SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

This article is part of the series Recent Advances in Operator Equations, Boundary Value Problems, Fixed Point Theory and Applications, and General Inequalities.

Open Access Research

An inverse coefficient problem for a quasilinear parabolic equation with nonlocal boundary conditions

Fatma Kanca1* and Irem Baglan2

Author Affiliations

1 Department of Management Information Systems, Kadir Has University, Istanbul, 34083, Turkey

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

For all author emails, please log on.

Boundary Value Problems 2013, 2013:213  doi:10.1186/1687-2770-2013-213


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/213


Received:7 June 2013
Accepted:27 August 2013
Published:30 September 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 the inverse problem of finding the time-dependent coefficient of heat capacity together with the nonlocal boundary conditions is considered. Under some natural regularity and consistency conditions on the input data, the existence, uniqueness and continuous dependence upon the data of the solution are shown. Some considerations on the numerical solution for this inverse problem are presented with an example.

1 Introduction

Denote the domain D by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M1">View MathML</a>

Consider the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M2">View MathML</a>

(1)

with the initial condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M3">View MathML</a>

(2)

the nonlocal boundary condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M4">View MathML</a>

(3)

and the overdetermination data

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M5">View MathML</a>

(4)

for a quasilinear parabolic equation with the nonlinear source term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M6">View MathML</a>.

The functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M7">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M8">View MathML</a> are given functions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M9">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M10">View MathML</a>, respectively.

The problem of finding the pair <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M11">View MathML</a> in (1)-(4) will be called an inverse problem.

Definition 1 The pair <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M11">View MathML</a> from the class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M13">View MathML</a>, for which conditions (1)-(4) are satisfied and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M14">View MathML</a> on the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M15">View MathML</a>, is called the classical solution of inverse problem (1)-(4).

The problem of identification of a coefficient in a nonlinear parabolic equation is an interesting problem for many scientists [1-3]. Inverse problems for parabolic equations with nonlocal boundary conditions are investigated in [4-6]. This kind of conditions arise from many important applications in heat transfer, life sciences, etc. In [7], also the nature of (3) type boundary conditions is demonstrated.

In [1] the boundary conditions are local, the solution is obtained locally and the authors obtained the solution in Holder classes using iteration method. In [5] the boundary condition is nonlocal but the problem is linear and the existence and the uniqueness of the classical solution is obtained locally using a fixed point theorem. In this paper, the existence and uniqueness of the classical solution is obtained locally using the iteration method.

The paper is organized as follows. In Section 2, the existence and uniqueness of the solution of inverse problem (1)-(4) is proved by using the Fourier method and the iteration method. In Section 3, the continuous dependence upon the data of the inverse problem is shown. In Section 4, the numerical procedure for the solution of the inverse problem is given.

2 Existence and uniqueness of the solution of the inverse problem

Consider the following system of functions on the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M16">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M17">View MathML</a>

The systems of these functions arise in [8] for the solution of a nonlocal boundary value problem in heat conduction. It is easy to verify that the system of functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M18">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M19">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M20">View MathML</a> , is biorthonormal on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M16">View MathML</a>. They are also Riesz bases in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M22">View MathML</a> (see [5,6]).

The main result on the existence and uniqueness of the solution of inverse problem (1)-(4) is presented as follows.

We have the following assumptions on the data of problem (1)-(4):

(A1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M23">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M24">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M25">View MathML</a>;

(A2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M26">View MathML</a>,

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M27">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M28">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M29">View MathML</a>,

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M30">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M31">View MathML</a> ;

(A3) Let the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M8">View MathML</a> be continuous with respect to all arguments in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M10">View MathML</a> and satisfy the following conditions:

(1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M34">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M35">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M36">View MathML</a>,

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M37">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M38">View MathML</a>,

(3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M39">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M40">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M41">View MathML</a>,

(4) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M42">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M43">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M44">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M45">View MathML</a>

By applying the standard procedure of the Fourier method, we obtain the following representation for the solution of (1)-(3) for arbitrary <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M46">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M47">View MathML</a>

(5)

Under conditions (A1)-(A3), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M48">View MathML</a>

(6)

Equations (5) and (6) yield

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M49">View MathML</a>

(7)

Definition 2 Denote the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M50">View MathML</a> of continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M51">View MathML</a> functions satisfying the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M52">View MathML</a> by B. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M53">View MathML</a> be the norm in B. It can be shown that B is the Banach space.

Theorem 3Let assumptions (A1)-(A3) be satisfied. Then inverse problem (1)-(4) has a unique solution for smallT.

Proof An iteration for (5) is defined as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M54">View MathML</a>

(8)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M55">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M56">View MathML</a>

From the conditions of the theorem, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M57">View MathML</a>, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M58">View MathML</a>.

Let us write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M59">View MathML</a> in (8).

