A Note on the Shifting Theorems for the Elzaki Transform

Transcription

1 Int. Jornal of Mh. Analysis, Vol. 8, 214, no. 1, HIKARI Ltd, A Note on the Shifting Theorems for the Elzaki Transform Hwajoon Kim Kyngdong University School of IT Engineering Goseong , Gangwon, Korea Corresponding hor Copyright c 214 Hwajoon Kim. This is an open access article distribted nder the Creive Commons Attribtion License, which permits nrestricted se, distribtion, and reprodction in any medim, provided the original work is properly cited. Abstract We have proposed the shifting theorems for the Elzaki transform which was proposed by Elzaki in 211 to solve initial vale problems in controll engineering problems. The proposed theorems are composed of the -shifting theorem and the time shifting, and the proof is done correctly when is compared with established ones. Additionally, we have dealt with the alternive form of it. Mhemics Sbject Classificion: 44A5, 34A2 Keywords: shifting theorem, Elzaki transform 1 Introdction The Elzaki transform[5-9], a kind of modified Laplace s/smd, was introdced by Elzaki in 211, and it is defined by T () = e t/ f(t)dt, for E[f(t)] = T (). Elzaki insists th the Elzaki transform shold be easily applied to the initial vale problems with less comptional work, and solve the varios examples which are not solved by the Laplace or the Smd

2 482 Hwajoon Kim transform[9]. We wold like to take a look into preceding researches of this topic. Wgla[12] proposed the Smd transform defined by G() = 1 e t/ f(t)dt for S[f(t)] = G(), Eltayeb[4] applied the transform to find the soltion of ODEs, Agwa[1] dealt with the transform on time scales. On the other hand, Ali[2] applied the Elzaki transform to finding the soltion of Abels integral eqions, and Hesameddini/Elzaki sed Homotopy pertrbion and Elzaki transform to obtain the approxime analytical soltion of Sine-Gorden and Klein-Gorden eqions/nonlinear partial eqions in [1]/[8], respectively. Cho[3] has tried to find the soltion of Bessel s eqion by sing the transform. Althogh this transform has strong points to find the soltion of differential eqions with variable coefficients, the shifting theorem does not have. Hence we wold like to propose the proof which is faithfl to the definition. In this article, we have proposed the shifting theorems for the Elzaki transform, and they are better when are compared with existing proofs. 2 The shifting theorems for the Elzaki transform To begin with, let s check the definition of the Elzaki transform. Definition 2.1 The Elzaki transform of the fnctions belonging to a class A, where A = {f(t) M,k 1,k 2 > sch th f(t) <Me t /k j,ift ( 1) j [, )} where f(t) is denoted by E[f(t)] = T () and defined as or eqivalently, T () = 2 f(t)e t dt, k 1,k 2 >, T () = f(t)e t/ dt, (k 1,k 2 ). The following reslts can be obtained from the definition and simple calclions. 1)E[f (t)] = T ()/ f() 2)E[f (t)] = T ()/ 2 f() f () 3)E[tf (t)] = 2 d [T ()/ f()] [T ()/ f()] d

3 A note on the shifting theorems for the Elzaki transform 483 4)E[t 2 f (t)] = 4 d2 [T ()/ f()] d2 5)E[tf (t)] = 2 d d [T ()/2 f() f ()] [T ()/ 2 f() f ()] 6)E[t 2 f (t)] = 4 d2 d [T 2 ()/2 f() f ()]. for E(f(t)) = T ()[6-7]. Elzaki proposed the shifting theorem in [6] which is T {e f(t)} = 1 1 a T ( 1 a ) for E{f(t)} = T (). Since this has a calclional problem, we wold like to clarify the shifting theorems for the Elzaki transform. The proposed proof is done by the different method, and make a comparison between the direct calclion and it. Additionally, we have checked the alternive form of the -shifting theorem, and the time shifting as well. Theorem 2.2 (-shifting theorem for the Elzaki transform) If f(t) has the transform T (), then e f(t) has the transform (1 a)t ( 1 a ) where a<1/ and E{f(t)} = T (). In formlas, T {e f(t)} =(1 a)t ( 1 a ) ( ) for a<1/. Proof. T ( 1 a )= 1 a for E{f(t)} = T (). Since f(t)e t 1 a dt we have t 1 a = t (1 a) = t +, T ( 1 a )= 1 1 a f(t)e t e dt = 1 1 a T {e f(t)}. Althogh the above theorem is very simple, it has a practical meaning. Let s consider some examples in order to check the validity of the theorem 2.2.

4 484 Hwajoon Kim Example 2.3 (The formlas of e sin wt and e cos wt) By the definition of the Elzaki transform, we have E(sin wt) = w3 1+w 2. 2 From this eqion and the (*) we immediely obtain the formla E(e sin wt) =(1 a) Organizing the eqality, we have Then =(1 a) E(e sin wt) w( 1 a )3 1+w 2 ( 1 a )2. w 3 (1 a) 3 + w 2 2 (1 a) w 3 = (1 a) 2 + w 2. ( ) 2 Next, we wold like to approach it by the direct calclion. Let s pt E(e sin wt) =I. I = = Using the integrion by parts, we have e t/ e sin wt dt e (a 1/)t sin wt dt. I = [ 1 w e(a 1 )t coswt] + w (a 1 ) cos wt e (a 1 )t dt = w + w (a 1 ) cos wt e (a 1 )t dt becase of a<1/. Once again, applying the above integral to the integrion by parts, we have I = w + w (a 1 )([ 1 w e(a 1 )t sin wt] (a 1 ) 1 w e (a 1 )t sin wt dt) = w 1 w 2 (a 1 ) 2I.

