a b s t r a c tThis article is devoted to the study of high order difference methods for the fractional diffusion-wave equation. The time fractional derivatives are described in the Caputo's sense. A compact difference scheme is presented and analyzed. It is shown that the difference scheme is unconditionally convergent and stable in L 1 -norm. The convergence order is Oðs 3Àa þ h 4 Þ.Two numerical examples are also given to demonstrate the theoretical results.
We propose second-order implicit-explicit (IMEX) time-stepping schemes for nonlinear fractional differential equations with fractional order 0 < β < 1. From the known structure of the non-smooth solution and by introducing corresponding correction terms, we can obtain uniformly second-order accuracy from these schemes. We prove the convergence and linear stability of the proposed schemes. Numerical examples illustrate the flexibility and efficiency of the IMEX schemes and show that they are effective for nonlinear and multi-rate fractional differential systems as well as multi-term fractional differential systems with non-smooth solutions.
We first develop a spectrally accurate Petrov-Galerkin spectral method for fractional delay differential equations (FDDEs). This scheme is developed based on a new spectral theory for fractional Sturm-Liouville problems (FSLPs), which has been recently presented in [M. Zayernouri and G. E. Karniadakis, J. Comput. Phys., 252 (2013), pp. 495-517]. Specifically, we obtain solutions to FDDEs in terms of new nonpolynomial basis functions, called Jacobi polyfractonomials, which are the eigenfunctions of the FSLP of the first kind (FSLP-I). Correspondingly, we employ another space of test functions as the span of polyfractonomial eigenfunctions of the FSLP of the second kind (FSLP-II). We prove the wellposedness of the problem and carry out the corresponding stability and error analysis of the PG spectral method. In contrast to standard (nondelay) fractional differential equations, the delay character of FDDEs might induce solutions, which are either nonsmooth or piecewise smooth. In order to effectively treat such cases, we first develop a discontinuous spectral method (DSM) of Petrov-Galerkin type for FDDEs, where the basis functions do not satisfy the initial conditions. Consequently, we extend the DSM scheme to a discontinuous spectral element method (DSEM) for possible adaptive refinement and long time-integration. In DSM and DSEM schemes, we employ the asymptotic eigensolutions to FSLP-I and FSLP-II, which are of Jacobi polynomial form, both as basis and test functions. Our numerical tests demonstrate spectral convergence for a wide range of FDDE model problems with different benchmark solutions.
In this work, a second-order approximation of the fractional substantial derivative is presented by considering a modified shifted substantial Grünwald formula and its asymptotic expansion. Moreover, the proposed approximation is applied to a fractional diffusion equation with fractional substantial derivative in time.With the use of the fourth-order compact scheme in space, we give a fully discrete Grünwald-Letnikov-formula-based compact difference scheme and prove its stability and convergence by the energy method under smooth assumptions. In addition, the problem with nonsmooth solution is also discussed, and an improved algorithm is proposed to deal with the singularity of the fractional substantial derivative. Numerical examples show the reliability and efficiency of the scheme.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.