APPLICABILITY AND APPLICATIONS OF THE METHOD OF FUNDAMENTAL SOLUTIONS

摘要

In the present work, we investigate the applicability of the method of fundamental solutions for the solution of boundary value problems of elliptic partial differential equations and elliptic systems. More specifically, we study. whether linear combinations of fundamental solutions can approximate the so- lutions of the boundary value problems under consideration. In our study, the singularities of the fundamental solutions lie on a prescribed pseudo_boundary _ the boundary of a domain which embraces the domain of the problem under consideration. We extend previous density results of Kupradze and Aleksidze, and of Bogomolny, to more general domains and partial differential opera_tors, and with respect to more appropriate norms. Our domains may possess holes and their boundaries are only required to satisfy a rather weak bound_ary requirement, namely the segment condition. Our density results .are with respect to the norms of the spaces C~l(__). Analogous density results are ob_tainable with respect to H_lder norms. We have studied approximation by fundamental solutions of the Laplacian, biharmonic and m_harmonic and modified Helmholtz and poly-Helmholtz operators. In the case of elliptic sys_tems, we obtain analogous density results for the Cauchy-Navier operator as well as for an operator which arises in the linear theory of thermo_elasticity. We also study alternative formulations of the method of fundamental solutions in cases when linear combinations of fundamental solutions of the equations under consideration are not dense in the solution space. Finally, we show that linear combinations of fundamental solutions of operators of order m > 4, with singularities lying on a prescribed pseudo-boundary, are not in general dense in the corresponding solution space.