Lattice-free finite difference method for numerical solution of inverse heat conduction problem

Abstract

Inverse heat conduction problem consists of finding an initial temperature distribution from the knowledge of a distribution of the temperature at the present time. Here, we assume that the associated boundary conditions are known. The heat conduction problem backward in time is a typical example of ill-posed problems in the sense that the solution exists only for regular functions of some kind describing the present temperature distribution and also the solution is unstable for the present temperature distribution function. Conventional numerical methods often suffer from instability of the problem itself when high accuracy is intended in the approximation. Our aim is to create a meshless method which is applicable to the ill-posed inverse heat conduction problem. We construct a high order finite difference method in which quadrature points do not need to have a lattice structure. In order to develop our new method we show a tool in using exponential functions in Taylor's expansion. From numerical experiments we confirmed that our method is effective for solving two-dimensional inverse heat conduction problem numerically subject to mixed boundary conditions.

Recovering the source and initial value simultaneously in a parabolic equation

“…For the classical heat conduction equation, it is called a backward heat conduction problem. Many authors have studied this kind of inverse problem; see [11,17,18,32,34,44,49,53,54,56,62]. As for the backward problem of a general parabolic equation, less study has been done.…”

Discretized Tikhonov regularization by reproducing kernel Hilbert space for backward heat conduction problem

“…There are quite a large number of works devoted to stable numerical methods for BHCP. The following is a partial list of articles which contain numerical tests: the method of fundamental solutions [22], boundary element method [10,29], iterative boundary element method [21], inversion methods [18,20], Tikhonov regularization by maximum entropy principle [23], operatorsplitting methods [14], lattice-free finite difference method [12], Fourier regularization [7,8], quasi-reversibility [15,35], quasi-boundary regularization [4], modified methods [16,26], group preserving scheme [17], regularization by semi-implicit finite difference method [30], nonlinear multigrid gradient method [36], approximate and analytic inversion formula [19]. Comparisons of some inverse methods can be found in [5,24].…”