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M60">View MathML</a>

Adding and subtracting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M61">View MathML</a> on both sides of the last equation, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M62">View MathML</a>

Applying the Cauchy inequality and the Lipschitz condition to the last equation and taking the maximum of both sides of the last inequality yields the following:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M63">View MathML</a>

Applying Cauchy’s inequality, Hölder’s inequality, Bessel’s inequality, the Lipschitz condition and taking maximum of both sides of the last inequality yields the following:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M64">View MathML</a>

Applying the same estimations, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M65">View MathML</a>

Finally, we have the following inequality:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M66">View MathML</a>

Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M67">View MathML</a>. In the same way, for a general value of N, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M68">View MathML</a>

From <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M69">View MathML</a> we deduce that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M70">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M71">View MathML</a>

An iteration for (7) is defined as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M72">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M73">View MathML</a> ,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M74">View MathML</a>

For convergence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M75">View MathML</a>

Applying Cauchy’s inequality, Hölder’s inequality, Bessel’s inequality, the Lipschitz condition and taking maximum of both sides of the last inequality yields the following:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M76">View MathML</a>

Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M77">View MathML</a>. In the same way, for a general value of N, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M78">View MathML</a>

We deduce that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M79">View MathML</a>.

Now we prove that the iterations <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M80">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M81">View MathML</a> converge in B as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M82">View MathML</a>.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M83">View MathML</a>

Applying Cauchy’s inequality, Hölder’s inequality, the Lipschitz condition and Bessel’s inequality to the last equation, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M84">View MathML</a>

Applying Cauchy’s inequality, Hölder’s inequality, the Lipschitz condition and Bessel’s inequality to the last equation, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M85">View MathML</a>

Applying Cauchy’s inequality, Hölder’s inequality, the Lipschitz condition and Bessel’s inequality to the last equation, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M86">View MathML</a>

For N, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M87">View MathML</a>

(9)

It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M88">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M82">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M90">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M82">View MathML</a>.

Therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M80">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M81">View MathML</a> converge in B.

Now let us show that there exist u and p such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M94">View MathML</a>

In the same way, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M95">View MathML</a>

(10)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M96">View MathML</a>

(11)

Applying Gronwall’s inequality to (10) and using (9) and (11), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M97">View MathML</a>

(12)

Here

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M98">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M82">View MathML</a>, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M100">View MathML</a>. Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M101">View MathML</a>.

For the uniqueness, we assume that problem (1)-(4) has two solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M102">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M103">View MathML</a>. Applying Cauchy’s inequality, Hölder’s inequality, the Lipschitz condition and Bessel’s inequality to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M104">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M105">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M106">View MathML</a>

(13)

Applying Gronwall’s inequality to (13), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M107">View MathML</a>. Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M108">View MathML</a>. □

The theorem is proved.

3 Continuous dependence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M109">View MathML</a> upon the data

Theorem 4Under assumptions (A1)-(A3), the solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M109">View MathML</a>of problem (1)-(4) depends continuously upon the dataφ, g.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M111">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M112">View MathML</a> be two sets of the data, which satisfy assumptions (A1)-(A3). Suppose that there exist positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M113">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M114">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M115">View MathML</a>

Let us denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M116">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M102">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M118">View MathML</a> be the solutions of inverse problem (1)-(4) corresponding to the data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M119">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M112">View MathML</a>, respectively. According to (5),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M121">View MathML</a>

(14)

Now, let us estimate the difference <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M122">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M123">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M124">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M125">View MathML</a> are constants that are determined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M126">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M127">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M128">View MathML</a>. Then we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M129">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M130">View MathML</a>. The inequality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M131">View MathML</a> holds for small T. Finally, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M132">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M133">View MathML</a>.

If we take this estimation in (14)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M134">View MathML</a>

applying Gronwall’s inequality, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M135">View MathML</a>

taking the maximum of the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M136">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M137">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M138">View MathML</a>. Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M139">View MathML</a>. □

4 Numerical procedure for nonlinear problem (1)-(4)

We construct an iteration algorithm for the linearization of problem (1)-(4) as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M140">View MathML</a>

(15)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M141">View MathML</a>

(16)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M142">View MathML</a>

(17)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M143">View MathML</a>

(18)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M144">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M145">View MathML</a>. Then problem (15)-(18) can be written as a linear problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M146">View MathML</a>

(19)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M147">View MathML</a>

(20)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M148">View MathML</a>

(21)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M149">View MathML</a>

(22)

We use the finite difference method to solve (19)-(22) with a predictor-corrector type approach which was explained in [9].

We subdivide the intervals <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M150">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M151">View MathML</a> into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M152">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M153">View MathML</a> subintervals of equal lengths <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M154">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M155">View MathML</a>, respectively. Then we add two lines <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M156">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M157">View MathML</a> to generate the fictitious points needed for dealing with the boundary conditions. We choose the implicit scheme, which is absolutely stable and has second-order accuracy in h and first-order accuracy in τ[10]. The implicit scheme for (1)-(4) is as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M158">View MathML</a>

(23)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M159">View MathML</a>

(24)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M160">View MathML</a>

(25)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M161">View MathML</a>

(26)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M162">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M163">View MathML</a> are the indices for the spatial and time steps, respectively, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M164">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M165">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M166">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M167">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M168">View MathML</a>. At <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M169">View MathML</a> level, adjustment should be made according to the initial condition and the compatibility requirements.

