Error Analysis of Chebyshev-Legendre Pseudo-spectral Method for a Class of Nonclassical Parabolic Equation

Abstract

Many physical phenomena are modeled by nonclassical parabolic initial boundary value problems which involve a nonclassical term u xxt in the governed equation. Combining with the Crank-Nicolson/leapfrog scheme in time discretization, ChebyshevLegendre pseudo-spectral method is applied to space discretization for numerically solving the nonclassical parabolic equation. The proposed approach is based on Legendre Galerkin formulation while the Chebyshev-Gauss-Lobatto (CGL) nodes are used in the computation. By using the proposed method, the computational complexity is reduced and both accuracy and efficiency are achieved. The stability and convergence are rigorously set up. The convergence rate shows 'spectral accuracy'. Numerical experiments are presented to demonstrate the effectiveness of the method and to confirm the theoretical results.

Applications of Fractional Differentiation Matrices in Solving Caputo Fractional Differential Equations

“…Therefore, more and more researchers turn to numerical methods, for example finite-difference methods [10][11][12] and spectral methods [13][14][15][16][17][18][19][20][21][22][23][24][25][26]. Spectral methods are well-known high-accuracy methods [27][28][29][30][31][32][33][34][35], normally carried out by a Galerkin, Tau, or collocation approach in practical problems. Because fractional derivative operators are nonlocal and spectral methods are global methods, it is very natural to apply spectral methods to solve FDEs.…”

Approximation Properties of Chebyshev Polynomials in the Legendre Norm

“…In [11,12], the approximation result of the Chebyshev interpolation operator without the Chebyshev weighted norm was first given. Some other valuable results related to Chebyshev polynomials can be referred to [2,[13][14][15][16][17][18][19] and references therein.…”