We propose a polynomial-based numerical scheme for solving some important nonlinear partial differential equations (PDEs). In the proposed technique, the temporal part is discretized by finite difference method together with θ-weighted scheme. Then, for the approximation of spatial part of unknown function and its spatial derivatives, we use a mixed approach based on Lucas and Fibonacci polynomials. With the help of these approximations, we transform the nonlinear partial differential equation to a system of algebraic equations, which can be easily handled. We test the performance of the method on the generalized Burgers–Huxley and Burgers–Fisher equations, and one- and two-dimensional coupled Burgers equations. To compare the efficiency and accuracy of the proposed scheme, we computed \(L_{\infty }\), \(L_{2}\), and root mean square (RMS) error norms. Computations validate that the proposed method produces better results than other numerical methods. We also discussed and confirmed the stability of the technique.

我们提出了一种基于多项式的数值方案，用于求解一些重要的非线性偏微分方程（PDE）。在提出的技术中，通过有限差分法和θ加权方案离散时间部分。然后，对于未知函数的空间部分及其空间导数的近似，我们使用基于Lucas和Fibonacci多项式的混合方法。在这些近似的帮助下，我们将非线性偏微分方程转换为可以轻松处理的代数方程组。我们在广义Burgers–Huxley和Burgers–Fisher方程以及一维和二维耦合Burgers方程上测试了该方法的性能。为了比较所提方案的效率和准确性，我们计算了\（L _ {\ infty} \），\（L_ {2} \）和均方根（RMS）误差范数。计算结果表明，所提出的方法比其他数值方法能产生更好的结果。我们还讨论并确认了该技术的稳定性。