Direct Solution Of Certain Ill-posed Boundary Value Problems By The Collocation Method
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A. H.-D. Cheng & J. J. S. P. Cabral
Numerical solution of ill-posed boundary value problems normally requires the conversion of the problems into well-posed ones and their solution by an iterative procedure. This paper offers a direct solution procedure to some illposed problems by utilizing the collocation method. Keywords: ill-posed problem, inverse problem, Cauchy problem, radial basis function, collocation method, meshless method. 1 Introduction Direct mathematical solution of engineering problems requires the problems to be set up as well-posed boundary value problems. Engineering problems, however, are not always set up as well-posed problems. Due to the issues of accessibility and cost of measurement, data can be missing on parts of the boundary. In other situations, the location of a boundary is itself not known. To make up for the missing information, a part of the boundary may be overspecified, or potential values can be given in interior points. There are many problems of these types. For example, in groundwater flow, boundary conditions are found along geological features that may not be connected to form a closed boundary. On the other hand, there often exist monitoring wells scattered in the domain of interest to provide the observation of piezometric head. In geophysical prospecting, underground features are inaccessible, but the surface location allows the imposition and measurement of electrical potential and flux such that both the Dirichlet and the Neumann conditions are provided. The same situation exists in the industrial application of
ill-posed problem, inverse problem, Cauchy problem, radial basis function, collocation method, meshless method.