Research Article Numerical Solutions of the Second-Order One-Dimensional Telegraph Equation Based on Reproducing Kernel Hilbert Space Method

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1 Hndaw Publshng Corporaton Abstract and Appled Analyss Volume 3, Artcle ID , 3 pages Research Artcle Numercal Solutons of the Second-Order One-Dmensonal Telegraph Equaton Based on Reproducng Kernel Hlbert Space Method Mustafa Inc, Al Akgül,,3 and Adem KJlJçman 4 Department of Mathematcs, Scence Faculty, Fırat Unversty, 39 Elazığ, Turkey Department of Mathematcs, Educaton Faculty, Dcle Unversty, 8 Dyarbakır, Turkey 3 Department of Mathematcs and Statstcs, Mssour Unversty of Scence and Technology, Rolla, MO 6549-, USA 4 Department of Mathematcs and Insttute for Mathematcal Research, Unverst Putra Malaysa, 434Serdang,Selangor,Malaysa Correspondence should be addressed to Adem Kılıçman; klcman@yahoo.com Receved 8 December ; Accepted 5 March 3 Academc Edtor: Mustafa Bayram Copyrght 3 Mustafa Inc et al. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. We nvestgate the effectveness of reproducng kernel method (RKM) n solvng partal dfferental equatons. We propose a reproducng kernel method for solvng the telegraph equaton wth ntal and boundary condtons based on reproducng kernel theory. Its exact soluton s represented n the form of a seres n reproducng kernel Hlbert space. Some numercal examples are gven n order to demonstrate the accuracy of ths method. The results obtaned from ths method are compared wth the exact solutons and other methods. Results of numercal examples show that ths method s smple, effectve, and easy to use.. Introducton The hyperbolc partal dfferental equatons model the vbratons of structures (e.g., buldngs, beams, and machnes). These equatons are the bass for fundamental equatons of atomc physcs. In ths paper, we consder the telegraph equaton of the form u (x, t) +α u t t (x, t) +β u (x, t) u (x, t) +f(x, t), x x,t,α>β, wth ntal condtons u (x, ) φ (x), u t (x, ) φ (x), () and approprate boundary condtons u (, t) g (t), u(, t) g (t), t (3) by usng reproducng kernel method (RKM). In recent years, much attenton has been gven n the lterature to () the development, analyss, and mplementaton of stable methods for the numercal soluton of () (3)[ 3]. Mohanty carred out a new technque to solve the lnear one-spacedmensonal hyperbolc equaton ()[4]. Hgh-order accurate method for solvng lnear hyperbolc equaton s presented n [5]. A compact fnte dfference approxmaton of fourth order for dscretzng spatal dervatve of lnear hyperbolc equaton and a collocaton method for the tme component are used n[6]. A numercal scheme s developed to solve the one-dmensonal hyperbolc telegraph equaton usng the collocaton ponts and approxmatng the soluton usng thn plate splnes radal bass functon [7]. Several test problems were gven, and the results of numercal experments were compared wth analytcal solutons to confrm the good accuracy of ther scheme. Yao [8] nvestgated a nonlnear hyperbolc telegraph equaton wth an ntegral condton by reproducng kernel space at α β. Yousef presented a numercal method for solvng the one-dmensonal hyperbolc telegraph equaton by usng Legendre multwavelet Galerkn method [9]. Dehghan and Lakestan presented a numercal technque for the soluton of the second-order

