determination of a diffusion coefficient in a quasilinear parabolic equation

© 2017 Kanca, published by De Gruyter Open This work is licensed under the Creative Commons Attribution NonCommercial NoDerivs 3 0 License Open Math 2017; 15 77–91 Open Mathematics Open Access Researc[.]

Open Math 2017; 15: 77–91 Open Mathematics Open Access Research Article Fatma Kanca* Determination of a diffusion coefficient in a quasilinear parabolic equation DOI 10.1515/math-2017-0003 Received August 2, 2016; accepted October 21, 2016 Abstract: This paper investigates the inverse problem of finding the time-dependent diffusion coefficient in a quasilinear parabolic equation with the nonlocal boundary and integral overdetermination conditions Under some natural regularity and consistency conditions on the input data the existence, uniqueness and continuously dependence upon the data of the solution are shown Finally, some numerical experiments are presented Keywords: Heat equation, Inverse problem, Nonlocal boundary condition, Integral overdetermination condition, Time-dependent diffusion coefficient MSC: 35K59, 35R30 Introduction In this paper, an inverse problem of determining of the diffusion coefficient a.t / has been considered with extra R1 integral condition u.x; t /dx which has appeared in various applications in industry and engineering [1] The mathematical model of this problem is as follows: u t D a.t /uxx C f x; t; u/; x; t / DT WD 0; 1/  0; T / u.x; 0/ D '.x/; (1) x Œ0; 1 ; (2) u.0; t / D u.1; t /; ux 1; t / D 0; t Œ0; T  ; (3) Z1 E.t / D u.x; t /dx;  t  T; (4) The functions '.x/ and f x; t; u/ are given functions The problem of a coefficient identification in nonlinear parabolic equation is an interesting problem for many scientists [2–5] In [6] the nature of (3)-type conditions is demonstrated In this study, we consider the inverse problem (1)-(4) with nonlocal boundary conditions and integral overdetermination condition We prove the existence, uniqueness and continuous dependence on the data of the solution by applying the generalized Fourier method and we construct an iteration algorithm for the numerical solution of this problem The plan of this paper is as follows: In Section 2, the existence and uniqueness of the solution of inverse problem (1)-(4) is proved by using the Fourier method and iteration method In Section 3, the continuous dependence upon the *Corresponding Author: Fatma Kanca: Department of Management Information Systems, Kadir Has University, 34083, Istanbul, Turkey, E-mail: fatma.kanca@khas.edu.tr © 2017 Kanca, published by De Gruyter Open This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License Unauthenticated Download Date | 3/2/17 10:56 AM 78 F Kanca data of the inverse problem is shown In Section 4, the numerical procedure for the solution of the inverse problem is given Existence and uniqueness of the solution of the inverse problem We have the following assumptions on the data of the problem (1)-(4) (A1 ) E.t / C Œ0; T ; E t /  0; (A2 ) 00 00 (1) '.x/ C Œ0; 1; '.0/ D '.1/; ' 1/ D 0; ' 0/ D ' 1/; (2) '2k  0; k D 1; 2; ::: (A3 ) (1) Let the function f x; t; u/ be continuous with respect to all arguments in DN T  1; 1/ and satisfy the following condition ˇ ˇ ˇ @.n/ f x; t; u/ @.n/ f x; t; u/ Q ˇˇ ˇ Q ; n D 0; 1; 2; ˇ ˇ  b.x; t / ju uj ˇ ˇ @x n @x n where b.x; t / L2 DT /; b.x; t /  0; (2) f x; t; u/ C Œ0; 1; t Œ0; T ; f x; t; u/jxD0 D f x; t; u/jxD1 ; fx x; t; u/jxD1 D 0; fxx x; t; u/jxD0 D fxx x; t; u/jxD1 ; (3) f2k t /  0; f0 t / > 0; 8t Œ0; T ; where Z1 'k D Z1 '.x/Yk x/dx; fk t / D f x; t; u/Yk x/dx; k D 0; 1; 2; ::: X0 x/ D 2; X2k x/ Y0 x/ D x; Y2k D cos 2kx; X2k x/ D 4.1 x/ x/ sin 2kx; k D 1; 2; ::: : D x cos 2kx; Y2k x/ D sin 2kx; k D 1; 2; :::: The systems of functions Xk x/ and Yk x/; k D 0; 1; 2; ::: are biorthonormal on Œ0; 1 They are also Riesz bases in L2 Œ0; 1 (see [7]) We obtain the following representation for the solution of (1)-(3) for arbitrary a.t / by using the Fourier method: Zt u.x; t / D 4'0 C f0  /d  X0 x/ C X 2k/2 4'2k e Rt C kD1 C X Zt a.s/ds f2k  /d  e 2k/2 Rt a.s/ds  d  X2k x/ 2k/2 4.'2k 4k'2k t / e Rt a.s/ds X2k x/ kD1 C X kD1 t Z f2k  / 4kf2k  /.t  // e 2k/2 Rt  a.s/ds d  X2k x/ (5) Differentiating (5) we obtain Z1 u t x; t /dx D E t /;  t  T: (6) Unauthenticated Download Date | 3/2/17 10:56 AM 79 Determination of a diffusion coefficient in a quasilinear parabolic equation (5) and (6) yield P E t / C 2f0 t / C kD1 a.t / D P 2k/2 8k 4'2k e Rt a.s/ds C kD1 Rt f t / k 2k 2k/2 f2k  /e Rt (7) a.s/ds  d5 Definition 2.1 fu.t /g D fu0 t /; u2k.t /; u2k t /; k D 1; :::; ng ;are continuous functions on Œ0; T  and satisfying  P max ju2k t /j C max ju2k t /j < 1: The set of these functions is the condition max ju0 t /j C 0t T 0t T kD1 0t T   P max ju2k t /j C max ju2k t /j : It denoted by B1 and the norm in B1 is ku.t /k D max ju0 t /j C 0t T kD1 0tT 0tT can be shown that B1 is the Banach space Theorem 2.2 If the assumptions A1 / one solution for small T .A3 / are satisfied, then the inverse coefficient problem (1)-(4) has at most Proof We define an iteration for Fourier coefficient of (5) as follows: C1/ u.N t / D u.0/ t / Zt Z1 C f ; ; u.N / ;  //d d  0 C1/ u.N t / 2k D u.0/ 2k t / Zt Z1 C f ; ; u N / ;  // sin 2k e 2k/2 Rt a.N / s/ds  d d  0 C1/ u.N 2k t / D u.0/ 2k t / Zt Z1 C f ; ; u N / ;  // cos 2k e 2k/2 Rt a.N / s/ds  d d  0 Zt Z1  /f ; ; u.N / ;  // sin 2k e t 4k 2k/2 Rt a.N / s/ds  d d  (8) 0 where N D 0; 1; 2; ::: and u.0/ t/ D '0 ; u.0/ 2k t / 2k/2 D '2k e Rt a.s/ds ; u.0/ 2k t / 2k/2 D '2k 4k t '2k 1/ e Rt a.s/ds : It is obvious that u.0/ t / B1 and a.0/ C Œ0; T : For N D 0, 0/ u.1/ t/ D u0 t / C Zt Z1 Œf ; ; u.0/ ;  // Zt Z1 f ; ; 0/d d  C 0 f ; ; 0/d d : 0 Let us apply Cauchy inequality, t 12 t Z