In this paper we consider a parabolic equation with a periodic boundary condition and we prove the stability of a solution on the data. We give a numerical example for the stability of the solution on the data.
Consider the following mixed problem:
for a quasilinear parabolic equation with the nonlinear source term .
The functions and are given functions on and respectively. Denote the solution of problem (1)-(4) by . The existence, uniqueness and convergence of the weak generalized solution of problem (1)-(4) are considered in . The numerical solution of problem (1)-(4) is considered .
In this study we prove the continuous dependence of the solution upon the data and . In , a similar iteration method is used with this kind of a local boundary condition for a nonlinear inverse coefficient problem for a parabolic equation. Then we give a numerical example for the stability.
2 Continuous dependence upon the data
Theorem 1Under the following assumptions, the solution depends continuously upon the data.
(A1) Let the function be continuous with respect to all arguments in and satisfy the following condition:
where , ,
(A2) , ,
Proof Let and be two sets of data which satisfy the conditions (A1)-(A3).
Let and be the solutions of problem (1)-(4) corresponding to the data ϕ and respectively, and
The solutions of (1)-(4), and , are presented in the following form, respectively:
From the condition of the theorem, we have and . We will prove that the other sequential approximations satisfy this condition.
where , , and , , .
First of all, we write in (6)-(7). We consider
Adding and subtracting
to both sides and applying the Cauchy inequality, Hölder inequality, Lipschitz condition and Bessel inequality to the right-hand side of (8) respectively, we obtain
In the same way, for a general value of N, we have
(The sequence is convergent, then we can write , ∀N.)
It follows from the estimation ([, pp.76-77]) that .
Then let for the last equation
If and , then . □
3 Numerical example
In this section we consider an example of numerical solution of (1)-(4) to test the stability of this problem. The numerical procedure of (1)-(4) is considered in .
Consider the problem
It is easy to see that the analytical solution of this problem is
In this example, we take and for different ε values.
The comparisons between the analytical solution and the numerical finite difference solution for , values when are shown in Figure 1.
Figure 1. The exact and numerical solutions of . The exact and numerical solutions of , for , for , for , the exact solution is shown with a dashed line.
The computational results presented are consistent with the theoretical results.
The authors declare that they have no competing interests.
FK conceived the study, participated in its design and coordination and prepared computing section. ISB participated in the sequence alignment and achieved the estimation.
Dedicated to Professor Hari M Srivastava.
Cannon, J, Lin, Y: Determination of a parameter in holder classes for some semilinear parabolic equations. Inverse Probl.. 4, 595–606 (1988). Publisher Full Text
Cannon, J, Lin, Y, Wang, S: Determination of a control parameter in a parabolic partial differential equation. J. Aust. Math. Soc. Ser. B, Appl. Math. 33, 149–163 (1991). Publisher Full Text
Kanca, F, Ismailov, M: Inverse problem of finding the time-dependent coefficient of heat equation from integral overdetermination condition data. Inverse Probl. Sci. Eng.. 20, 463–476 (2012). Publisher Full Text