## Boundary Value Problems

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

Boundary Value Problems20132013:28

DOI: 10.1186/1687-2770-2013-28

Accepted: 29 January 2013

Published: 14 February 2013

## 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 $f=f\left(x,t,u\right)$.

The functions $\phi \left(x\right)$ and $f\left(x,t,u\right)$ are given functions on $\left[0,\pi \right]$ and $\overline{D}×\left(-\mathrm{\infty },\mathrm{\infty }\right)$ respectively. Denote the solution of problem (1)-(4) by $u=u\left(x,t\right)$. 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 $u=u\left(x,t\right)$ upon the data $\phi \left(x\right)$ and $f\left(x,t,u\right)$. 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 $u=u\left(x,t\right)$ 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 $u=u\left(x,t\right)$ depends continuously upon the data.

(A1) Let the function $f\left(x,t,u\right)$ be continuous with respect to all arguments in $\overline{D}×\left(-\mathrm{\infty },\mathrm{\infty }\right)$ and satisfy the following condition:
$|f\left(t,x,u\right)-f\left(t,x,\stackrel{˜}{u}\right)|\le b\left(x,t\right)|u-\stackrel{˜}{u}|,$

where $b\left(x,t\right)\in {L}_{2}\left(D\right)$, $b\left(x,t\right)\ge 0$,

(A2) $f\left(x,t,0\right)\in {C}^{2}\left[0,\pi \right]$, $tϵ\left[0,\pi \right]$,

(A3) $\phi \left(x\right)\in {C}^{2}\left[0,\pi \right]$.

Proof Let $\varphi =\left\{\phi ,f\right\}$ and $\overline{\varphi }=\left\{\overline{\phi },\overline{f}\right\}$ be two sets of data which satisfy the conditions (A1)-(A3).

Let $u=u\left(x,t\right)$ and $v=v\left(x,t\right)$ be the solutions of problem (1)-(4) corresponding to the data ϕ and $\overline{\varphi }$ respectively, and
The solutions of (1)-(4), $u=u\left(x,t\right)$ and $v=v\left(x,t\right)$, are presented in the following form, respectively:
(5)
Let $Au\left(\xi ,\tau \right)=\frac{{u}_{0\left(\tau \right)}}{2}+{\sum }_{k=1}^{\mathrm{\infty }}\left[{u}_{ck}\left(\tau \right)cos2k\xi +{u}_{sk}\left(\tau \right)sin2k\xi \right]$.
(6)

Let $Av\left(\xi ,\tau \right)=\frac{{v}_{0}\left(\tau \right)}{2}+{\sum }_{k=1}^{\mathrm{\infty }}\left[{v}_{ck}\left(\tau \right)cos2k\xi +{v}_{sk}\left(\tau \right)sin2k\xi \right]$.

From the condition of the theorem, we have ${u}^{\left(0\right)}\left(t\right)$ and ${v}^{\left(0\right)}\left(t\right)\in B$. We will prove that the other sequential approximations satisfy this condition.
(7)
(8)

where ${u}_{0}^{\left(0\right)}\left(t\right)={\phi }_{0}$, ${u}_{ck}^{\left(0\right)}\left(t\right)={\phi }_{ck}{e}^{-{\left(2k\right)}^{2}t}$, ${u}_{sk}^{\left(0\right)}\left(t\right)={\phi }_{sk}{e}^{-{\left(2k\right)}^{2}t}$ and ${v}_{0}^{\left(0\right)}\left(t\right)={\overline{\phi }}_{0}$, ${v}_{ck}^{\left(0\right)}\left(t\right)={\overline{\phi }}_{ck}{e}^{-{\left(2k\right)}^{2}t}$, ${v}_{sk}^{\left(0\right)}\left(t\right)={\overline{\phi }}_{sk}{e}^{-{\left(2k\right)}^{2}t}$.

