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These equations may 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 $$[-1,1]$$<\/jats:tex-math>\n \n [<\/mml:mo>\n -<\/mml:mo>\n 1<\/mml:mn>\n ,<\/mml:mo>\n 1<\/mml:mn>\n ]<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> whereas the latter have global support. The global approximation space may contain different affine transformations of the basis, mapping $$[-1,1]$$<\/jats:tex-math>\n \n [<\/mml:mo>\n -<\/mml:mo>\n 1<\/mml:mn>\n ,<\/mml:mo>\n 1<\/mml:mn>\n ]<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> 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<\/jats:italic> independent sparse linear systems of size $$\\mathcal {O}(n)\\times \\mathcal {O}(n)$$<\/jats:tex-math>\n \n O<\/mml:mi>\n (<\/mml:mo>\n n<\/mml:mi>\n )<\/mml:mo>\n \u00d7<\/mml:mo>\n O<\/mml:mi>\n (<\/mml:mo>\n n<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, with $$\\mathcal {O}(n)$$<\/jats:tex-math>\n \n O<\/mml:mi>\n (<\/mml:mo>\n n<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> nonzero entries, where K<\/jats:italic> is the number of different intervals and n<\/jats:italic> is the highest polynomial degree contained in the sum space. This results in an $$\\mathcal {O}(n)$$<\/jats:tex-math>\n \n O<\/mml:mi>\n (<\/mml:mo>\n n<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> complexity solve. Applications to fractional heat and wave equations are considered.<\/jats:p>","DOI":"10.1007\/s10444-024-10164-1","type":"journal-article","created":{"date-parts":[[2024,7,10]],"date-time":"2024-07-10T07:02:31Z","timestamp":1720594951000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["A sparse spectral method for fractional differential equations in one-spatial dimension"],"prefix":"10.1007","volume":"50","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3522-8761","authenticated-orcid":false,"given":"Ioannis P. 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