Numerical approximation of the one-dimensional inverse Cauchy–Stefan problem using a method of fundamental solutions

Abstract

We investigate an application of the method of fundamental solutions (MFS) to the one-dimensional parabolic inverse Cauchy-Stefan problem, where boundary data and the initial condition are to be determined from the Cauchy data prescribed on a given moving interface. In [B.T. Johansson, D. Lesnic, and T. Reeve, A method of fundamental solutions for the one-dimensional inverse Stefan Problem, Appl. Math Model. 35 (2011), pp. 4367-4378], the inverse Stefan problem was considered, where only the boundary data is to be reconstructed on the fixed boundary. We extend the MFS proposed in show that the initial condition can also be simultaneously recovered, i.e. the MFS is appropriate for the inverse Cauchy-Stefan problem. Theoretical properties of the method, as well as numerical investigations, are included, showing that accurate results can be efficiently obtained with small computational cost.

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

“…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 this paper, we extend the approach considered in [5] to one and two-dimensional heat conduction backwards in time.…”

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].…”

An analytical method for solving the two-phase inverse Stefan problem

“…For this purpose we use conditions (5), (8) and (9). Thus, the first of above systems is completed by conditions of the form…”

“…(1) with conditions (2)- (9). All the other functions (ϕ k , θ 1 , ξ) and parameters (a k , λ k , κ, u * , s) are known.…”

“…Therefore for solving problems of that type the numerical methods of different kinds are the most often used [3][4][5][6][7][8][9][10][11][12][13][14][15]. Application of analytical or analytic-numerical methods is rather insignificant.…”