2 Abstract and Appled Analyss one-dmensonal lnear hyperbolc equaton []. Lakestan and Saray used nterpolatng scalng functons for solvng () (3) []. Dehghan provded a soluton of the second-order one-dmensonal hyperbolc telegraph equaton by usng the dualrecproctyboundaryntegralequaton(drbie)method [].Theproblemhasexplctsolutonthatcanbeobtanedby the method of separaton of varables n [3]. In ths paper, the problem s solved easly and elegantly by usng RKM. The technque has many advantages over the classcal technques. It also avods dscretzaton and provdes an effcent numercal soluton wth hgh accuracy, mnmal calculaton, and avodance of physcally unrealstc assumptons. In the next secton, we wll descrbe the procedure of ths method. The theory of reproducng kernels was used for the frst tme at the begnnng of the th century by Zaremba n hs work on boundary value problems for harmonc and bharmonc functons [4].Reproducng kernel theory has mportant applcaton n numercal analyss, dfferental equatons, probablty, and statstcs. Recently, usng the RKM, some authors dscussed telegraph equaton [5], Troesch s porblem [6], MHD Jeffery-Hamel flow [7], Bratu s problem [8], KdV equaton [9], fractonal dfferental equaton [], nonlnear oscllator wth dscontnuty [], nonlnear twopont boundary value problems [], ntegral equatons [3], and nonlnear partal dfferental equatons [4]. The paper s organzed as follows. Secton ntroduces several reproducng kernel spaces. The representaton n W(Ω) and a lnear operator are presented n Secton 3. Secton 4 provdes the man results. The exact and approxmate solutons of () (3) and an teratve method are developed for the knd of problems n the reproducng kernel space. We have proved that the approxmate soluton converges to the exact soluton unformly. Numercal experments are llustrated n Secton 5. Some conclusons are gven n Secton 6.. Reproducng Kernel Spaces In ths secton, some useful reproducng kernel spaces are defned. Defnton (reproducng kernel functon). Let E. A functon K : E E C s called areproducngkernel functon of the Hlbert space H f and only f (a) K(, t) H for all t E, (b) φ, K(, t) φ(t) for all t Eand all φ H. The last condton s called the reproducng property as the value of the functon φ at the pont t s reproduced by the nner product of φ wth K(, t). Defnton. Hlbert functon space H s a reproducng kernel space f and only f for any fxed x X,thelnear functonal I(f) f(x) s bounded [5,page5]. Defnton 3. We defne the space H [, ] by H [, ] {u AC[, ] :u L [, ]}. (4) The nner product and the norm n H [, ] are defned by u, g H u() g () + u (t) g (t) dt, u, g H [, ], u H u, u H, u H [, ]. (5) Lemma 4. The space H [, ] s a reproducng kernel space, and ts reproducng kernel functon q s s gven by [5,page3] q s (t) { +t, t s, + s, t > s. Defnton 5. We defne the space F 3 [, T] by F 3 [, T] {u AC[, T] :u,u AC[, T], u (3) L [, T],u() u () }. The nner product and the norm n F 3 [, T] are defned by u, g F 3 u () () g () () T + u (3) (t) g (3) (t) dt, u, g F 3 [, T], u F 3 u, u F 3, u F 3 [, T]. Lemma 6. The space F 3 [, T] s a reproducng kernel space, and ts reproducng kernel functon r s s gven by [5,page48] 4 s t + s t 3 4 st4 + t5, { t s, r s (t) { 4 s t + s3 t 4 ts4 + s5, { t>s. Defnton 7. We defne the space W 3 [, ] by W 3 [, ] {u AC[, ] :u,u AC[, ], u (3) L [, ],u() u() }. The nner product and the norm n W 3 [, ] are defned by u, g W 3 u () () g () () (6) (7) (8) (9) () + u (3) (x) g (3) (x) dx, u, g W 3 [, ], u W 3 u, u W 3, u W 3 [, ]. () The space W 3 [, ] s a reproducng kernel space, and ts reproducng kernel functon R y s gven by the followng theorem.

