Finite Difference Methods For The Fourth-order Time Multi-term Fractional Partial Differential Equations With Dirichlet Boundary Conditions

With the intersection of various fields and deepening of theoretical research in these fields such as physics and biology,it has been found that the fourth-order time multi-term fractional partial differential equations have a better simulation effect when describing some processes.Whereas,the analytical solution to this kind of problem can not be available in general,or its solution has a complex form,including special functions,so how to find its effective numerical solution has become one of the hot topics in recent years.This paper mainly focuses on the finite difference methods for solving space fourth-order time multi-term fractional diffusion-wave equations with the Dirichlet boundary conditions.At first,for the fourth-order time multi-term fractional sub-diffusion equations with the first Dirichlet boundary,using the method of order reduction,the original equation is converted into an equivalent lower-order system,then the system is considered at some special points.Next,performing an average operator on the each side of the resultant equality.With some novel techniques to handle the first Dirichlet boundary,the global convergence of the presented difference scheme reaches O(?2+h4),with ? and h the temporal and spatial step size,respectively.Numerical examples are used to further verify the above conclusions.Secondly,a difference scheme for solving the fourth-order mixed time-fractional diffusion-wave equation with the first Dirichlet boundary condition is discussed.Applying the method of order reduction to treat the fourth-order derivative in space,the equations are considered on two adjacent time levels.Performing an average operator on both hand sides of the resultant equation and using the L1 formula to discretize the fractional derivative,the difference scheme is derived.Numerical experiments are implemented to show the computational efficiency.Thirdly,the difference method for solving the fourth-order mixed time-fractional diffusion-wave equation with the second Dirichlet boundary condition is considered.The method of order reduction and the L1 formula are used to establish the difference scheme for such problem.Using Taylor's formula with integral remainder and Gronwall inequality,the energy method is used to prove that the convergence order of the scheme under the discrete maximum norm is O(?min(2-?,3-?}+h2)where ?(0<?<1)and y(1<?<2)are orders of time fractional derivatives.Finally,based on the above research,a compact difference scheme for solving the fourth-order mixed time-fractional diffusion-wave equation with the second Dirichlet boundary condition is built.Firstly the method of order reduction and the averaging of equations on two adjacent time levels are applied.Then performing another average operator on both hand sides of the equation and using Taylor's formula,a compact difference scheme is obtained.The energy method is used to give the stability and convergence analysis of the scheme.The convergence order of the scheme reaches O(?min{2?,3-?}+h4)in discrete maximum norm.