The diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles. In mathematics, it is related to Markov processes, such as random walks, and applied in many other fields, such as materials science, information theory, and biophysics. The present papers deals with the approximation of one and two dimensional multi-term time fractional wave diffusion equations. In this work a numerical method which combines Laplace transform with local radial basis functions method is presented. The Laplace transform eliminates the time variable with which the classical time stepping procedure is avoided, because in time stepping methods the accuracy is achieved at a very small step size, and these methods face sever stability restrictions. For spatial discretization the local meshless method is employed to circumvent the issue of shape parameter sensitivity and ill-conditioning of collocation matrices in global meshless methods. The bounds of the stability for the differentiation matrix of our numerical scheme are derived. The method is tested and validated against 1D and 2D wave diffusion equations. The 2D equations are solved over rectangular, circular and complex domains. The computational results insures the stability, accuracy, and efficiency of the method.

中文翻译：

---

基于变换的局部无网格法和正交积分法求解多项时间分数波扩散方程的数值解

扩散方程是抛物线偏微分方程。在物理学中，它描述了布朗运动中许多微粒的宏观行为，这是由于微粒的随机运动和碰撞引起的。在数学上，它与马尔可夫过程（例如随机游走）有关，并应用于许多其他领域，例如材料科学，信息论和生物物理学。本文涉及一维和二维多维时间分数波扩散方程的逼近。在这项工作中，提出了一种将拉普拉斯变换与局部径向基函数方法相结合的数值方法。拉普拉斯（Laplace）变换消除了避免使用经典时间步长程序的时间变量，因为在时间步长方法中，可以以非常小的步长实现精度，这些方法面临严重的稳定性限制。对于空间离散化，采用局部无网格方法来规避全局无网格方法中形状参数敏感性和搭配矩阵不适定的问题。推导了我们数​​值方案的微分矩阵的稳定性的界线。该方法针对一维和二维波扩散方程进行了测试和验证。在矩形，圆形和复杂域上求解二维方程。计算结果确保了该方法的稳定性，准确性和效率。推导了我们数​​值方案的微分矩阵的稳定性的界线。该方法针对一维和二维波扩散方程进行了测试和验证。在矩形，圆形和复杂域上求解二维方程。计算结果确保了该方法的稳定性，准确性和效率。推导了我们数​​值方案的微分矩阵的稳定性的界线。该方法针对一维和二维波扩散方程进行了测试和验证。在矩形，圆形和复杂域上求解二维方程。计算结果确保了该方法的稳定性，准确性和效率。