On Solvability of an Inverse Boundary Value Problem for Pseudo Hyperbolic Equation of the Fourth Order

Size: px

Start display at page:

Download "On Solvability of an Inverse Boundary Value Problem for Pseudo Hyperbolic Equation of the Fourth Order"

Paulina Morgan
5 years ago
Views:

1 Journal of Mathematics Research; Vol. 7, No. ; 5 ISSN E-ISSN Published by Canadian Center of Science and Education On Solvability of an Inverse Boundary Value Problem for Pseudo Hyperbolic Equation of the Fourth Order Yashar. Mehraliyev & Afaq F. Huseynova Department of differential and inteqral equations, Bau State University, Bau, Azerbaijan Correspondence: Yashar Mehraliyev, Department of differential and inteqral equations, Bau State University, Bau, AZ48, Azerbaijan. el: yashar aze@mail.ru Received: February 5, 5 Accepted: March 4, 5 Online Published: April 7, 5 doi:.5539/jmr.v7np URL: Abstract We analyze the solvability of the inverse boundary problem with an unnown coefficient depended on time for the pseudo hyperbolic equation of fourth order with periodic and integral conditions.he initial problem is reduced to an equivalent problem. With the help of the Fourier method, the equivalent problem is reduced to a system of integral equations. he existence and uniqueness of the solution of the integral equations is proved. he obtained solution of the integral equations is also the only solution to the equivalent problem. Basing on the equivalence of the problems, the theorem of the existence and uniqueness of the classical solutions of the original problem is proved. Keywords: inverse boundary problem, pseudo hyperbolic equation, method Fourier, classic solution. Introduction here are many cases the needs of the practice bring about the problems of determining coefficients or the right hand side of differential equations from some nowledge of its solutions. Such problems are called inverse boundary value problems of mathematical physics. Inverse boundary value problems arise in various areas of human activity such as seismology, mineral exploration, biology, medicine, quality control in industry etc., which maes them an active field of contemporary mathematics. he inverse problems are favorably developing section of up-to-date mathematics. Recently, the inverse problems are widely applied in various fields of science. Different inverse problems for various types of partial differential equations have been studied in many papers. First of all we note the papers of ( ihonov, 943), ( Lavrentyev, 94), ( Lavrentyev & Romanov, 98), ( Ivanov,Vasin & anina, 978), (Denisov, 994) and their followers. In searching of local and non-local boundary value problems for pseudohyperbolic equations practical and theoretical interests assume great importance and is more actively studied now days. In this paper, due to the ( Mehraliyev, )-( Mehraliyev, ), we proved the existence and uniqueness of the solution of the inverse boundary value problem for the pseudohyperbolic equation of fourth order with periodic and integral conditions.. Problem Statement and Its Reduction to Equivalent Problem Lets consider for the equation (Gabov & Orazov,98) u tt (x, t) u ttxx (x, t) u xx (x, t) = a(t)u(x, t) f (x, t) () in the domain D = {(x, t) : x, t } an inverse boundary problem with initial conditions u(x, ) = φ(x), u t (x, ) = ψ(x) ( x ), () the periodic condition u(, t) = u(, t) ( t ), (3)

2 the non-local integral condition and with additional condition u(x, t)dx = ( t ) (4) u(x, t) = h(t) ( t ), (5) x (, ) are the given number, f (x, t),φ(x),ψ(x),h(t) are the given functions, and u(x, t), a(t) are the required functions. he condition (4) is a non-local integral condition of first ind, i.e. the one not involving values of unnown functions at the domains boundary points. Definition. he classic solution of problem () (5) is the pair{u (x, t), a (t)} of the functions u(x, t) and a(t) with the following properties: )the function u(x, t) is continuous in D together with all its derivatives contained in equation (); )the function a(t) is continuous on [, ]; 3) all the conditions of () (5) are satisfied in the ordinary sense. he following lemma is valid. Lemma Let f (x, t) C(D ), f (x, t)dx = ( t ) φ(x), ψ(x) C [, ], h(t) C [, ], h(t) ( t ) and φ(x)dx =, φ () = φ (), ψ () = ψ (), ψ(x)dx =, φ(x ) = h(), ψ(x ) = h (). hen the problem on finding the classic solution of problem () (5) is equivalent to the problem on defining of the function u(x, t) and a(t), possessing the properties ) and ) of definition of the classic solution of problem () (5), from relations () (3) and satisfying u x (, t) = u x (, t) ( t ), () h (t) u ttxx (x, t) u xx (x, t) = a(t)h(t) f (x, t) ( t ). (7) Proof. Let {u (x, t), a (t)} be a classical solution to the problem () (5). Integrating equation () with respect to x from to, we have d u(x, t)dx d dt dt (u x(, t) u x (, t)) aing into account that By () and φ () = φ (), (u x (, t) u x (, t)) = a(t) u(x, t)dx f (x, t)dx = ( t ) and (4), we find that f (x, t)dx ( t ). (8) d dt (u x(, t) u x (, t)) (u x (, t) u x (, t)) = ( t ). (9) ψ () = ψ () we obtain u x (, ) u x (, ) = φ () φ () =, u tx (, ) u tx (, ) = ψ () ψ () =. () Since problem (9), ()has only a trivial solution, we have u x (, t) u x (, t) =, i.e. the condition () is fulfilled.

