EXISTENCE OF SOLUTIONS FOR ONE-DIMENSIONAL WAVE EQUATIONS WITH NONLOCAL CONDITIONS

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1 Eectronic Journa of Differentia Equations, Vo. 21(21), No. 76, pp ISSN: URL: or ftp ejde.math.swt.edu (ogin: ftp) EXISTENCE OF SOLUTIONS FOR ONE-DIMENSIONAL WAVE EQUATIONS WITH NONLOCAL CONDITIONS SERGEI A. BEILIN Abstract. In this artice we study an initia and boundary-vaue probem with a nonoca integra condition for a one-dimensiona wave equation. We prove existence and uniqueness of cassica soution and find its Fourier representation. The basis used consists of a system of eigenfunctions and adjoint functions. 1. Introduction Certain probems of modern physics and technoogy can be effectivey described in terms of nonoca probems for partia differentia equations. These nonoca conditions arise mainy when the data on the boundary cannot be measured directy. The first paper, devoted to second-order partia differentia equations with nonoca integra conditions goes back to Cannon [4]. Later, the probems with nonoca integra conditions for paraboic equations were investigated by Kamynin [1], Ionkin [9], Yurchuk [18], Bouziani [2]; probems for eiptic equations with operator nonoca conditions were considered by Mikhaiov and Guschin [7], Scubachevski [17], Paneiah [13]. Then, Gordeziani and Avaishvii [5], Bouziani [3] devoted a few papers to nonoca probems for hyperboic equations. Pukina [14, 15] studied the nonoca anaogue to cassica Goursat probem. In this paper we investigate the nonoca anaogue to cassica mixed probem, which invoves initia, boundary and nonoca integra conditions. In the rectanguar domain D = {(x, t) : < x <, < t < T }, we consider the equation with initia data Dirichet boundary condition and the nonoca condition LU U tt U xx = F (x, t) (1.1) U(x, ) = Φ(x), U t (x, ) = Ψ(x), (1.2) U(, t) = (1.3) U(x, t) dx =, (1.4) 2 Mathematics Subject Cassification. 35L99, 35L5, 35L2. Key words and phrases. Mixed probem, non-oca conditions, wave equation. c 21 Southwest Texas State University. Submitted August 21, 21. Pubished December 1, 21. 1

2 2 SERGEI A. BEILIN EJDE 21/76 where Φ(x), Ψ(x) are given, Φ(x) C[, ] C 2 (, ), Ψ(x) C[, ] C 1 (, ) and satisfy the compatibiity conditions Φ() =, Ψ() =, Φ(x) dx = Ψ(x) dx =. Note that we do not ose generaity by assuming that (1.3) and (1.4) are homogeneous. Indeed, if U(, t) = m(t) and U(x, t) dx = n(t), we introduce a new unknown function v(x, t) = U(x, t) W (x, t), where W (x, t) = (1 2x Then (1.1) is converted into the simiar equation )m(t) + 2x 2 n(t). v tt v xx = g(x, t), g(x, t) = F (x, t) LW, whie the Dirichet and integra conditions are now homogeneous. The presence of integra conditions compicates the appication of standard techniques. Therefore, we first reduce (1.1)-(1.4) to an equivaent probem. Lemma 1.1. Probem (1.1)-(1.4) is equivaent to (1.1)-(1.3) and U x (, t) U x (, t) = F (x, t) dx. (1.5) Proof. Let U(x, t) is a soution of (1.1)-(1.4). Integrating (1.1) with respect to x over (, ), and taking in account (1.4), we obtain U x (, t) U x (, t) = F (x, t) dx. Let now U(x, t) be a soution of (1.1)-(1.3), (1.5). We need ony to show that U(x, t) dx =. For this end we integrate again (1.1) and obtain d 2 dt 2 By virtue of the compatibiity conditions, U(x, ) dx =, U(x, t) dx =. U t (x, ) dx =. Then U(x, t) dx = is a unique soution to homogeneous Cauchy probem. x2 2 Introduce a new unknown function u(x, t) = U(x, t) w(x, t), where w(x, t) = F (x, t) dx. Then (1.1)-(1.3), (1.5) is transformed now into u tt u xx = g(x, t), (1.6) u(x, ) = ϕ(x), u t (x, ) = ψ(x), (1.7) u(, t) =, (1.8) u x (, t) = u x (, t), (1.9)

