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

TL;DR

This paper introduces an iterative BLUES function method for approximating solutions to nonlinear PDEs, demonstrating its effectiveness through several examples and a physical model of interface growth under shear, comparing favorably with existing methods.

Contribution

The paper extends the BLUES method to nonlinear PDEs with initial conditions acting as sources, providing a new approach for analytic approximants and comparing it with established techniques.

Findings

BLUES method offers a flexible alternative to ADM, VIM, and GVIM.

The method successfully approximates solutions for complex nonlinear PDEs.

Fourier analysis confirms the accuracy of the analytic coefficients.

Abstract

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