Inverse Coefficient Problem of the Parabolic Equation with Periodic Boundary and Integral Overdetermination Conditions

Fatma Kanca

doi:10.1155/2013/659804

Abstract

This paper investigates the inverse problem of finding a time-dependent diffusion coefficient in a parabolic equation with the periodic boundary and integral overdetermination conditions. Under some assumption on the data, the existence, uniqueness, and continuous dependence on the data of the solution are shown by using the generalized Fourier method. The accuracy and computational efficiency of the proposed method are verified with the help of the numerical examples.

Highlights

IntroductionThe parameter identification in a parabolic differential equation from the data of integral overdetermination condition plays an important role in engineering and physics [1,2,3,4,5,6,7]
Denote the domain DT by DT = {(x, t) : 0 < x < 1, 0 < t ≤ T} . (1)Consider the equation ut = a (t) uxx + F (x, t), (2)with the initial condition u (x, 0) = φ (x), 0 ≤ x ≤ 1, (3)the periodic boundary condition u (0, t) = u (1, t), ux (0, t) = ux (1, t), (4)0 ≤ t ≤ T, and the overdetermination condition∫ xu (x, t) dx = E (t), 0 ≤ t ≤ T. (5)The problem of finding a pair {a(t), u(x, t)} in (2)–(5) will be called an inverse problem
The inverse problem regarding the simultaneously identification of the time-dependent thermal diffusivity and the temperature distribution in one-dimensional heat equation with periodic boundary and integral overdetermination conditions has been considered

Summary

Introduction

The parameter identification in a parabolic differential equation from the data of integral overdetermination condition plays an important role in engineering and physics [1,2,3,4,5,6,7] This integral condition in parabolic problems is called heat moments [5]. Boundary value problems for parabolic equations in one or two local classical conditions are replaced by heat moments [8,9,10,11,12,13]. One heat moment is used with periodic boundary condition for the determination of thermal coefficient.

Existence and Uniqueness of the Solution of the Inverse Problem

Numerical Method

Numerical Examples and Discussions

Conclusions