Numerecal Solution For Two Classes Fractional Differential Equations

In the last three decades, fractional differential equations have drawn manyscholars’ attention due to their frequent appearance in various fields. Compared tothe classic integer order derivatives, it is found that fractional derivatives provide anexcellent instrument for the description of memory and hereditary properties ofvarious materials and processes since it’s non-local. But, getting the exact solutionof general fractional differential equations is as difficult as that of integer-orderdifferential equations. Hence people shift their attention to how to explore thenumerical solution. This paper provides numerical methods for a class of timefractional diffusion equations and a class of linear fractional delay differentialequations, which maybe beneficial to other fields.This paper consists of the following parts.In the first chapter, we introduce the background and significance of fractionaldifferential equations, the research status of our subject and the preliminaries thatmay be required. At the end of this chapter, we list main contents of this paper.In the second chapter, we use cubic spline theory to construct a new differencescheme of time fractional diffusion equations then we theoretically discuss the stability and convergence of this scheme by Fourieranalysis method. Finally, an example is used to confirm the effectiveness of themethod.In the third chapter, we discuss the fractional delay differential equationsand give a numerical method. Different from the previous article, we will apply themathematical induction to prove the stability and convergence of the method. Then,a numerical example illustrates validity of the method.At the end of this paper, both the innovative points and weak points of thispaper are demonstrated, which we hope to induce other scholars for further studies.Finally, all the references are listed.