Abstract
Linear combinations of fundamental solutions to the parabolic heat equation with source points fixed in time is investigated. The open problem whether these linear combinations generate a dense set in the space of square integrable functions on the lateral boundary of a space-time cylinder, is settled in the negative. Linear independence of the set of fundamental solutions is shown to hold. It is outlined at the end, for a particular example, that such linear combinations constitute a linearly independent and dense set in the space of square integrable functions on the upper top part (where time is fixed) of the boundary of this space-time cylinder.
Highlights
The method of fundamental solutions, being what is termed a meshless numerical method for partial differential equations, has gained popularity in recent years both for direct and inverse problems, see the surveys [7] and [19]
Research have in particular been prolific for stationary problems, where linear independence and denseness of fundamental solutions have been settled, see for example [1]
For time-dependent equations, a general strategy is to employ some transformation in time to reduce to the stationary case [12, Section 5]
Summary
The method of fundamental solutions, being what is termed a meshless numerical method for partial differential equations, has gained popularity in recent years both for direct and inverse problems, see the surveys [7] and [19]. Instead, following on from the stationary case, linear combinations of the fundamental solution of the heat equation are used with source points placed on a fictitious lateral boundary enclosing the lateral part of a given space-time cylinder in which the heat equation is posed. This method has been applied for various other direct and inverse heat problems, see, for example, [4, 17]. We show that distributing source points at a fixed time does not generate a dense set of approximations on the lateral boundary of the space-time solution cylinder in the space of square integrable functions.
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