The Use of Kamal Transform for Solving Partial Differential Equations

Transcription

1 Adances in Theoretical and Applied Mathematics ISSN Volume 12, Number 1 (217), pp Research India Publications The Use of Kamal Transform for Soling Partial Differential Equations 1,2 Abdelilah. K. Hassan Sedeeg and 3,4 Zahra. I. Adam Mahamoud 1 Mathematics Department Faculty of Education, Holy Quran and Islamic Sciences Uniersity, Sudan. 2 Mathematics Department Faculty of Sciences and Arts, Almikwah, Albaha Uniersity, Saudi Arabia. 3 Mathematics Department Faculty of Sciences and Arts, Bukirayh, Alqassim Uniersity, Saudi Arabia. 4 Mathematics Department Faculty of Education, Omdurman Islamic Uniersity, Sudan. Abstract Kamal Transform of partial deriatie is deried, and its applicability demonstrated using four different partial equations. In this paper we find the particular solutions of these equations. 1. INTRODUCTION The differential equation hae played a central role in eery aspect of applied mathematics for eery long time and with the adent of the computer, their importance has increased father. Thus inestigation and analysis of differential equations cruising in applications led to many deep mathematical problems; therefore, there are so many different techniques in order to sole differential equations. The integral transform were extensiely used and thus there are seeral words on the theory and applications of integral transforms such as the Laplace, Foureir, Mellin, Hankel and Sumudu, to name but a few. Recently, Abdelilah Kamal introduced a new integral transform, named the Kamal transform, and further applied it to the solution of ordinary and partial differential equations.

2 8 Abdelilah K. Hassan Sedeeg and Zahra. I. Adam Mahamoud In this paper we drie the formulate for Kamal transform of partial deriaties and apply them in soling some types of initial alue problems. Our purpose here is to show the applicability of this interesting new transform and its effecting in soling such problems. 2. DEFINITION AND DERIVATIONS KAMAL TRANSFORM OF DERIVATIVES: Kamal transform of the function f (t) is defined as [f(t)] = f(t)e t dt = G(), t >, k 1 k 2 (1) To obtain Kamal transform of partial deriaties we use integration by parts as follows: K [ f(x,t) ] = f(x,t) e t dt = lim ([e t p f(x, t)] p = f(x, ) + 1 G(x, ) p = lim e t p p f(x,t) dt + 1 e t f(x, t)dt ) Thus, K [ f(x,t) ] = 1 G(x, ) f(x, ) (2) To find K [ 2 f(x,t) ], let 2 f(x,t) = g(x, t), then by using Eq. (2) we hae: K [ 2 f(x, t) g(x, t) 2 ] = K [ ] = 1 g(x, t) K [ ] g(x, ) K [ 2 f(x,t) ] = 2 G(x, ) 1 f(x, ) f(x,) 2 (3) We can easily extend this result to the nth partial deriatie by using mathematical induction. Now, we assume the f(x, t) is piecewise continuous and is of exponential order. Then, f(x, t) K [ ] = e t x f(x, t) dt x

3 The Use of Kamal Transform for Soling Partial Differential Equations 9 Using the Leibniz rule Thus, Also we can find : In summary, K [ f(x,t) ] = x e t f(x,t) x dt = x e t f(x, t)dt K [ f(x,t) ] = d (G(x, )) (4) x dx K [ 2 f(x,t) x 2 K [ n f(x,t) x n ] = d2 2 (G(x, )) (5) dx ] = dn n (G(x, )) (6) dx 3. APPLICATION OF KAMAL TRANSFORM OF PARTIAL DIFFERENTIAL EQUATIONS: In this section we sole first order Partial differential Equations and the Second order partial differential equation, wae equation, heat equation, Laplace equation and Telegraphers equation which are known as four Fundamental equations in mathematical physics and occur in many branches of physics, in applied mathematics as well as in engineering. Example 3. 1: Find the solution of the first - order initial alue problem: u x (x, t) 2u t (x, t) = u(x, t), x >, t >, u(x, ) = e 3x (7) Taking Kamal transform of Eq. (7), we hae G (x, ) 2 1 G(x, ) + 2u(x, ) = G(x, ) Where G(x, ) is Kamal transform of u(x, t). By applying the initial condition, we get G (x, ) ( 2 + 1) G(x, ) = 2e 3x This is the linear ordinary differential equation, it has the integration factor F = e (2 +1)dx = e (2 +1)x Therefore, G(x, ) = 1+2 e 3x + ce ( 2 +1)x (8)

4 1 Abdelilah K. Hassan Sedeeg and Zahra. I. Adam Mahamoud Since G(x, ) is bounded, c Should be zero, if we take the inerse Kamal transform to Eq. (8), then the solution of Eq. (7) is: Example 3. 2: Let's consider the wae equation : u (x, t ) = e 3x 2t u tt u xx =, x π, t (9) u(x, ) = sin x, u(, t) =, u t (x, ) =, u(π, t) = Applying Kamal transform to both sides of this equation and using the differential property of Kamal transform, Eq. (9) can be written as: And, 2 G(x, ) 1 u(x, ) u t (x, ) G (x, ) = G (x, ) 1 sin x 2 G(x, ) = This is the second - order ordinary differential equation hae the solution in the form : G(x, ) = sinx (D 2 1 2) = sinx ( 1 1 2) = sinx 1+ 2 (1) Now, if we take the inerse Kamal transform to Eq. (1), then the particular solution of Eq. (9) is: u(x, t) = sinx cost Example 3. 3: Let's consider the homogeneous heat equation in one dimension in a normalized form: u t = u xx, (x, ) = sin π x, u(, t) = u(l, t) = (11) l Applying Kamal transform to both sides of this equation and using the differential property of Kamal transform, Eq. (11) can be written as: 1 G(x, ) u(x, ) = G (x, ) And, G (x, ) 1 G(x, ) = sin π l x This is the second - order ordinary differential equation hae the particular solution in the form G(x, ) = sinπ l x D 2 1 = sinπ l x = sinπ l x ( π l )2 1 1+( π l )2 (12)