First of all, we write $N=0$ in (6)-(7). We consider ${u}^{\left(1\right)}\left(t\right)-{v}^{\left(1\right)}\left(t\right)$
$\begin{array}{rcl}{u}^{\left(1\right)}\left(t\right)-{v}^{\left(1\right)}\left(t\right)& =& \frac{{u}_{0}^{\left(1\right)}\left(t\right)-{v}_{0}^{\left(1\right)}\left(t\right)}{2}\\ +\sum _{k=1}^{\mathrm{\infty }}\left[\left({u}_{ck}^{\left(1\right)}\left(t\right)-{v}_{ck}^{\left(1\right)}\left(t\right)\right)+\left({u}_{sk}^{\left(1\right)}\left(t\right)-{v}_{sk}^{\left(1\right)}\left(t\right)\right)\right]\\ =& \left({\phi }_{0}-\overline{{\phi }_{0}}\right)\\ +\frac{2}{\pi }{\int }_{0}^{t}{\int }_{0}^{\pi }\left[f\left(\xi ,\tau ,A{u}^{\left(0\right)}\left(\xi ,\tau \right)\right)-\overline{f}\left(\xi ,\tau ,A{v}^{\left(0\right)}\left(\xi ,\tau \right)\right)\right]\phantom{\rule{0.2em}{0ex}}d\xi \phantom{\rule{0.2em}{0ex}}d\tau \\ +\left({\phi }_{ck}-\overline{{\phi }_{ck}}\right){e}^{-{\left(2k\right)}^{2}t}\\ +\frac{2}{\pi }{\int }_{0}^{t}{\int }_{0}^{\pi }\left[f\left(\xi ,\tau ,A{u}^{\left(0\right)}\left(\xi ,\tau \right)\right)-\overline{f}\left(\xi ,\tau ,A{v}^{\left(0\right)}\left(\xi ,\tau \right)\right)\right]\\ ×{e}^{-{\left(2\pi k\right)}^{2}\left(t-\tau \right)}cos2\pi k\xi \phantom{\rule{0.2em}{0ex}}d\xi \phantom{\rule{0.2em}{0ex}}d\tau +\left({\phi }_{sk}-\overline{{\phi }_{sk}}\right){e}^{-{\left(2k\right)}^{2}t}\\ +\frac{2}{\pi }{\int }_{0}^{t}{\int }_{0}^{\pi }\left[f\left(\xi ,\tau ,A{u}^{\left(0\right)}\left(\xi ,\tau \right)\right)-\overline{f}\left(\xi ,\tau ,A{v}^{\left(0\right)}\left(\xi ,\tau \right)\right)\right]\\ ×{e}^{-{\left(2\pi k\right)}^{2}\left(t-\tau \right)}sin2\pi k\xi \phantom{\rule{0.2em}{0ex}}d\xi \phantom{\rule{0.2em}{0ex}}d\tau .\end{array}$
(9)
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 $N=1$,
$\begin{array}{rcl}|{u}^{\left(2\right)}\left(t\right)-{v}^{\left(2\right)}\left(t\right)|& \le & \frac{|{u}_{0}^{\left(2\right)}\left(t\right)-{v}_{0}^{\left(2\right)}\left(t\right)|}{2}+\sum _{k=1}^{\mathrm{\infty }}\left(|{u}_{ck}^{\left(2\right)}\left(t\right)-{v}_{ck}^{\left(2\right)}|+|{u}_{sk}^{\left(2\right)}\left(t\right)-{v}_{sk}^{\left(2\right)}\left(t\right)|\right)\\ \le & \left(\frac{\sqrt{3T}+\pi }{\sqrt{6}\pi }\right){\left({\int }_{0}^{t}{\int }_{0}^{\pi }{b}^{2}\left(\xi ,\tau \right)\phantom{\rule{0.2em}{0ex}}d\xi \phantom{\rule{0.2em}{0ex}}d\tau \right)}^{\frac{1}{2}}{A}_{T}\\ +\left(\frac{\sqrt{3T}+\pi }{\sqrt{6}\pi }\right){\left({\int }_{0}^{t}{\int }_{0}^{\pi }{\overline{b}}^{2}\left(\xi ,\tau \right)\phantom{\rule{0.2em}{0ex}}d\xi \phantom{\rule{0.2em}{0ex}}d\tau \right)}^{\frac{1}{2}}{A}_{T}.\end{array}$
For $N=2$,
In the same way, for a general value of N, we have
$\begin{array}{rcl}|{u}^{\left(N+1\right)}\left(t\right)-{v}^{\left(N+1\right)}\left(t\right)|& \le & \frac{|{u}_{0}^{\left(N+1\right)}\left(t\right)-{v}_{0}^{\left(N+1\right)}\left(t\right)|}{2}\\ +\sum _{k=1}^{\mathrm{\infty }}\left(|{u}_{ck}^{\left(N+1\right)}\left(t\right)-{v}_{ck}^{\left(N+1\right)}\left(t\right)|+|{u}_{sk}^{\left(N+1\right)}\left(t\right)-{v}_{sk}^{\left(N+1\right)}\left(t\right)|\right)\\ \le & {A}_{T}\cdot {a}_{N}={a}_{N}\phantom{\rule{0.1em}{0ex}}\left(\parallel \phi -\overline{\phi }\parallel +C\left(t\right)+{M}_{1}\parallel f-\overline{f}\parallel \right),\end{array}$
(10)
where
$\begin{array}{rcl}{a}_{N}& =& {\left(\frac{\sqrt{3T}+\pi }{\sqrt{6}\pi }\right)}^{N}\frac{{A}_{T}}{\sqrt{N!}}{\left[{\left({\int }_{0}^{t}{\int }_{0}^{\pi }{b}^{2}\left(\xi ,\tau \right)\phantom{\rule{0.2em}{0ex}}d\xi \phantom{\rule{0.2em}{0ex}}d\tau \right)}^{2}\right]}^{\frac{N}{2}}\\ +{\left(\frac{\sqrt{3T}+\pi }{\sqrt{6}\pi }\right)}^{N}\frac{{A}_{T}}{\sqrt{N!}}{\left[{\left({\int }_{0}^{t}{\int }_{0}^{\pi }{\overline{b}}^{2}\left(\xi ,\tau \right)\phantom{\rule{0.2em}{0ex}}d\xi \phantom{\rule{0.2em}{0ex}}d\tau \right)}^{2}\right]}^{\frac{N}{2}}\end{array}$
and
${M}_{1}={\left(\frac{\sqrt{3T}+\pi }{\sqrt{6}\pi }\right)}^{N}.$

