The Generalized Hermite Spectral Method Combined with Laplace Transform for Solution of the Time Fractional Fokker–Planck Equation

The paper proposes a method to solve the time fractional Fokker–Planck equation using the generalized Hermite spectral method and Laplace transform. By the Laplace transform, the ordinary differential equation about the orthogonal expansion coefficient is transformed into a linear algebraic equation system, with the coefficient matrix being a tridiagonal lower triangular matrix. The orthogonal expansion coefficients of frequency domain components are obtained through this transformation. The approximate solution at any time is then obtained by using the numerical inverse Laplace transform with the Talbot algorithm. Numerical experiments have been carried out to demonstrate the high accuracy and efficiency of the proposed method.