A numerical scheme based on polynomials and finite difference method is developed for numerical solutions of two-dimensional linear and nonlinear Sobolev equations. In this approach, finite difference method is applied for the discretization of time derivative whereas space derivatives are approximated by two-dimensional Lucas polynomials. Applying the procedure and utilizing finite Fibonacci sequence, differentiation matrices are derived. With the help of this technique, the differential equations have been transformed to system of algebraic equations, the solution of which compute unknown coefficients in Lucas polynomials. Substituting the unknowns constants in Lucas series, required solution of targeted equation has been obtained. Performance of the method is verified by studying some test problems and computing E₂, E\(_{\infty }\) and E_rms (root mean square) error norms. The obtained accuracy confirms feasibility of the proposed technique.

针对二维线性和非线性Sobolev方程的数值解，提出了一种基于多项式和有限差分法的数值格式。在这种方法中，有限差分法用于时间导数的离散化，而空间导数则由二维Lucas多项式近似。应用该程序并利用有限的斐波那契数列，导出微分矩阵。借助该技术，将微分方程转换为代数方程组，该方程组的解计算出卢卡斯多项式中的未知系数。代入卢卡斯级数的未知数常数，得到了目标方程的解。通过研究一些测试问题并通过计算E验证了该方法的性能。₂，  E \（_ {\ infty} \）和E _rms  （_均方根）误差范数。所获得的准确性证实了所提出技术的可行性。