On the NBVP for Semilinear Hyperbolic Equations

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1 Filomat 31: 217, DOI /FIL17999A Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: On the NBVP for Semilinear Hyperbolic Equations Necmettin Aggez a, Gulay Yucel a a Elit Gencler College, Bahcelievler 318, İstanbul, Turkey Abstract. This paper is concerned with establishing the solvability of the nonlocal boundary value problem for the semilinear hyperbolic equation in a Hilbert space. For the approximate solution of this problem, the first order of accuracy difference scheme is presented. Under some assumptions, the convergence estimate for the solution of this difference scheme is obtained. Moreover, these results are supported by a numerical example. 1. Introduction Second order linear and semilinear hyperbolic equations are of keen interest in fluid dynamics, acoustics, mathematical physics, electromagnetic, etc. The article [1] investigates the existence and uniqueness of weak solutions for the semilinear degenerate hyperbolic Goursat problem. The authors of [2] study the blow-up property of weak solutions to an initial and boundary value problem for models many real physical problems such as viscoelastic fluids, processes of filtration through a porous media, fluids with temperature-dependent viscosity, etc. In paper [3], in the conic domain, the existence or nonexistence of global solutions of a multidimensional version of the first Darboux problem for wave equations with power nonlinearity is investigated. For the approximate solution of such types of partial differential equations efforts are being made to develop efficient and high accuracy finite difference methods see []-[1] and the references therein. The authors of [11] investigate a nonlocal boundary value problem for semilinear hyperbolic-parabolic equations in a Hilbert space. The first and second order accuracy difference schemes approximately solving this problem are studied. The convergence estimates for the solution of these difference schemes are obtained. In paper [12], the unique solvability of local and nonlocal boundary value problems for the semilinear Schrödinger equation in a Hilbert space is investigated. The convergence estimates for the solution of difference schemes are established. The authors of [13] study the initial value problem for the semilinear integral-differential equation of the hyperbolic type. The convergence estimates for the solutions of the first and second order of accuracy difference schemes are obtained. In the present paper, we consider the nonlocal boundary value problem for semilinear hyperbolic equation d 2 ut + Aut dt 2 = f t, ut, t T, 1 u = ρu T + ψ, u = σut + ϕ 21 Mathematics Subject Classification. 35L71, 3B1, 65M12 Keywords. Semilinear hyperbolic equation, nonlocal boundary value problems, uniqueness, convergence Received: 28 December 215; Revised: 26 May 216; Accepted: 11 June 216 Communicated by Allaberen Ashyralyev Guest Editor addresses: necmiaggez@gmail.com Necmettin Aggez, gulayyucel.2@gmail.com Gulay Yucel

2 N. Aggez, G. Yucel / Filomat 31: 217, in a Hilbert space H with the self-adjoint positive definite linear operator A δi δ >, under the assumption σρ + 1 ρ > σ + 2 where f t, ut is the given function, ρ and σ are constants, ϕ DA and ψ DA 1/2 are given initial value functions. A function ut is the solution of problem 1 if the following conditions are satisfied: i ut is twice continuously differentiable on the interval, T, continuously differentiable on the segment [, T] and satisfies the equation and nonlocal boundary conditions of problem 1. ii The element ut belongs to DA for all t [, T], and the function Aut is continuous on the segment [, T]. In this work, we prove that problem 1 has a unique solution in C[, T], H. For the approximate solution of 1, the first order of accuracy difference scheme is presented. Under some assumptions, the convergence estimate for the solution of this difference scheme is obtained. To validate the main results, this difference scheme is applied to the one dimensional semilinear hyperbolic problem. 2. Existence and Uniqueness In this section, the uniqueness of the solution of semilinear hyperbolic problem 1 is considered. First of all, let us prove that the problem 1 has a unique solution in C[, T], H. Here, C[, T], H is the space of all continuous function, vt defined on the interval [, T] in Hilbert space H. We use the norm v C[,T],H = max t T vt H. 3 Strongly continuous cosine and sine operator functions {ct, st, t }, see [15, 16], are defined by formulas e ita1/2 + e ita1/2 c t = 2 eita1/2 /2, s t = A e ita1/2. 2i Lemma 2.1. The following estimates hold: ct 1, A 1/2 st 1, t. Lemma 2.2. Assume that the assumption 2 is satisfied. Then, operator P = [ 1 + σρ ] I ρ + σ ct has an inverse and P 1 σρ + 1 σ + ρ 5 is satisfied. Proof. Applying the estimates and triangle inequality, we obtain P = [ 1 + σρ ] I σ + ρ ct 1 + σρ σ + ρ >. Estimate 5 follows from this estimation. Theorem 2.3. Assume that f is continuous function in [, T] H and there exists K > such that f satisfies the Lipschitz condition: A 1 H 2 [ f t, v f t, u] K v u H, t [, T] 6

