| Time fractional diffusion or wave equations are the mainstream research topics in the theory of fractional calculus.With the wide application of the theory of fractional calculus in practical problems such as anomalous diffusion,viscoelastic mechanics,fluid mechanics and signal processing,there emerge a large number of reliability-related problems,and they have attracted great attention of scholars at home and abroad.Because fractional differential operators are non-local and they are suitable for describing physical behaviors with memory and genetic properties,they have become one of an important tools to describe such complex problems.At present,although researchers have achieved many breakthrough results in the numerical analysis of time fractional diffusion or wave equations,in some practical problems,using these two types of equations alone cannot be flexible and accurate to describe these behaviors.If the two types of equations are organically combined,that is,the time fractional mixed diffusion and wave equations introduced in this paper,these behaviors can be better simulated.Therefore,finding effective numerical simulation methods for this type of problems has important theoretical significance and application value.This paper firstly introduces the difference method based on L1 approximation for onedimensional problem.Using the discrete energy method,with the help of positive-definite condition,the existence,stability and convergence of the constructed difference scheme are strictly proved,which enriches the existing result.Finally,some numerical examples are used to verify the validity of the theoretical results.Secondly,the difference method based on L2-1σ approximation for the multi-dimensional multi-term time fractional mixed diffusion and wave equation is studied.The original problem is transformed into an equivalent problem by the method of order reduction,and the difference scheme,which has consistent second-order accuracy in time and space,has been established for the obtained problem.Using the discrete energy method,the existence,stability and convergence of the difference scheme are strictly proved.In addition,we also discuss the numerical algorithms of the distributed-order diffusion and wave equation and the multi-term time fractional mixed diffusion and wave equation with an initial singularity.Finally,some numerical examples are provided to verify the validity of the schemes.Subsequently,the fast compact ADI difference method based on the L2-1σ approximation for the two-dimensional problem is discussed.The corresponding fast approximation formula is established by using the sum-of-exponentials to approximate the kernel of the fractional operators,and the compact operator is used to approximate the spatial derivative to obtain the numerical scheme with fourth-order space accuracy.Finally,the ADI method is used to solve the problem to reduce the storage capacity.The discrete energy method is used to strictly prove the existence,stability and convergence of the fast compact ADI difference scheme.Finally,some numerical examples are used to verify the effectiveness of the scheme.Then,the compact difference method based on the L2-1σn approximation for the multidimensional variable-order time fractional diffusion equation is considered.Iterative method is used to solve the super-convergence point tn+σn around each time layer.At the superconvergence point,an approximation formula with second-order accuracy is established for the variable-order Caputo derivative,and a compact operator is used to approximate the spatial derivative to obtain a numerical formula with fourth-order accuracy in the spatial direction.The mathematical induction is used to strictly prove the existence,stability and convergence of the compact difference scheme.Finally,some numerical examples are used to verify the effectiveness of the scheme.Finally,we explore the difference method based on the L2-1σn approximation for twodimensional variable-order time fractional wave equation.A temporal second-order difference scheme is established for the equation using the method of order reduction.The discrete energy method is used to strictly prove the existence,stability and convergence of the difference scheme.Finally,some numerical examples are presented to verify the effectiveness of the scheme. |
