Properties of a method of fundamental solutions for the parabolic heat equation

Abstract

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. AbstractWe show that a set of fundamental solutions to the parabolic heat equation, with each element in the set corresponding to a point source located on a given surface with the number of source points being dense on this surface, constitute a linearly independent and dense set with respect to the standard inner product of square integrable functions, both on lateral-and time-boundaries. This result leads naturally to a method of numerically approximating solutions to the parabolic heat equation denoted a method of fundamental solutions (MFS). A discussion around convergence of such an approximation is included.

On non-denseness for a method of fundamental solutions with source points fixed in time for parabolic equations

“…with data ξ not growing faster than e c|x| 2 , for some positive constant c, has a unique solution among solutions satisfying the similar exponential growth bound. References for this result and the above are given in [15,Section 2]. Given initial data to the heat equation, it is possible to choose the boundary data to obtain a designated temperature profile at t = T , that is u(x, T ) is equal to a prescribed function (provided this function matches the outcome of a heat process, see [8,Theorem 4.1] for a class of admissible profiles u(x, T )).…”

“…A key fact to motivate the MFS in [16] is the linear independence and denseness of linear combinations of fundamental solutions of the heat equation. Proofs thereof are collected in [15] together with convergence of an MFS approximation.…”

Efficient numerical solution of one‐phase linear inverse Stefan and Cauchy–Stefan problems in two dimensions: A posteriori error control

“…Then, using the density arguments given in Refs. 27–29, the MFS approximation $$u_{MFS}$$ to the temperature distribution can be expressed as follows: $$\begin{equation} u_{MFS}(\mathbf {x},t) = \sum _{i=1}^{\mathcal {K}}\sum _{j=1}^{\mathcal {M}} c_{j}^{i} F(\mathbf {x},t; \mathbf {z}^{i},\tau _{j}),\;\;(\mathbf {x},t) \in \Omega _T, \end{equation}$$where $$c_{j}^{i}$$ are the unknown coefficients to be determined.…”

Sequential particle filter estimation of a time-dependent heat transfer coefficient in a multidimensional nonlinear inverse heat conduction problem

“…The MFS for the timedependent linear parabolic heat equation was proposed in [13], in which time-dependent fundamental solutions for parabolic PDEs were used. This approach can also be found in [14,15,16].…”