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This article is part of the series Proceedings of the International Congress in Honour of Professor Hari M. Srivastava.

Open Access Research

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

Fatma Kanca1* and Irem Sakinc Baglan2

Author Affiliations

1 Department of Information Technologies, Kadir Has University, Istanbul, 34083, Turkey

2 Department of Mathematics, Kocaeli University, Kocaeli, 41380, Turkey

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Boundary Value Problems 2013, 2013:28  doi:10.1186/1687-2770-2013-28


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/28


Received:7 January 2013
Accepted:29 January 2013
Published:14 February 2013

© 2013 Kanca and Baglan; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.

1 Introduction

Consider the following mixed problem:

(1)

(2)

(3)

(4)

for a quasilinear parabolic equation with the nonlinear source term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M5">View MathML</a>.

The functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M6">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M7">View MathML</a> are given functions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M8">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M9">View MathML</a> respectively. Denote the solution of problem (1)-(4) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M10">View MathML</a>. 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M10">View MathML</a> upon the data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M6">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M7">View MathML</a>. 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.

2 Continuous dependence upon the data

In this section, we will prove the continuous dependence of the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M10">View MathML</a> 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 solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M10">View MathML</a>depends continuously upon the data.

(A1) Let the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M7">View MathML</a> be continuous with respect to all arguments in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M9">View MathML</a> and satisfy the following condition:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M18">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M19">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M20">View MathML</a>,

(A2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M21">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M22">View MathML</a>,

(A3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M23">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M24">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M25">View MathML</a> be two sets of data which satisfy the conditions (A1)-(A3).

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M10">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M27">View MathML</a> be the solutions of problem (1)-(4) corresponding to the data ϕ and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M28">View MathML</a> respectively, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M29">View MathML</a>

The solutions of (1)-(4), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M10">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M27">View MathML</a>, are presented in the following form, respectively:

(5)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M33">View MathML</a>.

(6)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M35">View MathML</a>.

From the condition of the theorem, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M36">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M37">View MathML</a>. We will prove that the other sequential approximations satisfy this condition.

(7)

(8)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M40">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M41">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M42">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M43">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M44">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M45">View MathML</a>.

First of all, we write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M46">View MathML</a> in (6)-(7). We consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M47">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M48">View MathML</a>

(9)

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M51">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M52">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M53">View MathML</a>,

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M55">View MathML</a>

(10)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M56">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M57">View MathML</a>

(The sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M58">View MathML</a> is convergent, then we can write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M59">View MathML</a>, ∀N.)

It follows from the estimation ([[1], pp.76-77]) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M60">View MathML</a>.

Then let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M61">View MathML</a> for the last equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M62">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M63">View MathML</a>.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M64">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M65">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M66">View MathML</a>. □

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 [2].

Example 1

Consider the problem

(11)

(12)

(13)

It is easy to see that the analytical solution of this problem is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M70">View MathML</a>

In this example, we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M71">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M72">View MathML</a> for different ε values.

The comparisons between the analytical solution and the numerical finite difference solution for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M73">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M74">View MathML</a> values when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M75">View MathML</a> are shown in Figure 1.

thumbnailFigure 1. The exact and numerical solutions of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M76">View MathML</a>. The exact and numerical solutions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M77">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M78">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M79">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M80">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M81">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M82">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M83">View MathML</a>, the exact solution is shown with a dashed line.

The computational results presented are consistent with the theoretical results.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

FK conceived the study, participated in its design and coordination and prepared computing section. ISB participated in the sequence alignment and achieved the estimation.

Acknowledgements

Dedicated to Professor Hari M Srivastava.

References

  1. Ciftci, I, Halilov, H: Fourier method for a quasilinear parabolic equation with periodic boundary condition. Hacet. J. Math. Stat.. 37, 69–79 (2008). PubMed Abstract | PubMed Central Full Text OpenURL

  2. Sakinc, I: Numerical solution of the quasilinear parabolic problem with periodic boundary condition. Hacet. J. Math. Stat.. 39, 183–189 (2010)

  3. Cannon, J, Lin, Y: Determination of a parameter <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/28/mathml/M84">View MathML</a> in holder classes for some semilinear parabolic equations. Inverse Probl.. 4, 595–606 (1988). Publisher Full Text OpenURL

  4. 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 OpenURL

  5. 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 OpenURL