A sparse spectral method for fractional differential equations in one-spatial dimension
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866909221278187520 |
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| author | Papadopoulos, Ioannis P. A. Olver, Sheehan |
| author_facet | Papadopoulos, Ioannis P. A. Olver, Sheehan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_08247 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A sparse spectral method for fractional differential equations in one-spatial dimension Papadopoulos, Ioannis P. A. Olver, Sheehan Numerical Analysis 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. |
| title | A sparse spectral method for fractional differential equations in one-spatial dimension |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2210.08247 |