Purpose

This study aims to apply a numerical meshless method, namely, the boundary knot method (BKM) combined with the meshless analog equation method (MAEM) in space and use a semi-implicit scheme in time for finding a new numerical solution of the advection–reaction–diffusion and reaction–diffusion systems in two-dimensional spaces, which arise in biology.

Design/methodology/approach

First, the BKM is applied to approximate the spatial variables of the studied mathematical models. Then, this study derives fully discrete scheme of the studied models using a semi-implicit scheme based on Crank–Nicolson idea, which gives a linear system of algebraic equations with a non-square matrix per time step that is solved by the singular value decomposition. The proposed approach approximates the solution of a given partial differential equation using particular and homogeneous solutions and without considering the fundamental solutions of the proposed equations.

Findings

This study reports some numerical simulations for showing the ability of the presented technique in solving the studied mathematical models arising in biology. The obtained results by the developed numerical scheme are in good agreement with the results reported in the literature. Besides, a simulation of the proposed model is done on buttery shape domain in two-dimensional space.

Originality/value

This study develops the BKM combined with MAEM for solving the coupled systems of (advection) reaction–diffusion equations in two-dimensional spaces. Besides, it does not need the fundamental solution of the mathematical models studied here, which omits any difficulties.

中文翻译：

---

二维对流反应扩散和Brusselator方程解的边界结法

目的

这项研究旨在应用数值无网格方法，即边界结方法（BKM）与空间无网格模拟方程方法（MAEM）结合，并及时使用半隐式方案来寻找对流的新数值解。二维空间中的反应扩散和反应扩散系统，这是生物学中产生的。

设计/方法/方法

首先，将BKM应用于近似数学模型的空间变量。然后，本研究使用基于Crank–Nicolson思想的半隐式方案，导出了所研究模型的完全离散方案，该方案给出了线性方程组方程组，其每时间步长具有非平方矩阵，并通过奇异值分解来求解。 。所提出的方法使用特定和齐次解来近似给定偏微分方程的解，而不考虑所提出方程的基本解。

发现

这项研究报告了一些数值模拟，以显示所提出的技术解决生物学中出现的数学模型的能力。通过开发的数值方案获得的结果与文献报道的结果非常吻合。此外，在二维空间的黄油形状域上对提出的模型进行了仿真。

创意/价值

这项研究开发了结合MAEM的BKM来求解二维空间中（对流）反应扩散方程的耦合系统。此外，它不需要这里研究的数学模型的基本解决方案，而忽略了任何困难。