Research Article On the Solution of NBVP for Multidimensional Hyperbolic Equations

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1 e Scientific World Journal Volume 4, Article ID 846, pages Research Article On the Solution of BVP for Multidimensional Hyperbolic Equations Allaberen Ashyralyev and ecmettin Aggez Department of Mathematics, Fatih University, Buyukcekmece, 345 Istanbul, Turkey Correspondence should be addressed to ecmettin Aggez; Received 6 August 3; Accepted February 4; Published 5 May 4 Academic Editors: A Ibeas, L Kong, and F Mukhamedov Copyright 4 A Ashyralyev and Aggez This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited We are interested in studying multidimensional hyperbolic equations with nonlocal integral and eumann or nonclassical conditions For the approximate solution of this problem first and second order of accuracy difference schemes are presented Stability estimates for the solution of these difference schemes are established Some numerical examples illustrating applicability of these methods to hyperbolic problems are given Introduction In the last decades, for the development of numerical methods and theory of solutions of the hyperbolic problems with nonlocal integral, eumann and nonclassical conditions have been an important research topic in many natural phenomena Solutions of this type of hyperbolic problems were investigated in 3 These problems were studied in various directions: qualitative properties of solutions, spectral problems, various statements of boundary value problems, and numerical investigations For example, in 5thenonlocalboundaryvalueproblem u () = d u (t) dt Au(t) =f(t), t, n r= α r u(λ r )φ, u t () = n r= <λ λ λ n β r u t (λ r )ψ, was investigated The stability estimates for the solution of the problem were established The first order of accuracy difference schemes for the approximate solution of this problem waspresentedthestabilityestimatesforthesolutionofthese difference schemes were established Theoretical statements were supported by numerical examples () The well-posedness of the Cauchy problem, Goursat problem, and boundary value problem for multidimensional hyperbolic equations have been studied extensively in a large cycle of papers (see, eg, 4 and the references therein) Actually, in paper 4, the Goursat problem for a linear multidimensional hyperbolic equation was investigated Uniqueness of the solution and weak solvability of the Goursat problem were established In paper 5, the existence or nonexistence of global solutions of a multidimensional version of the first Darboux problem for wave equations with power nonlinearity in the conic domain was investigated In 6, 7, the solvability of an initial-boundary value problem for second order linear hyperbolic equations with a condition on the lateral boundary connecting the values of the solution or the conormal derivative of the solution with the values of some integral operator of the solution was studied The existence and uniqueness theorems for regular solutions were proved In 8, the difference schemes for multidimensional hyperbolic equations were investigated These methods were stable under the inequalities and contain the connection between the grid step sizes of time and space variables In, the authors develop a finite difference method (FDM)foramultidimensionalcoupledsystemofnonlinear parabolic and hyperbolic equations and prove the existence,

2 The Scientific World Journal stability, and uniqueness of its solution by a set of theorems Finally,theproposedmethodwasillustratedbyanumberof numerical experiments The study of difference schemes for hyperbolic equations with nonlocal conditions without using any necessary condition concerning the grid step sizes is of great interest Such a difference scheme for solving the initial-value problem for abstract hyperbolic equations was studied for the first time in In the present paper, the following multidimensional hyperbolic equation u (t, x) t m (a r (x) u xr ) xr σu(t, x) =f(t, x), r= x=(x,,x m ) Ω, <t<, () Stability of First Order of Accuracy Difference Scheme For approximately solving problem (), first order of accuracy difference scheme (u k u k u k )Au k =f k, f k =f(t k ), t k = (k), k, =, u = (I A) (u u )= α(t j )u j φ, β(t j )(u j u j )ψ (7) with nonlocal integral u (, x) = α (ρ) u (ρ, x) dρ φ (x), x Ω, u t (, x) = β(ρ)u t (ρ, x) dρ ψ (x), x Ω, (3) is considered A study of discretization of the nonlocal boundary value problem also permits one to include general difference schemes in applications, if differential operator in space variables A is replaced by difference operator A h that acts in a Hilbert space and is uniformly self-adjoint positive definite in h for <h h The stability estimates of solution of difference scheme (7) are established under the assumption and eumann or nonclassical conditions under the assumption u(t, x) = (4) n S u(t, x) S =, u(t, x) =, t, n S α (s) β (s) ds (5) > ( α (s) β (s) )ds (6) > α(t j) β(t j) α(t j) β(t j) Lemma The following estimates hold 3: where R H H, R H H, R R H H, RR H H, A/ R H H, A/ R H H, (8) (9) R=(IiA / ), R =(IiA / ) () Lemma The operator is considered Here, Ω is the unit open cube in the mdimensional Euclidean space R m with boundary S=S S, Ω=Ω S, a r (x) (x Ω), φ(x), ψ(x) (x Ω),andf(t, x) (t (, ), x Ω) are given smooth functions, and a r (x) a > n is the normal vector to Ω The first and second order of accuracy difference schemes for multidimensional hyperbolic problem () are presented The schemes are shown to be absolutely stable It is naturally seen that the second order difference schemes are much more advantageous than the first order ones T=I β(t j ) R R (R j R j ) α(t j ) Rj R j α(t j ) β(t j )(R R) j ()

