Spectral Methods For Burgers Equation And Fractional Burgers Equation On Full Line

Spectral method is an important numerical method for solving differential equations,which is widely used due to its high accuracy.In this thesis,we study spectral method for the Burgers equation and the fractional Burgers equation on the whole line.Firstly,we develop a spectral method for the Burgers equation on the whole line by the modified Legendre rational function.The stability and convergence of the scheme are proved.Numerical examples are provided to verify the theoretical analysis.The uniform weight function ?(x)? 1 in rational approximations makes it convenient in theoretical analysis and actual computation.Next,we propose a spectral method for the fractional Burgers equation on the whole line by the Hermite function and prove the boundedness of the solution.In order to overcome the difficulty in the computation of the fractional Laplacian operator,we introduce the generalized Hermite polynomial,construct the specific numerical scheme and design the corresponding algorithm.Numerical results show the efficiency of the proposed method.