On series solution for second order semilinear parabolic IBVPs

Abstract This paper presents a method to obtain semi-analytical solutions to second order semilinear IBVPs where the standard Adomian decomposition may fail to yield nontrivial solutions or fail to produce the correct partial solutions. In contrast to the standard Adomian decomposition method, the proposed solution method is distinguished by simultaneous inversion of the linear differential operators using eigenfunctions expansion representations. The proposed method is applied to several examples of initial boundary value problems — a linear Advection–Diffusion problem, a Burgers equation and the deterministic Kardar–Parisi–Zhang (KPZ) equation. It is shown that the dependence of the series solution on the truncation order ( N ), the number of eigenfunctions ( M ) and the diffusion coefficient is rather complex.