3 N. Aggez, G. Yucel / Filomat 31: 217, for all u, v H. Let the assumption 2 and KT 1 + σ ρ ρ + σ + σρ + 1 σ < ρ hold. Then, the nonlocal boundary value problem 1 has a unique solution in C[, T], H. Proof. For the solution of 1, we have see [17] T ut = ctp [I ρct] σ st λ f λ, uλ + ϕ +σst ρ ct λ f λ, uλ + ψ T +stp [I σct] ρ ct λ f λ, uλ + ψ ρast σ t st λ f λ, uλ + ϕ + st λ f λ, uλ. We denote the right-hand side by operator Fut, which maps C[, T], H into C[, T], H. We can prove that F is a contraction operator on C [, T], H. By using and Lipschitz condition 6, we get Fv t Fu t H { P 1 ρ + σ A /2 [ f λ, v λ f λ, u λ ] H for any t [, T]. So, + σ ρ A /2 [ f λ, v λ f λ, u λ ] H σ ρ A /2 [ f λ, v λ f λ, u λ ] H + σ ρ A /2 [ f λ, v λ f λ, u λ ] } H + Fu Fv C[,T],H α u v C[,T],H, A /2 [ f λ, v λ f λ, u λ ] H [ 1 + P ρ ρ ] σ + + σ A /2 [ f λ, v λ f λ, u λ ] H where α = KT 1 + σ ρ ρ + σ + σρ + 1 σ. + ρ From 7 it follows that F is a contraction operator on C [, T], H. Then by the fixed point theorem, the nonlocal boundary value problem 1 has a unique solution in C [, T], H.

4 3. The First Order of Accuracy Difference Scheme N. Aggez, G. Yucel / Filomat 31: 217, For the approximate solution of the nonlocal boundary value problem 1, we construct the first order of accuracy difference scheme 2 u k+1 2u k + u k + Au k+1 = f t k, u k, t k = k, 1 k N 1, N = T, u = σu N + ϕ, I + 2 A u 1 u = ρ u N u N + ψ. The following theorem is about the uniqueness of the solution u = {u k } N k=1 of difference scheme 8 in the space of grid functions C [, T], H with the norm 8 u C[,T],H = max 1 k N u k H. 9 Lemma 3.1. Let R, R, and P be defined as R = I + ia 1/2, R = I ia 1/2, P = I σ 2 The estimates hold: P R N + R N ρ 2 R N + R N + σρr N R N. 1 1 σ ρ σ ρ, R 1, R 1, A 1/2 R 1, A 1/2 R 1. 1 Theorem 3.2. Assume that f satisfies 6 in space C [, T], H and assumption 7 holds. Then the difference scheme 8 has a unique solution in C [, T], H. Proof. The proof of this theorem is similar to the Theorem 2.3 and based on the operator F which is defined on the space C [, T], H by the help of [17] Fu = P Fu 1 = P [ I ρ R R i A1/2 R N + R N] N R R σ 2i A/2 R N s R N s f t s, u s + ϕ + σ R R R N R N N 2 R R ρ R N s + R N s f t s, u s + ρr R f t N, u N + ψ, [ I ρ R R i A1/2 R N + R N] N R R σ 2i A/2 R N s R N s f t s, u s + ϕ + σ R R R N R N N 2 R R ρ R N s + R N s f t s, u s + ρr R f t N, u N + ψ [ [ 1 +P R R I σ 2 f t N, u N + ψ ] + ρ A1/2 2i R N + R N]] N 2 ρ R N R N N σ R N s + R N s f t s, u s + ρr R A /2 2i R N s R N s f t s, u s + ϕ,