3 The Scientific World Journal 3 has an inverse T = I β(t j ) R R (R j R j ) α(t j ) Rj R j α(t j ) and the following estimate is satisfied: β(t j )(R R) j T () ( α(t j) β(t j) () Theorem 3 Let φ D(A), ψ D(A / ),and(8) hold Then, for the solution of difference scheme (7) the following stability estimates hold: k u k H M A/ f s H A/ ψ H φ H, k (u k u k ) H k A/ u k H M (6) f s H ψ H A/ φ H, (7) k (u k u k u k ) H k Au k H M s= f s f s H f H A/ ψ H Aφ H, (8) β(t j) α(t j) ) (3) where M is independent of f s, s,andφ, ψ Proof First, we obtain formula for the solution of difference scheme (7) For the solution of difference scheme Proof Using formula () and the triangle inequality, we can write T H H α(t j) β(t j) (R R) j β(t j) R R ( Rj R j ) α(t j) Rj R j (4) (u k u k u k )Au k =f k, u =μ, f k =f(t k ), the following formulas k, u =μ, t k = (k), =, (I A) (u u )=ω, u =μr Rω, u k = Rk R k μ(r R) (R k R k )ω k i A/ R ks R ks f s, k, (9) () Applying the triangle inequality and estimates (9), we get T H H α(t j) β(t j) β(t j) α(t j) (5) Thus, estimate (3) follows from this estimate Lemma is proved The following theorem on the stability estimates for the solution of difference scheme (7)is established were obtained in Applying formula () and nonlocal boundary conditions in (7), we can write formula for μ and ω μ=t α () j j= α(t j ) i A/ R js R js f s φ I β(t j ) R R(R j R j )

4 4 The Scientific World Journal α(t j ) (R R) (R j R j ) R R β () β() j ω=t I R js R js f s β(t j ) β(t j ) i (R R) A / f j ψ, α(t j ) Rj R j β () β() β(t j ) j R js R js f s () k (u k u k u k ) H k Au k H M s= f s f s H f H A/ ω H Aμ H (5) were established Firstofall,letusfindestimatefor μ H Byusingformula ()andestimates(9), we obtain μ H M φ H A/ f j H A/ ψ H (6) And, applying A / to formula (), we get A / ω=t I α(t j ) Rj R j β () β() β(t j ) j R js R js A / f s β(t j ) i (R R) A / f j ψ β(t j ) i A/ R j R j α () j j= α(t j ) i A/ R js R js f s φ () Hence, for the solution of nonlocal boundary value problem (7)wehaveformulas(), (), and () Second, let us investigate stability of difference scheme (7) In, for the solution of (9)stabilityestimates k u k H M A/ f s H A/ ω H μ H, k (u k u k ) H k A/ u k H M (3) f s H ω H A/ μ H, (4) β(t j ) i (R R) A f j A / ψ β(t j ) i Rj R j α () j j= α(t j ) i Rjs R js A / f s φ (7) Using the triangleinequality,formula (7), and estimates (9), it follows that A/ ω M H A/ f j H A/ ψ H φ H (8) So, estimate (6) followsfromestimates(3), (6), and (8) Second, applying A / to formula ()andusingestimates(9), we get estimate A/ μ M H f j H ψ H A/ φ H (9)

5 The Scientific World Journal 5 By using formula ()and estimates(9), we obtain ω H M f j H ψ H A/ φ H (3) Using estimates (4), (9), and (3), we obtain estimate (7) for the solution of (7) Third, applying A to formula ()and using Abel s formula, we can write formula for Aμ Aμ = T α () j j= α(t j ) R js R js (f s f s ) s= ( R j R j )f ( R R) f j Aφ I β(t j ) R R(R j R j ) α(t j )(R j R j ) (R R) β () β() j s= R js R js (f s f s ) ( R j R j )f ( R R) f j β(t j )f j A / ψ It follows from formula (3)and estimates(9)that Aμ H M s= β(t j ) (3) f s f s H f H A/ ψ H Aφ H (3) Applying A / to formula () andusingabel sformula,we get A / ω =T I α(t j ) Rj R j β () β() j R js R js s= β(t j ) (f s f s )( R j R j )f ( R R) f j β(t j ) i (R R) f j A / ψ β(t j ) i Rj R j α () j j= α(t j ) R js R js (f s f s ) s= ( R j R j )f ( RR)f j Aφ, (33) and using the triangle inequality and estimates (9), we obtain the estimate A/ ω H M s= f s f s H f H A/ ψ H Aφ H (34) Thus, estimate (8) follows fromestimates(3), (3), and (34) This is the end of the proof of Theorem 3 3 Stability of Second Order of Accuracy Difference Scheme ow, we consider the second order accuracy difference scheme for approximate solution of boundary value problem () (u k u k u k ) Au k 4 A(u k u k )=f k, f k =f(t k ), t k =k, k, =, u = α(t j )u j u j φ,

