Fractional Differential Equations And Its Applications To Biochemical Reactions

Classical diffusion theory is widely applied in natural science such as physics,biology,chem-istry,has made a great achievement.However,phenomena of anomalous diffusion in fractal,porous,complex systems show that classical diffusion theory is no longer suitable.Experimen-tal results also present that this kind of diffusion tends to anomalous diffusion.The researchers have found that fractional order differential equation is an effective tool to characterize anoma-lous diffusion.Due to the nonlocality of fractional differential operators,the numerical methods and analytical studies(such as the well posedness and regularity of solution)are still in the stage of exploration.In this paper,we mainly study the bimolecular reaction under sub-diffusive cir-cumstance,the numerical method of the fractional sub-diffusion equation in fractal and the well posedness of the solution of fractional differential equations.The details are as follows:In Chapter 1,we introduce some research background on sub-diffusion and fractional calcu-lus.The main works of this paper are also given briefly.In Chapter 2,we introduce the basic definitions and properties of fractional calculus,and some useful special functions such as Mittag-Leffler function and Fox H function.In Chapter 3,we study the bimolecular chemical reaction system A + B(?)C under the sub-diffusion.The microscopic description of reaction diffusion process is given by CTRWs model.Under some appropriate assumptions,we derive the fractional reaction-subdiffusion equations at the macroscopic level from the CTRW model.Based on these equations,the system's statistical properties and stationary profiles are analyzed.It is illustrated that,besides that the integer second order moment under the normal Brown motion meets the Einstein's relation,some fractional order moments are also linearly proportional to time t.In Chapter 4,we study the the FBVP in fractal.The fractional singular Sturm-Liouville problem and the numerical method of a fractional sub-diffusion equation with Dirichlet condition are investigated respectively.We have given the series solution of equation and proved the stability and the convergence of the implicit numerical scheme.Through the robustness analysis,it is also found that the maximum absolute error are more sensitive to the spectral dimension ds than to the anomalous diffusion exponent d?.In Chapter 5,the first part,we study the local and nonlocal Cauchy problem for the following abstract nonlinear fractional differential equation 0CDt?u(t)= A(t,u)u(t)+ Bu(t)),0<?<1,0<t<T.By fixed point theory,the conditions for the existence of the classical solution and the mild solution of the equation are presented respectively.In addition,continuous dependence on initial values of the solution are proved.In Chapter 5,the second part,we study n-dimensional Riesz space fractional convection-diffusion equationBy Rothe method(so-called semi difference method),the existence and uniqueness of the weak solutions of the above equation are proved.