Numerical Methods For Multi-Term Time-Space Fractional Partial Differential Equations

In recent decades,fractional differential equations have been widely used in mathe-matics,engineering and other fields,and gradually become an important tool to solve problems in modern science.As is known that the different fractional orders in the complex phenomena and systems can indicate a potential source of anomalous relaxation,which can be described by multi-term fractional models.Therefore,the multi-term fractional partial differential equations have important applications in simulating anomalous diffusive processes and describing viscoelastic damping materials.However,it is very complex to solve the multi-term fractional partial differential equations.The analytical solutions of majority equations can not be obtained accurately,even if the analytical solutions of some equations can be obtained,they are expressed by special functions which are difficult to be calculated.Thus,scholars are committed to explore the numerical solutions of multi-term fractional partial differential equations.Nowadays,numerical methods for multi-term fractional partial differential equations have become one of the most popular research fields in the world.In this thesis,the numerical methods for four kinds of multi-term fractional partial differential equations with initial boundary conditions are studied and analyzed,and the specific work is as follows:In Chapter 2,a numerical scheme and an alternating direction implicit(ADI)scheme for the one-dimensional and two-dimensional time-space fractional vibration equations are constructed,respectively.Firstly,the considered differential equations are equivalently transformed into their partial integro-differential forms with the classical first-order integrals and the Riemann-Liouville derivative.Secondly,the difference schemes are obtained by discretizing the equations.Then,we prove the proposed schemes are convergent and unconditionally stable.Both of the schemes are convergent with the second-order accuracy in time and space.Finally,two numerical examples are provided to verify the theoretical results.In Chapter 3,this thesis extends the research content of Chapter 2 to the fractional nonlinear vibration equations(FNVEs)with time Caputo derivatives of fractional order ??(1,2).A linearized difference scheme and a linearized ADI scheme are constructed for the one-dimensional and two-dimensional problem,respectively.Firstly,we use the classical central difference formula and a new 3-?-order formula to approximate the second-order derivative and the Caputo derivative in temporal direction,respectively.Meanwhile,the central difference quotient and fractional central difference formula are applied to deal with the spatial discretizations.Further,an ADI scheme is presented for the two-dimensional case.Then,the proposed schemes are proved stable and convergent with the 3-?-order accuracy in time and the second-order accuracy in space.Finally,the effectiveness of the proposed schemes and the theoretical fndings are illustrated by numerical simulation.In Chapter 4,we further study the paitial differential equations with multi-term time Caputo derivatives and spatial Riesz derivatives.An ADI scheme is constructed for the equations based on their partial integro-differential equations and the fast implement of the proposed ADI scheme is discussed by the sum-of-exponentials technique for both Caputo derivatives and Riemann-Liouville integrals.Then,the solvability,convergence and unconditional stability of the proposed scheme are strictly established.At last,two numerical examples are given to demonstrate the computational performances of the fast ADI scheme.In Chapter 5,a superlinear convergence scheme for the multi-term and distribution-order fractional wave equation with initial singularity is proposed.To reduce the smoothness requirement in time and discretize the equations on uniform mesh,the proposed scheme is constructed based on the equivalent partial integro-differential equation.Then,the superlinear convergence and the unconditional stability of the proposed scheme are proved by the energy method.Finally,the correctness of the theory is verified by numerical experiments.