Solutions of Klein - Gordan equations, using Finite Fourier Sine Transform
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1 IOSR Journl of Mthemtics (IOSR-JM) e-issn: , p-issn: X. Volume 13, Issue 6 Ver. IV (Nov. - Dec. 2017), PP Solutions of Klein - Gordn equtions, using Finite Fourier Sine Trnsform Abdllh.H.A Deprtment of Mthemtics-college of Science, Sudn University of Science & Technology Abstrct: In this pper the finite Fourier sine trnsform is presented for obtining solutions for Klein- Gordn equtions. The initil-boundry vlue problems for the Klein-Gordn equtions re solved on the hlf rnge, using finite Fourier sine trnsform. Such problems posed on time-depend domin. The results revel tht the finite Fourier sine trnsform is very effective, simple, convenient nd flexible. Keywords: differentil equtions, dispersive, perturbtion, decomposition Dte of Submission: Dte of cceptnce: I. Introduction In this pper we consider two importnt equtions of mthemticl physics, the homogeneous Klein- Gordn eqution[10] u tt x, t u xx x, t + u x, t = 0 (1) nd the non-homogeneous Keilin-Gordn eqution u tt x, t u xx x, t + u x, t = 0 (2) which pper in quntum field theory, reltivistic physics, dispersive wve-phenomen, plsm physics, nd pplied physicl sciences. Severl techniques including finite difference, colloction, finite element, scttering, decomposition nd vrition itertion using Adomin's polynomils hve been used to hndle such equtions [1,2,10,11]. He [4,13] developed the homotopy perturbtion technique for solving such physicl problems. In recent yers, mny reserch workers hve been pid ttention, to study the solutions of prtil differentil equtions by using vrious methods. Among these re the Adonin decomposition method (ADM) [3], He's semi-inverse method [3], the tnh method, the homotopy perturbtion method (HPM), the differentil trnsform method nd the vritionl itertion method (VIP) [5,8]. He [6,7] developed the homotopy perturbtion method (HPM) by merging the stndrd homotopy nd perturbtion for solving vrious physicl problems. Vrious wys hve been proposed recently to del with these prtil differentil equtions, such s Adomin decomposition method. In this work we pply the finite Fourier sine trnsform method to solve homogeneous nd non-homogeneous liner Klein-Gordn equtions[12]. II. The finite Fourier sine trnsform Definition (1): The finite Fourier sine trnsform of function u(x, t) denoted u n, t is defined by[9]: F s u x, t = u n, t = u x, t sin nπx 0 dx (3) where n is n integer. The function u(x, t) is then clled the inverse finite Fourier trnsform nd is given by :. u x, t = 2 n=1 u n, t sin nπx (4) DOI: / Pge
2 Solutions of Klein - Gordn equtions, using Finite Fourier Sine Trnsform Definition (2): If u is some function of x nd t, then finite Fourier of t for 0 < x < nd t > 0 is given by [9]: F ð2 u(x, t) s x 2 = n2 π 2 2 u n, t + nπ u 0, t u, t cos(nπ) (5) To illustrte the bsic ide of this method, we consider generl non-homogeneous liner prtil differentil eqution of the form: with boundry conditions: Au tt x, t + Bu xx x, t + Cu x, t = x, t (6) u 0, t = u, t = 0 (7) nd initil coditions: u x, 0 = f x (8) u t x, 0 = g x (9) where A, B,nd C re constnts. Tking the finite Fourier sine trnsform of both sides of Eq(6), we obtin A d2 dt 2 u x, t + C B n2 π 2 nπ u n, t + 2 u 0, t u, t cos(nπ) + Cu(n, t) = H n, t (10) where H n, t = F s ( x, t ),using the boundry condition (7) nd ssociting like terms, Eq(10)becomes A d2 dt 2 u x, t + C B n2 π 2 u n, t = H n, t (11) 2 which is second order ordinry differentil eqution, nd hs the following solutions: Cse1: if C > B n 2 π 2 Cse2: if C < B n 2 π 2 2 then, solution of Eq(11)is: u n, t = c 1 cos β 0 t + c 2 sin β 0 t + λ( t) (12) 2 then, solution of Eq(11) is: u n, t = c 1 cosh β 0 t + c 2 sin β 0 t + λ( t) (13) where β 0 = C2 Bn 2 π 2 nd λ(t) is the prticulr solution of Eq(11). A 2 It is esy to show tht c 1 = λ 0 + f(x) nd c 2 = initil conditions (8) nd (9). Cse3: if C = B n 2 π 2, then the solution of Eq(11) is: 2 λ 0 +g(x) β n by pplying the finite Fourier trnsform to the DOI: / Pge
3 Solutions of Klein - Gordn equtions, using Finite Fourier Sine Trnsform u n, t = f x + tg x 1 t τ H n, t dt A 0 Tking the inverse finite Fourier sine trnsform to get the finl solution using Eq(4). t (14) III. Applictions The finite Fourier trnsforms re used to solve differentil equtions rising in boundry vlue problems of physics nd mechnics [9]. In this section we will pply the finite Fourier sine trnsform to solve homogeneous nd non-homogeneous liner Klein-Gordn equtions [12]: 3.1 Illustrtive exmple Consider the following boundry vlue problem [9] u t = 2 2 t x 2 (15) with boundry conditions: nd initil condition: u 0, t = u 5, t = 0, (16) u x, 0 = 10 sin(4πx), 0 < x < 5 (17) Tking finite Fourier sine trnsform of both sides of Eq(15), nd using Eq(5) (with = 5), we obtin F s u(x, t) t = 2n2 π 2 u n, t + 2nπ 25 5 u 0, t u 5, t cos(nπ) (18) then using the boundry conditions (16), Eq(18) becomes d dt u n, t = 2n2 π 2 u n, t (19) 25 which is seprble ordinry differentil eqution, nd hs solution u n, t = Ce n 2 π 2 25 t (20) Tking finite Fourier sine trnsform of the initil condition (17),combining with Eq(20), we hve Hence, Eq(20) becomes u 0, t = C = 10 sin nπ x dx = 25, for n = 20 0, for ll oter vlues of n (21) u n, t = 25e n 2 π 2 25 t (22) Tking inverse Fourier sine trnsform of Eq(22), we obtin which is the required solution. u n, t = 10e 2n 2t sin nπx (23) 3.2 The homogeneous liner Klein-Gordn eqution we next investigte the Klein-Gordn eqution[12]: u tt x, t u xx x, t + u(x, t) = 0 (24) with boundry conditions: DOI: / Pge
4 Solutions of Klein - Gordn equtions, using Finite Fourier Sine Trnsform u 0, t = u, t = 0 (25) nd initil condition: u x, 0 = 0 (26) u t x, 0 = x (27) Tking finite Fourier sine trnsform of both sides of Eq(24), nd using Eq(5), we obtin F s u tt (x, t) = n2 π 2 2 u n, t + nπ u 0, t u, t cos nπ + u n, t = 0 (28) Then using the boundry conditions (25) nd ssociting like terms, we get d 2 dt 2 u n, t + β 0 2 u n, t = 0 (29) which is second order homogeneous ordinry differentil equtions, nd hs solution: u n, t = c 1 cos β n t + c 2 sin β n t (30) where β n = n2 π (31) c 1 nd c 2 re rbitrry constnts of integrtion, tking finite Fourier sine trnsform of the initil conditions (26), we get then Eq(30) becomes: c 1 = 0 (32) u n, t = c 2 sin β n t (33) tking finite Fourier sine trnsform of the initil conditions (27), we obtin then Eq(33) becomes c 2 = 2 nπβ n cos(nπ) (34) u n, t = 2 nπβ n cos(nπ) sin β n t (35) Tking inverse Fourier sine trnsform of Eq(35), we get u x, t = 2 n=0 2 cos(nπ) sin β nπβ n t sin nπ n x (36) Substituting β n from Eq(31), we obtin u x, t = 22 π n=0 cos(nπ) n n 2 π 2 + sin n2 π t sin nπ x (37) DOI: / Pge
