Numerical Methods For Fractional Integro-Differential Equations

The fractional calculus differential is an important branch of mathematical analysis, is a special study of important mathematical properties of arbitrary order integrator and differentiator and its application field. Fractional differential equations can be applied to memory materials, viscoelastic mechanics, seismic analysis and fractional capacitance theory etc. The fractional order differential equations for the derivative is nonlocal, can effectively describe some physical memory and genetic materials. On the other hand, impulsive differential equations can be used to describe the physical model of the original state.Therefore, it is of great significance to their research. At present, although many scholars have done research on the theory, the study of numerical calculation was a lack.This paper describes a detailed study of the numerical solution of fractional ordinary differential equations,two-dimensional Volterra integral equations and impulsive differential equations, as follows:Firstly,We propose in this chapter a new high order scheme to solve ordinary fractional differential equations. The proposed scheme is based on modified block-by-block approach.In our approach, it avoids the coupling of the unknown solutions at each block step with an exception at first three steps.Secondly, based on the classical block-by-block approach, a modified block-by-block numerical scheme is proposed for two-dimensional fractional Volterra integral equations. The advantage of this approach is that it is necessary to derivethe coupling of the unknonwn solutions at each block step with an exception only at u?x₁,y?,u?x₂,y?,u?x,y₁? and u?x,y₂?. Numerical examples show that thenumerical solution approximates to the exact solution well.Lastly, the impulsive differential equation was transformed into an equivalent integral equation, and a higher order numerical scheme for the equivalent integral equation was proposed by using the block-by-block method and the modified block-by-block method,which convergence and stability analysis also have been given. A numerical example was carried out to verify the correctness and the efficiency of the theory analysis.