In this paper a numerical technique is developed for the one-dimensional telegraph equation, we prove the existence, uniqueness, and continuous dependence upon the data of solution to a telegraph equation with purely integral conditions. The proofs are based on a priori estimates and Laplace transform method. Finally, we obtain the solution by using a simple and efficient algorithm for numerical solution.Publisher's Versio

Bouziani, ‪Abdelfatah

Merad, ‪Ahcene

TWMS Journal of Applied and Engineering Mathematics

English

Isik University Academic Open Access

TWMS J. App. Eng. Math. V.3, N.1, 2013, pp. 117-125SOLVABILITY THE TELEGRAPH EQUATION WITH PURELYINTEGRAL CONDITIONSA. MERAD1, A. BOUZIANI1 §Abstract. In this paper a numerical technique is developed for the one-dimensionaltelegraph equation, we prove the existence, uniqueness, and continuous dependenceupon the data of solution to a telegraph equation with purely integral conditions. Theproofs are based on a priori estimates and Laplace transform method. Finally, we obtainthe solution by using a simple and efficient algorithm for numerical solution.Keywords: Telegraph equation, purely integral conditions, a priori estimates, Laplacetransform method.AMS Subject Classification: 35B45, 35A35, 44A101. IntroductionIn the rectangular domain D = {(x, t) : 0 < x < 1, 0 < t ≤ T} , we consider a secondorder telegraph equation∂2u∂t2− c2∂2u∂x2+ a∂u∂t+ bu = f (x, t) , 0 < x < 1, 0 < t ≤ T, (1)subject to the initial conditionsu(x, 0) = φ (x) , 0 < x < 1, (2)∂u(x, 0)∂t= ψ (x) , 0 < x < 1, (3)and the purely integral conditions∫ 10u (x, t) dx = 0, 0 < t ≤ T, (4)∫ 10xu (x, t) dx = 0, 0 < t ≤ T, (5)where f, φ, and ψare known functions, c, a, b, and T are known positives constants.The first investigation of this type of problems goes back to [4] in 1996, in which theauthor proved the existence, uniqueness, and continuous dependence of the solution uponthe data of certain hyperbolic problems with only integral boundary conditions. Later,similar problems have been studied in [7, 10, 11, 13, 21] by using the energetic methodand the Rothe time-discretization method. We refer the reader to [3, 4, 6, 8, 9, 10, 12, 15,1 Department of Mathematics, Faculty of Sciences, Larbi Ben M’hidi University, Oum El Bouaghi,04000, ALGERIA.e-mail: merad ahcene@yahoo.fr and aefbouziani@yahoo.fr§ Manuscript received October 16, 2012.TWMS Journal of Applied and Engineering Mathematics Vol.3 No.1 c⃝ Işık University, Departmentof Mathematics 2013; all rights reserved.117118 TWMS J. APP. ENG. MATH. V.3, N.1, 201320, 22, 23, 24] for hyperbolic equations with Neumann and integral condition. For otherproblems with nonlocal conditions, related to other equations, we refer to [2, 4, 10, 11, 12]and references therein.In this paper a Laplace transform method is presented for the problem of obtainingnumerical approximations.The main tool used in this paper is the Laplace transform andthen used the numerical technique for the inverse Laplace transform to obtain the numer-ical solution. We use a numerical method for inverting the Laplace transform to get thesolution.The paper is organized as follows. In Section 2, we begin by introducing certain functionspaces which are used in the next sections, and we reduce the posed problem to one withhomogeneous integral conditions. In Section 3, we first establish the existence of thesolution by the Laplace transform. In Section 4, we establish a priori estimates, whichgive the uniqueness and continuous dependence.2. Preliminaries and NotationsDefinition 2.1. Denote by L2(0, T ;H) the set of all