Boundary Element Methods Applied to the Helmholtz Operator Coupled with a Urysohn Operator

This substantially revised study develops a mathematical and computational framework for Boundary Element Methods (BEM) applied to boundary value problems governed by the Helmholtz operator and coupled with a nonlinear Urysohn integral operator. The differential problem is transformed into a boundary integral equation using the Helmholtz fundamental solution. The nonlinear volume source is represented by a Urysohn-type operator, producing a nonlinear fixed-point formulation suitable for existence, uniqueness, compactness, stability and convergence analysis. Illustrative theorems with proofs, coloured figures, numerical surfaces, convergence tables and MATLAB-oriented algorithms support theoretical interpretation and reproducible computation