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Comput."],"published-print":{"date-parts":[[2026,5]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    This paper introduces a high-order efficient algorithm for approximating a nonlinear time-fractional biharmonic equation with an initial singularity. The Caputo fractional derivative is employed, and a second-order scheme is developed to discretize the time derivative on nonuniform time steps, effectively addressing the initial singularity. For the spatial derivative, a high-order non-polynomial parametric quintic spline method is considered. The proposed approach efficiently handles the initial singularity and reduces computational cost through a fast nonuniform time discretization scheme. The resulting method is computationally efficient, with a complexity of approximately\n                    <jats:inline-formula>\n                      <jats:tex-math>$$\\mathcal{O}(MN \\log^2 N)$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    and storage requirements of\n                    <jats:inline-formula>\n                      <jats:tex-math>$$\\mathcal{O}(M \\log^2 N)$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    , where\n                    <jats:inline-formula>\n                      <jats:tex-math>$$N$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    and\n                    <jats:inline-formula>\n                      <jats:tex-math>$$M$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    denote the total number of grid points in the time and spatial directions, respectively. Furthermore, the method is proven to be unconditionally stable and convergent, with an error of order\n                    <jats:inline-formula>\n                      <jats:tex-math>$$\\mathcal{O}\\left(h^{4.5} + N^{-\\min\\{r\\mu, 2\\}}+\\epsilon\\right)$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    , where\n                    <jats:inline-formula>\n                      <jats:tex-math>$$\\mu$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    is the fractional derivative order,\n                    <jats:inline-formula>\n                      <jats:tex-math>$$r$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    is the mesh grading parameter,\n                    <jats:inline-formula>\n                      <jats:tex-math>$$h$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    is the spatial mesh size and\n                    <jats:inline-formula>\n                      <jats:tex-math>$$\\epsilon$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    represents the approximation tolerance introduced by the fast sum-of-exponentials technique. Numerical experiments are presented to validate the theoretical analysis and demonstrate the method\u2019s effectiveness in achieving high accuracy and computational efficiency.\n                  <\/jats:p>","DOI":"10.1007\/s12190-026-02810-2","type":"journal-article","created":{"date-parts":[[2026,4,24]],"date-time":"2026-04-24T14:35:59Z","timestamp":1777041359000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Efficient higher-order approximations for a nonlinear time-fractional biharmonic equation with initial singularity"],"prefix":"10.1007","volume":"72","author":[{"given":"Richa","family":"Singh","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Sunil","family":"Kumar","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2791-6230","authenticated-orcid":false,"given":"Higinio","family":"Ramos","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,4,24]]},"reference":[{"key":"2810_CR1","volume-title":"Fractional Differential Equations: an Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications","author":"I. 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