Determination of a diffusion coefficient in a quasilinear parabolic equation

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.


Introduction
In this paper, an inverse problem of determining of the diffusion coefficient a.t / has been considered with extra integral condition R 1 0 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.0; t / D u.1; t /; u x .1; t / D 0; t 2 OE0; T ; 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][3][4][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 data of the inverse problem is shown. In Section 4, the numerical procedure for the solution of the inverse problem is given.
Theorem 2.2. If the assumptions .A 1 / .A 3 / are satisfied, then the inverse coefficient problem (1)-(4) has at most one solution for small T.
Proof. We define an iteration for Fourier coefficient of (5) as follows: Let us apply Cauchy inequality, If we take the maximum of the last inequality, we get the following estimation for u .1/ 0 .t /: Let us apply Cauchy inequality, and take the sum of the last inequality and partial derivative of f with respect to and apply Hölder inequality, By applying Bessel inequality we obtain If we use Lipschitzs condition and take the maximum of the last inequality, we get the following estimation for Similarly, let us apply Cauchy inequality, and take the sum of the last inequality and partial derivative of f with respect to and apply Hölder inequality and Bessel inequality, If we use Lipschitzs condition and take the maximum of the last inequality, we get the following estimation for Finally we obtain the following inequality: In the same way, for N we have fu.t /g D fu 0 .t /; u 2k .t /; u 2k 1 .t /; k D 1; 2; :::g 2 B 1 : We define an iteration for (7) as follows: It is clear that Hence a .1/ .t/ 2 C OE0; T . In the same way, for N; we have Now let us prove that the iterations u .N C1/ .t / and a .N C1/ .t / converge in B 1 and C OE0; T , respectively, as N ! 1:  Applying the same estimations we obtain: If we apply the Cauchy inequality, the Hölder Inequality, the Lipschitz condition and the Bessel inequality to the last equation, we obtain: If we use the same estimations, we get If we apply the Cauchy inequality, the Hölder Inequality, the Lipshitzs condition and the Bessel inequality tǒ u u .N C1/ˇa ndˇa a .N /ǔ and the Gronwall inequality to the last inequality and using inequality (9), we have Then N ! 1 we obtain u .N C1/ ! u; a .N C1/ ! a: Let us prove the uniqueness of these solutions. Assume that problem (1)-(4) has two solution pair .a; u/ ; .b; v/ : Applying the Cauchy inequality, the Hölder Inequality, the Lipshitzs condition and the Bessel inequality to ju.t / v.t /j and ja.t / b.t /j, we obtain:  Proof. LetˆD f'; E; f g andˆD˚'; E; f « be two sets of the data, which satisfy the assumptions .A 1 / .A 3 / : Suppose that there exist positive constants M i ; i D 0; 1; 2 such that

Numerical method for the problem (1)-(4)
In order to solve problem (1)-(4) numerically, we need the linearization of the nonlinear terms: In this step, we use the implicit finite difference approximation for the discretizing problem (14)-(17): Let us integrate the equation (1) with respect to x and use (3) and (4) to obtain The finite difference approximation of (21) is f .x; t j /dx; j D 0; 1; :::; N t : We mention that R 1 0 e f .x; t j /dx is numerically calculated using Simpson's rule of integration. a j.s/ ; v j.s/ i are the values of a j ; v j i at the s-th iteration step; respectively. At each .s C 1/-th iteration step, a j C1.sC1/ is as follows The iteration of (18)-(20) is for the given functions   In order to investigate the stability of the numerical solution, noise is added to the overdetermination data (4) as follows where is the percentage of noise and Â are random variables generated from a uniform distribution in the interval OE 1; 1: Figure 3 shows the exact and numerical solutions of a.t / when the input data (4) are contaminated by D 1%; 5% and 10% noise. From these figures it can be seen that the numerical solution becomes unstable as the input data is contaminated with noise. We use wavelet decomposition and thresholding to remove noise and we obtain Figure  4.
Example 4.2 (discontinuous diffusion coefficient). In the previous Example 4.1, a smooth function given by a.t / D t 2 C 1 is considered. In Example 4.2, a more severe discontinuous test function is given: ( t 2 C 2 ; t 2 OE0; 1/ t 2 C 2 ; t 2 OE1; 2 Let us apply the scheme above for the step sizes h D 0:01, D 0:005. Figure 5 shows the exact and the numerical solutions of a.t/ when T D 2 .

Some discussions
In the previous section, in Example 4.1, the man-made noise in the measured output data is added to show the stability of the numerical method. Unstable numerical solution is obtained and wavelet decomposition and thresholding are used to remove noise. Also in Example 4.2, discontinuous source function is given to show the efficiency of the present method. From Figure 5 it can be seen that the agreement between the numerical and exact solutions for a.t / is excellent.
In future the fractional problem of this inverse problem can be studied [8][9][10].