Now, let us construct the predicting-correcting mechanism. First, differentiating equation (1) with respect to x and using (3) and (4), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M170">View MathML</a>

(27)

The finite difference approximation of (27) is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M171">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M172">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M173">View MathML</a>.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M174">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M175">View MathML</a>

and the values of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M176">View MathML</a> allow us to start our computation. We denote the values of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M177">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M178">View MathML</a> at the sth iteration step <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M179">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M180">View MathML</a>, respectively. In numerical computation, since the time step is very small, we can take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M181">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M182">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M183">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M184">View MathML</a>. At each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M185">View MathML</a>th iteration step, we first determine <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M186">View MathML</a> from the formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M187">View MathML</a>

Then from (15)-(18) we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M188">View MathML</a>

(28)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M189">View MathML</a>

(29)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M190">View MathML</a>

(30)

The system of equations (28)-(30) can be solved by the Gauss elimination method and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M191">View MathML</a> is determined. If the difference of values between two iterations reaches the prescribed tolerance, the iteration is stopped, and we accept the corresponding values <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M186">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M191">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M184">View MathML</a>) as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M195">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M196">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M184">View MathML</a>), at the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M198">View MathML</a>th time step, respectively. By virtue of this iteration, we can move from level j to level <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M199">View MathML</a>.

5 Numerical example

Example 1 Consider inverse problem (1)-(4) with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M200">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M201">View MathML</a>

(31)

Let us apply the scheme which was explained in the previous section for the step sizes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M202">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M203">View MathML</a>.

In the case when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M204">View MathML</a>, the comparisons between the analytical solution (31) and the numerical finite difference solution are shown in Figures 1 and 2.

thumbnailFigure 1. The analytical and numerical solutions of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M205">View MathML</a>when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M206">View MathML</a>. The analytical solution is shown with dashed line.

thumbnailFigure 2. The analytical and numerical solutions of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M207">View MathML</a>at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M206">View MathML</a>. The analytical solution is shown with dashed line.

Next, we will illustrate the stability of the numerical solution with respect to the noisy overdetermination data (4) defined by the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M209">View MathML</a>

(32)

where γ is the percentage of noise and θ are random variables generated from uniform distribution in the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M210">View MathML</a>. Figure 3 shows the exact and the numerical solution of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M211">View MathML</a> when the input data (4) is contaminated by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M212">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M213">View MathML</a> and 5% noise.

thumbnailFigure 3. The numerical solutions of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M205">View MathML</a>(a) for1%noisy data, (b) for3%noisy data, (c) for5%noisy data. In Figure 3(a)-(c) the analytical solution is shown with dashed line.

It is clear from these results that this method has shown to produce stable and reasonably accurate results for these examples. Numerical differentiation is used to compute the values of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M215">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M216">View MathML</a> in the formula <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M217">View MathML</a>. It is well known that numerical differentiation is slightly ill-posed and it can cause some numerical difficulties. One can apply the natural cubic spline function technique [11] to get still decent accuracy.

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. IB participated in the sequence alignment and achieved the estimation.

References

  1. Cannon, J, Lin, Y: Determination of parameter <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/213/mathml/M217">View MathML</a> in Holder classes for some semilinear parabolic equations. Inverse Probl.. 4, 595–606 (1988)

  2. Pourgholia, R, Rostamiana, M, Emamjome, M: A numerical method for solving a nonlinear inverse parabolic problem. Inverse Probl. Sci. Eng.. 18, 1151–1164 (2010)

  3. Gatti, S: An existence result for an inverse problem for a quasilinear parabolic equation. Inverse Probl.. 14, 53–65 (1998)

  4. Namazov, G: Definition of the unknown coefficient of a parabolic equation with nonlocal boundary and complementary conditions. Trans. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci.. 19, 113–117 (1999)

  5. Ismailov, M, Kanca, F: An inverse coefficient problem for a parabolic equation in the case of nonlocal boundary and overdetermination conditions. Math. Methods Appl. Sci.. 34, 692–702 (2011)

  6. 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)

  7. Nakhushev, AM: Equations of Mathematical Biology, Vysshaya Shkola, Moscow (1995)

  8. Ionkin, N: Solution of a boundary-value problem in heat conduction with a nonclassical boundary condition. Differ. Equ.. 13, 204–211 (1977)

  9. Cannon, J, Lin, Y, Wang, S: Determination of source parameter in a parabolic equations. Meccanica. 27, 85–94 (1992)

  10. Samarskii, AA: The Theory of Difference Schemes, Dekker, New York (2001)

  11. Atkinson, KE: Elementary Numerical Analysis, Wiley, New York (1985)