3 Abstract and Appled Analyss 3 Theorem 8. The space W 3 [, ] s a reproducng kernel space, and ts reproducng kernel functon R y s gven by c (y) x, { R y (x) 5 { d (y) x, { 5 x y, x>y, c (y) 5 56 y4 56 y5 5 6 y 5 78 y y, c (y) 5 64 y4 64 y5 + 4 y 5 3 y3 5 6 y, c 3 (y) 5 87 y4 87 y y y y, () c 4 (y) y y y y3 4 y, 5 c 5 (y) 3744 y4 87 y5 64 y 87 y3 56 y+, d (y) y5, d (y) 4 y4 56 y5 5 6 y 5 78 y y, d (y) 7 4 y y4 64 y5 + 4 y 5 6 y, d 3 (y) 5 87 y4 87 y5 5 3 y y y, d 4 (y) y y y y y, d 5 (y) 3744 y4 87 y5 64 y 87 y3 56 y. (3) Proof. Let u W 3 [, ] and let y.bydefnton 7 and ntegratng by parts two tmes, we obtan that u, R y W 3 u () () R () y () + u (3) (x) R (3) y (x) dx u() R y () +u () R y () +u () R y () +u () R (3) y () u () R (3) y () u () R (4) y () +u () R (4) y () + u (x) R (5) y (x) dx. (4) After substtutng the values of R y (), R y (), R y (), R(3) y (), R (4) y (), R(3) y (),andr(4) y () nto the above equaton, we get u, R y W 3 u() +u () ( 3 3 y 5 78 y3 5 6 y y4 56 y5 ) +u () ( 5 3 y 5 56 y3 + 5 y y4 3 y5 ) +u () u () ( 5 3 y 5 56 y3 + 5 y y4 3 y5 ) u () +u () ( 3 3 y y y 5 56 y y5 ) + u (x) R (5) y (x) dx; (5) thus we obtan that u, R y W 3 u (x) R (5) y (x) dx y u (x) R (5) y (x) dx+ y u (x) R (5) y (x) dx y u (x) ( 3 y 5 78 y3 5 6 y y4 56 y5 ) dx + u (x) ( y 3 y 5 78 y3 5 6 y y4 56 y5 ) dx (u(y) u()) ( 3 y 5 78 y3 5 6 y y4 56 y5 )

4 4 Abstract and Appled Analyss +(u() u (y)) ( 3 y 5 78 y3 5 6 y y4 56 y5 ) u(y) u() ( 3 y 5 78 y3 5 6 y y4 56 y5 ) +u() ( 3 y 5 78 y3 5 6 y y4 56 y5 ). By Defnton 7,wehaveu() u().so Ths completes the proof. (6) u, R y W 3 u(y). (7) Defnton 9. We defne the bnary space W(Ω) by W (Ω) {u : 4 u x s completely contnuous n t Ω[, ] [, ], u (x, ), u (x, ) t 6 u x 3 t 3 L (Ω),, u (, t),u(, t) }. The nner product and the norm n W(Ω) are defned by u, g W + [ 3 t 3 j j u (, t) x 3 t 3 g (, t)] dt x t j u (x, ), j t j g (x, ) W 3 + [ 3 3 x 3 u (x, t) t3 3 3 x 3 g (x, t)] dt dx, t3 (8) (9) u W u, u W, u W(Ω). () Lemma. W(Ω) s a reproducng kernel space, and ts reproducng kernel functon K (y,s) s gven by [5,page48] K (y,s) R y r s. () Defnton. We defne the bnary space W(Ω) by W (Ω) {u : u s completely contnuous n Ω[, ] [, ], u x t L (Ω)}. The nner product and the norm n W(Ω) are defned by u, g W [ t u (, t) g (, t)] dt t u W u, u W, u W (Ω). () + u(x, ),g(x, ) H + [ (3) u (x, t) g (x, t)] dt dx, x t x t Lemma. W(Ω) s a reproducng kernel space, and ts reproducng kernel functon G (y,s) s gven as [5,page48] G (y,s) Q y q s. (4) Remark 3. Hlbert spaces can be completely classfed: there s a unque Hlbert space up to somorphsm for every cardnalty of the base. Snce fnte-dmensonal Hlbert spaces are fully understood n lnear algebra and snce morphsms of Hlbert spaces can always be dvded nto morphsms of spaces wth Aleph-null (χ ) dmensonalty, functonal analyss of Hlbert spaces mostly deals wth the unque Hlbert space of dmensonalty Aleph-null and ts morphsms. One of the open problems n functonal analyss stoprovethateveryboundedlnearoperatoronahlbert space has a proper nvarant subspace. Many specal cases of ths nvarant subspace problem have already been proven [6]. 3. Soluton Representaton n W(Ω) In ths secton, the soluton of () s gven n the reproducng kernel space W(Ω).On defnng the lnear operatorl : W(Ω) W(Ω) by LV V t V x +α V t +β V (x, t), (5) after homogenzng the ntal and boundary condtons, model problem () (3) changes to the problem LV M(x, t), (x, t) [, ] [, ], V (x, ) V t (x, ) V (, t) V (, t), (6)