6 6 The Scientific World Journal (I A 4 ) (u u ) (f Au ) = β(t j )u j u j ψ (35) The stability of solutions of this difference scheme is investigated under the assumption > α(t j ) β(t j ) α(t j ) β(t j ) Lemma 4 The following estimates hold 3: where (I ± ia/ ), H H R H H, R H H, (I ± ia/ ) H H, A/ (I ± ia / ) H H, R=(I ia/ R=(I ia/ )(I ia/ ), )(I ia/ ) (36) (37) (38) Lemma 5 Supposethatassumption(36) holds Then, the operator T=I α( )α( )α(3 α( 3 R )(R ) ) (R R ) α(t j ) R j R j R j R j β ( )β(3 ) (I A 4 ) β( 3 )ia/ R R ia / β(t j ) Rj R j R j R j β ( )β(3 ) (I A 4 ) β( 3 )ia/ R R ia / β(t j ) Rj R j R j R j α( )α( )α(3 ) ( R R )α( 3 R )(R ) α(t j ) R j R j R j R j β ( )β(3 ) (R R ) β( )β(3 ) R R β(t j ) Rj R j R j R j α ( )α(3 ) α (3 R R )A/ i A / α(t j ) Rj R j R j R j i (39) has an inverse T and the following estimate is satisfied: T α(t j ) β(t j ) β(t j ) α(t j ) (4) Proof Using formula (39),the triangle inequality,and estimates (37), we obtain T H H α( ) α( ) α(3 ) R R

7 The Scientific World Journal 7 α(3 ) α(t j ) R R R j R j R j R j β( ) β(3 ) (I A 4 ) β(3 ) A / R R β(t j ) A / R j R j R j R j β( ) β(3 ) (I A 4 ) β(3 ) A / R R β(t j ) A / R j R j R j R j α( ) α( ) α(3 ) R R α(3 ) α(t j ) R R R j R j R j R j β( ) β(3 ) R R β( ) β(3 ) β(t j ) α( ) α(3 ) R R R j R j R j R j α(3 ) A / R R α(t j ) A / R j R j R j R j α( ) α(3 ) α(t j ) β( ) β(3 ) β(t j ) β( ) β(3 ) β(t j ) α( ) α(3 ) α(t j ) α(t j ) β(t j ) α(t j ) β(t j ) (4) Estimate (4) follows from this estimate Lemma 5 is proved Theorem 6 Let φ D(A), ψ D(A / ), and assumption (36) hold For the solution of difference scheme (3) the following stability estimates k u k H M A/ f s H A/ ψ H φ H, s= k (u k u k ) H k A/ u k H M s= (4) f s H A/ φ H ψ H, (43) k (u k u k u k ) H k Au k u k M H f s f s H f H A/ ψ H Aφ H (44) are valid, where M is independent of f s, s,and φ, ψ

8 8 The Scientific World Journal Proof We obtain formula for the solution of difference scheme (3) For the solution of difference scheme (u k u k u k ) Au k 4 A(u k u k )=f k, u =μ, the following formulas u =μ, f k =f(t k ), k, t k =k, =, (I A 4 ) (u u ) (Au f )=ω, u =(I A 4 ) (I A 4 )μω f, u k = Rk R k μ i A/ R k R k (ω f ) (45) (46) j i A/ R js R js R js R js f s i (R R) f j φ α ( )α(3 ) α (3 R R )A/ i A / β( )β(3 α(t j ) Rj R j R j R j i ) (I A 4 ) ia/ β( 3 ) R R β(t j )ia/ k i A/ R ks R ks f s, k, Rj R j R j R j f were obtained in Applying formula (46) and nonlocal boundary conditions in (3), we obtain formulas μ=t Iβ( )β(3 ) (I A 4 ) β( 3 )ia/ R R β(t j )ia/ Rj R j R j R j α( )α(3 )α(3 R R )A/ i α(t j )A/ Rj R j R j R j i 4 f α( 3 ) 4i A/ (R R) f j α(t ) ia / R R β( 3 )f i A/ β(t j ) j β(t j )f j R js R js R js R js f s ψ, ω=t I α( )α( )α(3 α( 3 R )(R ) ) (R R ) α(t j ) R j R j R j R j β( )β(3 ) (I A 4 ) ia/ β( 3 ) R R (47)

9 The Scientific World Journal 9 β(t j )ia/ ia / R R Rj R j R j R j f j i A/ R js R js R js R js f s i (R R) f j φ (48) β( 3 )f i A/ β(t j ) j R js R js R js R js f s ψ β(t j )f j β ( )β(3 ) (I A 4 ) (I A 4 )β( ) β( 3 ) R R β(t j ) Rj R j R j R j α( )α(3 ) α( 3 )A/ R R i α(t j )A/ Rj R j R j R j i 4 f α( 3 ) 4i A/ (R R) f j α(t ) Hence, for the solution of nonlocal boundary value problem (3)wehaveformulas(46), (47), and (48) ow, let us investigate the stability of difference scheme (3) In, for the solution of (45) thefollowingstability estimates k u k H M A/ f s H A/ ω H μ H, s= k (u k u k ) H k A/ u k H M s= (49) f s H ω H A/ μ H, (5) k (u k u k u k ) H k Au k u k M H f s f s H f H A/ ω H Aμ H (5) were established ow, from formula (47)andestimates(37) it follows that μ H M A/ f s H A/ ψ H φ H (5) s= Applying A / to formula (48), we get A / ω =T I α ( )α( )α(3 α( 3 R )(R ) α(t j ) ) (R R ) R j R j R j R j

