分数阶扩散方程组斑图生成问题的研究，一直是偏微分方程理论中的一项困难工作。近年来由于计算机运算能力的飞速发展，分数阶扩散方程组的分析结果可以得到数值模拟的验证。计算机不仅是数值模拟工具，也为方程中非线性现象和理论分析提供新的思想。本项目以一类普遍的分数阶扩散方程组为研究对象，从理论分析和数值模拟两方面研究斑图生成理论。一方面用微扰分析、多尺度分析、中心流形理论推导方程组的振幅方程，为斑图生成的条件以及几何性质做出理论预测；另一方面用Faedo-Galerkin方法、自适应高分辨率有限体积法建立方程组高效收敛的数值解，检验振幅方程的精确性。由于目前的多尺度分析工具只能做到有限阶的慢时间尺度， 数值模拟能够研究大时间尺度下斑图的稳定性，进一步开拓理论分析所不能达到的研究范围。 本项目的研究将进一步完善和发展分数阶扩散方程组的非线性动力学理论和方法。

The investigations for pattern formation of fractional diffusion equations have been always difficult work in the field of partial differential equations. In recent years due to the rapid development of computing power, the analytical results of fractional diffusion equations can be verified by numerical simulations. Computer is not only a numerical simulation tool, but also provides new ideas for nonlinear phenomena and theoretical analysis in the equation. This project takes a general class of fractional diffusion equations as the research object, and studies the theory of pattern generation from two aspects of theoretical analysis and numerical simulation. On the one hand, perturbation analysis, multi-scale analysis and central manifold theory are used to deduce the equations of the amplitude of the equations for the theoretical prediction of the conditions and geometrical properties of the generated regular patterns; On the other hand, Faedo-Galerkin method and adaptive multi-resolution finite volume method are used to establish the numerical solution of the efficient convergence of the equations and to verify the accuracy of the amplitude equation. Because the current multi-scale analysis tools can only achieve a finite-order slow time scale, numerical simulation can study the stability of the pattern on a large time scale and further open up the range of research that can not be achieved by theoretical analysis. The research of this project will further improve and develop the nonlinear dynamics theory and method of fractional diffusion equations.