Application of Radon Transform for Fast Solution of Boundary Value Problems for Elliptic PDE in Domains With Complicated Geometry [chapter]

Alexandre I. Grebennikov
2010 Matrix Methods: Theory, Algorithms and Applications  
New General Ray (GR) method for solution of the Dirichlet boundary value problem for the class of elliptic Partial Differential Equations (PDE) is proposed. GRmethod consists in application of the Radon transform directly to the PDE and in reduction PDE to assemblage of Ordinary Differential Equations (ODE). The class of the PDE includes the Laplace, Poisson and Helmgoltz equations. GRmethod presents the solution of the Dirichlet boundary value problem for this type of equations by explicit
more » ... ytical formulas that use the direct and inverse Radon transform. Proposed version of GR-method justified theoretically, realized by fast algorithms and MATLAB software, which quality we demonstrate by numerical experiments. Here we extend the proposed approach and construct another version of GRmethod based on application of the direct Radon transform [Radon (1917) ] to the PDE [Sigurdur (1999); Gelfand and Shapiro (1955) ]. This version of GR-method is justified theoretically, realized as algorithms and program package in MATLAB system, illustrated by numerical experiments.
doi:10.1142/9789812836021_0030 fatcat:2yzqk73l65ginihnqhbeelsi3e