This article describes a numerical method based on the dual reciprocity boundary elements method (DRBEM) for solving some well-known nonlinear parabolic partial differential equations (PDEs). The equations include the classic and generalized Fisher's equations, Allen-Cahn equation, Newell-Whithead equation, Fitz-HughNagumo equation and generalized Fitz-HughNagumo equation with time-dependent coefficients. The concept of the dual reciprocity is used to convert the domain integral to the boundary that leads to an integration free method. We employ the time stepping scheme to approximate the time derivative, and the linear radial basis functions (RBFs) are used as approximate functions in presented method. The nonlinear terms are treated iteratively within each time step. The developed formulation is verified in some numerical test examples. The results of numerical experiments are compared with analytical solution to confirm the accuracy and efficiency of the presented scheme.

翻译：本文阐述了一种基于双重互易边界元方法（DRBEM）的数值方法，用于求解若干著名的非线性抛物型偏微分方程（PDEs）。这些方程包括经典及广义Fisher方程、Allen-Cahn方程、Newell-Whitehead方程、Fitz-Hugh-Nagumo方程以及具有时间依赖系数的广义Fitz-Hugh-Nagumo方程。双重互易概念被用于将域积分转化为边界积分，从而实现无积分方法。我们采用时间步进格式对时间导数进行近似，并在所提方法中使用线性径向基函数（RBFs）作为近似函数。非线性项在每个时间步内通过迭代方式处理。所建立的公式通过若干数值测试算例进行了验证。数值实验的结果与解析解进行了比较，以确认所提方案的准确性和效率。