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Solving the Laplace equation across a screen by radial basis functions

Quintero, Lucas E.
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2016
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We consider the problem of solving for a function that satisfies the Laplace equation everywhere with the normal derivative of the unknown function given on a screen centered about the origin. This problem is found in fracture analysis, seismology, and electromagnetic wave scattering. We will use the Green function of the Laplacian to develop a relation for how the unknown function changes across the origin. By use of Fourier transforms, we develop an integral equation from the partial differential equation. Introducing radial basis functions on a square-grid to approximate our solution creates a matrix problem which can be solved numerically. We find that the matrix has a block symmetric-Toeplitz structure which can be solved with quick and efficient solvers, like LQ-decomposition. We use Gaussian radial basis functions to create a spline approximation. We use Chebyshev polynomials to find an exact solution for a certain screen shape, giving values with which we may compare to the approximation. After, we modify the problem to have a radial-symmetric grid we compare the errors produced and investigate the behavior of the solution.
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