A method of fundamental solutions for two-dimensional heat conduction

Abstract

International audienceWe investigate an application of the method of fundamental solutions (MFS) to heat conduction in two-dimensional bodies, where the thermal diffusivity is piecewise constant. We extend a recent MFS for one-dimensional heat conduction with the sources placed outside the space domain of interest (Engng. Anal. Boundary Elements 32, 697-703, 2008), to the two-dimensional setting. Theoretical properties of the method, as well as numerical investigations, are included, showing that accurate results can be obtained efficiently with small computational cost

A comparative study on applying the method of fundamental solutions to the backward heat conduction problem

“…Denseness results also hold when we have Neumann boundary conditions, with the appropriate modifications made to the proofs of Theorem 3.1 in [5,6].…”

“…Fortunately, the MFS is versatile and can also be applied to inverse problems. To justify the use of the MFS for this problem, we generalize the denseness result for initial data, given in [6], and show that it holds for any t ≥ 0. In particular, a certain set of linear combinations of fundamental solutions are dense at t = T , see Theorem 3.2.…”

“…This technique has been used to solve stationary heat flow problems governed by elliptic partial differential equations, see [3,4], and recently, in [5,6], an MFS for the time-dependent linear heat equation in one and two-dimensions, respectively, was proposed and investigated. This method has also been extended to heat conduction in one-dimensional layered materials in [7], free surface Stefan problems in [8], inverse Stefan problems [9], and inverse Cauchy-Stefan problems [10].…”

“…In [6] heat conduction was considered in two-dimensions for a direct, well-posed problem, and in the present paper, we investigate an inverse problem; namely the backward heat conduction problem (BHCP). Instead of an initial condition being given at time t = 0, we have an ''upper base'' final condition given at time t = T > 0, and we wish to recover the initial temperature at t = 0.…”

A method of fundamental solutions for the radially symmetric inverse heat conduction problem

“…Recently, however, investigations into the application, accuracy, and the placement of source points have been carried out for time-dependent problems, see, for example, [3,9,11,17,21,24]. The method has been applied to direct problems, as well as to inverse problems, for example, heat conduction in one-dimensional layered materials [12], the free surface Stefan problem [4], heat conduction in two-dimensional domains [15], the inverse Stefan problem [14], the inverse Cauchy-Stefan problem [16] and the backward heat conduction problem [13].…”