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

Plane wave functions e λ 〈 x , w θ 〉 in R 2 , where λ > 0 , x = ( x , y ) , w θ = ( cos θ , sin θ ) , and 〈 x , w θ 〉 ≔ x cos θ + y sin θ , are used as basis functions to solve boundary value problems of modified Helmholtz equations Δ u ( x ) - λ 2 u ( 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 R 2 . 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.