5 R k + R k P N. Aggez, G. Yucel / Filomat 31: 217, { [ I ρ R R i A1/2 R N + R N] R R Fu k = 1 2 N σ 2i A/2 R N s R N s f t s, u s + ϕ R R R N R N R R N 2 ρ R N s + R N s f t s, u s + ρr R f t N, u N + ψ + R R R R R k R k P [ 1 I σ 2 R N + R N] N 2 ρ R N s + R N s f t s, u s + ρr R f t N, u N + ψ ] + ρ 1 2i A1/2 R N R N N σ 2i A/2 R N s R N s f t s, u s + ϕ ]} k 2i A/2 R k s R k s f t s, u s, 2 k N. By using triangle inequality, 6 and 1 we get Fv Fu H P ρ N σ A /2 [ f t s, v s f t s, u s ] H KT P ρ σ vk u k H, Fv 1 Fu 1 H P ρ ρ N σ + + σ A /2 [ f t s, v s f t s, u s ] H KT P ρ ρ σ + + σ vk u k H, Fv k Fu k H [ 1 + P ρ ρ ] N σ + + σ A /2 [ f t s, v s f t s, u s ] H KT 1 + P ρ ρ σ + + σ vk u k H. Theorem 3.3. Let u k represent the solution of difference scheme 8 at t k and u t k be the exact solution of 1. If u t and Au t are continuous functions and f satisfies the Lipschitz condition 6, the convergence estimate u k u t k H L is satisfied, where L is independent of. Proof. Using problem 1 and first order difference scheme 8, we get { 2 z k+1 2z k + z k + Az k+1 = a k+1, 1 k N 1, z = σz N + a, I + 2 A z 1 z ρ z N z N = a 1, 11 where z k = u t k u k, a = u ut σu N ut N =,

6 N. Aggez, G. Yucel / Filomat 31: 217, a 1 = u ρu T [ I + 2 A u u ρ ut ut ] = O, and a k+1 = f t k, u k f t k, u t k + d2 ut k dt 2 2 u t k+1 2u t k + u t k +Au t k Au t k+1, 1 k N 1. For the solution of 11, we have [ z = P I ρ R R R N + R N i A1/2 R R ] N σ [ 2i A/2 R N s R N s] a s+1 +σ R R R N R N N 2 R R ρ [ R N s R N s] a s+1 + ρr Ra N + a 1, and [ z 1 = P R R I σ 2 σρ A1/2 2i R N + R N] N 2 ρ [ R N s R N s] a s+1 + ρr Ra N + a 1 N R N R N [ 2i A/2 R N s R N s] a s+1 + z, z k = 1 2 R k + R k z + R R R k R k k z 1 z [ 2i A/2 R k s R k s] a s+1, 2 k N. By using triangle inequality and 6, we obtain A /2 H a k+1 A /2 f t k, u k f t k, u t k H + M δ Au t k Au t k+1 H +M δ d 2 ut k 2 u t dt 2 k+1 2u t k + u t k K u k u t k H + M, 1 k N 1, A /2 H a 1 L 1, where L 1 does not depend on. Applying the estimates 1 and Lipschitz condition 6, we get z H P ρ N σ A /2 H a s+1 + M 1, 12 H and z 1 H P ρ ρ N σ + + σ A /2 H a s+1 + M 2, 13 z k H [ 1 + P ρ ρ ] N σ + + σ A /2 H a s+1 + M 3, 2 k N. 1

