Research Article Finite Difference Method for Hyperbolic Equations with the Nonlocal Integral Condition

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1 Discrete Dyamics i Nature ad Society Volume 011 Article ID pages doi: /011/56385 Research Article Fiite Differece Method for Hyperbolic Equatios with the Nolocal Itegral Coditio Allabere Ashyralyev 1 ad Necmetti Aggez 1 1 Departmet of Mathematics Fatih Uiversity Istabul Turkey Departmet of Mathematics ITTU Ashgabat Turkmeista Correspodece should be addressed to Necmetti Aggez aggez@fatih.edu.tr Received 1 April 010; Accepted 4 Jauary 011 Academic Editor: Leoid Shaikhet Copyright q 011 A. Ashyralyev ad N. Aggez. This is a ope access article distributed uder the Creative Commos Attributio Licese which permits urestricted use distributio ad reproductio i ay medium provided the origial work is properly cited. The stable differece schemes for the approximate solutio of the olocal boudary value problem for multidimesioal hyperbolic equatios with depedet i space variable coefficiets are preseted. Stability of these differece schemes ad of the first- ad secod-order differece derivatives is obtaied. The theoretical statemets for the solutio of these differece schemes for oe-dimesioal hyperbolic equatios are supported by umerical examples. 1. Itroductio Nolocal problems have bee a major research area i moder physics biology chemistry ad egieerig whe it is impossible to determie the boudary values of the ukow fuctio. Numerical methods ad theory of solutios of the olocal boudary value problems for partial differetial equatios of variable type were carried out i for example 1 10 ad the refereces therei. Hyperbolic equatios with olocal itegral coditios are widely used for chemical heterogeeity plasma physics thermoelasticity ad so forth. The solutios of hyperbolic equatios with olocal itegral coditios were ivestigated i The method of operators as a tool for ivestigatio of the solutio to hyperbolic equatios i Hilbert ad Baach spaces has bee studied extesively see e.g I the preset paper the olocal boudary value problem for the multidimesioal hyperbolic equatio with olocal itegral coditio u t x a t r x u xr xr f t x x x 1...x m Ω 0 <t<1

2 Discrete Dyamics i Nature ad Society u 0x 1 0 α ρ ) u ρ x ) dρ ϕ x u t 0x ψ x x Ω u t x 0 0 <t<1 x S 1.1 is cosidered. Here Ω is the uit ope cube i the m-dimesioal Euclidea space R m {x x 1...x m : 0 < x j < 1 1 j m} with boudary S Ω Ω S a r x x Ω ϕ x ψ x x Ω adf t x t 0 1 x Ω are give smooth fuctios ad a r x a>0. The first ad secod orders of approximatio i t ad the secod order of approximatio i space variables differece schemes for the approximate solutio of olocal boudary value problem 1.1 are preseted. Stability of these differece schemes ad of the first- ad secod-order differece derivatives established. Error aalysis is obtaied by umerical solutios of oe-dimesioal hyperbolic equatios with itegral coditio.. Differece Schemes ad Stability Estimates The discretizatio of problem 1.1 is carried out i two steps. I the first step let us defie the grid sets Ω h { x x r h 1 r 1...h m r m r r 1...r m 0 r j N j h j N j 1 j 1...m } Ω h Ω h Ω S h Ω h S..1 We itroduce the Hilbert space L h L Ω h of the grid fuctios ϕ h x { ϕ h 1 r 1...h m r m }. defied o Ω h equipped with the orm ϕ h L Ω h x Ω h ϕ h x h 1 h m 1/..3 To the differetial operator A x geerated by problem 1.1 we assig the differece operator A x by the formula h ) A x h uh x a r x u h xr x r r j.4

