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.

中文翻译：

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BLUES 函数方法应用于偏微分方程和剪切下界面生长的解析逼近

提出了一种基于BLUES（超越线性方程叠加）函数方法的迭代序列，用于计算非线性偏微分方程解的解析近似值。这扩展了以前使用这种方法处理具有外部源项的非线性常微分方程的工作。现在，初始条件起到源的作用。该方法在三个示例上进行了测试：反应-扩散-对流方程、具有增长或衰减的多孔介质方程以及非线性 Black-Scholes 方程。与其他三种方法进行了比较：Adomian 分解法（ADM）、变分迭代法（VIM）和带有格林函数的变分迭代法（GVIM）。作为物理应用，结合 Burgers 和 Kardar-Parisi-Zhang 非线性，提出了确定性微分方程用于剪切下的界面生长。热噪声被忽略。该模型使用高斯和空间周期初始条件进行研究。进行了详细的傅立叶分析，并将解析系数与 ADM、VIM、GVIM 和标准微扰理论的系数进行了比较。事实证明，BLUES 方法是其他方法的一种有价值的替代方法。它提供的优势来自于明智地选择线性部分、相关的格林函数和包含手头微分算子的非线性部分的残差的自由。进行了详细的傅立叶分析，并将解析系数与 ADM、VIM、GVIM 和标准微扰理论的系数进行了比较。事实证明，BLUES 方法是其他方法的一种有价值的替代方法。它提供的优势来自于明智地选择线性部分、相关的格林函数和包含手头微分算子的非线性部分的残差的自由。进行了详细的傅立叶分析，并将解析系数与 ADM、VIM、GVIM 和标准微扰理论的系数进行了比较。事实证明，BLUES 方法是其他方法的一种有价值的替代方法。它提供的优势来自于明智地选择线性部分、相关的格林函数和包含手头微分算子的非线性部分的残差的自由。