The solutions of partial differential equations with

The solutions of partial differential equations with variable coefficient by Sumudu Transform Method Hasan Bulut, H. Mehmet Baskonus, and Seyma Tuluce Citation: AIP Conf. Proc. 1493, 91 (2012); doi: 10.1063/1.4765475 View online: http://dx.doi.org/10.1063/1.4765475 View Table of Contents: http://proceedings.aip.org/dbt/dbt.jsp?KEY=APCPCS&Volume=1493&Issue=1 Published by the American Institute of Physics.

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The Solutions of Partial Differential Equations with Variable Coefficient by Sumudu Transform Method Hasan BULUT 1 , H. Mehmet BASKONUS 2 and Seyma TULUCE 1 1 2

Department of Mathematics, University of Firat, 23119, Elazig, Turkey

Department of Computer Engineering, University of Tunceli, 62100, Tunceli, Turkey [email protected], [email protected], [email protected]

Abstract. In this paper, we studied to obtain numerical solutions of partial differential equations with variable coefficient by Sumudu Transform Method (STM). We applied to three examples this method. Then, we ploted 2D and 3D graphics of these equations by means of programming language Mathematica. In order to solve these differential equations which is ordinary and partial, the integral transforms such as the Laplace, Hankel and Fourier were unusually used and therefore there are several works on literature and applications of them. But it has few been given by using STM. And thus, in recent years STM has been an important places in applied sciences such as especially engineering, physics, physical sciences. Moreover, it is important for solving ordinary and partial differential equations which is linear and nonlinear. Keywords: Sumudu Transform Method, Partial Differential equations with variable coefficient.

over the set of functions, t j A f t M , 1 , 2 0, f t Me i , if t 1 0,

INTRODUCTION STM was firstly proposed by G. K. Watugala in order to obtain solutions of differential equations [1]. Partial differential equations with variable coefficient have arisen from daily activities such as physic, chemistry, applied sciences, technology of space, communication systems, nonlinear science, numerical simulation and control engineering problems. The coefficients in a partial differential equation depend on the structure of physical problem. G. K. Watugala applied so as to study solutions of various differential equations by STM [2,3]. F. M. Belgacem, and A. A. Karaballi introduced the fundamental properties of sumudu transform [4,5]. It was given that a number of studies in this field [6-14]. In this paper, we used STM in order to find solution of partial differential equation [15].

the Sumudu transform is

G u S f t f ut e t dt , u 1 , 2 . 0

To illustrate the basic ideas of this method, we consider a general linear form of partial differential equations; u x, t 2 u x, t (1) x, t t x 2 with subject to initial condition (2) x, 0 f x , where f (r) is a known analytical function.

ANALYSIS OF THE METHOD

Now, we know that sumudu transform of partial derivatives is as follows [2]:

The Sumudu transform is defined in [4] as following:

9th International Conference on Mathematical Problems in Engineering, Aerospace and Sciences AIP Conf. Proc. 1493, 91-95 (2012); doi: 10.1063/1.4765475 © 2012 American Institute of Physics 978-0-7354-1105-0/$30.00

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dF x, t 1 S F x, u f x, 0 dt u

Where initial conditions for Eq.(4) are

u x,0 u0 x 2 , ut x,0 0.

(2.a)

d 2 F x, t 1 f x,0 S 2 F x, u f x,0 u 2 t dt u

2u 1 f S 2 2 F x, u f x,0 u x,0 , t t u

(2.b)

2 u x, t d 2 F x , u ' x 2 dx 2

d 2 F x, t d 2 F x , u S . 2 dx 2 dx

(2.c) When we regulate Eq.(1) according to Eq.(2.a), Eq.(2.b) and Eq.(2.c), we obtain Sumudu transform of Eq.(1) as following: d 2 F x, u 1 1 F x, u f x, 0 0. dx 2 u x, u u x, u (3) F x, u is founded from Eq.(3) by known methods

f 0 x 2 " F x , u f x , 0 u F t 2 2 2 (6) F " 2 2 F 2 0, xu u

1 u2

where is

F"

1 F x, u x 2 . 2 1 u

transform table in [4], we get solution of Eq.(1) by STM.

(7)

When we take inverse sumudu transform of Eq.(7) by using inverse transform table in [4], we get solution of Eq.(4) by STM as following;

THE APPLICATIONS OF STM

u x, t x 2 cosh t .

Example 1 We consider partial differential equation with variable coefficient

x2 uxx , 0 x 1, t 0. 2

d 2F . If we solve to Eq.(6) by dx 2

known methods [16], we obtain sumudu transform of Eq.(4) as following;

such as HPM, ADM, VIM. Then, when we take inverse sumudu transform of F x, u by using inverse

utt

(5)

We construct a sumudu transform for Eq.(5) as following;

(8)

(4)

1.0 0.8 0.6 0.4 0.2 0.0 0.0 0.2 Exact Sol .

