Simultaneous determination of time and space-dependent coefficients in a parabolic equation

Abstract

This paper investigates a couple of inverse problems of simultaneously determining time and space dependent coefficients in the parabolic heat equation using initial and boundary conditions of the direct problem and overdetermination conditions. The measurement data represented by these overdetermination conditions ensure that these inverse problems have unique solutions. However, the problems are still ill-posed since small errors in the input data cause large errors in the output solution. To overcome this instability we employ the Tikhonov regularization method. The finite-difference method (FDM) is employed as a direct solver which is fed iteratively in a nonlinear minimization routine. Both exact and noisy data are inverted. Numerical results for a few benchmark test examples are presented, discussed and assessed with respect to the FDM mesh size discretisation, the level of noise with which the input data is contaminated, and the chosen regularization parameters.

A Local Meshless Method for Two Classes of Parabolic Inverse Problems

“…So the inverse problem is transformed to a direct problem, then we use the local meshless method described in Section 2 solving the problem (8)- (10).…”

“…The inverse problem of parabolic equations appears naturally in a wide variety of physical and engineering settings; many researchers solved this problem using different methods [1]- [10]. An important class of inverse problem is reconstructing the source term in parabolic equation, and it has been discussed in many papers [11]- [19].…”

Identification of a Time-Dependent Coefficient in Heat Conduction Problem by New Iteration Method

“…Nonlocal boundary specifications like (6) arise from many important applications in heat transfer, thermoelasticity, control theory, life science, etc. For example, in a heat transfer process, if we let represent the temperature distribution, then (1)- (4) and (6) can be regarded as a control problem with source control. A source control parameter ( ) needs to be determined so that a desired thermal energy can be obtained for a portion of the spatial domain.…”

Multiple time-dependent coefficient identification thermal problems with a free boundary

“…To give more physical meaning to the inverse problem, we have in mind a process in which a finite slab is undertaking radioactive decay such that its diffusivity, convection and reaction coefficients are unknown but they depend on time [1,Chap.13], [16]. We finally mention that extensions to cases when the time-dependent heat source is also unknown or when some unknown coefficients may depend on space as well have recently been considered elsewhere, [7,8]. The initial condition is u(x, 0) = ϕ(x), 0 ≤ x ≤ h(0) =: h 0 ,…”

“…For this, we differentiate with respect to y equations (7) and (8), apply (7) at y ∈ {0, 1} and use (9) to obtain ∂w ∂t (y, t) = a(yh(t), t) h 2 (t)…”