5 Abstract and Appled Analyss 5 M (x, t) Z t (x, t) V (x, t) x +α Z t (x, t) +β Z (x, t) +f(x, t) ; for convenence, we agan wrte u nstead of V n (6) Lemma 4. L s a bounded lnear operator. (7) Now, choose a countable dense subset {(x,t ), (x, t ),...}n Ω and defne φ G (x,t ), Ψ L φ, (3) L s the adjont operator of L.Theorthonormalsystem { Ψ } of W(Ω) canbedervedfromtheprocessofgram- Schmdt orthogonalzaton of {Ψ } as Proof. Let u W(Ω) and let (x, t) Ω. ByLemma, we have and thus u (x, t) u,k (x,t) W, (8) Lu (x, t) u,lk (x,t) W, Ψ k β k Ψ k. (33) Theorem 5. Suppose that {(x,t )} s dense n Ω. Then {Ψ } s a complete system n W(Ω),and x Lu (x, t) u, x LK, (x,t) W t Lu (x, t) u, t LK, (x,t) W t x Lu (x, t) u, t x LK. (x,t) W Hence there exst a,b,a,b >such that Therefore, Lu (x, t) a u W, t Lu (x, t) b u W, x Lu (x, t) a u W, t x Lu (x, t) b u W. Lu W [ dt t Lu (, t)] + Lu (x, ),Lu(x, ) H + [ x Ths completes the proof. t Lu (x, t)] dt dx [ t Lu (, t)] dt+[lu (, )] + [ x Lu (x, )] dx + [ x t Lu (x, t)] dt dx (a +a +b +b ) u W. (9) (3) (3) Ψ LK (x,t ) (x, t). (34) Proof. We have Ψ L φ L φ,k (x,t) W φ,lk (x,t) W LK (x,t),g (x,t (35) ) W LK (x,t) (x,t ) LK (x,t ) (x, t). Clearly, Ψ W(Ω).Foreachfxedu W(Ω),f u, Ψ W,,,..., (36) then u,ψ W u,l φ W (37) Lu,φ W Lu(x,t ),,,... Note that {(x,t )} s dense n Ω.Hence,Lu.Fromthe exstence of L,tfollowsthatu.Theproofscompleted. Theorem 6. If {(x,t )} s dense n Ω, then the soluton of (6) s gven by u β k M(x k,t k ) Ψ. (38) k

6 6 Abstract and Appled Analyss Proof. By Theorem 5, {Ψ (x, t)} s a complete system n W(Ω).Thus, u u, Ψ W Ψ β k u, Ψ k W Ψ k u(x) 5 5 β k u, L φ k W Ψ k β k Lu, φ k W Ψ k β k Lu, G (xk,t k ) W Ψ k (39) 4 RKM ES 4 Fgure : Graph of numercal results for Example (α, β, t.5). x β k Lu (x k,t k ) Ψ k β k M(x k,t k ) Ψ. k Ths completes the proof. Now the approxmate soluton u n canbeobtanedfrom the n-term ntercept of the exact soluton u and Obvously n u n β k M(x k,t k ) Ψ. (4) k u n u W, n. (4) Theorem 7. If u W(Ω),then u n u W, n. (4) Moreover, a sequence u n u W s monotoncally decreasng n n. Proof. From (38) and(4), t follows that Thus, u n u W n+ β k M(x k,t k ) Ψ (x,. (43) t) W k u n (x, t) u(x, t) W, n. (44) In addton u n u W n+ n+ ( β k M(x k,t k ) Ψ (x, t) k k W β k M(x k,t k ) Ψ (x, t)). (45) Clearly, u n (x, t) u(x, t) W s monotoncally decreasng n n. 4. The Method Implementaton () If (6) s lnear, then the analytcal soluton of (6) can be obtaned drectly by (38). () If (6) s nonlnear, then the soluton of (6) canbe obtaned by the followng teratve method. We construct an teratve sequence u n,puttng A A n any fxed u W 3 [, ], u n n A Ψ, A β M(x k,t k,u(x k,t k )) k n k β k M(x k,t k,u k (x k,t k )),. β nk M(x k,t k,u k (x k,t k )). (46) (47) Next we wll prove that u n gven by the teratve formula (46) converges to the exact soluton. Theorem 8. Suppose that u n defned by (46) s bounded and (6) has a unque soluton. If {(x,t )} s dense n Ω, then u n converges to the analytcal soluton u of (6),and A s gven by (47). u A Ψ, (48)

