Continuous dependence on data for a solution of the quasilinear parabolic equation with a periodic boundary condition
© Kanca and Baglan; licensee Springer. 2013
Received: 7 January 2013
Accepted: 29 January 2013
Published: 14 February 2013
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.
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
In this section, we will prove the continuous dependence of the solution using an iteration method. The continuous dependence upon the data for linear problems by different methods is shown in [4, 5].
Theorem 1 Under the following assumptions, the solution depends continuously upon the data.
where , ,
(A2) , ,
Proof Let and be two sets of data which satisfy the conditions (A1)-(A3).
where , , and , , .
(The sequence is convergent, then we can write , ∀N.)
It follows from the estimation ([, pp.76-77]) that .
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 .
In this example, we take and for different ε values.
The computational results presented are consistent with the theoretical results.
Dedicated to Professor Hari M Srivastava.
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