10 The Scientific World Journal β( )β(3 ) (I A 4 ) ia/ β( 3 ) R R β(t j )ia/ ia / R R Rj R j R j R j A / f α(t j ) 4i (R R) f j j α(t ) j i A/ R js R js R js R js f s φ (53) β( 3 )A/ f i j β(t j ) β(t j )A/ f j R js R js R js R js A / f s A / ψ β ( )β(3 ) R R A/ β( ) β( 3 )A/ R R β(t j )A/ Rj R j R j R j α( )α(3 ) α( 3 )A/ R R i α(t j )A/ Rj R j R j R j i 4 f α( 3 ) 4i A/ (R R) f and using estimates (37), we obtain A/ ω M H A/ f s H A/ ψ H φ H s= (54) So, using estimates (49), (5), and (54), we obtain (4)forthe solution of (3) Applying A / to formula (47), we get A / μ =T Iβ( )β(3 ) (I A 4 ) β( 3 )ia/ R R β(t j )ia/ Rj R j R j R j α ( )α(3 )α(3 ) R R α(t j ) i Rj R j R j R j i 4 f α( 3 ) 4i (R R) f α(t j ) 4i (R R) f j A / φ

11 The Scientific World Journal j α(t ) j i Rjs R js R js R js f s α( )α(3 ) α (3 ) R R α(t j ) Rj R j R j R j i β ( )β(3 ia/ β( 3 ) R R β(t j )ia/ i ) (I A 4 ) Rj R j R j R j ia / R R β( 3 )f i A/ β(t j ) j R js R js R js f β(t j )f j Thus, estimate (43) follows from estimates(5), (56), and (57) Applying A to (47)andusingAbel sformula,weobtain Aμ =T I β ( )β(3 ) (I A 4 ) β( 3 ) ia/ R R β(t j )ia/ Rj R j R j R j α( )α(3 )α(3 )A/ R R α(t j )A/ i Rj R j R j R j i 4 f α( 3 ) 4i A/ (R R) f α(t j ) 4i A/ (R R) f j Aφ j α(t ) 4 j (I ia/ )R js R js f s ψ From the last formula and estimates (37)it follows that (55) (I ia/ ) R js (f s f s )(I ia/ )R js A/ μ M H f s H ψ H A/ φ H (56) s= (I ia/ ) R js f Using formula (48), the triangle inequality, and estimates (37), we obtain ω H M f s H ψ H A/ φ H (57) s= f j j α(t ) 4

12 The Scientific World Journal j (I ia/ )R js (I ia/ ) R js (f s f s ) (I ia/ )R js (I ia/ ) R js f f j α ( )α(3 ) α( 3 ) R R i α(t j ) Rj R j R j R j i β ( )β(3 iβ (3 ) R R β(t j )i i R R Rj R j R j R j β( 3 )f j β(t ) 4 j (I ia/ )R js ) A/ (I A 4 ) f β(t j )f j A / ψ (I ia/ ) R js (f s f s ) (I ia/ )R js (I ia/ ) R js f f j j β(t ) 4 j (I ia/ )R js (I ia/ ) R js (f s f s ) (I ia/ )R js f j (I ia/ ) R js f From the last formula and estimates (37)it follows that Aμ H M (58) f s f s H f H A/ ψ H Aφ H Applying A / to (48)andusingAbel sformula,weget A / ω =T I α ( )α( )α(3 ) ( R R )α( 3 R )(R ) α(t j ) R j R j R j R j (59)

13 The Scientific World Journal 3 β( )β(3 iβ (3 R ) R β(t j )i i R R ) A/ (I A 4 ) Rj R j R j R j f β( 3 )f j β(t ) 4 j (I ia/ )R js β(t j )f j A / ψ (I ia/ ) R js (f s f s ) (I ia/ )R js (I ia/ ) R js f f j j β(t ) 4 j (f s f s ) (I ia/ )R js (I ia/ ) R js (I ia/ )R js (I ia/ ) R js f f j β( )β(3 β( 3 ) R R ) R R β( ) β(t j ) Rj R j R j R j α( )α(3 ) α( 3 )A/ R R i α(t j )A/ Rj R j R j R j i 4 f α( 3 ) 4i A/ (R R) f i A/ (R R) f j Aφ j α(t ) 4 j (I ia/ )R js (I ia/ ) R js (f s f s ) (I ia/ )R js (I ia/ ) R js f f j j α(t ) 4 j (I ia/ )R js (I ia/ ) R js (f s f s ) (I ia/ )R js (I ia/ ) R js f f j (6)

