Three Kinds Of Difference Schemes For Solving The Fourth-order Multi-term Time-fractional Mixed Diffusion And Wave Equations

In recent years,the applications of fractional partial differential equations in various fields of natural science have attracted extensive attention.With the development of the research,scholars have found that some practical problems cannot be described separately by time-fractional sub-diffusion equations or time-fractional wave equations.Therefore,a mathematical model containing both time-fractional diffusion terms and time-fractional wave terms is a good option,which is usually called the time-fractional mixed diffusion and wave equations.As an extension of single-term time-fractional partial differential equations,the multi-term time-fractional partial differential equations also have attracted much attention.Therefore,designing numerical methods for solving multi-term time-fractional mixed diffusion and wave equations has certain theoretical and application value.This paper mainly focuses on three kinds of difference schemes for the solving the two-dimensional fourth-order multi-term time-fractional mixed diffusion and wave equations.At first,an H2N2 difference scheme for solving the two-dimensional fourth-order multi-term time-fractional mixed diffusion and wave equations is discussed.The Hermite and Newton quadratic interpolation polynomials are applied for time discretization and the method of order reduction together with the central quotient approximation are used in space direction.The difference scheme is developed.It is proved that the convergence order of the scheme is O(τ^3-α₀+h₁²+h₂²)by the energy method,whereα₀ is the maximum one of the orders of time-fractional derivatives.The accuracy and effectiveness of the proposed numerical schemes are verified by two numerical examples.Secondly,a second-order difference scheme for solving the two-dimensional fourth-order multi-term time-fractional mixed diffusion and wave equations is established.Using the method of order reduction,the time multi-term fractional diffusion and wave terms are converted into the time multi-term fractional integral and diffusion terms,respectively.Then,the L2-1_σformula and the piecewise linear interpolation are applied for the approximation in time and the fourth-derivative term in space is also treated using the method of order reduction.The finite difference scheme is established.The stability and convergence of the scheme are rigorously analyzed by the energy method.It is proved that the scheme is unconditionally stable and convergent,with the convergence order of two in both time and space.The accuracy and effectiveness of the numerical scheme are verified by two numerical examples.Finally,the fast H2N2 difference scheme and the fast L2-1_σdifference scheme for solving the two-dimensional fourth-order multi-term time-fractional mixed diffusion and wave equations are discussed based on the efficient sum-of-exponentials（SOE）approximation for kernels in Caputo fractional derivatives.Some numerical examples are calculated to verify the convergence accuracy of the proposed fast schemes.Compared with the direct scheme,it shows the advantage of the fast difference scheme in the speed of calculation.