0.4

0.6

0.8

1.0

Figure 1. The 2D and 3D graphics of the solution u x, t of Eq.(4) for F4 when t 0.5 by means of STM


2 u x, t d 2 F x , u . x 2 dx 2 f x 2 1 2 F x, u f x , 0 u 0 F " , t 12 u

Example 2 We consider partial differential equation with variable coefficient

utt

x2 uxx , 0 x 1, t 0. 12

(9)


F"

u x,0 u0 0, ut x,0 x . 4

(11)

(10)

We construct a sumudu transform for Eq.(9) as following;

2u 1 S 2 2 t u

12 x2 F 12 0, x 2u 2 u

where is F "

f 0 F x, u f x, 0 u t

d 2 F . If we solve to Eq.(11) by known dx 2

methods [16], we obtain sumudu transform of Eq.(9) as following;

u F x, u x 4 . 2 1 u

and

(12)

When we take inverse sumudu transform of Eq.(12) by using inverse transform table in [4], we get solution of Eq.(9) by STM as following;

u x, t x 4 sinh t .

(13)

0.5 0.4 0.3 0.2 0.1 0.0 0.0 0.2 Exact Sol .

0.4

0.6

0.8

1.0


u x,0 u0 x3 , ut x,0 0.

Example. 3 We consider partial differential equation with variable coefficient

utt

x2 uxx , 0 x 1, t 0. 6

We construct a sumudu transform for Eq.(14) as following; (14)



(15)

1 F x, u x 3 . 2 1 u

u 1 1 1 f S 2 2 F x, u 2 f x , 0 x, 0 , u u t t u 2

When we take inverse sumudu transform of Eq.(17) by using inverse transform table in [4], we get solution of Eq.(14) by STM as following; (18) u x, t x3 cosh t .

2u x, t d 2 F x, u x 2 dx 2 1 1 1 f x2 2 F x, u 2 f x,0 x,0 F " , 6 u u u t F" where is

F"

6 x F 6 2 0, x 2u 2 u

(17)

(16)

d 2F . If we solve to Eq.(16) by dx 2

known methods [16], we obtain sumudu transform of Eq.(14) as following;

35 30 25 20 15 10 5 0 0.0 0.5 Exact Sol .

1.0

1.5

2.0

2.5

3.0


CONCLUSION REFERENCES We studied to obtain solutions of many partial differential equations with variable coefficient by STM. This new approach applied began to be used in the a lot of different area. We hope that this method will substitute with Laplace transform because it has more suitable and new one than Laplace transform.

1. G. K. Watugala, International Journal of Mathematical Education in Science and Technology, 24, 35-43 (1993). 2. G. K. Watugala, Mathematical Engineering in Industry, 6, 319-329 (1998). 3. G. K. Watugala, Mathematical Engineering in Industry, 8, 293-302 (2002).

As a consequently, we showed that analytical solutions of partial differential equations with variable coefficient reached by STM.

4. F. M. Belgacem, A. A. Karaballi, Journal of Applied Mathematics and Stochastics Analysis, 2006, 1–23 (2006). 5. F. B. M. Belgacem, A. A. Karaballi, Mathematical Problems in Engineering, 2003, 103-118 (2003).


6. H. Eltayeb and A. Kilicman, Applied Mathematical Sciences, 4, 1089-1098 (2010). 7. M. A. Asiru, International Journal of Mathematical Education in Science and Technology, 32, 906-910 (2001).

8. M. A. Asiru, International Journal of Mathematical Education in Science and Technology, 34, 944-949 (2003). 9. M. A. Asiru, International Journal of Mathematical Education in Science and Technology, 33, 441-449 (2002). 10. V. G. Gupta, Bhavna Shrama, and A. Kilicman, Journal of Applied Mathematics, 2010, 9 pages (2010). 11. H. Eltayeb, A. Kilicman, and B. Fisher, Integral Transforms and Special Functions, 21, 367-379 (2010). 12. V. G. Gupta and B. Sharma, Applied Mathematical Sciences, 4, 435-446 (2010). 13. A. Kilicman, H. Eltayeb, and P. R. Agarwal, Abstract and Applied Analysis, 2010, 11 pages (2010). 14. J. Singh, D. Kumar, and Sushila, Adv. Theor. Mech. 4, 165-175 (2011). 15. A. Wazwaz, Partial Differantial Equations: Methods and Applications, 2002, USA pp.139 16. H. Bulut, H. M. Baskonus and S. Tuluce, International Journal of Basic & Applied Sciences, 12(1), 6-16 (2012).
