Lattice Boltzmann Method For Two Classes Of Spatial Fractional Partial Differential Equations

In recent years,the fractional differential model has been widely used in describing abnormal phenomena due to its time memory and global dependence.However,the analytical solution of the fractional differential equation is difficult to obtain,so it is of great significance to develop numerical methods to solve the fractional differential equation.In this paper,the lattice Boltzmann method is used to solve the spatial fractional Burgers equation and Telegraph equation.The main work of this paper is as follows:For one-dimensional and two-dimensional fractional Burgers equations,the lattice Boltzmann models of D1Q3 and D2Q9 with correction terms are established.The integral part of the fractional operator is discretized.The Taylor expansion and multi-scale expansion techniques are used to select the appropriate equilibrium distribution function to restore it to the macroscopic equation.The numerical simulation is carried out through examples.Firstly,the case of degradation to integer order is calculated,and then the fractional order example is calculated.The error comparison diagrams and tables of the numerical solution and the exact solution at diferent times are given.The convergence order in one-dimensional case is calculated by numerical method,and the applicability,effectiveness and accuracy of the model are verified.For one-dimensional and two-dimensional fractional Telegraph equations,the lattice Boltzmann models of D1Q3 and D2Q5 with correction terms are established.The integral part of the fractional operator is discretized,and the Taylor expansion and multi-scale expansion techniques are used to select the appropriate equilibrium state distribution function to restore it to the macroscopic equation,and the central difference scheme of time is used to restore the macroscopic quantity.The numerical simulation is carried out through an example.Firstly,the case of degradation to integer order is calculated,and then the fractional order example is calculated.The error comparison diagram and table of numerical solution and accurate solution at different times arc given.The convergence order in one-dimensional case is calenlated by numerical method,and the applicability,effectiveness and accuracy of the model are verified.Finally,according to the lattice Boltzmann evolution equation and the rules of migration and collision,the difference schemes of the two kinds of equations under lattice Boltzmann method are derived.The first kind of Burgers equation is a iterative method with 4-layer finite difference scheme,and the second kind of Telegraph equation is a iterative method with 6-layer finite difference scheme.It is revealed that this method can can correspond to different iterative schemes by giving different equilibrium distribution functions for different partial differential equations.It has the advantage of adaptability in solving the numerical solution of the equation.