3 EJDE 21/76 WAVE EQUATION WITH A NONLOCAL CONDITION 3 where g(x, t) = F (x, t) + x2 2 ϕ(x) = Φ(x) + x2 2 ψ(x) = Ψ(x) + x2 2 F tt (x, t) dx 1 F (x, ) dx, F t (x, ) dx. F (x, t) dx, 2. Uniqueness Theorem 2.1. There exists at most one soution to (1.6)-(1.9). Proof. Let u 1 (x, t), u 2 (x, t) be two different soutions of (1.6)-(1.9). Then u(x, t) = u 1 (x, t) u 2 (x, t) is a nontrivia soution to the homogeneous probem u tt u xx =, u(x, ) =, u t (x, ) =, u(, t) =, u x (, t) = u x (, t). As u C 1 ( D) C 2 (D), then u(x, t) takes on certain vaue for x =. Let u(, t) = µ(t). Consider mixed probem for the equation u tt u xx = with homogeneous initia data and the boundary conditions u(, t) =, u(, t) = µ(t). Note, that µ(t) is required to satisfy the compatibiity conditions µ() = and µ () =. For a conditions to be homogeneous, we et ũ = u x µ(t). Then, taking in account the compatibiity conditions for µ(t), we obtain ũ tt ũ xx = x µ (t), ũ(x, ) =, ũ t (x, ) =, ũ(, t) =, ũ(, t) =. It is we known that there exists unique soution ũ(x, t) to this probem [1], hence u(x, t) assumes the form u(x, t) = 2 ( 1) k ( ) π 2 k 2 µ kπ(t τ) (τ) sin dτ sin kπx + x µ(t). Now we find that and consider u x (, t) = 2 π u x (, t) = 2 π ( 1) k u x (, t) u x (, t) = 4 π k 1 k µ (τ) µ (τ) sin µ (τ) sin m=1 kπ(t τ) kπ(t τ) dτ + 1 µ(t), dτ + 1 µ(t) 1 (2m 1)π(t τ) sin dτ. 2m 1

4 4 SERGEI A. BEILIN EJDE 21/76 As in [6] m=1 Then by (1.9) we can write sin (2m 1)x 2m 1 = = u x (, t) u x (, t) = { π/4, if < x < π π/4, if π < x < 2π. µ (τ) dτ. Taking into account the compatibiity conditions µ() = µ () =, we easiy obtain µ(t). Now from the uniqueness theorem [1], we obtain u(x, t). 3. Existence Obviousy, the soution to the probem (1.6)-(1.9), if it exists, is a sum of soutions to the foowing two probems: Probem H Probem N H u(x, ) = ϕ(x), u tt u xx =, u t (x, ) = ψ(x), u(, t) =, u x (, t) = u x (, t) u tt u xx = g(x, t), u(x, ) = u t (x, ) =, u(, t) =, u x (, t) = u x (, t). First consider probem H and use separation of variabes. Let u(x, t) = X(x)T (t). Substituting in the equation u tt u xx = and taking into account (1.8), (1.9), we obtain X (x) + λx(x) =, X() =, X () = X (). (3.1) Note that probem (3.1) is not sef-adjoint: The adjoint probem is Y (x) + λy (x) =, Y () =, Y () = Y (). (3.2) The eigenvaues and eigenfunctions of probem (3.1) are λ k = ( 2πk ) 2, k = 1, 2,... (3.3) X = x, X k = sin 2πkx (3.4) respectivey. Note, that for k > the functions (3.4) are not orthonorma with X. To construct a basis in L 2, we compete (3.4) by using adjoint functions. Foowing M. Kedysh [11], we define an adjoint function X k, corresponding eigenvaue λ k from (3.3), as a soution to the boundary-vaued probem X k (x) + λ k Xk (x) = 2 λ k X k (x), Xk () =, X k () = X k(). (3.5) We obtain X k (x) = x cos 2πkx, k = 1, 2,... Rewrite now a system of eigenvaue and adjoint functions of (3.1) as X = x, X 2k 1 (x) = x cos 2πkx, X 2k (x) = sin 2πkx. (3.6)

