Continuous dependence on data for a solution of the quasilinear parabolic equation with a periodic boundary condition

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 [1]. The numerical solution of problem (1)-(4) is considered [2].

In this study we prove the continuous dependence of the solution upon the data and . In [3], 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.

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 1Under the following assumptions, the solutiondepends continuously upon the data.

(A₁) Let the function be continuous with respect to all arguments in and satisfy the following condition:

where , ,

(A₂) , ,

(A₃) .

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:

Let .

Let .

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

For ,

For ,

In the same way, for a general value of N, we have

where

and

(The sequence is convergent, then we can write , ∀N.)

It follows from the estimation ([[1], pp.76-77]) that .

Then let for the last equation

where .

If and , then . □

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 [2].

Example 1

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.

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.

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