1,721 research outputs found
Second-order Stable Finite Difference Schemes for the Time-fractional Diffusion-wave Equation
We propose two stable and one conditionally stable finite difference schemes
of second-order in both time and space for the time-fractional diffusion-wave
equation. In the first scheme, we apply the fractional trapezoidal rule in time
and the central difference in space. We use the generalized Newton-Gregory
formula in time for the second scheme and its modification for the third
scheme. While the second scheme is conditionally stable, the first and the
third schemes are stable. We apply the methodology to the considered equation
with also linear advection-reaction terms and also obtain second-order schemes
both in time and space. Numerical examples with comparisons among the proposed
schemes and the existing ones verify the theoretical analysis and show that the
present schemes exhibit better performances than the known ones
An Efficient Iteration Method for Toeplitz-Plus-Band Triangular Systems Generated from Fractional Ordinary Differential Equation
It is time consuming to numerically solve fractional differential equations. The fractional ordinary differential equations may produce Toeplitz-plus-band triangular systems. An efficient iteration method for Toeplitz-plus-band triangular systems is presented with OMlogM computational complexity and OM memory complexity in this paper, compared with the regular solution with OM2 computational complexity and OM2 memory complexity. M is the discrete grid points. Some methods such as matrix splitting, FFT, compress memory storage and adjustable matrix bandwidth are used in the presented solution. The experimental results show that the presented method compares well with the exact solution and is 4.25 times faster than the regular solution
Rational spectral methods for PDEs involving fractional Laplacian in unbounded domains
Many PDEs involving fractional Laplacian are naturally set in unbounded
domains with underlying solutions decay very slowly, subject to certain power
laws. Their numerical solutions are under-explored. This paper aims at
developing accurate spectral methods using rational basis (or modified mapped
Gegenbauer functions) for such models in unbounded domains. The main building
block of the spectral algorithms is the explicit representations for the
Fourier transform and fractional Laplacian of the rational basis, derived from
some useful integral identites related to modified Bessel functions. With these
at our disposal, we can construct rational spectral-Galerkin and direct
collocation schemes by pre-computing the associated fractional differentiation
matrices. We obtain optimal error estimates of rational spectral approximation
in the fractional Sobolev spaces, and analyze the optimal convergence of the
proposed Galerkin scheme. We also provide ample numerical results to show that
the rational method outperforms the Hermite function approach
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