Generalized Iteration Method for First-Kind Integral Equations

An iteration method is described to solve one-dimensional, first-kind integral equations with finite integration limits and difference kernel, K(x − x ′), that decays exponentially. The method relies on deriving via the Wiener–Hopf factorization and solving by suitable iterations in the Fourier complex plane a pair of integral relations, where each iteration accounts for all end point singularities in x of the exact solution. For even and odd kernels, this pair reduces to decoupled, 2nd-kind Fredholm equations, and the iteration yields Neumann series subject to known convergence criteria. This formulation is applied to a classic problem of steady advection-diffusion in the two-dimensional (2D) potential flow of concentrated fluid. The remarkable overlap of recently derived asymptotic expansions for the flux in this case is shown to be intimately related to the analyticity of the kernel Fourier transform.