14 4 The Scientific World Journal Usingthetriangleinequalityandestimates (37), we obtain A/ ω H M f s f s H f H A/ ψ H Aφ H (6) As a result, estimate (44) follows from estimates(5), (59), and (6) Theorem 6 is proved 4 Application The discretization of hyperbolic equation ()witheumann and integral or nonclassical and integral boundary conditions is carried out in two steps First, let us define the grid sets Ω h =x=x r =(h r,,h m r m ), r=(r,,r m ), for an infinite system of ordinary differential equations Second, we replace problem (66) by the difference scheme u h k (x) uh k (x) uh k (x) A x h uh k (x) =fh k (x), f h k (x) =fh (t k,x), t k = (k), k, =, x Ω h, u h (x) = α(t m )u h m (x) φh (x), x Ω h, m= (I A x h )(uh (x) uh (x)) = β(ρ j )u h j (x) uh j (x)ψh (x), x Ω h (67) r j j, h j j =,,,m, Ω h = Ω h Ω, S h = Ω h S (6) of the first order accuracy in t For the stability of first order of accuracy difference scheme, the following theorem is presented We introduce the Banach space L h = L ( Ω h ) of the grid functions φ h (x) =φ(h r,,h m r m ) (63) defined on Ω h, equipped with the norm φh =( L ( Ω h ) φh (x) h h m ) x Ω h / (64) To the differential operator A x generated by (), we assign the difference operator A x h by the formula A x m h uh x = (a r (x)u h x r ) xr σu h (x) (65) r j r= acting in the space of grid functions u h (x), satisfyingthe condition D h u h (x) = for all x S h or u h (x) =, x S h and D h u h (x) =, x S h, S h = S h S h D h u h is the approximation of ( u/ n)itisknownthata x h is a self-adjoint positive definite operator in L ( Ω h ) With the help of A x h we arrive at the nonlocal boundary value problem d V h (t, x) dt A x h Vh (t, x) =f h (t, x), <t<, x Ω h, V h (, x) = α(ρ)v h (ρ, x) dρ φ h (x), x Ω h, dv h (, x) dt = β(ρ) dvh (ρ, x) dρ ψ h (x), x Ω dt h (66) Theorem 7 Let and h be sufficiently small numbers Then, the solutions of difference scheme (67) satisfy the following stability estimates: k uh k L h k (u h k uh k ) L h k (uh k ) x L h M k fh k L h φh x L h ψh L h, k (u h k uh k uh k ) L h k (uk x ) x L h M fh L h k (f h k fh k ) L h (φh x ) x L h ψh x L h (68) Here, M is independent of, h,φ h (x), ψ h (x), andf h k (x), k< The proof of Theorem 7 is based on the symmetry property of difference operator A x h defined by formula (65) and on the following theorem on coercivity inequality of the elliptic difference problem Theorem 8 For the solutions of the elliptic difference problem A x h uh (x) =ω h (x), x Ω h, D h u h (x) =, x S h or u h (x) =, x S h, D h u h (x) =, x S h (69)

15 The Scientific World Journal 5 the following coercivity inequality holds 3: k (u h k uh k uh k ) L h m uh x r x r r j Lh M ωh (7) L h r= (u k x ) x (uk x ) x k L h In addition, the second order of accuracy difference scheme u h k (x) uh k (x) uh k (x) Ax h uh k (x) 4 Ax h uh k (x) uh k (x) =fh k (x), x Ω h, f h k =fh k (t k,x), t k =k, k, =, u h (x) = α(ρ j )uh j (x) uh j (x) φ h (x), x Ω h, (I A x h ) u h (x) uh (x) = fh (x) Ax h uh (x) β(ρ j )uh j (x) uh j (x)ψh (x), f h =fh (, x), f h =fh (, x), x Ω h, (7) for approximately solving hyperbolic equation ()withnonlocal integral and eumann or nonclassical conditions is presented The following theorem on the stability of (7) is obtained Theorem 9 Let and h be sufficiently small numbers Then, the solution of difference scheme (7) satisfies the following stability estimates: k uh k L h k (u h k uh k ) L h k (uh k ) x L h M fh L h k (fh k fh k ) L h φh x L h ψh L h, M fh L h (f h fh ) L h k (f h k fh k ) L h (φh x ) x L h ψh x L h (7) Here, M does not depend on, h, φ h (x),andf h k, k< The proof of Theorem 9 isbasedonthesymmetryproperty of difference operator A x h defined by formula (65) and on Theorem 8 on coercivity inequality of elliptic difference problem (69) 5 umerical Examples In this section, we apply finite difference schemes (67) and (7) to three examples which are one-dimensional hyperbolic equation with nonlocal and nonclassical conditions Example The nonlocal boundary value problem u (t, x) t (x) u (t, x) x =f(t, x), u x (t, x) u(t, x) f (t, x) = (x) cos xsin x (e t t)e t cos x, <t<, u (, x) = e s u (s, x) ds φ (x), φ (x) =(3e ) cos x, u t (, x) = e s u s (s, x) ds ψ (x), ψ (x) =e cos x, x π, <x<π, u x (t, ) =u x (t, π) =, t, (73) for one-dimensional hyperbolic equation with variable coefficients is considered The exact solution of this problem is u (t, x) =(e t t)cos x (74) For the approximate solution of the problem (73), we apply finite difference schemes (67)and(7)