7 N. Aggez, G. Yucel / Filomat 31: 217, Applying 1, we get z k H Kβ N z s H + M, 1 k N, where β = α/kt. It follows that max z k H αmax z s H + M. 1 k N 1 s N Therefore, using 7 we have max z k H 1 1 k N 1 α M = M 5, z H L 2. That shows us that the unique solution of difference scheme 8 converges to the unique solution of 1 in a Hilbert space H. To obtain the unique solution of the difference scheme 8, the recursive relation 2 mu k+1 2 m u k + m u k + Am u k+1 = f t k, m u k, tk = k, 1 k N 1, N = T, mu = σ m u N + ϕ, I + 2 A m u 1 m u = ρ m u N m u N + ψ 15 is used, where m u = { mu k } N k=1, m = 1, 2, 3,... and u is given. The sequence of the solutions converges to the unique solution u of difference scheme 8.. Numerical Analysis In this section, we apply the first order of accuracy difference scheme 8 to the semilinear hyperbolic problem. Example.1. Consider the problem 2 ux, t t 2 2 ux, t x 2 u x x, t + ux, t = f x, t, u, < t < 1, < x < π, f x, t, u =.5 sin e t t 1 sin x + 3e t 2t 2 sin x e t t 1 cos x sin u 2, ux, = 1 2 e ux, 1 + ϕx, ϕx = sin x, x π, 2 2 u t x, = 1 2 u tx, 1 + ψx, ψx = 1 e 2 u, t = uπ, t =, t 1 sin x, x π, 16 which has exact solution u = e t t 1 sin x. Note that, the inequality 7 holds with K < 1 12, σ = ρ = 1, the function f t, x, u satisfies the Lipschitz 2 condition 6. For the approximate solutions of nonlocal boundary value problem 16, we apply 15.

8 N. Aggez, G. Yucel / Filomat 31: 217, mu k+1 n 2 m u k n + m u k k+1 n mun+1 2 mun k+1 + m u k+1 k+1 n mun+1 m u k+1 n 2 h 2 2h + m u k+1 n =.5 sin e t k tk 1 sin x n + 3e t k 2tk 2 sin x n e t k tk 1 cos x n.5 sin m u k n, N = 1, x n = nh, 1 n M 1, Mh = π, t k = k, 1 k N 1, mu n 1 u N n = 2 e sin x n, n M, 2 m 2 17 mu 1 n m u n + 1 u N n 1 2 m 2 mu N n = 1 e sin x n, n M, 2 u k n = sin x, m u k = m u k =, k N, M n M where m = 1, 2, 3,... To solve the difference scheme 17 modified Gauss elimination method is used. The solution of difference scheme 17 is recorded for different values of M and N. The errors are computed as following: me N M = max 1 n M 1 k N m u k n ux n, t k where m u k+1 n denotes the numerical solution, ux n, t k denotes the exact solution at x n, t k. Results of simulations are shown in the following table. We observe that the error decreases as N and M increases. Note that, error is approximately cut in half when N and M are doubled. Table 1: Error analysis for the first order of accuracy difference scheme N/M 2/2 / 8/8 16/16 Error To summarize, for the solution of the problem 1, the uniqueness of the solution is established. The convergence estimates are established for the solution of first order of accuracy difference scheme 8. Finally, this difference scheme is applied to the one dimensional semilinear hyperbolic problems. Furthermore, this technique can be applied to the higher order of accuracy difference schemes. Without proof extended abstract of this work was printed in [1]. 5. Acknowledgement The authors would like to thank Professor Allaberen Ashyralyev for the helpful suggestions to the improvement of this paper. Competing interests The authors declare that they have no competing interests. References [1] D. Lupo, K. Payne, N. I. Popivanov, On the degenerate hyperbolic Goursat problem for linear and nonlinear equations of Tricomi type, Nonlinear Analysis

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