Finite Volume Method For A Space Distributed-Order Time-Fractional Diffusion Equation

In recent years,distributed-order differential equations have attracted a lot of atten-tion from many scholars,mainly because distributed-order differential equations are more effective for modeling complex processes,which usually follow the mixture of power laws or flexible variations in space.Therefore,the study of the distributed-order equation is necessary.In this thesis,we mainly propose two numerical methods to solve the space distributed-order time-fractional diffusion equation.Firstly,the finite difference method is used to discretize the time-fractional deriva-tive of the space distributed-order time-fractional diffusion equation.For the space distributed-order term,we use the midpoint quadrature method to transform it into mul-tiple space fractional terms.Next,the nodal basis functions are constructed by piecewise-linear polynomials.Then,the finite volume method is used to discretize the transformed multi-term fractional equation to obtain the Crank-Nicolson iterative scheme of the e-quation.After that we prove the stability and convergence of the iterative scheme,and obtain that the convergence order is O(τ2-β+σ2+h2).In the end,we give two numerical examples to prove the effectiveness of the numerical method is shown.Since the spacial convergence order of the above iterative scheme is second order,in order to increase the spacial convergence order,we modify the above numerical method and propose a higher order numerical method to solve the space distributed-order time-fractional equation,so that the spacial convergence order can reach the third order.Compared with the first method,the second method does not change the discrete format of the time fractional derivative.We mainly replace the piecewise-linear polynomi-als with the piecewise-quadratic polynomials to construct the node basis functions.Next,the finite volume method is used to discretize the multi-term fractional equation,and a Crank-Nicolson iterative scheme is also obtained.Then,the iterative scheme is proved to be unconditionally stable and convergent with the accuracy of O(τ2-β+σ2+h3).Finally,we also give an example to prove this higher order numerical method is effective.