5 EJDE 21/76 WAVE EQUATION WITH A NONLOCAL CONDITION 5 In a simiar way we find the system of eigenvaue and adjoint functions (3.2): Y (x) = 2 2, Y 2k 1(x) = 4 2πkx cos, Y 2 2k (x) = 4( x) 2 sin 2πkx, (3.7) where for every λ k with k >, X 2k (x), Y 2k (x) are eigenvaue functions, X 2k 1 (x), Y 2k 1 (x) are adjoint functions of the probems (3.1) and (3.2) respectivey. Direct cacuations show that (3.6) and (3.7) form a biorthogona system for x (, ): (X i, Y j ) = X i (x)y j (x) dx = δ ij. As it was shown in [8] the system (3.6) is compete and forms a basis in L 2 (, ). Hence, an arbitrary function f(x) L 2 (, ) may be expanded as where f(x) = A X (x) + (A 2k X 2k (x) + A 2k 1 X 2k 1 (x)), A i = f(x)y i (x) dx. (3.8) Returning to the separation variabes technique, for T (t) we obtain T k (t) = a k sin 2πkt + b k cos 2πkt. We assume now that a soution to H is of the form u(x, t) = A X + ( (A 2k X 2k + A 2k 1 X 2k 1 )T k t 2πk A 2k 1X 2k T k Substitute T k (t) and rewrite the coefficients. Then u(x, t) =C X + (X 2k (C 2k sin 2πkt + X 2k 1 (C 2k 1 sin 2πkt tx 2k (C 2k 1 cos 2πkt + D 2k cos 2πkt ) + D 2k 1 cos 2πkt ) D 2k 1 sin 2πkt )). The initia data (1.7) give us the foowing two equaities ψ(x) = ϕ(x) = C X + ( ( 2πk (D 2k X 2k + D 2k 1 X 2k 1 ), C 2k C 2k 1 )X 2k + 2πk ) C 2k 1 X 2k 1, ). (3.9) (3.1) and the coefficients can be found via formua (3.8). Assume a soution to the probem N H is of the form u(x, t) = V (t)x (x) + (V 2k (t)x 2k (x) + V 2k 1 (t)x 2k 1 (x)), (3.11)

6 6 SERGEI A. BEILIN EJDE 21/76 where V i (t) are unknown coefficients satisfying the initia conditions V i () = V i () =. Substitute (3.11) into the equation u tt u xx = g(x, t), where g(x, t) has been expanded as a biorthogona series: g(x, t) = g (t)x (x) + (g 2k (t)x 2k (x) + g 2k 1 (t)x 2k 1 (x)), with coefficients We obtain V (t)x g i (t) = ( V g(x, t)y i (x) dx, i =, 1,... 2k(t) + 4π2 k 2 2 ) V 2k (t) sin 2πkx ( V 2k 1(t) + 4π2 k 2 ) 2 V 2k 1 (t) x cos 2πkx V 2k 1 (t) 4πk = g (t)x (x) + sin 2πkx (g 2k (t)x 2k (x) + g 2k 1 (t)x 2k 1 (x)). Thus we have a Cauchy probem for the system of ordinary differentia equations with initia data V V (t) = g (t) 2k + 4πk ( πk V 2k(t) + V 2k 1 (t)) = g 2k (t) V 2k 1(t) + 4π2 k 2 V 2k 1 (t) = g 2k 1 (t) 2 V () = V () =, V 2k () = V 2k() =, V 2k 1 () = V 2k 1() =, which has a unique soution V (t) = V 2k 1 (t) = 1 kπ V 2k (t) = 1 kπ Theorem 3.1. Let: (t τ)g (τ) dτ, g 2k 1 (τ) sin kπ(t τ) (g 2k (τ) 4πkV 2k 1 (τ)) sin dτ, kπ(t τ) (1) g(x, t) C 2 (D), g x (x, t) C[, ] for a t (, T ), g(x, t) P, (x, t) D (2) ϕ C[, ] C 2 (, ), ψ C[, ], ϕ() =, ϕ () = ϕ (), ψ() =. Then there exists the soution to (1.6) (1.9), u(x, t) C( D) C 1 ( D \ {t = T }) C 2 (D) which has the form of a sum of (3.9) and (3.11). dτ.

