Solving Ordinary differential equations with variable coefficients

Transcription

1 Jornal of Progreive Reearch in Mathematic(JPRM) ISSN: SCITECH Volme 1, Ie 1 RESEARCH ORGANISATION Pblihe online: November 3, 216 Jornal of Progreive Reearch in Mathematic Solving Orinary ifferential eqation with variable coefficient 1,3 Abelbagy.A.Alhikhan 2,3 Mohan M. Abelrahim. Mahgob 1 Mathematic Department Faclty of Ecation-AlzaeimAlazhari Univerity-San 2 Department of Mathematic, Faclty of Science & technology, Omrman Ilamic Univerity, Khartom, San 3 Mathematic Department Faclty of Science an Art-Almikwah-Albaha Univerity- Sai Arabia Abtract ZZ tranform, whoe fnamental propertie are preente in thi paper, i till not wiely known, nor e, ZZ tranform may be e to olve problem withot reorting to a new freqency omain. In thi paper, we e ZZ Tranform to olve Orinary Differential eqationwith variable coefficient.the relt reveal that the propoe metho i very efficient, imple an can be applie to linear an nonlinear ifferential eqation. Keywor: ZZ tranform- ifferential eqation. Introction Integral tranform [1-2] play an important role in many fiel of cience. In literatre, integral tranform are wiely e in mathematical phyic, optic, engineering mathematic an, few other. Among thee tranform which were extenively e an applie on theory an application arelaplace Tranform, Forier Tranform, SmTranform [3], Elzaki Tranform [4-6], ZZ Tranform, Natral Tranform, an Abooh Tranform [7-9. Of thee the mot wiely e tranform i Laplace Tranform. New integraltranform, name a ZZ Tranformation [1-13] introce by ZainUlAbainZafar[216], ZZ tranform wa cceflly applie to integral eqation, partial ifferential eqation, orinary ifferential eqationan ytem of all thee eqation. The main objective i to introce oltion oforinary Differential Eqation with Variable Coefficient by ing a ZZtranform.The plane of the paper i afollow: In ection 2, we introce the baic iea of ZZtranform, application in 3 an conclion in 4, repectively. 2. Definition an Stanar Relt The ZZ Tranform: Definition: Let (t) be a fnction efine for all t. The ZZ tranform of f t i the fnction Z, Define by Volme 1, Ie 1 available at

2 Z, = H f t = f t e t t Jornal of Progreive Reearch in Mathematic(JPRM) ISSN: Provie the integral on the right ie exit. The niqe fnction f t in (1) i calle the invere Tranform of Z, i inicate by Eqation (1) can be written a H f t = f t e t t ZZ tranform of ome fnction: H 1 = 1 H in(at) =, H t n n = n!, H n eat = a f t = H 1 Z, a 2, H co(at) = 2 +a a 2 2. (1) (2) ZZ tranform of erivative: Theorem I If ZZ tranform of the fnction f t given byh f t = Z,, then: 1) let H f t = Z, then H f n (t) = n Z, n 2) (i) H tf(t) = 2 Z, + Z, (ii) H tf (t) = 2 Z, + Z, n 1 n k n k k= (iii) H tf (t) = Z, Z, + f() f k Proof : 2) (i) z, = H f t = Z, = f t e t t f t e t t Z, =. 2 = e t tf t e t t 2 f t t 2 f t e t t f t e t t Volme 1, Ie 1 available at

3 Jornal of Progreive Reearch in Mathematic(JPRM) ISSN: Z, = 2 H tf t 1 z, 2 H tf t = Z, + 1 z, H tf t = 2 Z, + The proof of ii an iii are imilar to the Proof of i. z, Now we apply the above theorem to fin ZZtranform for ome ifferential eqation: 4 Application : Example 4.1 Solve the ifferential eqation: y + ty y = With the initial conition, y =, y = 1 (4) oltion Uing the ifferential property of ZZ tranform Eq.(3) can be written a 2 2 Z, + 2 Z, Z, Z, + Z, Z, = 2 2 Z, Z, = Z, = (5) 2 Thi i a linear ifferential eqation for nknown fnction Z,, have the Soltion in the form Z, = + c an C, then:z, = By ing the invere ZZ tranform we obtain the Soltion in the form of y t = t Example 4.2 Solve the ifferential eqation: y + 2ty 4y = 6 With the initial conition, y =, y = (9) Soltion Uing the ifferential property of ZZ tranform Eq.(8) can be written a 2 2 Z, Z, Z, + 2Z, 4Z, = 6 2 (3) (6) (7) (8) Volme 1, Ie 1 available at