3 Discrete Dyamics i Nature ad Society 3 actig i the space of grid fuctios u h x satisfyig the coditio u h x 0 for all x S h.it is kow that A x h is a self-adjoit positive defiite operator i L Ω h. With the help of A x h we arrive at the olocal boudary value problem d v h t x A x dt h vh t x f h t x 0 <t<1 x Ω h v h 0x dv h 0x dt 1 0 ψ h x α ρ ) v h ρ x ) dρ ϕ h x x Ω h x Ω h.5 for a ifiite system of ordiary differetial equatios. I the secod step we replace problem.5 by the differece scheme u h k 1 x uh k x uh k 1 x A x h uh k 1 x f h k 1 x τ f h k 1 x f h t k 1 x t k 1 k 1 τ 1 k N 1 Nτ 1 x Ω h N u h 0 x α t m u h m x τ ϕ h x x Ω h m 1 ) I τ A x h )u h 1 x uh 0 x τ 1 ψ h x x Ω h.6 of the first order accuracy i t. Theorem.1. Let τ ad h be sufficietly small umbers. The the solutios of the differece scheme.6 satisfy the followig stability estimates: max u h k 0 k N max 1 k N 1 max M 1 [ 0 k N max f h k 1 k N 1 u h k ) τ u h k 1 uh k uh k 1 [ f h M 1 1 x r r j ψ h ϕ h x r r j ] max k N 1 ) max 0 k N τ 1 f h k f h k 1 ) u h k ) x r x r r j ψ h x r r j ϕ h x r x r r j ].7 where M 1 does ot deped o τ h ϕ h x ψ h x ad f h x 1 k<n. k

4 4 Discrete Dyamics i Nature ad Society The proof of Theorem.1 is based o the symmetry property of differece operator A x defied by the formula.4 ad o the followig theorem o coercivity iequality of the h elliptic differece problem. Theorem.. For the solutios of the elliptic differece problem A x h uh x ω h x x Ω h u h x 0 x S h.8 the followig coercivity iequality holds [9]: u h x r x r r j Mω h..9 Moreover the secod order of accuracy differece schemes τ [ ] u h k 1 x uh k x uh k 1 x 1 Ax h uh k x 1 ] 4 Ax h [u h k 1 x uh k 1 x f h k x x Ω h N u h 0 x τ j 1 ) I τ A x h f h k f h k t kx t k kτ 1 k N 1 Nτ 1 [ α t j ) u h t j x ) α t j 1 ) u h t j 1 x )] ϕ h x x Ω h τ 1[ u h 1 x uh 0 x ] τ ] [f h 0 x Ax h uh 0 x ψ h x f h 0 f h 0x f h N f h 1x x Ω h.10 ad τ ) u h k 1 uh k uh k 1 A x h uh k τ ) A x u h h k 1 4 f h k x f h k f h k t kx t k kτ 1 k N 1 Nτ 1 x Ω h N u h 0 x τ [ α ) t j u h t j x ) α ) t j 1 u h t j 1 x )] ϕ h x x Ω h j 1 I τ )[ A x h I τ ) A x h 4 4 τ 1[ u h 1 x uh 0 x ] τ [ f h 0 x Ax h uh 0 x ] ] ψ h x.11 f h 0 f h 0x f h N f h 1x x Ω h for approximately solvig the boudary value problem 1.1 is preseted.

5 Discrete Dyamics i Nature ad Society 5 We have the followig theorem. Theorem.3. Let τ ad h be sufficietly small umbers. The for the solutio of the differece schemes.10 ad.11 the stability iequalities max u h k 0 k N max 1 k N 1 max [ M max 0 k N f h k 0 k N u h k ) τ u h k 1 uh k uh k 1 [ f h M 0 x r j r ψ h ϕ h x r j r ] max 1 k N 1 ) τ 1 f h k f h k 1 ) ] ψ h x r j r ψ h x r x r j r.1 hold where M is idepedet of τ h ϕ h x ψ h x ad f h x 0 k N. k The proof of Theorem.3 is based o the symmetry property of differece operator A x defied by formula.4 ad o Theorem. o coercivity iequality of elliptic differece h problem.8. I Theorems.1 ad.3 the costats M 1 ad M caot be obtaied sharply. Therefore i the followig sectio we will study the accuracy of these differece schemes for solvig the oe-dimesioal hyperbolic equatios with the itegral coditio. Moreover the method is supported by umerical experimets. 3. Numerical Aalysis 3.1. The First Order of Accuracy i Time Differece Scheme I this sectio the olocal boudary value problem u t x t 1 x u t x x u x t x u t x f t x f t x x si x cos x e t 1 t ) e t si x u 0x 1 0 e s u s x ds ψ x ψ x u t 0x 0 0 x π 0 <t<1 0 <x<π 1 3e 1) si x 3.1 u t 0 u t π 0 0 t 1 for oe dimesioal hyperbolic equatio is cosidered. The exact solutio of problem 3.1 is u t x e t 1 t ) si x. 3.

