A sparse spectral method for fractional differential equations in one-spatial dimension
We develop a sparse spectral method for a class of fractional differential equations, posed on $\mathbb{R}$, in one dimension. These equations can include sqrt-Laplacian, Hilbert, derivative and identity terms. The numerical method utilizes a basis consisting of weighted Chebyshev polynomials of the...
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Main Authors | , |
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Format | Journal Article |
Language | English |
Published |
15.10.2022
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Subjects | |
Online Access | Get full text |
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Summary: | We develop a sparse spectral method for a class of fractional differential
equations, posed on $\mathbb{R}$, in one dimension. These equations can include
sqrt-Laplacian, Hilbert, derivative and identity terms. The numerical method
utilizes a basis consisting of weighted Chebyshev polynomials of the second
kind in conjunction with their Hilbert transforms. The former functions are
supported on $[-1,1]$ whereas the latter have global support. The global
approximation space can contain different affine transformations of the basis,
mapping $[-1,1]$ to other intervals. Remarkably, not only are the induced
linear systems sparse, but the operator decouples across the different affine
transformations. Hence, the solve reduces to solving $K$ independent sparse
linear systems of size $\mathcal{O}(n)\times \mathcal{O}(n)$, with
$\mathcal{O}(n)$ nonzero entries, where $K$ is the number of different
intervals and $n$ is the highest polynomial degree contained in the sum space.
This results in an $\mathcal{O}(n)$ complexity solve. Applications to
fractional heat and wave equations are considered. |
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DOI: | 10.48550/arxiv.2210.08247 |