4 Z, Jornal of Progreive Reearch in Mathematic(JPRM) ISSN: Z, + 2 Z, 4Z, = 6 Z, = 3 Thi i a linear ifferential eqation for nknown fnction Z,, have the Soltion in the form 1 2 Z, = Ce 4 2 an C, then: Z, = 2 (11) 2 By ing the invere ZZ tranform we obtain the Soltion in the form of y t = 3t 2 (12) Example 4.3Conier the econ-orer ifferential eqation ty t + t + 1 y t + 2y t = e t (13) With the initial conition, y =, y = 4 (14) Soltion: Applying the ZZ tranform of both ie of Eq. (13),: H ty t H t + 1 y t + H 2y = H, o (15) Uing the ifferential property of ZZ tranform Eq.(15) can be written a: Z, Z, + 2 y + 2Z, = + Uing initial conition (14), Eq. (16) can be written a Z, + Z, + Z, y + + Z, + 2Z, = + 2 Z, + Z, = + (1) (16) + 2 (17) Thi i a linear ifferential eqation for nknown fnction Z,, have the Soltion in the form Z, = c, an c By ing the invere ZZ tranform we obtain the Soltion in the form of Y(t) = te t (18) Example 4.4 Conier the initial vale problem ty t + y t + ty t = (19) With the initial conition y = 1, y = (2) Soltion: Volme 1, Ie 1 available at

5 Applying the ZZ tranform to both ie of (19) we have Jornal of Progreive Reearch in Mathematic(JPRM) ISSN: H t y t + H y t + H ty t = (21) Uing the ifferential property of ZZ tranform Eq.(21) can be written a: Z, Z, + y + Z, 2 y + Now applying the initial conition to obtain Therefore Z, = c 2 Z, + Z, = Z, Z, = 2 Z, + 2 Z, + Z, = (22) (23) Now applying the invere ZZ tranform, we get y t = J t Example 4.5 Conier the initial vale problem ty t ty t + y t = 2 (25) With the initial conition y = 2, y = 1 (26) Soltion: Applying the ZZ tranform to both ie of (25) we have H t y t H ty t + H y t = H 2 (27) Uing the ifferential property of ZZ tranform Eq.(27) can be written a Z, Z, + f + Now applying the initial conition to obtain Z, 2 Z, + 2 (24) Z, + Z, + Z, = 2 (28) Z, Z, = 2 2 Z, 1 Z, = 2 (29) Eqation (29) i a linear ifferential eqation, which ha oltion in the form Z, = 2 + c (3) Now applying the invere ZZ tranform, we get y t = 2 + ct (31) Volme 1, Ie 1 available at

6 Conclion Jornal of Progreive Reearch in Mathematic(JPRM) ISSN: In thi paper, we apply a new integral tranform ''ZZ tranform'' to olve ome orinary ifferential eqation with variable coefficient, The relt reveal that the propoe metho i very efficient, imple an can be applie to linear an nonlinear ifferential eqation. Reference [1] J.W. Mile, Integral Tranform in Applie Mathematic, Cambrige: Cambrige Univerity Pre, [2] Lokenath Debnath an D. Bhatta. Integral tranform an their Application econ Eition, Chapman & Hall /CRC (26). [3] Haan Eltayeb an Aemkilicman, A Note on the Sm Tranform an ifferential Eqation, Applie Mathematical Science, VOL, 4,21, no.22, [4] Tarig M. Elzaki, (211), The New Integral Tranform Elzaki Tranform Global Jornal of Pre an Applie Mathematic, ISSN , Nmber 1, pp [5] Abelbagy A. Alhikh an Mohan M. Abelrahim Mahgob, A Comparative Sty Between Laplace Tranform an Two New Integral ELzaki Tranform an Abooh Tranform, Pre an Applie Mathematic Jornal216; 5(5): [6] Mohan M. Abelrahim Mahgob an Tarig M. Elzaki Soltion of Partial Integro- Differential Eqation by Elzaki Tranform Metho Applie Mathematical Science, Vol. 9, 215, no. 6, [7] K. S. Abooh, The New Integral Tranform Abooh Tranform Global Jornal of pre an Applie Mathematic, 9(1), 35-43(213). [8] Mohan M. Abelrahim Mahgob an Abelbagy A. Alhikh On The Relationhip Between Abooh Tranform an New Integral Tranform " ZZ Tranform, Mathematical Theory an Moeling, Vol.6, No.9, 216. [9] Abelilah K. Haan Seeeg an Mohan M. Abelrahim Mahgob, Abooh Tranform Homotopy Pertrbation Metho For Solving Sytem Of Nonlinear Partial Differential Eqation, Mathematical Theory an Moeling Vol.6, No.8, 216. [1] ZainUlAbainZafar, ZZ Tranform metho, IJAEGT, 4(1), , Jan (216). [11] ZainUlAbainZafar, M.O. Ahma, A. Pervaiz, Nazir Ahma: ZZ Forth Orer Compact BVM for the Eqation of Lateral Heat Lo,Pak. J. Engg. & Appl. Sci. Vol. 11, Jly., 212 (p ). [12] Zafar, H. Khan, Waqar. A. Khan, N-Tranform- Propertie an Application, NUST Jor. Of Engg. Science, 1(1): , 28. [13] ZainUlAbainZafar, et al., Soltion of Brger Eqation with the help of Laplace Decompoition metho. Pak. J. Engg. & Appl. Sci. 12: 39-42, 213. Volme 1, Ie 1 available at