Abstract
In this paper, we solve the variable-coefficient fractional diffusion-wave equation in a bounded domain by the Legendre spectral method. The time fractional derivative is in the Caputo sense of order gamma in (1,2). We propose two fully discrete schemes based on finite difference in temporal and Legendre spectral approximations in spatial discretization. For the first scheme, we discretize the time fractional derivative directly by the L_{1} approximation coupled with the Crank–Nicolson technique. For the second scheme, we transform the equation into an equivalent form with respect to the Riemann–Liouville fractional integral operator. We give a rigorous analysis of the stability and convergence of the two fully discrete schemes. Numerical examples are carried out to verify the theoretical results.
Highlights
Fractional differential equations (FDEs) have a long history of applications in physics, chemistry, biology, engineering, economics, and many other scientific and engineering fields [1,2,3,4,5,6,7,8,9,10,11]
Chen et al [33] considered a fractional diffusion-wave equation with damping; by using the method of separation of variables the analytical solution is derived, an implicit difference scheme is constructed, and the stability and convergence of the scheme are proved by the energy method
Since its exact solution belongs to H5(Λ), but not to H6(Λ), we can see from Fig. 7 that the convergence rate is between N–4 and N–5, which conforms with our theoretical analysis
Summary
Fractional differential equations (FDEs) have a long history of applications in physics, chemistry, biology, engineering, economics, and many other scientific and engineering fields [1,2,3,4,5,6,7,8,9,10,11]. We consider the following variable-coefficient fractional diffusion-wave equation:. Pskhu [32] constructed a fundamental solution for a fractional diffusion-wave equation with Dzhrbashyan–Nersesyan fractional differential operator with respect to t, with Riemann–Liouville and Caputo derivatives as its particular cases. Chen et al [33] considered a fractional diffusion-wave equation with damping; by using the method of separation of variables the analytical solution is derived, an implicit difference scheme is constructed, and the stability and convergence of the scheme are proved by the energy method. For variable-coefficient case, Wang [37] developed a compact difference scheme for a class of variable-coefficient time fractional convection–diffusion-wave equations with convergence rate O(τ 3–γ + h4).
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