16 6 The Scientific World Journal First, we obtain the first order of accuracy difference scheme u k n u k n uk n (x n ) uk n uk n u k n h uk n uk n u k n =f(t h k,x n ), f(t k,x n )= (x n )cos x n sin x n (e t k t k ) e t k cos x n, u n e k u k n =φ(x n), k= =, x n =nh, n M, Mh = π, φ(x n )=(3e ) cos x n, t k =k, k, u n u n e k (u k n uk n )=ψ(x n ), k= ψ(x n )=e cos x n, n M, n M, u k =uk, uk M =uk M, k (75) e n e n C n = d e n e n () () a n = (x n) h h, b =, c =, d n = (x n) h, e n = (x n) h h, at = e e, ap = e e 3, an = e () e, f n = f n f n f n f n (),, The system can be written in the matrix form A n u n B n u n C n u n =Dφ n, n M, u =u, u M =u M Here, (76) f k n =f(t k,x n )=(x n )cos x n sin x n (e t k t k )e t k cos x n, f n =(3e ) cos x n, n M, f n =e cos x n, n M, k, (77) B n = a n a n A n = d a n a n () () e e e () e b c d n b c b d d n c d n e at ap an e () (),, and D=I is the identity matrix Consider U s = u s u s u s u s (), s=n,n,n (78) This type system was used by 4 fordifferenceequations For the solution of matrix equation (76), we will use modified Gauss elimination method We seek a solution of the matrix equation by the following form: u n =α n u n β n, n=m,,,, (79)

17 The Scientific World Journal 7 where u M =(Iα M ) β M, α j (j =,,M)are ( ) ( ) square matrices and β j (j =,,M)are ( ) column matrices α is identity and β is zero matrices, and f k n =(x n )cos x n sin x n (e t k t k )e t k cos x n, Mh = π, x n =nh, n M, =, t k =k, k, α n =(B n C n α n ) A n, β n =(B n C n α n ) (D n φ n C n β n ), n=,,3,,m (8) u n = ek u k n e(k) u k n φ(x n ), k= n M, φ(x n )=(3e ) cos x n, n M, Second, applying formulas u n u n 4 ( x n) u () 5u(h) 4u(h) u(3h) h u () =O(h ), u (π) 5u(πh) 4u(πh) u(π3h) h u (π) =O(h ) (8) andusingthesecondorderofaccuracyimplicitdifference scheme (7), we get second order of accuracy difference scheme = u n u n u n h u n u n 4h 4 u n u n 4 f n k= u n u n u n h u n u n 4h e k(/) u k n uk n ψ(x n ), n M, ψ(x n )=e cos x n, n M, u u = h u 5u 4u u3 u φ u k n u k n uk n (x n ) uk n uk n u k n 4h uk n uk n uk n h uk n uk n u k n 4h uk n uk n uk n uk n 8h 4h uk n uk n u k n 4 =f k n, uk n uk n 8h u u = h u 5u 4u u 3 u φ u k uk = h uk u k uk h u k φk, k, (8) for the approximate solutions of nonlocal boundary value problem (73) We have again () ()system of linear equations We can write the system as a matrix equation (76) Here, a n a n a n a n a n a n E n = d a n a n a n w n w n () (),

18 8 The Scientific World Journal e e e 3 e () e b n d n b n b n d n b n F n = d b n d n b n y n t n λe 3/ λe 5/ λe (3)/ e (/) c n c n c n c n c n c n G n = d c n c n c n z n z n a n = (x n) 4h 8h, c n = (x n) 4h 8h, () () b n = x n h 4, d n = x n h, y n = 4 ( x n) h e /, t n = 4 ( x n) h (e )e /,, () () w n = 4h ( x n) h, z n = 4h ( x n) h, λ = (e ) f n = f n f n f n () f k n =f(t k,x n )=(x n )cos x n sin x n,, (e t k t k )e t k cos x n, k, f n =(3e ) cos x n, n M, f n =e cos x n, n M, λ λ λ 3 λ 4 a b a a b a T= d a b a a b a λ 4 λ 3 λ λ () (),

