Numerical Solutions For The Fourth-order Time Fractional Integro-differential Equations

Fractional integro-differential equations are important tools for studying physics,biology,chemistry,economics,electromagnetism,epidemiology,etc.Since the analytical solutions of such equations are difficult to obtain,we use two numerical methods to get the numerical solutions of fourth-order time fractional integro-differential equations in this dissertation.In chapter three,the Sinc-Galerkin method is considered and analyzed for solving the fourth-order time fractional integro-differential equation with weakly singular kernels.Firstly,for the temporal direction,we use L1 scheme to approximate the Caputo derivative and the trapezoidal convolution quadrature rule to discretize the Riemann-Liouville fractional integral term,respectively.While for the spatial direction,we utilize SincGalerkin method to deal with fourth-order partial derivative and obtain the fully discrete scheme.Then we deduce that the scheme has exponentially convergent accuracy in space and(1+δ)order accuracy in time,where δ=min{1+α,2-β}.Finally we present a numerical example to verify the accuracy and effectiveness of the proposed method.In chapter four,the fourth-order time fractional integro-differential equation is solved by mixed finite element method.Firstly,for the temporal direction,the second-order formula and the fractional convolution quadrature formula are used to discretize the partial derivative term and the Riemann-Liouville fractional integral,respectively.Secondly,for the spatial direction,we convert the original equation into a coupled system with the help of auxiliary variable σ,and construct a finite element space to obtain the fully discrete format.Finally,we prove the stability and optimal convergence order of the fully discrete scheme and give an example to prove that the method is effective.