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An iteration sequence based on the BLUES (beyond linear use of equationsuperposition) function method is presented for calculating analyticapproximants to solutions of nonlinear partial differential equations. Thisextends previous work using this method for nonlinear ordinary differentialequations with an external source term. Now, the initial condition plays therole of the source. The method is tested on three examples: areaction-diffusion-convection equation, the porous medium equation with growthor decay, and the nonlinear Black-Scholes equation. A comparison is made withthree other methods: the Adomian decomposition method (ADM), the variationaliteration method (VIM), and the variational iteration method with Greenfunction (GVIM). As a physical application, a deterministic differentialequation is proposed for interface growth under shear, combining Burgers andKardar- Parisi-Zhang nonlinearities. Thermal noise is neglected. This model isstudied with Gaussian and space-periodic initial conditions. A detailed Fourieranalysis is performed and the analytic coefficients are compared with those ofADM, VIM, GVIM, and standard perturbation theory. The BLUES method turns out tobe a worthwhile alternative to the other methods. The advantages that it offersensue from the freedom of choosing judiciously the linear part, with associatedGreen function, and the residual containing the nonlinear part of thedifferential operator at hand.

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