5 The Use of Kamal Transform for Soling Partial Differential Equations 11 Now, if we take the inerse Kamal transform to Eq. (12), then the solution of Eq. (11) is: u(x, t) = sin π l x e (π l )2 t Example 3. 4: Let's consider the homogeneous Laplace equation: u xx + u tt = u(x, ) =, u t (x, ) = cos x x, t > (13) Applying Kamal transform to both sides of this equation and using the differential property of Kamal transform, Eq. (13) can be written as: G (x, ) + 2 G(x, ) 1 u(x, ) u t (x, ) = G (x, ) + 2 G(x, ) = cos x This is the second - order ordinary differential equation hae the particular solution in the form: G(x, ) = cos x = D cos x = 2 cos x (14) Now, if we take the inerse Kamal transform to Eq. (14), then the solution of Eq. (13) is : u(x, t) = cos x sinh t Example 3. 5: Consider the telegraphers equation: u tt (x, t) + 2αu t (x, t) = α 2 u xx (x, t), < x < 1, t > (15) With the initial conditions: u(x, ) = cosx, u t (x, ) = Take Kamal transform of Eq. (15), we get: 2 G(x, ) 1 u(x, ) u t (x, ) + 2α 1 G(x, ) 2αu(x, ) = α 2 G (x, ) Applying the initial conditions, we get α 2 G (x, ) ( 1+2α 2 ) G(x, ) = cosx ( 1+2α )

6 12 Abdelilah K. Hassan Sedeeg and Zahra. I. Adam Mahamoud This is the second - order ordinary differential equation hae the particular solution in the form : G(x, ) = cosx (1+2α) α 2 D 2 ( 1+2α ) = cosx( + 2α 2 ) α 2 2 (1 + 2α) = cosx( + 2α2 ) (1 + α) 2 2 Or G(x, ) = cosx ( α2 + ) (16) (1+α) 2 1+α Now, if we take the inerse Kamal transform to Eq. (16), then the solution of Eq. (15) is : u(x, t) = cosx (1 + αt)e αt REFERENCES [1] Abdelilah Kamal. H. Sedeeg, The New Integral Transform Kamal Transform, Adances in Theoretical and Applied Mathematics, Vol. 11, No. 4, 216, pp [2] Tarig M. Elzaki and Salih M. Elzaki, Applications of New Transform ELzaki Transform to Partial Differential Equations, Global Journal of Pure and Applied Mathematics,,211,pp65-7. [3] Abdelilah. K. Hassan Sedeeg and Mohand M. Abdelrahim Mahgob Comparison of New Integral Transform '' Aboodh Transform" and Adomian Decomposition Method, International Journal of Mathematics And its Applications Volume 4, Issue 2-B (216), p ISSN: [4] K. Sankara Rao, Introduction to Partial Differential Equations, 1995 [5] Lokenath Debnath and D. Bhatta. Integral transform and their Application second Edition, Chapman & Hall /CRC (26). [6] G. K. watugala, simudu transform- anew integral transform to Sole differential equation and control engineering problems. Math. Engrg Induct. 6 (1998), no 4, [7] A. Kilicman and H. E. Gadain. An application of double Laplace transform and sumudu transform, Lobacheskii J. Math. 3 (3) (29), pp [8] J. Zhang, Asumudu based algorithm m for soling differential equations, Comp. Sci. J. Moldoa 15(3) (27), pp [9] Christian Constanda, Solution Techniques for Elementary Partial differential Equations, New York, 22. [1] Dean G. Duffy, Transform Methods for soling partial differential Equations, 2nd Ed, Chapman & Hall / CRC, Boca Raton, FL, 24. [11] SunethraWeera Koon, Application of Sumudu transform to partial differential equation. INT. J. MATH. EDUC. Scl. TECHNOL, 1994, Vol. 25, No2,

7 The Use of Kamal Transform for Soling Partial Differential Equations 13 [12] Hassan Eltayeb and Ademkilicman, A Note on the Sumudu Transforms and differential Equations, Applied Mathematical Sciences, VOL, 4,21, no. 22, [13] Kilicman A. & H. ELtayeb. A note on Integral transform and Partial Differential Equation, Applied Mathematical Sciences, 4(3) (21), PP [14] Hassan ELtayeh and Ademkilicman, on Some Applications of a new Integral Transform, Int. Journal of Math. Analysis, Vol, 4, 21, no. 3, [15] A. Aghili, B. Salkhordeh Moghaddam, Laplace transform Pairs of N- dimensions and second order Linear partial differential equations with constant coefficients, Analysis Mathematicae et Informaticae, 35 (28),pp, 3-1. [16] Hassan ELtayeb, Ademkilicman and Brian Fisher, A new integral transform and associated distributions, Integral Transform and special Functions. Vol, co, No, Month 29, [17] M. G. M. Hussain, F. B. M. Belgacem. Transient Solution of Maxwell's Equations Based on Sumudu Transform, Progress in Electromagnetic Research, PIER, , (27).

8 14 Abdelilah K. Hassan Sedeeg and Zahra. I. Adam Mahamoud