An operator–theoretic analysis of the Adomian–Neumann series for the forced damped wave equation

Abstract

We establish an operator–theoretic framework for the Adomian–Neumann series applied to the forced damped wave equation with homogeneous Dirichlet boundary conditions in this paper. The second–order time derivative being inverted gives rise to a Volterra integral equation in time, by means of showing the Dirichlet Laplacian. Using this formulation, we demonstrate that the Adomian decomposition recursion is equivalent to the Neumann series expansion of the corresponding Volterra operator. As a result, the solution can be represented in terms of iterated time integrals and thus adopts a resolvent-type representation. In appropriate Sobolev spaces, we prove a convergence theorem, where factorial decay of successive terms arises naturally from repeated time integration. Furthermore, the resulting expansion is shown to reconstruct the Taylor series of the solution with respect to time. A Fourier mode example is presented to illustrate the analytical results.