6 6 Discrete Dyamics i Nature ad Society Applyig the formulas u x 1 u x 1 h u τ u 0 τ u x 1 u x u x 1 h u x O h ) u 0 O τ u x O h ) 3.3 ad usig the first order of accuracy i t implicit differece scheme.6 weobtaithe differece scheme first order of accuracy i t ad secod order of accuracy i x u k 1 u k u k 1 1 x τ uk 1 1 uk 1 u k 1 1 uk 1 1 uk 1 1 u k 1 h f t k 1 x h f t k 1 x x si x cos x e t ) k 1 1 t k 1 e t k 1 si x u 0 Nτ 1 x h 1 M 1 Mh π N e kτ τu k ψ x t k kτ 1 k N 1 k 1 ψ x 1 3e 1) si x 0 M 3.4 u k 0 uk M 0 u1 u k N 1 M 1 for approximate solutios of olocal boudary value problem 3.1. It ca be writte i the matrix form A u 1 B u C u 1 Dϕ 1 M 1 u 0 0 u M Here a a 0 0 A a a N 1 N 1

7 Discrete Dyamics i Nature ad Society 7 1 τe τ τe τ τe 3τ τe N 1 τ τe Nτ b c d b c d b c 0 0 B d c d e e 0 0 C e e N 1 N 1 a 1 x 1 h h b 1 c τ τ d 1 τ 1 x 1 e h 1 x 1 h h ϕ 0 ϕ 1 ϕ ϕ.. N 1 N 1 ϕ N N 1 1 ϕ k 1 f t k 1 x x si x cos x e t k ) 1 t k e t k si x x h t k kτ 1 k N ad D I N 1 is the idetity matrix. u 0 s u 1 s U s u s.. s u N s N 1 1

8 8 Discrete Dyamics i Nature ad Society This type system was used by Samarskii ad Nikolaev 30 for differece equatios. For the solutio of the matrix equatio 3.5 we will use the modified Gauss elimiatio method. We seek a solutio of the matrix equatio by the followig form: u α 1 u 1 β 1 M where u M 0 α j j 1...M 1 are N 1 N 1 square matrices β j j 1...M 1 are N 1 1 colum matrices α 1 β 1 are zero matrices ad α 1 B C α 1 A β 1 B C α 1 ) 3.9 D ϕ C β M The Secod Order of Accuracy i Time Differece Scheme Applyig 3.3 ad usig the secod order of accuracy i t implicit differece scheme.10 we obtai the secod order of accuracy differece scheme i t ad i x u k 1 [ u k u k 1 1 u k 1 1 x τ u 1 u 0 τ [ u k 1 1 uk 1 4h 1 uk 1 1 uk 1 uk 1 8h 4h u k 1 1 uk 1 uk u k 1 h uk 1 1 ] uk h uk uk 1 uk 1 1 uk 1 4h u k 1 4 ϕ k x si x cos x e t k 1 t k ) e t k si x u 0 k 1 Mh π x h 1 M 1 Nτ 1 t k kτ 1 k N 1 N τ [ e kτ u k ψ x e k 1 τ u k 1 ] ψ x 0 M 1 3e 1) si x 0 M ϕ k u k 1 1 [ u 0 1 u0 1 1 x u0 1 ] u0 u 0 1 u 0 h h si x 1 M 1 ] u k 0 uk M 0 0 k N 3.10 for approximate solutios of the olocal boudary value problem 3.1. We have agai N 1 M 1 system of liear equatios. We ca write the system as a matrix equatio 3.5.

