In this study, a new composite algorithm with the help of the finite difference and the modified cubic trigonometric B-spline differential quadrature method is developed. The developed method was applied to two-dimensional coupled Burgers’ equation with initial and Dirichlet boundary conditions for computational modeling. The established algorithm is better than the traditional differential quadrature algorithm proposed in literature due to more smoothness of cubic trigonometric B-spline functions. In the development of the algorithm, the first step is semidiscretization in time with the forward finite difference method. Furthermore, the obtained system is fully discretized by the modified cubic trigonometric B-spline differential quadrature method. Finally, we obtain coupled Lyapunov systems of linear equations, which are analyzed by the MATLAB solver for the system. Moreover, comparative study of these solutions with the numerical and exact solutions which are appeared in the literature is also discussed. Finally, it is found that there is good suitability between exact solutions and numerical solutions obtained by the developed composite algorithm. The technique can be extended for various multidimensional Burgers’ equations after some modifications.

中文翻译：

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二维耦合伯格斯方程数值解的复合算法

在这项研究中，利用有限差分和修正三次三角B样条微分求积法开发了一种新的复合算法。将开发的方法应用于具有初始和狄利克雷边界条件的二维耦合 Burgers 方程，以进行计算建模。建立的算法优于文献中提出的传统微分正交算法，因为三次三角B样条函数更平滑。在算法的开发中，第一步是使用前向有限差​​分法进行时间上的半离散化。此外，所获得的系统通过改进的三次三角 B 样条微分正交方法完全离散化。最后，我们得到线性方程的耦合李雅普诺夫系统，MATLAB 求解器对系统进行分析。此外，还讨论了这些解与文献中出现的数值解和精确解的比较研究。最后发现，所开发的复合算法得到的精确解与数值解具有良好的适用性。经过一些修改后，该技术可以扩展到各种多维 Burgers 方程。