A second-order accurate numerical method for a fractional wave equation
type: Item
instance of: scholarly article
title: A second-order accurate numerical method for a fractional wave equation
zbMATH Open document ID: 1111.65113
author: William Mclean
author: Kassem Mustapha
published in: Numerische Mathematik
publication date: 2007-01-30T00:00:00Z
review text: For \(0<\alpha<1\), let \(J^\alpha\) denote the Riemann-Liouville integral operator of order \(\alpha\) and consider the generalized fractional wave equation \(\partial u / \partial t + J^\alpha A u = f\) for \(t>0\) with initial condition \(u(0) = u_0\). Here \(A\) denotes a selfadjoint and positive semidefinite linear operator with a complete eigensystem in a Hilbert space. In the classical example where \(A\) is the negative Laplacian, this problem reduces to the standard version of the fractional wave equation. After an introductory section, the authors provide a detailed discussion of the regularity of the solution of the problem described above. Based on these findings, a scheme for the numerical solution of the problem is derived and analyzed. The scheme can be considered as a generalization of the well known Crank-Nicolson method for the classical heat equation. However, the properties of the exact solution make it necessary to use a non-uniformly spaced mesh with respect to the time variable. This means that standard methods for fractional differential or integral equations like convolution quadrature methods cannot be used. The key result is that the numerical method is unconditionally stable and that, under appropriate smoothness assumptions on the given data and assuming sufficient regularity of the spatial domain, the convergence is of order \(O(h^2 + k^2)\) where \(h\) and \(k\) are the parameters describing the time and space meshes, respectively.
reviewed by: Kai Diethelm
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 65M15
Mathematics Subject Classification ID: 26A33
Mathematics Subject Classification ID: 45K05
Mathematics Subject Classification ID: 65M12
Mathematics Subject Classification ID: 65M60
zbMATH DE Number: 5119707
zbMATH Keywords: Crank-Nicolson method
zbMATH Keywords: stability
zbMATH Keywords: convergence
MaRDI profile type: Publication
full work available at URL: s00211-006-0045-y
OpenAlex ID: W1978354679
cites work: Piecewise Polynomial Collocation Methods for Linear Volterra Integro-Differential Equations with Weakly Singular Kernels
cites work: Product Integration-Collocation Methods for Noncompact Integral Operator Equations
cites work: Convolution quadrature time discretization of fractional diffusion-wave equations
cites work: Spline Collocation Methods for a Class of Hyperbolic Partial Integro-Differential Equations
cites work: Integrodifferential equation which interpolates the heat equation and the wave equation
cites work: Multiple (multiindex) Mittag-Leffler functions and relations to generalized fractional calculus
cites work: On the numerical inversion of the Laplace transform of certain holomorphic mappings
cites work: A Spectral Order Method for Inverting Sectorial Laplace Transforms
cites work: A Difference Scheme for a Nonlinear Partial Integrodifferential Equation
cites work: Discretized Fractional Calculus
cites work: Convolution quadrature and discretized operational calculus. I
cites work: Nonsmooth data error estimates for approximations of an evolution equation with a positive-type memory term
cites work: Numerical solution of an evolution equation with a positive-type memory term
cites work: Time discretization of an evolution equation via Laplace transforms
cites work: Discretization with variable time steps of an evolution equation with a positive-type memory term
cites work: The random walk's guide to anomalous diffusion: A fractional dynamics approach
cites work: AnH1-Galerkin Mixed Finite Element Method for an Evolution Equation with a Positive-Type Memory Term
cites work: Collocation Methods for Weakly Singular Second-kind Volterra Integral Equations with Non-smooth Solution
cites work: Fractional diffusion and wave equations
cites work: A Numerical Method for a Partial Integro-Differential Equation
cites work: Q3313404
cites work: Orthogonal Spline Collocation Methods for Some Partial Integrodifferential Equations
DOI: 10.1007/S00211-006-0045-Y
description: scientific article; zbMATH DE number 5119707