Study On Three Numerical Methods For Solving Fractional Partial Differential Equations

Fractional partial differential equations have been widely used in many fields,such as physics,biology,atmosphere and finance,because of the good properties of fractional derivatives such as memory,integrity and heredity.However,most fractional partial differential equations cannot obtain the exact solution.In order to promote the development of scientific research in all directions,it is necessary to simulate their numerical solutions.At present,there are many kinds of numerical methods to solve fractional partial differential equations,but each of them has its own advantages and limitations.Carry on the corresponding analysis to the specific problem,choose an appropriate method to play the strengths and expect to get satisfactory results.In this paper,the matrix method,Fourier spectral method and reproducing kernel method are studied to solve fractional partial differential equations.Firstly,the matrix method is used to solve a class of time-space fractional diffusion equations.Compared with other methods in literature,it is found that this method has the advantages of simple program,small error and fast operation speed for solving time-space fractional diffusion equations.Secondly,the Fourier spectral method is extended to the fractional Gardner problem,and compared with other methods in the literature,it can solve the fractional Gardner equation with high accuracy,low computational cost and little memory,which shows the feasibility of this method in solving the fractional Gardner problem.Finally,the two methods of reproducing kernel(based on Taylor formula and Jacobi polynomials to construct thought of combining the reproducing kernel function respectively and divided two kinds of reproducing kernel method)applied to the two-sided space-fractional partial differential equations,from the point of the numerical results,the two methods of reproducing kernel in solve fractional order partial differential equation,which not only has the validity,it has the advantages of high precision and short running time.In conclusion,the three numerical methods for solving fractional partial differential equations have their own advantages.In this paper,three kinds of fractional partial differential equations are solved by matrix method,Fourier spectral method and reproducing kernel method respectively,and good numerical results are obtained.It provides a mathematical basis for the application.