7 EJDE 21/76 WAVE EQUATION WITH A NONLOCAL CONDITION 7 Series Proof. It is sufficient to prove uniform convergence of the series (3.9) and (3.11) and the series, obtained with forma differentiation. Let ϕ (x) M 1, ϕ (x) M 2, ψ(x) N, ψ (x) N 1, g x P 1, g xx P 2. Integrating C i, D i, V i by parts and taking in account the abovementioned assumptions, we obtain: D 2k 1 k 2 (M 2 + 2M 1 ) π 2, D 2k 1 1 k 2 M 2 π 2, C 2k 1 k 2 (N 1 + 2N) 2π 2, C 2k 1 1 k 2 N 1 π 2, V 2k 1 k 2 4T 2 (2p 1 + P 2 ) π 2, V 2k 1 1 k 2 2T P 1 π 2, and hence the series (3.9) and (3.11) and the series, obtained with forma differentiation, converge uniformy. References [1] A. V. Bitzadze, Urawnenija matematicheskoj fiziki, M., Nauka, [2] A. Bouziani, On a cass of paraboic equations with nonoca boundary conditions, Buetin de a Casse des Sciences, Académie Royae de Begique, T.X, 1999, p [3] A. Bouziani, Soution forte d un probème mixte avec conditions non ocaes pour une casse d équations hyperboiques, Buetin de a Casse des Sciences, Académie Royae de Begique, T. VIII, 1997, p [4] J. R. Cannon, The soution of the heat equation subject to specification of energy. Quart. App. Math., 21, n2, , [5] D. G. Gordeziani, G. A. Avaishvii, On the constructing of soutions of the nonoca initia boundary probems for one-dimentiona medium osciation equations, Matem. Modeirovanie, 12, N. 1, 2, [6] I. S. Gradshtein, I. M. Ryzhik, Tabes of integras, sums, series and products, M., [7] A. K. Gushin, V. P. Mikhaiov, On sovabiity of nonoca probems for second-odered eiptic equation, Matem. Sbornik, V. 185, 1994, p [8] V. A. Iyin, Necessary and sufficient properties of being a basis of a subsystem of eigenfunctions and adjoint functions for Kedysh bunde for odrinary differentia operators, Dokady Acad. Nauk SSSR, 227, N. 4, 1976, p [9] N. I. Ionkin, Soutions of boundary vaue probem in heat conductions theory with nonoca boundary conditions, Differents. Uravn., Vo. 13, N2, 1977, p [1] L. I. Kamynin, A boundary vaue probem in the theory of the heat conduction with noncassica boundary condition, Z. Vychis. Mat. Fiz., 4, N6, 1964, p [11] M. V. Kedysh, On eigenvaues and eigenfunctions of certain casses of not sef-adjoint equations, Dokady Acad. Nauk SSSR, 87, 1951, p [12] S. Mesoub, A. Bouziani, On a cass of singuar hyperboic equation with a weighted integra condition, Internat. J. Math. & Math. Sci., Vo. 22, No. 3, 1999, pp [13] B. P. Paneiah, On certain nonoca boundary probemsfor inear differentia operators, Matem. zametki., 1984, v. 35, N.3, p [14] L. S. Pukina, A nonoca probem with integra conditions for hyperboic equations, Eectron. J. Ddiff. Eqns., Vo. 1999(1999), No. 45, 1-6. [15] L. S. Pukina, On sovabiity in L 2 of nonoca probem with integra conditions for a hyperboic equation, Differents. Uravn., V. N. 2, 2. [16] A. A. Samarskii, Some probems in differentia equations theory, Differents. Uravn., Vo. 16, n11, 198, p [17] A. L. Skubachevski, G. M. Stebov, On spectrum of differentia operators with domain nondense in L 2, Dokadi AN USSR, v. 321, N.6, p [18] N. I. Yurchuk, Mixed probem with an integra condition for certain paraboic equations, Differents. Uravn., Vo. 22, N. 12, 1986, p

8 8 SERGEI A. BEILIN EJDE 21/76 Sergei A. Beiin Department of Mathematics, Samara State University, 1, Ac.Pavov st., Samara Russia E-mai address: