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

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.