5 which is the required solution. Solutions of Klein - Gordn equtions, using Finite Fourier Sine Trnsform 3.3 The in-homogeneous liner Klein-Gordn eqution We next consider the in-homogeneous liner Klein-Godn eqution [12]: with boundry conditions: u tt x, t u xx x, t + u x, t = 2 sin x (38) u 0, t = u, t = 0 (39) nd initil conditions: u x, 0 = sin x, 0 < x < (40) u t x, 0 = 1, 0 < x < (41) Tking finite Fourier sine trnsform of both sides of Eq(38), nd using Eq(5), we obtin F s u tt (x, t) = n2 π 2 2 u n, t + nπ u 0, t u, t cos nπ + u n, t = F s (2 sin x ) (42) Then using the boundry conditions (39) nd ssociting like terms, we get d 2 dt 2 u n, t + β 0 2 u n, t = nπ ξ 0 cos nπ sin () (43) which is second order non-homogeneous ordinry differentil equtions, where ξ 0 = Eq(43) hs solution 2 2 n 2 π 2 2, n2 π 2 2 (44) u n, t = c 1 cos β n t + c 2 sin β n t + λ n (45) where β n is s defined in Eq(31), nd λ n = ξ 0 cos nπ sin () (46) β2 n Upon using the initil condition (40) gives c 1 = λ n ξ 0 (47) Then, using the initil condition (41), we hve c 2 = Therefore Eq(43) becomes nπβ n cos(nπ) 1 (48) DOI: / Pge
6 Solutions of Klein - Gordn equtions, using Finite Fourier Sine Trnsform u x, t = n=1 λ n cos β ξ n t + 0 nπβ n cos(nπ) 1 sin β n t + λ n sin nπ x (49) which is the required solution. IV. Conclusion After the direct ppliction of finite Fourier sine trnsform method nd from the results obtined, we cn sy tht this method is esy to implement nd effective. As result, the conclusion tht comes through this work, is tht the finite Fourier sine trnsform method cn be pplied to other prtil differentil eqution, due to the efficiency in the ppliction to get the possible results. References [1]. S. Abbsbndy. Numericl solutions of nonliner Klein-Gordon eqution by vritionl itertion method. Internt. [2]. A.M. Wzwz. The modified decomposition method for nlytic tretment of differentil equtions. Appl. Mth. Comput. 2006, pp [3]. G. Adomin. Solving Frontier Problems of Physics: The Decomposition Method. Boston: Kluwer Acd. Publ [4]. J.H. He. Homotopy perturbtion technique. Computer Methods in Applied Mechnics nd Engineering [5]. J.H. He. Vritionl itertion method kind of nonliner nlyticl technique: some exmples. Interntionl Journl of Nonliner Mechnics [6]. J.H. He. Homotopy perturbtion method: new nonliner nlyticl technique. Applied Mthemtics nd Computtion [7]. E. Hesmeddini, H. Ltifizdeh. A new vision of the He s homotopy perturbtion method. Interntionl Journl of Nonliner Sciences nd Numericl Simultion, [8]. E. Hesmeddini, H. Ltifizdeh. Reconstruction of vritionl itertion lgorithms using the Lplce trnsform. Interntionl Journl of Nonliner Sciences nd Numericl Simultion [9]. S.L.Kkni, C.Hemjni, Mthemticl physics,cbs Publisher nd Distributors, New Dlhi (Indi),2004. [10]. S.T. Mohyud-Din, M.A. Noor, K.I. Noor. some reltively new techniques for nonliner problems. Mth. Porb.Eng. Article ID , 25 pges, doi: /2009/234849, [11]. S.T. Mohyud-Din, A. Yildirim. Vritionl itertion method for solving Klein- Gordon equtions. Journl of Applied Mthemtics, Sttistics nd Informtics, [12]. Jgdev Singh, Devendr Kumr nd Sushil Rthore, Appliction of Homotopy Perturbtion Trnsform Method for Solving Liner nd Nonline Klein- Gordon Equtions, Journl of Informtion nd Computing [13]. Science,ISS Englnd, UK,Vol. 7, No. 2, 2012, pp [14]. A. Yildirim. An Algorithm for Solving the Frctionl Nonliner Schröndinger Eqution by Mens of the Homotopy Perturbtion Method. Interntionl Journl of Nonliner Science nd Numericl Simultion Abdllh.H.A "Solutions of Klein - Gordn equtions, using Finite Fourier Sine Trnsform." IOSR Journl of Mthemtics (IOSR-JM) 13.6 (2017): DOI: / Pge
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