5 A note on the shifting theorems for the Elzaki transform 485 Organizing the eqality, we have and so, [1 + 1 w 2 (a 1 ) 2 ] I = w [ w2 2 +(a 1) 2 ] I = w 2 2 w. Ths, we obtain the desired reslt I = w 3 w 2 2 +(a 1) 2 for a<1/. Of corse, this is the same reslt with (**). Similarly, we can obtain the formla E(e cos wt) = (1 a) 2 (1 a) 2 + w 2 2. Next, let s check the alternive form T ( a) of the -shifting theorem. Srely, it is practical for solving differential eqions with variable coefficients by sing the Elzaki transform. Example 2.4 Find the inverse of the Elzaki transform E(y(t)) = Soltion. 2 y(t) =3E 1 3 2(1 ) 2 [ ( 1) ( 1) 2 +2 ] 2 =3e t sin 2t +3 2 e t cos 2t. Corollary 2.5 (alternive form of -shifting theorem) If f(t) has the transform T (), then holds for E{f(t)} = T (). T ( a) =(1 a ) T [e ( a) f(t)]

6 486 Hwajoon Kim Proof. From the definition of the Elzaki transform, Since T ( a) =( a) t a =( t ) a =( t )(1 + e t a f(t)dt. a a )= t ( a), we have T ( a) = Expanding the eqality, we have T ( a) =T [e ( a) e t e ( a) a f(t)] T [e f(t)dt. ( a) f(t)] =(1 a )T [e ( a) f(t)] for E{f(t)} = T (). By the corollary 2.5. T () =(1 a + a ) T [e (+a) f(t)] ( ) holds, and this idea with respect to T () can be applied to another transform sch as Smd. Corollary 2.6 If ( a) = 1, T ( 1 )= ( )T () 2 ( ) 4 holds for E{f(t)} = T (). Proof. It follows from the (*) and the (***). The corollary 2.6 ste th if a>2, (****) holds becase of 1/. On the other hand, we proposed the time shifting theorem in [11]. In formlas, where H(t) is Heaviside fnction. For example, let s check E{f(t a)h(t a)} = e a/ T () E{a sin(t 2)H(t 2)} = ae 2/

7 A note on the shifting theorems for the Elzaki transform 487 for H(t) is Heaviside fnction. From the example 2.3 and the time shifting theorem, the above eqality is held. Next, let s check the above example by a direct calclion. Ptting we have I = E{a sin(t 2)H(t 2)}, I = a e t/ sin(t 2)dt 2 = a{[ e t/ sin(t 2)] 2 + e t/ cos(t 2)dt}. Since the first term is, let s integre by part to the second term. Then we get I = a 2 {[ e t/ cos(t 2)] 2 e t/ sin(t 2)dt} Ths we obtain the reslt = a 2 (e 2/ 1/5I). I = ae 2/ References [1] H. A. Agwa, F. M. Ali and A. Kilicman, A new integral transform on time scales and its applicions, Adv. Differ. eqs., 6 (212), [2] S. S. Ali and M. S. Chadhary, On a new integral transform and soltion of some integral eqions, Int. J. of Pre & Appl. Mh., 73 (211), [3] Ig. Cho and Hj. Kim, The soltion of Bessel s eqion by sing integral transforms, Appl. Mh. Sci., 7 (213), [4] H. Eltayeb and A. kilicman, On Some Applicions of a New. Integral Transform, Int. J. of Mh. Anal., 4 (21), [5] T. M. Elzaki and S. M. Ezaki, Applicions of new transform ELzaki Transform to partial differential eqions, Glob. J. of Pre & Appl. Mh., 7 (211), [6] T. M. Elzaki and S. M. Ezaki, On the connections between Laplace and ELzaki transforms, Adv. in Theo. & Appl. Mh., 6 (211), 1-1.

8 488 Hwajoon Kim [7] T. M. Elzaki and S. M. Ezaki, On the ELzaki transform and ordinary differential eqion with variable coefficients, Adv. in Theo. & Appl. Mh., 6 (211), [8] T. M. Elzaki and J. Biazar, Homotopy pertrbion method and Elzaki transform for solving systems of nonlinear partial differential eqions, Wor. Appl. Sci. J., 24 (213), [9] T. M. Elzaki, S. M. Ezaki and E. M. A. Hilal, ELzaki and Smd Transform for Solving some Differential Eqions, Glob. J. of Pre & Appl. Mh., 8 (212), [1] E. Hesameddini and N. Abdollahy, Homotopy pertrbion and Elzaki transform for solving Sine-Gorden and Klein-Gorden eqions, Iran. J. nt. J. of Nmer. Anal. & Otimi., 3 (213), [11] Hj. Kim, The time shifting theorem and the convoltion for Elzaki transform, Int. J. of Pre & Appl. Mh. 87 (213), [12] G. K. Wgla, Smd Transform: a new integral transform to solve differential eqions and control engineering problems, Int. J. of Mh. Ed. in Sci. & Tech., 24 (1993), Received: Febrary 19, 214