{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/a-sparse-spectral-method-for-fractional","title":"A sparse spectral method for fractional differential equations in one-spatial dimension","arxiv_id":"2210.08247","date":"2022-10-15","proceeding":null,"authors":["Ioannis P. A. Papadopoulos","Sheehan Olver"],"abstract":"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.","url_abs":"https://arxiv.org/abs/2210.08247v3","url_pdf":"https://arxiv.org/pdf/2210.08247v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"a-sparse-spectral-method-for-fractional","repo_url":"https://github.com/ioannisPApapadopoulos/SumSpaces.jl","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}