19 The Scientific World Journal 9 a= h, b = h h h, λ = h h ( ), λ = 5 h, λ 3 = h, λ 4 = h, D=I, U s = u s u s u s u s (), s=n,n,n (83) For the solution of the matrix equation (76), we used the same algorithm as in the first order of accuracy difference scheme Here, u M =(α M α M 4α M 3I) (4β M α M β M β M ), α =T ( I), β =T φ k, k (84) Example The nonlocal boundary value problem u (t, x) t (x) u (t, x) x =f(t, x), u x (t, x) u(t, x) f (t, x) = (x) cos xsin x (e t t) e t (cos x), <t<, <x<π, u (, x) = e s u (s, x) ds φ (x), φ (x) =(3e ) (cos x), u t (, x) = e s u s (s, x) ds ψ (x), ψ (x) =e (cos x), x π, u (t, ) =u x (t, π) =, t, (85) is considered Here, we use the same procedure as in the first example The exact solution of this problem is u (t, x) =(e t t)(cos x) (86) Using the same manner, we can construct first order of accuracydifferenceschemeanditcanbewritteninthematrix form Here, matrices A n, B n, C n,andd aregivenintheprevious example, and f k n f n = f n f n f n f n () =f(t k,x n )=(x) cos xsin x (e t t)e t (cos x), f n =(3e ) (cos x),, k, n M, f n =e (cos x), n M (88) For the solution of matrix equation (87), we will use modified Gauss elimination method We seek a solution of the matrix equation by the following form: u n =α n u n β n, n=m,,,, (89) where u M =(Iα M ) β M, α j (j =,,M)are ( ) ( ) square matrices and β j (j =,,M)are () column matrices α and β are zero matrices, and α n =(B n C n α n ) A n, β n =(B n C n α n ) (D n φ n C n β n ), n =,, 3, M (9) By using the second order of accuracy implicit difference scheme (7), we can write the matrix form A n u n B n u n C n u n =Df n, u =, u M =u M n M, (87) E n u n F n u n G n u n =Df n, n M, u =, u M 4u M 3u M = (9)

20 The Scientific World Journal Here, matrices E n, F n, G n,andd aregivenintheprevious example, and also f n isgiveninthefirstorderaccuracy difference scheme For the solution of the matrix equation (9),weusedthesamealgorithmasinthefirstorderof accuracy difference scheme, where u M =(α M α M 4α M 3I) (4β M α M β M β M ), α and β arezeromatrices Example In this example, the nonlocal boundary value problem u (t, x) t (x) u (t, x) x =f(t, x), u x (t, x) u(t, x) f (t, x) = (x) cos xsin x (e t t) e t (cos x), <t<, <x<π, u (, x) = e s u (s, x) ds φ (x), φ (x) =(3e ) (cos x), u t (, x) = e s u s (s, x) ds ψ (x), ψ (x) =e (cos x), u x (t, ) =u(t, π) =, t, x π (9) for one-dimensional hyperbolic equation is considered The exact solution of this problem is u (t, x) =(e t t)(cos x) (93) First, we use the first order of accuracy implicit difference scheme (67) for the approximate solutions of nonlocal boundary value problem (9) and we obtain the matrix equation A n u n B n u n C n u n =Df n, u M =, u =u n M, (94) Here, matrices A n, B n, C n,andd arethesameasinthefirst example, and f k n f n = f n f n f n f n () =f(t k,x n )=(x) cos xsin x (e t t)e t (cos x),, k, f n =(3e ) (cos x), n M, f n =e (cos x), n M (95) For the solution of matrix equation (94), we will use modified Gauss elimination method We seek a solution of the matrix equation by the following form: u n =α n u n β n, n=m,,,, (96) where u M =, α is identity and β is zero matrices, and α n =(B n C n α n ) A n, β n =(B n C n α n ) (D n φ n C n β n ), n=,,3,,m (97) Second, for the approximate solutions of nonlocal boundary value problem (9), we use second order of accuracy difference scheme (7)andtheformulas 3u k 4uk uk =, u k 5uk 6uk uk 3 =, u k M 5uk M 4uk M uk M3 =, u k M =, k, k k, k, Thesystemcanbewritteninthefollowingmatrixform: P n u n E n u n F n u n G n u n R n u n =Df n, u 4u 3u =, u k 5uk 6uk uk 3 =, u M =, u k M 5uk M 4uk M uk M3 =, n M (98) (99) Here, P n and R n zero matrices and E n, F n, G n, and D are given in the first example, and also f n is given in the first order accuracy difference scheme For the solution of matrixequation (99), we will use modified Gauss elimination method We seek a solution of the matrix equation in the following form: u n =α n u n β n u n γ n, u M =(I 4 5 α M 5 α Mα M ) n=m,,, ( 4 5 γ M 5 α Mγ M 5 γ M) u M =, ()

