The Generalized Laguerre Matrix Method or Solving Linear Differential-Difference Equations with Variable Coefficients

In this paper, a new and efficient approach based on the generalized Laguerre matrix method for numerical approximation of the linear differential-difference equations (DDEs) with variable coefficients is introduced. Explicit formulae which express the generalized Laguerre expansion coefficients for the moments of the derivatives of any differentiable function in terms of the original expansion coefficients of the function itself are given in the matrix form. In the scheme, by using this approach we reduce solving the linear differential equations to solving a system of linear algebraic equations, thus greatly simplify the problem. In addition, several numerical experiments are given to demonstrate the validity and applicability of the method

Effect of the nodes near boundary points on the stability analysis of sixth-order compact finite difference ADI scheme for the two-dimensional time fractional diffusion-wave equation

In this paper, the aim is to present a high-order compact alternating direction implicit (ADI) scheme for the two-dimensional time fractional diffusion-wave (FDW) equation. The time fractional derivative which has been described in the Caputo’s sense is approximated by a scheme of order O(τ3−α), 1<α<2 and the space derivatives are discretized with a sixth-order compact procedure. The solvability, stability and H1norm of the scheme are proved. Numerical results are provided to verify the accuracy and efficiency of the proposed method of solution. The sixth-order accuracy in the space directions has not been achieved in previously studied schemes. MSC: 35R11, 65M06, 65M12, Keywords: Fractional diffusion-wave equation, Sixth-order compact ADI scheme, Stability, Convergenc

The Jacobi Spectral Method for Nonlinear Magneto- Hydrodynamic Jeffery-Hamel Problem

Abstract In this paper, using the collocation method based on Jacobi polynomials, we obtain the approximate solution of the magneto-hydrodynamic (MHD) Jeffery-Hamel problem. In addition, the sensitive analysis is studied. The method reduces solving the nonlinear ordinary differential equation to solve a system of nonlinear algebraic equations. The comparison of the results with the other numerical methods, show the efficiency and accuracy of the present method