High-order Numerical Schemes And Their Theoretic Analysis Of Fractional Partial Differential Equations

Fractional difusion equations (FDEs) which are derived from the basicrandom walk models are widely used in mathematics, physics, engineering andapplication in the recent year, thus it has attracted a considerable interest.Many complex disordered systems including phenomena of anomalous difu-sion through fractal media can be well described as FDEs. Along with thefurther practical application, analytic and numerical methods for FDEs aredeveloped rapidly.In this paper, we investigate the higher-order numerical schemes for frac-tional order Stokes’ frst problem for a heated generalized second grade fluidand the difusion equation with derivative of fractional order in time, respec-tively. The solvability, stability, convergence and improvement for the schemesare also discussed. The paper is consist of four parts as follows.In chapter one, we give a brief description on the origin of fractional calcu-lus and current situation. The classic defnitions together with their propertiesof fractional derivatives, namely Gru¨nwald-Letnikov, Riemann-Liouville, Ca-puto derivatives are listed. In addition, the overview on numerical methodsfor fractional diferential equations is introduced.In chapter two, we use a simple discrete method on the difusion equationwith derivative of fractional in time to obtain a new higher-order numericalscheme. Similar to Chapter one, we discuss the theoretic analysis for the nu-merical scheme. The linear interpolation method is employed to improve thetemporal accuracy for the scheme.In chapter three, we concern the fractional Stokes’ frst problem (FOSFP)with initial and boundary value condition. The fourth-order compact difer-ence operator is used to approximate spatial derivative of second order, bywhich we derive a new higher-order implicit schemes for (FOSFP). The un-conditional stability and convergence are analysed via Fourier method and thematrix methods. The improvement of the mentioned scheme is also presented.In chapter four, three numerical examples are presented to illustrate theadvantage of the schemes and the validity of the theoretic analysis. Fromthe tables and fgures produced by IFDS, IIFDS, IDS and IIDS, we can conclude that our schemes show higher accuracy than INAS (from literature[33]) and IDAS (from literature [43]).