三角五次B样条置信算法求解反应扩散方程组
Numerical Solutions of Reaction-Diffusion Equation Systems with Trigonometric Quintic B-spline Collocation Algorithm
AI总结：
本文提出三角五次B样条置信算法用于求解反应扩散方程组，通过时间离散化和空间离散化方法，验证了该算法在不同问题中的准确性与有效性，适用于高阶微分方程的求解。
摘要：
本文研究了利用三角五次B样条有限元置信方法求解反应扩散系统数值解的问题。这些方程出现在多个学科中，用于描述某些物理现象，如图案形成、自催化化学反应和种群动态。Schnakenberg、Gray-Scott和Brusselator模型作为数值示例被考虑。在数值计算中，Crank-Nicolson公式用于时间离散化，所得系统通过泰勒展开线性化。在有限元方法中，构建了解域的均匀划分用于空间离散化。在上述网格上，Dirac-delta函数和三角五次B样条函数分别被选为加权函数和基函数。因此，反应扩散系统转化为代数系统，可以表示为矩阵方程，使得系数为包含一定数量非零元素的块矩阵。该方法在不同问题上进行了测试。为了说明精度，线性问题计算了误差范数，而非线性问题给出了相对误差。由于非线性问题的特性，出现的空间模式由依赖变量的轨迹形成。基多项式的次数允许该方法用于高阶微分方程的求解。该算法即使在时间步长较大时也能产生准确的结果。因此，所提出的三角五次B样条置信方法是一种有效的方法，能够为反应扩散系统的求解提供可接受的结果。
英文摘要：
In this study, the numerical solutions of reaction-diffusion systems are investigated via the trigonometric quintic B-spline nite element collocation method. These equations appear in various disciplines in order to describe certain physical facts, such as pattern formation, autocatalytic chemical reactions and population dynamics. The Schnakenberg, Gray-Scott and Brusselator models are special cases of reaction-diffusion systems considered as numerical examples in this paper. For numerical purposes, Crank-Nicolson formulae are used for the time discretization and the resulting system is linearized by Taylor expansion. In the finite element method, a uniform partition of the solution domain is constructed for the space discretization. Over the mentioned mesh, dirac-delta function and trigonometric quintic B-spline functions are chosen as the weighted function and the bases functions, respectively. Thus, the reaction-diffusion system turns into an algebraic system which can be represented by a matrix equation so that the coeffcients are block matrices containing a certain number of non-zero elements in each row. The method is tested on different problems. To illustrate the accuracy, error norms are calculated in the linear problem whereas the relative error is given in other nonlinear problems. Subject to the character of the nonlinear problems, the occurring spatial patterns are formed by the trajectories of the dependent variables. The degree of the base polynomial allows the method to be used in high-order differential equation solutions. The algorithm produces accurate results even when the time increment is larger. Therefore, the proposed Trigonometric Quintic B-spline Collocation method is an effective method which produces acceptable results for the solutions of reaction-diffusion systems.
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