7 Abstract and Appled Analyss 7 Table : Numercal results for Example for t.5. RE RE RE x CPU tme (s) α α α α α 5 α 5 β β β5 β5 β β Table : Numercal results for Example for t.. RE RE RE x CPU tme (s) α α α α α 5 α 5 β β β5 β5 β β Table 3: A comparson between nterpolatng scalng functon method [] and RKM for dfferent values of α, β, and t for Example. x CPU tme (s) [] α t.5 RKM α t.5 [] β t RKM β t Table 4: A comparson between nterpolatng scalng functon method [] and RKM for dfferent values of α, β, and t for Example. x [] α t.5 RKM α t.5 [] β5 t RKM β5 t Proof. Frst,weprovetheconvergenceofu n.from(46) and the orthonormalty of { Ψ },wenferthat u n+ n+ A n A +A n+ u n +A n+ u n. (49) By (49), u n s nondecreasng, and by the assumpton, u n s bounded. Thus u n s convergent. By (49), there exsts a constant c such that A c. (5) Ths mples that If m>n,then u m u n k+ u k knu {A } l. (5) m m u k+ u k kn m A k+ kn, m, n. (5)

8 8 Abstract and Appled Analyss Table 5: Numercal results for Example for t.5. RE RE RE x CPU tme (s) α α α α α5 α5 β β β5 β5 β β Table 6: Numercal results for Example for t.. RE RE RE x CPU tme (s) α α α α α5 α5 β β β5 β5 β β u(x) RKM ES 4 6 Fgure : Graph of numercal results for Example (α, β, t.5). The completeness of W(Ω) shows that there exsts u W(Ω) such that u n u as n.now,weprovethat u solves (6). Takng lmts n (4), we get Note that u L u (L u) (x k,t k ) x A Ψ. (53) A L Ψ, (54) A L Ψ (x k,t k ) A L Ψ,G (xk,t k ) W In vew of (47), we have A L Ψ,φ k W A Ψ,L φ k W A Ψ,Ψ k W. (55) L u (x l,t l ) M(x l,t l,u l (x l,t l )). (56) Snce {(x,t )} s dense n Ω,foreach(y, s) Ω, there exsts asubsequence{(x nj,t nj )} j such that We know that (x nj,t nj ) (y, s), j. (57) L u(x nj,t nj )M(x nj,t nj,u nj (x nj,t nj )). (58) Let j.bythecontnutyoff,wehave (L u) (y, s) M(y, s, u (y, s)), (59) whch ndcates that u satsfes (6). Remark 9. Inthesamemanner,tcanbeprovedthat u n x u, n, (6) x u x Ψ A x, and A s gven by (47). n u n x Ψ A x, (6)

9 Abstract and Appled Analyss 9 Table 7: RMS errors for Example. [] RKM [] RKM [] RKM N α α α α α α β β β β β5 β5 t.5 t.5 t. t. t.5 t Table 8: Numercal results for Example for t.5. RE RE RE x α α CPU tme (s) α α α 5 α 5 β β β5 β5 β β Numercal Results To test the accuracy of the present method, some numercal experments are presented n ths secton. Usng our method, we chose 36 ponts n Ω and obtaned the approxmate soluton u 36. The comparson between nterpolatng scalng functon method [] and RKM for dfferent values of α, β, and t sgvenntables4 and 5. We solve these examples for a set of ponts {x,...,x ( ) h,...,x N }, h N. (6) In Tables 7 and we calculate the RMS error by the followng formula: RMS error N+ (u u 36). (63) N+ ItcanbeseenfromTables4 and 5 and 7 that the results obtaned