3 Assume now that h(t) C [, ]. Differentiating (5) twice, we get It follows from () that u t (x, t) = h (t), u tt (x, t) = h (t) ( t ). () u tt (x, t) u ttxx (x, t) u xx (x, t) = a(t)u(x, t) f (x, t) ( t ). () Hence, taing into account (5) and (), we conclude that (7) is fulfilled. Now suppose that {u (x, t), a (t)} is a solution of problem () (3), (), (7), then from (8) and () we find that d dt u(x, t)dx a(t) By () and φ(x)dx =, ψ(x)dx =, it is obvious that u(x, t)dx = ( t ). (3) u(x, )dx = φ(x)dx =, u t (x, )dx = ψ(x)dx =. (4) Since the problem (3), (4) has only a trivial solution, fulfilled. From (7) and () we obtain By () and φ(x ) = h(), ψ(x ) = h () we have u(x, t)dx = ( t ), i.e. the condition (4) is d dt (u(x, t) h(t)) = a(t)(u(x, t) h(t)) ( t ). (5) { u(x, ) h() = φ(x ) h() =, u t (x, ) h () = ψ(x ) h () =. () From (5) and () we conclude that the condition (5) is fulfilled. he lemma is proved. 3. Investigation of the Existence and Uniqueness of the Classic Solution of the Inverse Boundary Value Problem It is nown (Buda, Samarsii & ihonov, 97) that the system, cos λ x, sin λ x,..., cos λ x, sin λ x,... (7) is a basis in L (, ), λ = π ( =,,...). herefore, it is obvious that for each solution {u(x, t), a(t)} to the problem () (3), (), (7) its first component u(x, t) has the form: u (t) = u(x, t) = u (t) cos λ x u (t) sin λ x (λ = π), (8) = u (t) = u(x, t) cos λ xdx, u (t) = u(x, t)dx, u(x, t) sin λ xdx ( =,,...). hen, applying the formal scheme of the Fourier method, from () and () we have u (t) = F (t; u, a) ( t ), (9) 3

4 ( λ ) u i (t) λ u i (t) = F i (t; u, a) ( t ; i =, ; =,,...), () u () = φ, u () = ψ, () u i () = φ i, u i () = ψ i (i =, ; =,,...), () f (t) = φ = F (t; u, a) = a(t)u (t) f (t) ( =,,...), f (x, t)dx, f (t) = φ = φ(x)dx, ψ = φ(x)cosλ xdx, ψ = F (t; u, a) = a(t)u (t) f (t), f (t) = φ = φ(x) sin λ xdx, ψ = f (x, t) cos λ xdx ( =,,...), ψ(x)dx, ψ(x)cosλ xdx ( =,,...), f (x, t) sin λ xdx ( =,,...), ψ(x) sin λ xdx ( =,,...). Solving problem (9) (), we find u (t) = φ tψ (t τ)f (τ; u, a)dτ ( t ), (3) u i (t) = φ i cos β t ψ i β sin β t β ( λ ) F i (τ; u, a) sin β (t τ)dτ (i =, ; =,,...; t ), (4) β = λ λ (i =, ; =,,...). (5) After substituting the expressions u (t) ( =,,...) and u (t) ( =,,...) into (8), for the component u(x, t) of the solution {u(x, t), a(t)} of problem () (3), (), (7) we get u (x, t) = φ tψ β ( λ ) (t τ)f (τ; u, a)dτ { φ cos β t ψ β sin β t F (τ; u, a) sin β (t τ)dτ cos λ x { φ cos β t ψ β sin β t 4