9 Discrete Dyamics i Nature ad Society 9 Here a a a A 0 a a a a a a w N 1 N 1 1 τ τe τ τe τ τe N 1 τ τ e Nτ b d b b d 0 0 B b d b y c c c c C c c c c c z y 1 1 x τ h τ a 1 x 4h 1 8h c 1 x 4h 1 8h N 1 N 1 b 1 τ 1 x h 1 4 d τ 1 x h 1 N 1 N 1 w τ 4h 1 x τ z h τ 4h 1 x τ h ϕ 0 ϕ 1 ϕ ϕ N N 1 1 ϕ k f t k x x si x cos x e t k ) 1 t k e t k si x 1 k N 1 u 0 s u 1 s D I N 1 U s u s s u N s N 1 1

10 10 Discrete Dyamics i Nature ad Society For the solutio of the matrix equatio 3.5 we used the same algorithm as i the first order of accuracy differece scheme The Secod Order of Accuracy i Time Differece Scheme Geerated by A Applyig 3.3 ad formulas u x 4u x 1 6u x 4u x 1 u x h 4 u 0 5u h 4u h u 3h h u π 5u π h 4u π h u π 3h h u iv x O h ) u 0 O h ) u π O h ) 3.1 ad usig differece scheme.11 we obtai the secod order of accuracy differece scheme u 1 u 0 τ u k 1 u k u k 1 τ τ 4 [ 1 x uk 1 uk u k 1 h 1 x uk 1 4uk 1 1 6uk 1 h 4 uk 1 uk 1 h 4u k 1 1 uk x uk 1 uk 1 1 uk 1 1 uk 1 h 3 uk 1 1 ] uk 1 1 u k 1 ϕ k h ϕ k x si x cos x e t k 1 t k ) e t k si x Mh π x h M Nτ 1 t k kτ 1 k N 1 u k u k 1 1 x uk 1 u k 1 1 h N u 0 τ [ ] e kτ u k e k 1 τ u k 1 ψ x 0 M k 1 ψ x 1 3e 1) si x 0 M [ u 0 1 u0 1 1 x u0 1 ] u0 u 0 1 u 0 h h si x M u k 0 uk M 0 0 k N u k uk 1 5 uk 3 0 k N u k M uk M 1 5 uk M 3 0 k N 3.13

11 Discrete Dyamics i Nature ad Society 11 for approximate solutios of problem 3.1. Oe ca write the N 1 M 1 system of liear equatios 3.13 as the matrix equatio A u B u 1 C u D u 1 E u Rϕ u 0 0 u M 0 M u u 1 5 u u M u M 1 5 u M 3. Here A B C D E adr are N 1 N 1 square matrices: a b c a b 0 0 A B a c b c y τ τe τ τe τ τe N τ τe N 1 τ τ e Nτ d e f d e C e f d e f w g l g D g l g l z

12 1 Discrete Dyamics i Nature ad Society m m 0 0 E m m We deote a τ 1 x 4 h 4 1 x ) b h 3 1 x 1 h h d 1 τ e τ 1 x 1 g h 1 h 1 x h c τ 4 1 x 4 1 x x 4 h 4 h 3 h 1 ) h l τ 4 1 x 4 1 x x 4 h 4 h 3 h 1 ) h f 1 τ τ 6 1 x 4x ) 4 h 4 h 1 m τ 1 x 1 x ) 4 h 3 h 4 z τ 4h 1 x τ h w 1 1 x τ τ h y τ ϕ 0 ϕ 1 ϕ.. ϕ N N 1 1 4h 1 x τ h ϕ k f t k x x si x cos x e t k ) 1 t k e t k si x 1 k N 1 U 0 s U 1 s R I N 1 is the idetity matrix U s. U N s N where s ± ± 1.