by the RKM are more accurate than those obtaned by the methods n [, ]. Ths ndcates that RKM s a relable method. The CPU tme (s) s gven n Tables,, 3, 4, 5, 6, 7, 8, 9,and. Numercal solutons are descrbed n the extended doman [ 4, 4] [ 3, 3]. ThecomparsonofRMSerrors gven for our method and Chebyshev method. Example. Consder the followng telegraph equaton wth ntal and boundary condtons: u (x, t) +α u t t (x, t) +β u (x, t) u +f(x, t), x u (x, ) φ (x) snh (x), u t (x, ) φ (x) snh (x), u (, t) g (t), t, u (, t) g (t) exp ( t) snh (), t, (64) f (x, t) (3 4α+β ) exp ( t) snh (x). (65) The exact soluton of (64)s gven by[] If we apply u (x, t) exp ( t) snh (x). (66) V (x, t) u(x, t) xsnh () (exp ( t) +t) + snh (x)(t ) to (64), then the followng equaton (68)sobtaned: V t V x +α V t +β V (x, t) M(x, t), V (x, ) V (x, ) V (, t) V (, t), t M (x, t) (3 4α+β ) exp ( t) snh (x) snh (x)(t ) αsnh () ( exp ( t) +)x +4αsnh (x) β snh () (exp ( t) +t)x (67) (68) +β snh (x)(t ) 4xexp ( t) snh (). (69)

10 Abstract and Appled Analyss Table 9: Numercal results for Example for t.. RE RE RE x CPU tme (s) α α α α α 5 α 5 β β β5 β5 β β Table : RMS errors for Example. [] RKM [] RKM [] RKM N α α α5 α5 α5 α5 β5 β5 β β β β t. t. t.5 t.5 t. t After homogenzng the ntal and boundary condtons and usng the above method, we obtan Tables 4 and Fgure. Example. Consder the followng telegraph equaton wth ntal and boundary condtons: u t +α u t +β u (x, t) u +f(x, t), x u (x, ) φ (x) sn (x), u t (x, ) φ (x), u (, t) g (t), t, u (, t) g (t) cos (t) sn (), t, (7) f (x, t) αsn (t) sn (x) +β cos (t) sn (x). (7) The exact soluton of (7)s gven by[] If we apply u (x, t) cos (t) sn (x). (7) V (x, t) u(x, t) xsn ()(cos (t) ) sn (x) (73) to (7), thenthefollowng (74)sobtaned: V t (x, t) V (x, t) +α V x t +β V (x, t) M(x, t), V (x, ) V (x, ) V (, t) V (, t), t (74) M (x, t) αsn (t) sn (x) +β cos (t) sn (x) sn (x) αsn (t) snh () x+xcos (t) sn () β x sn ()(cos (t) ) β sn (x). (75) After homogenzng the ntal and boundary condtons and usng the above method, we obtan Tables 5 7 and Fgures and 3. Example. Consder the followng telegraph equaton wth ntal and boundary condtons: u t +α u t +β u (x, t) u +f(x, t), x u (x, ) φ (x) tan ( x ), u t (x, ) φ (x) u (, t) g (t) tan ( t ), ( + tan (x/)), t, u (, t) g (t) tan ( +t ), t, (76) f (x, t) α(+tan ( x+t )) +β tan ( x+t ). (77) The exact soluton of (76)sgvenby[] u (x, t) tan ( x+t ). (78)

11 Abstract and Appled Analyss (a) (b) Fgure 3: Plots of numercal results for Example (α 6,β) If we apply V (x, t) u(x, t) + (x ) tan ( t ) xtan ( +t ) tan (x ) (+( +xtan ( t tan (x/) )) tan (/) )(+(t )) to (76), then the followng equaton (8)sobtaned V t (x, t) V (x, t) +α V x t +β V (x, t) M(x, t), V (x, ) V (x, ) V (, t) V (, t), t M (x, t) α(+tan ( x+t )) + β tan ( x+t ) tan (x/) +( + t 4 + 3t 4 tan ( x )) (+tan ( x )) + (x ) α(+tan ( t )) αx(+tan ( t+ )) α tan ( x ) +αxtan ( )+β (x ) tan ( t ) β x tan ( +t ) β tan ( x ) (79) (8) (+ +β x tan ( x x t tan (x/) ) t tan (/) )(+ ) tan ( t )(+tan ( t )) tan (+t )(+tan ( +t )). (8) After homogenzng the ntal and boundary condtons and usng the above method, we obtan Tables 8 and Fgure 4. Remark 3. In Tables 9, weabbrevatetheexactsoluton and the approxmate soluton by AS and ES, respectvely. stands for the absolute error, that s, the absolute value of the dfference of the exact soluton and the approxmate soluton, whlerendcatestherelatveerror,thats,theabsoluteerror dvded by the absolute value of the exact soluton. 