21 The Scientific World Journal Table : Errors for the approximate solution of problem (73) Difference schemes =M= =M= =M=4 Difference scheme (67) Difference scheme (7) Table : Errors for the approximate solution of problem (87) Difference schemes =M= =M= =M=4 Difference scheme (67) Difference scheme (7) Table 3: Errors for the approximate solution of problem (94) Difference schemes =M= =M= =M=4 Difference scheme (67) Difference scheme (7) where α j, β j are () ()square matrices and γ n (j =,,M)are ( ) column matrices defined by T n =F n G n α n R(β n α n α n ), α n =(T n ) (E n G n β n Rα n α n ), β n =(T n ) P n, γ n =(T n ) (Df n G n γ n R n (α n γ n γ n )), α = 4 3 I, β = 3 I, γ =, α = 8 5 I, β = 3 5 I, γ = () Using formulas ()and(), we can compute U n, k M ow,letusgivetheresultsofnumericalanalysisthe numerical solutions are recorded for different values of = M,andu(t k,x n ) represents the exact solution, and u k n represents the numerical solution at (t k,x n )Fortheir comparison, the errors are computed by E M = k, n M u(t k,x n )un k () Thus, the results given in Tables,, and3 show that the second order of accuracy difference scheme (7) is more accurate comparing with the first order of accuracy difference scheme (67) 6 Conclusion In this paper, we presented first and second order stable difference schemes for solving the second order multidimensional hyperbolic equation with nonlocal integral and eumann or nonclassical boundary conditions Stability of the difference schemes do not depend on any additional condition between h and The numerical results given in the previous sections demonstrate the efficiency and good accuracy of these schemes Finally we would like to mention that this technique can be applied to get the highest order stable difference schemes Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper References A Ashyralyev and P E Sobolevskii, ew Difference Schemes for Partial Differential Equations, Operator Theory: Advances and Applications, Birkhauser, Basel, Switzerland, 4 D G Gordeziani and G A Avalishvili, Time-nonlocal problems for schrodinger type equations I Problems in abstract spaces, Differential Equations,vol4,no5,pp73 7,5 3 A Ashyralyev and O Gercek, onlocal boundary value problems for elliptic-parabolic differential and difference equations, Discrete Dynamics in ature and Society, vol8,articleid 9484, 6 pages, 8 4ASBerdyshevandETKarimov, Somenon-localproblems for the parabolic-hyperbolic type equation with noncharacteristic line of changing type, Central European Journal of Mathematics,vol4,no,pp83 93,6 5 A Ashyralyev and O Yildirim, On multipoint nonlocal boundary value problems for hyperbolic differential and difference equations, Taiwanese Journal of Mathematics,vol4,no,pp 65 94, 6 A Ashyralyev and Aggez, A note on the difference schemes of the nonlocal boundary value problems for hyperbolic equations, umericalfunctionalanalysisandoptimization,vol5, no 5-6, pp , 4 7 P Shi, Weak solution to evolution problem with a nonlocal constraint, SIAM Journal on Mathematical Analysis, vol 4, pp 46 58, L S Pulkina, A non-local problem with integral conditions for hyperbolic equations, Electronic Journal of Differential Equations,vol999,pp 6,999 9 M Dehghan, On the solution of an initial-boundary value problem that combines neumann and integral condition for the wave equation, umerical Methods for Partial Differential Equations,vol,no,pp4 4,5 M Ashyraliyev, A note on the stability of the integraldifferential equation of the hyperbolic type in a Hilbert space, umerical Functional Analysis and Optimization,vol9,no7-8, pp , 8 D Orlovsky and S Piskarev, The approximation of Bitzadze- Samarsky type inverse problem for elliptic equations with eumann conditions, Contemporary Analysis and Applied Mathematics,vol,no,pp8 3,3 L S Pulkina, On solvability in L of nonlocal problem with integral conditions for hyperbolic equations, Differentsial nye Uravneniya,vol36,pp79 8, 3 Aggez and A Ashyralyev, Finite difference method for hyperbolic equations with the nonlocal integral condition, Discrete Dynamics in ature and Society, vol,articleid 56385, 5 pages, 4 H Soltanov, A note on the Goursat problem for a multidimensional hyperbolic equation, Contemporary Analysis and Applied Mathematics,vol,no,pp98 6,3

22 The Scientific World Journal 5 S Kharibegashvili, On the solvability of one multidimensional version of the first Darboux problem for some nonlinear wave equations, onlinear Analysis, Theory, Methods and Applications,vol68,no4,pp9 94,8 6 A I Kozhanov and L S PulKina, On the solvability of boundary value problems with a nonlocal boundary condition of integral form for multidimensional hyperbolic equations, Differential Equations,vol4,no9,pp33 46,6 7 L S Pulkina, onlocal problem with a first-kind integral condition for a multidimensional hyperbolic equation, Doklady Mathematics,vol76,no,pp74 743,7 8 A A Samarskii, Local one-dimensional difference schemes for multi-dimesional hyperbolic equations in an arbitrary region, Zhurnal Vychislitel noi Matematiki i Matematicheskoi,vol4,no 4, pp , R K Mohanty, Stability interval for explicit difference schemes for multi-dimensional second-order hyperbolic equations with significant first-order space derivative terms, Applied Mathematics and Computation,vol9,no,pp683 69,7 R L Higdon, Absorbing boundary conditions for difference approximations to the multi-dimensional wave equation, Mathematics of Computation,vol47,no76,pp ,986 X Liu, X Cui, and J Sun, FDM for multi-dimensional nonlinear coupled system of parabolic and hyperbolic equations, Journal of Computational and Applied Mathematics,vol86,no, pp , 6 A Ashyralyev and P E Sobolevskii, A note on the difference schemes for hyperbolic equations, Abstract and Applied Analysis,vol6,pp63 7, 3 P E Sobolevskii, Difference Methods for the Approximate Solution of Differential Equations, Voronezh State University Press, Voronezh, Russia, A A Samarskii and E S ikolaev, umerical Methods for Grid Equations, Iterative Methods, vol,birkhauser,basel, Switzerland, 989

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