13 Discrete Dyamics i Nature ad Society 13 Table 1 Differece schemes N M 0 N M 40 N M 80 Differece schemes Differece schemes Differece schemes For the solutio of matrix equatio 3.14 we will use the modified Gauss elimiatio method. We seek a solutio of the matrix equatio by the followig formula: U α 1 u 1 β 1 u γ 1 M where α j β j j 1...M 1 are N 1 N 1 square matrices γ j j 1...M 1 are N 1 1 colum matrices ad α 1 β 1 γ 1 γ are zero matrices. α β are α 4 5 I N 1 β 1 5 I N 1 F C D α E α 1 α E β 1 α 1 F 1 [ ] B D β E α 1 β β 1 F 1 A 3.18 γ 1 F 1 [ Rϕ D γ E α 1 γ E γ 1 ]. For solutio of the last differece equatio we eed to fid u M u M 1 u M 0 u M 1 β M 5I ) 4I α M α M 1 ) 1 4I αm γ M 1 γ M Error Aalysis The errors are computed by E N M max 1 k N 11 M 1 u t k x u k 3.0 of the umerical solutios where u t k x represets the exact solutio ad u k represets the umerical solutio at t k x ad the results are give i Table 1. Thus the results show that the secod order of accuracy differece schemes 3.10 ad 3.13 are more accurate comparig with the first order of accuracy differece scheme 3.4. Ackowledgmet The authors would like to thak Professor P. E. Sobolevskii Jerusalem Israel for the helpful suggestios to the improvemet of this paper.

14 14 Discrete Dyamics i Nature ad Society Refereces 1 N. Gordeziai P. Natalii ad P. E. Ricci Fiite-differece methods for solutio of olocal boudary value problems Computers & Mathematics with Applicatios vol. 50 o. 8-9 pp R. Dautray ad J.-L. Lios Aalyse Mathématique et Calcul Numérique pour les Scieces et les Techiques vol. 1 Masso Paris Frace D. Gordeziai H. Meladze ad G. Avalishvili O oe class of olocal i time problems for firstorder evolutio equatios Zhural Obchyslyuval oï ta Prykladoï Matematyky vol. 88 o. 1 pp D. G. Gordeziai ad G. A. Avalishvili Time-olocal problems for Schrödiger-type equatios I. Problems i abstract spaces Differetsial ye Uraveiya vol. 41 o. 5 pp R. P. Agarwal M. Boher ad V. B. Shakhmurov Maximal regular boudary value problems i Baach-valued weighted space Boudary Value Problems vol. 005 o. 1 pp A. Ashyralyev ad O. Gercek Nolocal boudary value problems for elliptic-parabolic differetial ad differece equatios Discrete Dyamics i Nature ad Society vol. 008 Article ID pages M. Dehgha ad M. Lakestai The use of cubic B-splie scalig fuctios for solvig the oedimesioal hyperbolic equatio with a olocal coservatio coditio Numerical Methods for Partial Differetial Equatios vol. 3 o. 6 pp A. Ashyralyev I. Karatay ad P. E. Sobolevskii O well-posedess of the olocal boudary value problem for parabolic differece equatios Discrete Dyamics i Nature ad Society vol. 004 o. pp A. Ashyralyev ad O. Yildirim O multipoit olocal boudary value problems for hyperbolic differetial ad differece equatios Taiwaese Joural of Mathematics vol. 14 o. 1 pp A. Ashyralyev ad H. A. Yurtsever The stability of differece schemes of secod-order of accuracy for hyperbolic-parabolic equatios Computers & Mathematics with Applicatios vol. 5 o. 3-4 pp M. Dehgha O the solutio of a iitial-boudary value problem that combies Neuma ad itegral coditio for the wave equatio Numerical Methods for Partial Differetial Equatios vol. 1 o. 1 pp S. Mesloub ad A. Bouziai O a class of sigular hyperbolic equatio with a weighted itegral coditio Iteratioal Joural of Mathematics ad Mathematical Scieces vol. o. 3 pp L. S. Pulkia O the solvability i L of a olocal problem with itegral coditios for a hyperbolic equatio Differetsial ye Uraveiya vol. 36 o. pp A. Saadatmadi ad M. Dehgha Numerical solutio of the oe-dimesioal wave equatio with a itegral coditio Numerical Methods for Partial Differetial Equatios vol. 3 o. pp M. Ramezai M. Dehgha ad M Razzaghi Combied fiite differece ad spectral methods for the umerical solutio of hyperbolic equatio with a itegral coditio Numerical Methods for Partial Differetial Equatios vol. 4 o. 1 pp A. Ashyralyev ad N. Aggez A ote o the differece schemes of the olocal boudary value problems for hyperbolic equatios Numerical Fuctioal Aalysis ad Optimizatio vol.5o.5-6 pp P. E. Sobolevskiĭ ad S. M. Semeov O some approach to ivestigatio of sigular hyperbolic equatios Doklady Akademii Nauk SSSR vol. 70 o. 3 pp P. E. Sobolevskii ad L. M. Chebotaryeva Approximate solutio by method of lies of the Cauchy problem for a abstract hyperbolic equatios Izvestiya Vysshikh Uchebykh Zavedeii. Matematikao. 5 pp Russia. 19 A. Ashyralyev M. Martiez J. Pastor ad S. Piskarev Weak maximal regularity for abstract hyperbolic problems i fuctio spaces further progress i aalysis i Proceedigs of the 6th Iteratioal ISAAC Cogress pp World Scietific Akara Turkey B. A. Kosti O aalytic semigroups ad cosie fuctios Doklady Akademii Nauk SSSR vol. 307 o. 3 pp Russia.