5 β ( λ ) F (τ; u, a) sin β (t τ)dτ sin λ x. () Now, from (7) and (8) we have a (t) = [h (t)] { h (t) f (x, t) [ λ u (t) λ u (t) ] cos λ x [ λ u (t) λ u (t) ] sin λ x. (7) By () and (4) we have λ u i (t) λ u i(t) = F i (t; u, a) u i (t) = = β F i(t; u, a) β φ i cos β t β ψ i sin β t β F λ i (τ; u, a) sin β (t τ)dτ(i =, ; =,,...). (8) o obtain the equation for the second component a(t) of the solution {u (x, t), a (t)} to the problem () (3), (), (7), substitute expression (8) into (7) and have a(t) = [h(t)] h (t) f (x, t) { β F (t; u, a) β φ cos β t β ψ sin β t β F λ (τ; u, a) sin β (t τ)dτ cos λ x { β F (t; u, a)β φ cos β t β ψ sin β t β λ F (τ; u, a) sin β (t τ)dτ sin λ x (9) hus, the problem () (3), (), (7) is reduced to solving the system (), (9) with respect to the unnown functions u(x, t) and a(t). Similarly to (Mehraliyev, ) it is possible to prove the following lemma. Lemma If {u (x, t), a (t)} is any solution of problem () (3), (), (7), then the functions u (t) = satisfy system (3), (4) in [, ]. u (t) = u(x, t) cos λ xdx, u (t) = u(x, t)dx, u(x, t) sin λ xdx ( =,,...). Remar It follows from lemma that to prove the uniqueness of the solution to the problem () (3), (), (7), it suffices to prove the uniqueness of the solution to the system (), (9). 5

6 In order to investigate problem () (3), (), (7), consider the following spaces:. Denote by B 3, ( Mehraliyev, ) the set of all functions u(x, t) of the form u(x, t) = u (t) cos λ x u (t) sin λ x = (λ = π), defined on D such that the functions u (t) ( =,,...), u (t) ( =,,...) are continuous on [, ] and J (u) u (t) C[,] (λ 3 u (t) C[,] ) he norm on this set is given by (λ 3 u (t) C[,] ) <. u(x, t) B 3, = J (u).. Denote by E 3 the space B3, C[, ] of the vector-functions z(x, t) = {u(x, t), a(t)} with the norm It is nown that B 3, and E3 are Banach spaces. Now, in the space E 3 consider the operator z E 3 = u(x, t) B 3, a(t) C[,]. Φ(u, a) = {Φ (u, a), Φ (u, a)}, Φ (u, a) = ũ(x, t) ũ (t) cos λ x ũ (t) sin λ x, = Φ (u, a) = ã(t), ũ (t), ũ i (t), (i =, ; =,,...) and ã(t) equal to the right hand sides of (3),(4) and (9), respectively. It is easy to see that < β <. aing into account these relations, by means of simple transformations we find (λ 3 ũ i(t) C[,] ) ũ (t) C[,] φ ψ f (τ) dτ (λ f i (τ) ) dτ a(t) C[,] u (t) C[,], (3) (λ 3 φ i ) (λ 3 ψ i ) a(t) C[,] (λ 3 u i(t) C[,] ), (3)

7 ã(t) C[,] [h(t)] C[,] { h (t) f (x, t) C[,] i= (λ 3 φ i ) i= (λ f i (τ) ) dτ ( ) a(t) C[,] i= (λ 3 ψ i ) i= (λ f i (t) C[,] ) i= (λ 3 u i(t) C[,] ). (3) Suppose that the data of problem () (3), (), (7) satisfy the following conditions.φ(x) C [, ], φ (x) L (, ), φ() = φ(), φ () = φ (), φ () = φ ()..ψ(x) C [, ], ψ (x) L (, ), ψ() = ψ(), ψ () = ψ (), ψ () = ψ (). 3. f (x, t) C(D ), f x (x, t) L (D ), f (, t) = f (, t) ( t ). 4. h(t) C [, ], h(t) ( t ). hen, from (3) (3), we get ũ(x, t) B 3, A () B () a(t) C[,] u(x, t) B 3,, (33) ã(t) C[,] A () B () a(t) C[,] u(x, t) B 3,, (34) A () = φ(x) L (,) ψ(x) L (,) f (x, t) L (D ) 4 φ (x) L (,) 4 ψ (x) L (,) 4 f x (x, t) L (D ), B () =, A () = [h(t)] { C[,] h (t) f (x, t)) C[,] φ (x) L (,) ψ (x) L f (,) x (x, t) L (D ) f x (x, t) C[,] L (,), It follows from the inequalities (33), (34) that B () = [h(t)] C[,] ( ). ũ(x, t) B 3, ã(t) C[,] A() B() a(t) C[,] u(x, t) B 3,, (35) A() = A () A (), B() = B () B (). Now we can prove the following theorem. heorem. Let the conditions -4 be fulfilled and (A() ) B() <. (3) 7

