Purpose

The purpose of this study is to construct quartic trigonometric tension (QTT) B-spline collocation algorithms for the numerical solutions of the Coupled Burgers’ equation.

Design/methodology/approach

The finite elements method (FEM) is a numerical method for obtaining an approximate solution of partial differential equations (PDEs). The development of high-speed computers enables to development FEM to solve PDEs on both complex domain and complicated boundary conditions. It also provides higher-order approximation which consists of a vector of coefficients multiplied by a set of basis functions. FEM with the B-splines is efficient due both to giving a smaller system of algebraic equations that has lower computational complexity and providing higher-order continuous approximation depending on using the B-splines of high degree.

Findings

The result of the test problems indicates the reliability of the method to get solutions to the CBE. QTT B-spline collocation approach has convergence order 3 in space and order 1 in time. So that nonpolynomial splines provide smooth solutions during the run of the program.

Originality/value

There are few numerical methods build-up using the trigonometric tension spline for solving differential equations. The tension B-spline collocation method is used for finding the solution of Coupled Burgers’ equation.

中文翻译：

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非线性偏微分方程组的四次三角张力b样条算法

目的

本研究的目的是为耦合伯格斯方程的数值解构建四次三角张力 (QTT) B 样条搭配算法。

设计/方法/方法

有限元法 (FEM) 是一种用于获得偏微分方程 (PDE) 近似解的数值方法。高速计算机的发展使得开发 FEM 能够解决复杂域和复杂边界条件下的偏微分方程。它还提供了由系数向量乘以一组基函数组成的高阶近似。具有 B 样条的 FEM 是有效的，因为它提供了具有较低计算复杂度的较小代数方程组，并且根据使用高度 B 样条提供了更高阶的连续近似。

发现

测试问题的结果表明了获得 CBE 解决方案的方法的可靠性。QTT B样条搭配方法在空间上收敛为3阶，在时间上收敛为1阶。以便非多项式样条在程序运行期间提供平滑的解决方案。

原创性/价值

使用三角张力样条求解微分方程的数值方法很少。张力B样条搭配法用于寻找耦合伯格斯方程的解。