| Fractional differential equation is a generalization of the traditional integral calculus equation theory,but integer differential operators are different from fractional differential operators.Non-locality is a property of fractional differential operators,which can be used to deal with the related problems of materials with memory and genetic characteristics.So how to accurately solve the fractional equation has become a hot topic for many scholars.The complex format used by the macroscopic finite difference method in solving fractional partial differential equations reduces the effectiveness of the algorithm.However,the lattice Boltzmann model of artificial system,which is between statistical mechanics theory and fluid mechanics,is simple enough and suitable for numerical simulation.It provides a great idea for solving fractional order equation in the future,especially for solving fractional order equation with branch time,which provides very great reference value.In this paper,two methods of solving time fractional order equations are given.One is the conventional time fractional order equation.When solving this type of time fractional order equation,we use the treatment method of Caputo type fractional order equation to discretize time fractional order equation.Then,based on the lattice Boltzmann method,the time fractional equation is solved.First,the Chapman-Enskog multi-scale expansion technique and Taylor expansion are used.The lattice Boltzmann method can accurately restore the macro equation we need.And calculate the equilibrium distribution function expression in each direction.Then error analysis is carried out to determine the accuracy of calculation.Finally,numerical simulation will be carried out.In this chapter,two one and one two-dimensional examples will be given to verify the effectiveness of lattice Boltzmann method in solving time fractional equation.The other is the solution of the modified fractional time equation.By including a quadratic fractional time derivative acting on the diffusion operator,a model is proposed to describe how the modified fractional time diffusion equation becomes less abnormal with the passage of time.Compared with the conventional fractional time equation,this equation has certain similarities.For example,both of them are Caputo fractional order equations,so the discretization methods are the same.However,because the fractional order is located in different positions,new mathematical symbols should be introduced to represent macroscopic quantities when dealing with the modified time fractional order equation,which leads to certain differences in the following studies on how to use lattice Boltzmann method to solve the equation. |
