On solving boundary value problems of modified Helmholtz equations by plane wave functions

Authors:
Xin Li
Affiliations:
Department of Mathematical Sciences, University of Nevada, Las Vegas, Las Vegas, NV
Venue:
Journal of Computational and Applied Mathematics - Special issue: The international symposium on computing and information (ISCI2004)
Year:
2006

Citing 2
Cited 2

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Quasi-reversibility and truncation methods to solve a Cauchy problem for the modified Helmholtz equation

Mathematics and Computers in Simulation
Numerical implementation of the EDEM for modified Helmholtz BVPs on annular domains

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Abstract

Plane wave functions eλ〈x, wθ〉 in R2, where λ 0, x = (x, y), wθ = (cos θ, sin θ), and 〈x, wθ〉 := x cos θ + y sin θ, are used as basis functions to solve boundary value problems of modified Helmholtz equations Δu(x) - λ2u(x) = 0, x ∈ Ω, u(x)= h(x) x ∈ ∂Ω, where Δ is the Laplace operator and Ω a bounded and simply connected domain in R2. Approximations of the exact solution of the above problem by plane wave functions are explicitly constructed for the case that Ω is a disc, and the order of approximations is derived. A computational algorithm by collocation methods based on a simple singular decomposition of circular matrices is proposed, and numerical examples are shown to demonstrate the efficiency of the methods.