measurable abstract functions u (., t)from (0, T ) into H equiped with the norm∥u∥L2(0,T ;H) =(∫ T0∥u (., t)∥2H dt)1/2<∞. (6)Definition 2.2. We denote by C0 (0, 1) the vector space of continuous functions withcompact support in (0, 1). Since such functions are Lebesgue integrable with respect to dx,we can define on C0(0, 1) the bilinear form given by((u,w)) =∫ 10ℑmx u · ℑmx wdx, m ≥ 1, (7)whereℑmx u =∫ x0(x− ξ)m−1(m− 1)!u (ξ, t) dξ; for m ≥ 1. (8)The bilinear form (2.2) is considered as a scalar product on C0 (0, 1) for which C0 (0, 1) isnot complete.Definition 2.3. Denote by Bm2 (0, 1) , the completion of C0 (0, 1) for the scalar product(2.2), which is denoted (., .)Bm2 (0,1), introduced in [5]. By the norm of function u fromBm2 (0, 1) , m ∈ N∗, we understand the nonnegative number :∥u∥Bm2 (0,1)=(∫ 10(ℑmx u)2 dx)1/2= ∥ℑmx u∥ ; for m ≥ 1. (9)Lemma 2.1. For all m ∈ N∗, the following inequality holds:∥u∥2Bm2 (0,1) ≤12∥u∥2Bm−12 (0,1). (10)Proof. See [5]. Corollary 2.1. For all m ∈ N∗, we have the elementary inequality∥u∥2Bm2 (0,1) ≤(12)m∥u∥2L2(0,1) . (11)A. MERAD, A. BOUZIANI: SOLVABILITY THE TELEGRAPH EQUATION ... 119Definition 2.4. We denote by L2(0, T ;Bm2 (0, 1)) the space of functions which are squareintegrable in the Bochner sense, with the scalar product(u,w)L2(0,T ;Bm2 (0,1))=∫ T0(u (., t) , w (., t))Bm2 (0,1)dt. (12)Since the space Bm2 (0, 1) is a Hilbert space, it can be shown that L2(0, T ;Bm2 (0, 1)) is aHilbert space as well. The set of all continuous abstract functions in [0, T ] equipped withthe normsup0≤t≤T∥u (., t)∥Bm2 (0,1)is denoted C(0, T ;Bm2 (0, 1)).Corollary 2.2. For every u ∈ L2 (0, 1) , from which we deduce the continuity of theimbedding L2 (0, 1) −→ Bm2 (0, 1), for m ≥ 1.3. Existence of the SolutionIn this section we shall apply the Laplace transform technique to find solutions of partialdifferential equations, we have the Laplace transformU (x, s) = L{u (x, t) ; t −→ s} =∫ ∞0u (x, t) exp (−st) dt, (13)where s is positive reel parameter. Taking the Laplace transforms on both sides of (1.1) ,we have−c2 d2dx2[U (x, s)] +(s2 + as+ b)U (x, s) = F (x, s) + (s+ a)φ (x) + ψ (x) , (14)where F (x, s) = L{f (x, t) ; t −→ s} . Similarly, we have∫ 10U (x, s) dx = 0, (15)∫ 10xU (x, s) dx = 0, (16)Thus, considered equation is reduced in boundary value problem governed by second orderinhomogeneous ordinary differential equation. We obtain a general solution of (3.2) asU (x, s) = − c√s2 + as+ b∫ x0[F (τ, s) + (s+ a)φ (τ) + ψ (τ) ] sinh(√s2 + as+ bc[x− τ ])dτ+C1 (s) exp(−√s2 + as+ bcx)+ C2 (s) exp(√s2 + as+ bcx), (17)where C1and C2 are arbitrary functions of s. Substitution of (3.5) into (3.3) − (3.4) , wehaveC1 (s)∫ 10exp(−√s2 + as+ bcx)dx+ C2 (s)∫ 10exp(√s2 + as+ bcx)dx=c√s2 + as+ b∫ 10[F (τ, s) + (s+ a)φ (τ) + ψ (τ)∫ 1τsinh(√s2 + as+ bc[x− τ ])dx]dτ,120 TWMS J. APP. ENG. MATH. V.3, N.1, 2013C1 (s)∫ 10x exp(−√s2 + as+ bcx)dx+ C2 (s)∫ 10x exp(√s2 + as+ bcx)dx=c√s2 + as+ b∫ 10[F (τ, s) + (s+ a)φ (τ) + ψ (τ)∫ 1τx sinh(√s2 + as+ bc[x− τ ])dx]dτ,where (C1 (s)C2 (s))=(a11 (s) a12 (s)a21 (s) a22 (s))−1×(b1 (s)b2 (s)), (18)anda11 (s) =∫ 10exp(−√s2 + as+ bcx)dx,a12 (s) =∫ 10(√s2 + as+ bcx)dx,a21 (s) =∫ 10x(−√s2 + as+ bcx)dx,a22 (s) =∫ 10x(√s2 + as+ bcx)dx,b1 (s) =c√s2 + as+ b∫ 10[F (τ, s) + (s+ a)φ (τ) + ψ (τ)∫ 1τsinh(√s2 + as+ bc[x− τ ])dx]dτ,b2 (s) =c√s2 + as+ b∫ 10[F (τ, s) + (s+ a)φ (τ) + ψ (τ)∫ 1τx sinh(√s2 + as+ bc[x− τ ])dx]dτ,(19)It is possible to evaluate the integrals in (3.5) and (3.7) exactly. In general, one mayhave to resort to numerical integration in order to compute them, however. For example,the Gauss’s formula (25.4.30) given in Abramowitz and stegun [1] may be employed tocalculate these integrals numerically, we have∫ 10exp(±√s2 + as+ bcx)dx≃ 12N∑i=1wi exp(±√s2 + as+ b2c[xi + 1]),∫ 10x exp(±√s2 + as+ bcx)dx≃ 12N∑i=1wi(12[xi + 1])exp(±√s2 + as+ b2c[xi + 1]),A. MERAD, A. BOUZIANI: SOLVABILITY THE TELEGRAPH EQUATION ... 