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.
Highlights
It is a challenge, in exact sciences and theoretical physics in particular, to obtain useful analytical approximations to solutions of nonlinear differential equations (DEs)
We extend the approach to nonlinear partial differential equations (PDEs), e.g., in time t and one space coordinate x, which cannot be reduced to ordinary differential equations (ODEs)
In previously reported applications to ODEs, a source term had to be added to the differential equation, corresponding to a physical input external to the problem and inevitably somewhat ad hoc
Summary
In exact sciences and theoretical physics in particular, to obtain useful analytical approximations to solutions of nonlinear differential equations (DEs). We extend the approach to nonlinear partial differential equations (PDEs) and present an application to the physics of interface growth in a soft condensed matter system under shear flow. To situate this development, we briefly recall the history of the BLUES function method. In [8] it was noted that an exponential tail solution may simultaneously solve the nonlinear ODE and a related linear ODE, both with a co-moving Dirac delta source This led to an analytic method that uses the Green function beyond the.
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