ?‘Ľ0 +

1

đ?œ• 2đ?‘“

2!

đ?œ•đ?‘Ľ 2

(đ?‘Ľ02 ) + 2

đ?œ• 2đ?‘“ đ?œ•đ?‘Ľ 2

U''(x0)=

đ?‘Ľ0 +

đ?œ•2đ?‘“đ?œ•đ?‘Ľ2(đ?‘Ľ1)+13!đ?œ•2đ?‘“đ?œ•đ?‘Ľ2(đ?‘Ľ0)3+3đ?œ•2đ?‘“đ?œ•đ?‘Ľ2(đ?‘Ľ1)2+đ?œ•2đ?‘“đ?œ•đ?‘Ľ2 đ?‘Ľ2+‌‌ (13)

Applications In this section, we solve nonlinear heat equation by DTM and TSM Example Consider the following nonlinear heat equation đ?‘˘đ?‘Ą đ?‘Ľ, đ?‘Ą = đ?‘˘đ?‘Ľđ?‘Ľ − 2đ?‘˘3 1 + 2đ?‘Ľ đ?‘˘0 đ?‘Ľ = 2 đ?‘Ľ +đ?‘Ľ+1

U'''(x0)= (� 2 +�+1)4 From (13), � �, � =

Example Consider the equation, đ?‘Ľ 1 − đ?‘Ľđ?‘Ś Solution with DTM đ?‘˘0 đ?‘Ľ =

đ?‘Ľ

đ??š0 đ?‘Ľ = [

đ??š0 đ?‘Ľ = [ đ?‘˘1 đ?‘Ľ =

−6đ?‘Ľ

đ??š1 đ?‘Ľ =

−18đ?‘Ľ 3 (1 − đ?‘Ľđ?‘Ś)2

36đ?‘Ľ

đ?‘˘2 đ?‘Ľ = (1−đ?‘Ľđ?‘Ś )3 216đ?‘Ľ 3 đ??š2 đ?‘Ľ (14) = (1 − đ?‘Ľđ?‘Ś)5 −216 đ?‘Ľ

đ?‘˘3 đ?‘Ľ = (1−đ?‘Ľđ?‘Ś )4

1 + 2đ?‘Ľ 3 ] +đ?‘Ľ+1

−6(1+2đ?‘Ľ)

đ?‘Ľ 6đ?‘Ľ 36đ?‘Ľ 216đ?‘Ľ − + − 1 − đ?‘Ľđ?‘Ś (1 − đ?‘Ľđ?‘Ś)2 (1 − đ?‘Ľđ?‘Ś)3 (1 − đ?‘Ľđ?‘Ś)4 +â‹Ż

Solution with TSM � � �0 =

−18(1+2đ?‘Ľ)3

(16)

1−đ?‘Ľđ?‘Ś

(đ?‘Ľ 2 +đ?‘Ľ+1)2 36(1+2đ?‘Ľ)

−6đ?‘Ľ

U'(x0)=(1−đ?‘Ľđ?‘Ś )2

�2 � = (� 2 +�+1)3

3

đ??š2 đ?‘Ľ = −216 (1+2đ?‘Ľ)

From (7)

đ?‘Ľ ]3 1 − đ?‘Ľđ?‘Ś

(1−đ?‘Ľđ?‘Ś )2

đ?‘Ľ2

�1 � = (� 2 +�+1)2

(đ?‘Ľ 2 +đ?‘Ľ+1)4

(13)

1−đ?‘Ľđ?‘Ś

� �, � =

�3 � =

1 + 2đ?‘Ľ 6 1 + 2đ?‘Ľ − 2 đ?‘Ą + đ?‘Ľ + 1) (đ?‘Ľ + đ?‘Ľ + 1)2 36 1 + 2đ?‘Ľ 216 1 + 2đ?‘Ľ 3 + 2 đ?‘Ą2 − 2 đ?‘Ą 3 (đ?‘Ľ + đ?‘Ľ + 1) (đ?‘Ľ + đ?‘Ľ + 1)4 +â‹Ż

(đ?‘Ľ 2

From (7)

Solution with DTM

đ??š1 đ?‘Ľ =

−6(1+2đ?‘Ľ)

(đ?‘Ľ 2 +đ?‘Ľ+1)2 36(1+2đ?‘Ľ)

(đ?‘Ľ 2 +đ?‘Ľ+1)3 −216 (1+2đ?‘Ľ)

(11)

A Taylor’s series can also written in terms, � � = � �0 + � ′ �0 + �"(�0 ) + ⋯(12) For a function of Taylor series derivatives of heat equation, � � = � �0 +

1 + 2đ?‘Ľ 6 1 + 2đ?‘Ľ − 2 đ?‘Ą + đ?‘Ľ + 1) (đ?‘Ľ + đ?‘Ľ + 1)2 36 1 + 2đ?‘Ľ 216 1 + 2đ?‘Ľ 3 + 2 đ?‘Ą2 − 2 đ?‘Ą 3 (đ?‘Ľ + đ?‘Ľ + 1) (đ?‘Ľ + đ?‘Ľ + 1)4 +â‹Ż

(đ?‘Ľ 2

216(1 + 2đ?‘Ľ) (đ?‘Ľ 2 + đ?‘Ľ + 1)5

U''(x0)=

(17)

36đ?‘Ľ

(1−đ?‘Ľđ?‘Ś )3 −216 đ?‘Ľ

U'''(x0)=(1−đ?‘Ľđ?‘Ś )4 From (13), đ?‘˘ đ?‘Ľ, đ?‘Ą =

(18)

đ?‘Ľ 6đ?‘Ľ 36đ?‘Ľ 216đ?‘Ľ − + − 1 − đ?‘Ľđ?‘Ś (1 − đ?‘Ľđ?‘Ś)2 (1 − đ?‘Ľđ?‘Ś)3 (1 − đ?‘Ľđ?‘Ś)4 +â‹Ż

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R.Selvi and T.Ramesh : Solution of Non linear Heat Equation Using Differential Transform Method and Taylor Series Method II. CONCLUSION This paper proposes the application of DTM and TSM for solving the nonlinear heat equations. The two series methods were applied separately to nonlinear heat equations.The study showed that the DTM is simple and easy to use and produces reliable results. The method also minimizes the computational difficulties of the TSM in that the components are determined elegantly by recurrence equations.

REFERENCES [1] A.Yazdani(2016) comparison between differential transform method and Taylor series method for solving linear and nonlinear ordinary differential equations vol.6, Apr pp 2872-2877.

[2] Shawqi Malek Alhaddad (2017) Adomian decomposition method for solving the nonlinear heat equation vol.7,issue.4,Apl,pp.97-100.

[3] Ekaterina kutafina (2008) Taylor series for adomian decomposition method jan,pp.1-8 [4] ZahraAdabiFiroozjae (2015) The comparison adomian decomposition method for solving some nonlinear partial differential equations, vol.3, issue.3, june,pp.90-94

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