A second-order accurate numerical method for a fractional wave equation
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.
scientific article; zbMATH DE number 5119707
scientific article