(The sequence ${a}_{N}$ is convergent, then we can write ${a}_{N}\le M$, N.)

It follows from the estimation ([[1], pp.76-77]) that ${lim}_{N\to \mathrm{\infty }}{u}^{\left(N+1\right)}\left(t\right)=u\left(t\right)$.

Then let $N\to \mathrm{\infty }$ for the last equation
$|u\left(t\right)-v\left(t\right)|\le M\parallel \phi -\overline{\phi }\parallel +{M}_{2}\parallel f-\overline{f}\parallel ,$

where ${M}_{2}=M\cdot {M}_{1}$.

If $\parallel f-\overline{f}\parallel \le \epsilon$ and $\parallel \phi -\overline{\phi }\parallel \le \epsilon$, then $|u\left(t\right)-v\left(t\right)|\le \epsilon$. □

## 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
$u\left(x,t\right)=sin2xexp\left(-t\right).$

In this example, we take $f\left(x,t,u\right)=f\left(x,t,u\right)+\epsilon$ and $\phi \left(x\right)=\phi \left(x\right)+\epsilon$ for different ε values.

The comparisons between the analytical solution and the numerical finite difference solution for $\epsilon =0,01$, $\epsilon =0,05$ values when $T=1$ are shown in Figure 1.

The computational results presented are consistent with the theoretical results.

## Declarations

### Acknowledgements

Dedicated to Professor Hari M Srivastava.

## Authors’ Affiliations

(1)
Department of Information Technologies, Kadir Has University
(2)
Department of Mathematics, Kocaeli University

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