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

Plane wave functions eλ〈x,wθ〉eλ〈x,wθ〉 in R2R2, where λ>0λ>0, x=(x,y)x=(x,y), wθ=(cosθ,sinθ)wθ=(cosθ,sinθ), and 〈x,wθ〉≔xcosθ+ysinθ〈x,wθ〉≔xcosθ+ysinθ, 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 ΔΔ is the Laplace operator and ΩΩ a bounded and simply connected domain in R2R2. Approximations of the exact solution of the above problem by plane wave functions are explicitly constructed for the case that ΩΩ 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.