15 Discrete Dyamics i Nature ad Society 15 1 H. O. Fattorii Secod Order Liear Differetial Equatios i Baach Spaces vol. 108 of North-Hollad Mathematics Studies North-Hollad Amsterdam The Netherlads S. Piskarev ad Y. Shaw O certai operator families related to cosie operator fuctios Taiwaese Joural of Mathematics vol. 1 o. 4 pp A. A. Samarskii I. P. Gavrilyuk ad V. L. Makarov Stability ad regularizatio of three-level differece schemes with ubouded operator coefficiets i Baach spaces SIAM Joural o Numerical Aalysis vol. 39 o. pp A. Ashyralyev ad M. E. Koksal O the umerical solutio of hyperbolic PDEs with variable space operator Numerical Methods for Partial Differetial Equatios vol. 5 o. 5 pp M. Ashyraliyev A ote o the stability of the itegral-differetial equatio of the hyperbolic type i a Hilbert space Numerical Fuctioal Aalysis ad Optimizatio vol. 9 o. 7-8 pp D.GuidettiB.Karasöze ad S. Piskarev Approximatio of abstract differetial equatios Joural of Mathematical Scieces vol. 1 o. pp A. Ashyralyev ad Y. Ozdemir Stability of differece schemes for hyperbolic-parabolic equatios Computers & Mathematics with Applicatios vol. 50 o. 8-9 pp W.-D. Li Z.-Z. Su ad L. Zhao A aalysis for a high-order differece scheme for umerical solutio to u tt A x t u xx F x t u u t u x Numerical Methods for Partial Differetial Equatios vol. 3 o. pp P. E. Sobolevskii Differece methods for the approximate solutio of differetial equatios Izdatelstvo Voroezhskogo Gosud Uiversiteta Voroezh 1975 Russia. 30 A. A. Samarskii ad E. S. Nikolaev Numerical Methods for Grid Equatios vol. of Iterative Methods Birkhäuser Basel Switzerlad 1989.

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