High Order Multiple Method For Fractional Ordinary Differential Equation And Numerical Method For Variable Order Fractional Diffusion Equation

Fractional calculus has a history of 300 years, its application is verybroad,including the memory of many kinds of material,characterization of mechan-ics and electricity,earthquake analysis, robots,electric fractal network,fractional sineoscillator,Robot, electronic circuits, electrolysis chemical,fractional capacitance the-ory,electrode electrolyte interface description, fractal theory, especially in the dynamicprocess description of porous structure ,fractional controller design, vibration controlof viscoelastic system and pliable structure objects, fractional biological neurons andprobability theory ,etc. The characteristic of fractional order differential equation iscontaining the non-integer order derivative. It can effectively describe the memory andtransmissibility of many kinds of material, and plays an increasingly important role inengineering, physics, finance, hydrology and other fields.This thesis consists of the four chapters.Introduction gives some concerning fractional calculus to prepare the knowledgeand present basic definitions and properties of fractional calculus.In Chapter 2, starting from the basic fractional ordinary differential equations,weapply a high order approximation of fractional derivative advanced by Lubich to frac-tional differential equation, construct a high numerical difference scheme to solve thefractional differential equation, present error analysis of the algorithms theoretically,and prove the consistency ,convergency and stability.In Chapter 3, considering fractional relaxation equation, we make use of directlythe Grunwald-Letnikov definition to discrete fractional derivative, obtain a numericalmethod of fractional relaxation equation,and give the proof of consistency ,conver-gency and stability.In Chapter 4, we consider more complex fractional nonlinear differential equation,also using the high order approximation presented by Lubich to construct correspond-ing numerical scheme and giving the error analysis of the algorithms. In Chapter 5, noticing that recently in some models the order of fractional deriva-tive can variety with time or space, we discussed variable order fractional diffusionequation based on Riesz fractional derivative. We presented a numerical method tosolve this kind of equation,and prove its consistency,convergency and stability.