Finite Difference Methods For Solving A Class Of Multi-term Time Fractional Diffusion-wave Equations With The Spatial Fourth-order Derivative

The fractional differential equation?FDE?is the generalization of the integral order differential equation.Due to the global correlation,it can be well used to describe the specific process of various physical models.Therefore,the theory and numerical methods of the FDE become a hot topic at present.Here we mainly focus on finite difference methods for solving a class of multi-term time fractional diffusion-wave equations with the spatial fourth-order derivative.Firstly,a finite difference method is studied for solving a class of multi-term time fractional sub-diffusion equations with the spatial fourth-order derivative,where the L1 formula is used to approximate the time fractional derivatives and the method of order reduction is applied to deal with the spatial fourth-order derivative.Using the discrete energy method,the unconditional stability and convergence of the scheme are proved and the convergence order is O(?^2-?₂+h²)in maximum norm,wheretand h are the temporal step size and the spatial step size,respectively,a₂ is the maximum order of time fractional derivatives.Finally,numerical examples are carried out to demonstrate the accuracy and efficiency of the present method.Secondly,we discuss a high-order finite difference scheme for solving the above multi-term time fractional sub-diffusion equations with the spatial fourth-order derivative.At first,the method of order reduction is used to reduce the original equations into an equivalent lower-order system.Then an average operator is performed on both sides of the corresponding discretized equations.The L1formula is used to discretize the time fractional derivatives and the compact approximation for the spatial derivative is developed.A high-order finite difference scheme is derived.Using the discrete energy method,the unconditional stability and convergence of the scheme are proved and the convergence order is O(?^2-?₂+h⁴)in L_?-norm.We demonstrate the convergence order and effectiveness of the compact difference scheme with numerical examples.Thirdly,an effective finite difference scheme for solving a class of multi-term time fractional wave equations with the spatial fourth-order derivative is considered.By averaging the equation on two adjacent time levels,the finite difference scheme is established.By using the Taylor expansion with the integral remainder,the Cauchy-Schwarz inequality and the discrete energy method,the unconditional stability and convergence of the scheme are shown and the convergence order is O(?^3-?₂+h²)in maximum norm,where ?₂ is the maximum order of time fractional derivatives in the governing equations.Numerical experiments demonstrate the accuracy and efficiency of the presented scheme.Finally,a high-order accurate finite difference scheme for solving the above multi-term time fractional wave equations with the spatial fourth-order derivative is taken into account.The method of order reduction is firstly used to deal with the original equations,then the time fractional derivatives are discretized by the L1 formula,and the compact approximation of the spatial derivative is investigated.The compact finite difference scheme is presented.By the discrete energy method,the scheme is proved to be unconditionally stable and convergent.And its convergence order is O(?^3-?₂+h⁴)in maximum norm.Finally,numerical examples are calculated to show the accuracy and efficiency of the present method.