8 hen the problem () (3), (), (7) has a unique solution in the ball K = K R (z E 3 R = A() ) of the space E 3. Remar. Inequality (3) is satisfied for sufficiently small values at h (t) C[,]. Proof. In the space E 3 consider the equation z = Φz, (37) z = {u, a} and the components Φ i (u, a) (i =, ) of the operator Φ(u, a) are given by the right hand sides of the equations (), (9). Consider the operator Φ(u, a) in the ball K = K R from E 3. Similar to (35), we see that for any z, z, z K R the following estimates hold: Φz E 3 A() B() a(t) C[,] u(x, t) B 3,, (38) Φz Φz E 3 B()R(a (t) a (t) C[,] u (x, t) u (x, t) B 3, ). (39) hen, it follows from (3) together with the estimates (38) and (39) that the operator Φ acts in the ball K = K R and is contractive. herefore, in the ball K = K R the operator Φ has a unique fixed point {u, a}, that is a unique solution of equation (37) in the ball K = K R, i.e. it is a unique solution of system (), (9) in the ball K = K R. he function u(x, t), as an element of the space B 3, is continuous and has continuous derivatives u x(x, t) and u xx (x, t) in D. Further, from () it follows that u i (t)(i =, ; =,,...) is continuous in [, ] and consequently we have: (λ 3 u i (t) C[,] ) (λ 3 u i(t) C[,] ) f (x, t) a (t) u (x, t)c[,] L (,) (i =, ). From the last relation it is obvious that u tt (x, t), u ttx (x, t), u ttxx (x, t) is continuous in D. It is easy to verify that the equation () and conditions (), (3), (), (7) are satisfied in the ordinary sense. Consequently, {u (x, t), a (t)} is a solution of problem () (3), (), (7), and by lemma it is unique in the ball K = K R. he theorem is proved. By lemma the unique solvability of the initial problem () (5) follows from the theorem. heorem. Let all the conditions of theorem be fulfilled and f (x, t)dx = ( t ), φ(x)dx =, ψ(x)dx =, φ(x ) = h(), ψ(x ) = h (). hen the problem () (5) has a unique classical solution in the ball K = K R (z E 3 R = A() ) of the space E 3. References Buda, B.M., Samarsii, A. A., & ihonov, A. N. (97). Collector of tass in mathematical physics, M.: Naua (in Russian) Denisov, A. M. (994). Introduction to theory of inverse problems. MSU. Gabov, S. A., & Orazov, B. B. (98). he equation t [u xx u] u xx = and several problems associated with it. Dol. Computational Mathematics and Mathematical Physics, Vol., No, pp. 9 (in Russian) 8

9 Ivanov, V. K., Vasin V. V., & anina, V. P. (978). heory of linear ill-posed problems and its applications. M.: Naua (in Russian) Lavrent ev, M. M. (94). On an inverse problem for a wave equation. Dol. AN SSSR, Vol.57, No 3, pp. 5 5 (in Russian) Lavrent ev, M. M., Romanov, V. G., & Shishatsy, S.. (98). Ill-posed problems of mathematical physics and analysis. M.: Naua (in Russian) Mehraliyev, Ya.. (). Inverse boundary problem for a partial differential equation of fourth order with integral condition. Dol. South Ural State University Herald. Series: Mathematics, Mechanics, Physics, Vol.3(49), No 5, pp. 5 5 (in Russian) Mehraliyev, Ya.. (). On one inverse boundary value problem for hyperbolic equations of second order with integral condition of the first ind. Dol. he Bryans State University Herald No 4, pp. 8 (in Russian) Mehraliyev, Ya.. (). On solvability of an inverse boundary value problem for a second order elliptic equation. Dol. Vestnic versogo Gosudarstvennogo Universiteta. Ser. priladnaya matematia No 3, pp (in Russian) Mehraliyev, Ya.. (). Inverse boundary value problem for a second order elliptic equation with additional integral condition. Dol. Vestni of Udmurtsogo Universiteta. Mathematia. Mehania. Komputerniye Naui, Issue, pp. 3 4 (in Russian) ihonov, A. N. (943). On stability of inverse problems. Dol. AN SSSR, 39(5), (in Russian) Copyrights Copyright for this article is retained by the author(s), with first publication rights granted to the journal. his is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license ( 9

MATH 220: Problem Set 3 Solutions

MATH 220: Problem Set 3 Solutions Problem 1. Let ψ C() be given by: 0, x < 1, 1 + x, 1 < x < 0, ψ(x) = 1 x, 0 < x < 1, 0, x > 1, so that it verifies ψ 0, ψ(x) = 0 if x 1 and ψ(x)dx = 1. Consider (ψ j )