121∫ x0[F (τ, s) + (s+ a)φ (τ) + ψ (τ) ] sinh(√s2 + as+ bc[x− τ ])dτ≃ x2N∑i=1wi[F(x2[xi + 1] ; s)+ (s+ a)φ(x2[xi + 1])+ ψ(x2[xi + 1])]×× sinh(√s2 + as+ bc[x− x2[xi + 1]]),∫ 10[F (τ, s) + (s+ a)φ (τ) + ψ (τ)∫ 1τsinh(√s2 + as+ bc[x− τ ])dx]dτ≃ 14N∑i=1wi[F(12[xi + 1] ; s)+ (s+ a)φ(12[xi + 1])+ ψ(12[xi + 1])](1− 12[xi + 1])××N∑i=1wj sinh(√s2 + as+ bc[12[(1− 12[xi + 1])xj +(1 +12[xi + 1])]− 12(xi + 1)]),∫ 10[F (τ, s) + (s+ a)φ (τ) + ψ (τ)∫ 1τx sinh(√s2 + as+ bc[x− τ ])dx]dτ≃ 14N∑i=1wi[F(12[xi + 1] ; s)+ (s+ a)φ(12[xi + 1])+ ψ(12[xi + 1])](1− 12[xi + 1])××(12[(1− 12[xi + 1])xj +(1 +12[xi + 1])])N∑i=1wj sinh(√s2 + as+ bc[12[(1− 12[xi + 1])xj +(1 +12[xi + 1])]− 12(xi + 1)])(20)where xi and wi are the abscissa and weights, defined asxi : ith zero of Pn (x) , ωi = 2/(1− x2i) [P′n (x)]2.Their tabulated values can be found in [1] for different values of N.Numerical inversion of Laplace transform. Sometimes, an analytical inversion of a Laplacedomain solution is difficult to obtain; therefore a numerical inversion method must be used.A nice comparison of four frequently used numerical Laplace inversion algorithms is givenby Hassan Hassanzadeh, Mehran Pooladi-Darvish [16]. In this work we use the Stehfest’salgorithm [25] that is easy to implement. This numerical technique was first introducedby Graver [14] and its algorithm then offered by [25].Stehfest’s algorithm approximatesthe time domain solution asu (x, t) ≈ ln 2t2m∑n=1βnU(x;n ln 2t), (21)where, m is the positive integer,βn = (−1)n+mmin(n,m)∑k=[n+12 ]km (2k)!(m− k)!k! (k − 1)! (n− k)! (2k − n)!, (22)122 TWMS J. APP. ENG. MATH. V.3, N.1, 2013and [q] denotes the integer part of the real number q.The parameter m is a free parameterthat should be optimized by trial and error. It was seen that with increasingm accuracy ofresult increases up to a point and then owing to the rounding errors it decreases [25]. Thus,for choosing optimum m, it is beneficial to apply an algorithm repeatedly for differentvalues of m and study its effect on the solution. The other way to choose optimal value ofm could be, to apply the Stehfest’s algorithm for inverting the Laplace transform of someelementry functions which are known.Remark 3.1. 1) Stehfest’s method gives accurate results for many problems includingdiffusion problem, fractional functions in the Laplace domain. However, it fails to predictet type functions or those with oscillatory behavior such as sine and wave functions (see[16]). 2) Note that more than one numerical inversion algorithm can also be performed tocheck the accuracy of the result.4. Uniqueness and Continuous dependence of the SolutionWe first establish an a priori estimate, the uniqueness and continuous dependence ofthe solution with respect to the data are immediate consequences.Theorem 4.1. If u (x, t) is a solution of problem (1.1)− (1.5) and f ∈ C(D), then wehave∥u (., τ)∥2L2(0,1)≤ c1(∫ τ0∥f (., t)∥2B12(0,1) dt+ ∥φ∥2L2(0,1) + ∥ψ∥2B12(0,1)), (23)∥∥∥∥∂u (., t)∂t∥∥∥∥2B12(0,1)≤ c2(∫ τ0∥f (., t)∥2B12(0,1) dt+ ∥φ∥2L2(0,1) + ∥ψ∥2B12(0,1)), (24)wherec1 =1(b+ 2c2)max(1,12a,(b+ 2c2)2), c2 = max(1,12a,(b+ 2c2)2)and 0 ≤ τ ≤ T.Proof. Taking the scalar product in B12 (0, 1) of both sides of equation (1.1) with∂u∂t , andintegrating over (0, τ), we have∫ τ0(∂2u (., t)∂t2,∂u (., t)∂t)B12(0,1)dt− c2∫ τ0(∂2u (., t)∂x2,∂u (., t)∂t)B12(0,1)dt+a∫ τ0(∂u (., t)∂t,∂u (., t)∂t)B12(0,1)+ b∫ τ0(u (., t) ,∂u (., t)∂t)B12(0,1)=∫ τ0(f (., t) ,∂u (., t)∂t)B12(0,1)dt. (25)A. MERAD, A. BOUZIANI: SOLVABILITY THE TELEGRAPH EQUATION ... 