The High-accuracy Numerical Algorithms For The Time Fractional Diffusion-wave Equation

In this paper,two kinds of high-accuracy numerical differentiation formulas to ap-proximate the Caputo fractional derivative of order??1<?<2?is developed,the truncation errors of two formulas are discussed in detail.Then the numerical algorithms for the time-fractional diffusion wave equations are obtained,and the convergence and stability of the difference scheme are analyzed.Finally numerical examples are carried out to verify the effectiveness of the difference schemes.The first chapter gives the definition and properties of several types of fractional calculus,and the research background and research status of the numerical solution of fractional partial differential equations are given,finally lists the main contents of the article structure.In the second chapter,a new high-accuracy L1-2 difference scheme is constructed for the Caputo fractional derivative of??1<?<2?by using the reduced order method without adding nodes,and the coefficient distribution and properties were studied,and its truncation error was proved to be O(?t^4-?+h²),which is the highest numerical difference scheme so far.As an application of this formula,a semi-discrete and full-discrete high-accuracy difference scheme for the Caputo-type time-fraction diffusion wave equation is constructed,under certain conditions,the formula is strictly proved to be unconditionally stable and convergent,and the convergence is O(?t^4-?+h²).The numerical examples are computed to verify the formula is a high precision,simple,reliable and efficient formula.In the third chapter,another new consistent second-order difference scheme is con-structed for the Caputo fractional derivative of??1<?<2?,and the truncation error estimate is given.Finally,numerical examples are used to verify the convergence order and validity of the differential scheme.The fourth chapter summarizes the paper and prospects the follow-up research.