A method of fundamental solutions for heat and wave propagation from lateral Cauchy data

Abstract

We derive a method of fundamental solutions (MFS) for the numerical solution of an ill-posed lateral Cauchy problem for the hyperbolic wave equation in bounded planar annular domains. The Laguerre transform is applied to reduce the time-dependent lateral Cauchy problem to a sequence of elliptic Cauchy problems with a known set of fundamental solutions termed a fundamental sequence. The solution of the elliptic problems is approximated by linear combinations of the elements in the fundamental sequence. Source points are placed outside of the solution domain, and by collocating on the boundary of the solution domain itself a sequence of linear equations is obtained for finding the coefficients in the MFS approximation. It is shown that the fundamental solutions used constitute a linearly independent and dense set on the boundary of the solution domain with respect to the L2-norm. Tikhonov regularization is applied to get a stable solution to the obtained systems of linear equations in combination with the L-curve rule for selecting the regularization parameter. Numerical results confirm the efficiency and applicability of the proposed strategy for the considered lateral Cauchy problem both in the case of exact and noisy data. Adjusting the coefficients in the sequence of elliptic equations, the similar strategy and results apply also to the parabolic lateral Cauchy problem as verified by an included numerical example. It is also shown that by adjusting the coefficients further the method of Rothe can be applied as an alternative to the Laguerre transformation in time.