123Integrating by parts on the left-hand side of (4.3) , we obtain12∥∥∥∥∂u (., τ)∂t∥∥∥∥2B12(0,1)+(b2+ c2)∥u (., τ)∥2B12(0,1) + a∫ τ0∥∥∥∥∂u (., t)∂t∥∥∥∥2B12(0,1)dt ≤∫ τ0(f (., t) ,∂u (., t)∂t)B12(0,1)dt+12∥ψ∥2B12(0,1) +(b+ 2c24)∥φ∥2L2(0,1) (26)By the ε−Cauchy inequality, the first term in the right-hand side of (4.4) is bounded byε2∫ τ0∥f (., t)∥2B12(0,1) dt+12ε∫ τ0∥∥∥∥∂u (., t)∂t∥∥∥∥2B12(0,1)dt. (27)We choose ε = 12a so that the second term will be simplified by the third term in theleft-hand sid. Thus we have∥∥∥∥∂u (., τ)∂t∥∥∥∥2B12(0,1)+(b+ 2c2)∥u (., τ)∥2L2(0,1)≤ 12a∫ τ0∥f (., t)∥2B12(0,1) dt+ ∥ψ∥2B12(0,1)+(b+ 2c22)∥φ∥2L2(0,1) . (28)From (4.6) , we obtain estimates (4.1) and (4.2) . Corollary 4.1. If problem (1.1) − (1.5) has a solution, then this solution is unique anddepends continuously on (f, φ, ψ).5. ConclusionIn this work we study a Telegraph equation with purely integral conditions. The exis-tence and uniqueness of the solution are proved. The proof is based on a priori estimatesand Laplace transform method, sometimes, an analytical inversion of a Laplace domainsolution is difficult to obtain; therefore a numerical inversion method must be used. Weuse the Stehfest’s algorithm that is easy to implement to obtain approximate solution.References[1] Abramowitz, M., Stegun. I.A.,(1972), Hand book of Mathematical Functions, Dover, New York.[2] Ang, W.T., (2002) ,A Method of Solution for the One-Dimentional Heat Equation Subject to NonlocalConditions, Southeast Asian Bulletin of Mathematics 26 , 185-191.[3] Bëılin, S. A., (2001) , Existence of solutions for one-dimentional wave nonlocal conditions, Electron.J. 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Sci. 22 ), no. 3, 511–519.[21] Mesloub S. and Bouziani, A., (2001), Mixed problem with integral conditions for a certain class ofhyperbolic equations, Journal of Applied Mathematics, Vol. 1 , no. 3, 107-116.[22] Pul’kina,L. S.,(1999), A non-local problem with integral conditions for hyperbolic equations, Electron.J. Differential Equations , no. 45, 1–6.[23] Pul’kina,L. S.,(2000), On the solvability in L2 of a nonlocal problem with integral conditions for ahyperbolic equation, Differ. Equ. 36 , no. 2, 316–318.[24] Pul’kina,L. S., (2003) ,A mixed problem with integral condition for the hyperbolic equation, Matem-aticheskie Zametki, vol. 74, no. 3, , pp. 435–445.[25] Stehfest,H., (1970), Numerical Inversion of the Laplace Transform, Comm. ACM 13, 47-49.[26] Shruti A.D.,(2010), Numerical Solution for Nonlocal Sobolev-type Differential Equations, ElectronicJournal of Differential Equations, Conf. 19 , 75-83.Ahcene Merad got his M.Sc. degree from Jijel University-Algeria. Heis assistant professor at Oum El Bouaghi University-Algeria. His area ofresearch is nonlocal PDE, numerical solution of linear PDE and fractionalpartial differential equations.A. MERAD, A. BOUZIANI: SOLVABILITY THE TELEGRAPH EQUATION ... 125Abdelfatah Bouziani got his Ph.D. degree from Constantine University-Algeria. He is professor at Oum El Bouaghi University-Algeria. His area ofresearch is nonlocal PDE, numerical solution of linear and nonlinear PDE.

Solvability the telegraph equation with purely integral conditions

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Solvability the telegraph equation with purely integral conditions

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