| Anomalous diffusion phenomenon widely exist in physics, mathematics, biology, and financial and many other fields. Fractional order calculus method was found to can effectively simulate the anomalous diffusion process. In recent years, the fractional order differential equation for describing complex physical process of model caused the wide attention of scholars both at home and abroad, the fractional order calculus in the field of nonlinear application status is also growing. Using fractional derivative as the model can deal with dynamics about the anomalous diffusion. It has been confirmed that fractional diffusion equation model is superior to the standard integer order model, the reason is that the former is a good way to describe the abnormal index or inherent heavy-tailed decay process, and the fractional order model can more accurately describe with memory and physical process of nonlocal. This paper adopted the finite difference method as numerical method, Main work of this article is the application space of fractional order reaction diffusion equation, the numerical simulation of a simple geometric scale and boundary conditions of space fractional order reaction diffusion system, the main contents and conclusions are as follows:The first part: Mainly introducing relevant theories and research anomalous diffusion and fractional order calculus. As the same time, we briefly introduce the motivation and contents of research.The second part: Using the Allen-Cahn as the mathematical model to research the dynamics behavior of the one-dimensional fractional order reaction and diffusion system. We find that the fractional order and the diffusion coefficient will influence the dynamics of system. If the fractional order is changed, then the wave shape of the wave solution will to varying degrees changed. We can find that the greater the fractional order, the faster the propagation velocity, and the propagation velocity will be faster and faster with different amplitude. We also find that the propagation velocity will mainly depend on the occurrence of fractional order or the system coefficient k.The third part: Using the fractional FHN model(the diffusion term has the fractional derivative) to research the dynamics behavior of the single travelling wave and the travelling wave pulse chain in the two-dimensional fractional order system. For the only one disturbance, the system will produce a single travelling wave, and the propagation velocity of it will be influenced by the fractional order, and we find that there is a linear proportional relationship between the fractional order or the arithmetic square root of the diffusion coefficient and the propagation velocity of the single travelling wave. When the fractional order is different, then the propagation of the single travelling wave which would need the minimum value will be different, there is an exponential attenuation relationship of between the minimum value and the fractional order. For the travelling wave pulse chain, which is effected by the periodic disturbances, in the special disturbed period, the period of the travelling wave pulse chain will be locked. The best significant locking belt is 1:1 and 2:1, and the relevant disturbed period is called 1:1 and 2:1 locking belt. We also find that the fractional order will influence the location of the locking belt on the pT axes. locking belt will move along the increasing direction of the pT axes with the decreased fractional order. And when 2.0, 95pα=T=,locking belt is 1:1; if the fractional order is decreased continue, when the fractional order is equal to 1.7, the locking belt is 2:1. In addition, the degree of closeness of the travelling wave pulse chain will be influenced by the locking belt.The fourth part: Using the fractional order FHN model(the diffusion term has the fractional derivative), we research the movement of spiral wave in the two-dimensional fractional order reaction and diffusion system. We find that tip drift will be influenced by the fractional order diffusion of system. When the diffusion coefficient is less than 0.00465, the normal diffusion system will support the wandering spiral wave; its tip is made up of petal units. If the fractional order is decreased, then the spiral wave will drift along a direction. If the diffusion is closed to the normal one, the path of local region is still the petal, but the next petal path unit will have a departure than the previous. The direction of the departure is identical; at last, the whole departure will be produced. If the fractional order is decreased continue, the departure will be intense, the two petal units will be connected by round roll forming path unit, and the direction of the round roll forming path unit is identical. If the fractional order is equal to 2.0, then the tip will move to the boundary by round roll forming path. In the process of the fractional order decreased, the drift velocity will be faster and faster, when the diffusion coefficient is greater than 0.00465, the normal system will support the rigid rotation spiral wave, the spiral wave will drift to the boundary by round roll forming path with the fractional order decreased, and the drift velocity will be greater if the fractional order is decreased. We also research the change of the small region of suppressor variable over time, and simply analyse the process and mechanism of tip drift in the fractional reaction-diffusion system.In the end, we come to a brief conclusion for our research work. |