6. Concluson In ths study, a second-order one-dmensonal telegraph equaton wth ntal and boundary condtons was solved by reproducng kernel Hlbert space method. We descrbed the method and used t n some test examples n order to show ts applcablty and valdty n comparson wth exact and other numercal solutons. The obtaned results show that ths approach can solve the problem effectvely and need few computatons. The satsfactory results that we obtaned were compared wth the results that were obtaned by [, ]. Numercal experments on test examples show that our proposed schemes are of hgh accuracy and support thetheoretcalresults.asshownntables7 and our results are better than the results that were obtaned by []. Accordng to these results, t s possble to apply RKM to lnear and nonlnear dfferental equatons wth ntal and

12 Abstract and Appled Analyss (a) (b) Fgure 4: Plots of numercal results for Example (α, β 5). boundary condtons. It has been shown that the obtaned results are unformly convergent and the operator that was used s a bounded lnear operator. From the results, RKM can be appled to hgh dmensonal partal dfferental equatons, ntegral equatons, and fractonal dfferental equatons wthout any transformaton or dscretzaton, and good results can be obtaned. References [] M. Dehghan, On the soluton of an ntal-boundary value problem that combnes Neumann and ntegral condton for the wave equaton, Numercal Methods for Partal Dfferental Equatons,vol.,no.,pp.4 4,5. []R.K.Mohanty,M.K.Jan,andK.George, Ontheuseof hgh order dfference methods for the system of one space second order nonlnear hyperbolc equatons wth varable coeffcents, Journal of Computatonal and Appled Mathematcs,vol.7,no.,pp.4 43,996. [3] E. H. Twzell, An explct dfference method for the wave equaton wth extended stablty range, BIT, vol.9,no.3,pp , 979. [4] R. K. Mohanty, An uncondtonally stable dfference scheme for the one-space-dmensonal lnear hyperbolc equaton, Appled Mathematcs Letters,vol.7,no.,pp. 5,4. [5] A. Mohebb and M. Dehghan, Hgh order compact soluton of the one-space-dmensonal lnear hyperbolc equaton, Numercal Methods for Partal Dfferental Equatons,vol.4,no.5,pp. 35, 8. [6] M. Dehghan, Fnte dfference procedures for solvng a problem arsng n modelng and desgn of certan optoelectronc devces, Mathematcs and Computers n Smulaton,vol.7,no., pp. 6 3, 6. [7] M. Dehghan and A. Shokr, A numercal method for solvng the hyperbolc telegraph equaton, Numercal Methods for Partal Dfferental Equatons,vol.4,no.4,pp.8 93,8. [8] H. Yao, Reproducng kernel method for the soluton of nonlnear hyperbolc telegraph equaton wth an ntegral condton, Numercal Methods for Partal Dfferental Equatons,vol.7,no. 