BLUES function method applied to partial differential equations and analytic approximants for interface growth under shear

An iteration sequence based on the BLUES (beyond linear use of equation superposition) function method is presented for calculating analytic approximants to solutions nonlinear partial differential equations. This extends previous work using this ordinary equations with an external source term. Now, initial condition plays role source. The tested three examples: a reaction-diffusion-convection equation, porous medium growth or decay, and Black-Scholes equation. A comparison made other methods: Adomian decomposition (ADM), variational (VIM), Green (GVIM). As physical application, deterministic proposed interface under shear, combining Burgers Kardar-Parisi-Zhang nonlinearities. Thermal noise neglected. model studied Gaussian space-periodic conditions. detailed Fourier analysis performed coefficients are compared those ADM, VIM, GVIM, standard perturbation theory. turns out be worthwhile alternative methods. advantages that it offers ensue from freedom choosing judiciously part, associated function, residual containing part operator at hand.