4, pp ,. [9] S. A. Yousef, Legendre multwavelet Galerkn method for solvng the hyperbolc telegraph equaton, Numercal Methods for Partal Dfferental Equatons, vol.6,no.3,pp ,. [] M. Dehghan and M. Lakestan, The use of Chebyshev cardnal functons for soluton of the second-order one-dmensonal telegraph equaton, Numercal Methods for Partal Dfferental Equatons,vol.5,no.4,pp ,9. [] M. Lakestan and B. N. Saray, Numercal soluton of telegraph equaton usng nterpolatng scalng functons, Computers & Mathematcs wth Applcatons, vol.6,no.7,pp ,. [] M. Dehghan and A. Ghesmat, Soluton of the second-order one-dmensonal hyperbolc telegraph equaton by usng the dual recprocty boundary ntegral equaton (DRBIE) method, Engneerng Analyss wth Boundary Elements,vol.34,no.,pp. 5 59,. [3] A. N. Tkhonov and A. A. Samarskĭ, Equatons of Mathematcal Physcs,Dover,NewYork,NY,USA,99. [4] N. Aronszajn, Theory of reproducng kernels, Transactons of the Amercan Mathematcal Socety,vol.68,pp ,95. [5] M. Inc, A. Akgül, and A. Kılıçman, Explct soluton of telegraph equaton based on reproducng kernel method, Journal of Functon Spaces and Applcatons,vol.,p.3,. [6] M. Inc and A. Akgül, The reproducng kernel Hlbert space method for solvng Troesch s problem, Journal of the Assocaton of Arab Unverstes for Basc and Appled Scences.Inpress. [7] M. Inc, A. Akgül, and A. Kılıçman, A new applcaton of the reproducng kernel Hlbert space method to solve MHD Jeffery- Hamel flows problem n nonparallel walls, Abstract and Appled Analyss, vol. 3, Artcle ID 39454, pages, 3. [8] M. Inc, A. Akgül, and F. Geng, Reproducng kernel Hlbert space method for solvng Bratu s problem, Bulletn of the Malaysan Mathematcal Scences Socety.Inpress. [9] M. Inc, A. Akgül, and A. Kılıçman, On solvng KdV equaton usng reproducng kernel Hlbert space method, Abstract and Appled Analyss, vol. 3, Artcle ID 57894, pages, 3. [] W. Jang and Y. Ln, Representaton of exact soluton for the tme-fractonal telegraph equaton n the reproducng kernel

13 Abstract and Appled Analyss 3 space, Communcatons n Nonlnear Scence and Numercal Smulaton,vol.6,no.9,pp ,. [] Y. Wang, L. Su, X. Cao, and X. L, Usng reproducng kernel for solvng a class of sngularly perturbed problems, Computers & Mathematcs wth Applcatons, vol. 6, no., pp. 4 43,. [] F. Geng and M. Cu, A novel method for nonlnear two-pont boundary value problems: combnaton of ADM and RKM, Appled Mathematcs and Computaton,vol.7,no.9,pp ,. [3] F. Geng and F. Shen, Solvng a Volterra ntegral equaton wth weakly sngular kernel n the reproducng kernel space, Mathematcal Scences Quarterly Journal, vol.4,no.,pp.59 7,. [4] F. Geng and M. Cu, Homotopy perturbaton-reproducng kernel method for nonlnear systems of second order boundary value problems, Journal of Computatonal and Appled Mathematcs, vol. 35, no. 8, pp. 45 4,. [5] M. Cu and Y. Ln, Nonlnear Numercal Analyss n the Reproducng Kernel Space, Nova Scence, New York, NY, USA, 9. [6] A. M. Krall, Hlbert Space, Boundary Value Problems and Orthogonal Polynomals, vol.33ofoperator Theory: Advances and Applcatons, Brkhäuser, Basel, Swtzerland,.