{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T10:32:00Z","timestamp":1775817120424,"version":"3.50.1"},"reference-count":39,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2025,8,14]],"date-time":"2025-08-14T00:00:00Z","timestamp":1755129600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,8,14]],"date-time":"2025-08-14T00:00:00Z","timestamp":1755129600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100001333","name":"Rhodes University","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100001333","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Nonlinear Sci"],"published-print":{"date-parts":[[2025,10]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>Singularly perturbed differential equations pose significant challenges for many numerical and semi-analytical solution methods in classical calculus. The complexity of these equations increases when dealing with fractional differential equations, which exhibit unique properties, including the nonlocal nature of the arbitrary non-integer order differential operators. This study presents an adaptive multidomain Chebyshev pseudospectral method to effectively approximate the solutions of singularly perturbed fractional differential equations. In each subdomain, the fractional differential operator in the Caputo sense accounts for both local derivative effects and memory by dividing the discrete fractional operator into two distinct components, one representing local behavior and the other capturing memory effects. The adaptivity of the method is controlled by ensuring that the maximum residual error in each domain remains below a predetermined tolerance value. If the maximum error exceeds the set tolerance, the size of the subinterval is reduced by a specified ratio, and the solution is recomputed within that interval. To assess the performance of the adaptive method, a comparative study with the multidomain pseudospectral method with uniform step length is used. The accuracy of the adaptive multidomain Chebyshev pseudospectral method is validated through a series of numerical tests on various fractional differential equations, thus demonstrating the effectiveness of the approach. The adaptive multidomain pseudospectral method offers a robust solution for handling the challenges posed by singularly perturbed differential equations common in fields such as wave propagation, astrophysics, and quantum mechanics.\n<\/jats:p>","DOI":"10.1007\/s00332-025-10199-8","type":"journal-article","created":{"date-parts":[[2025,8,14]],"date-time":"2025-08-14T12:07:53Z","timestamp":1755173273000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Adaptive Multidomain Numerical Solution for Singularly Perturbed Fractional Differential Equation: Chebyshev Pseudospectral Method"],"prefix":"10.1007","volume":"35","author":[{"given":"Yusuf Olatunji","family":"Tijani","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Olumuyiwa","family":"Otegbeye","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shina Daniel","family":"Oloniiju","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,8,14]]},"reference":[{"key":"10199_CR1","doi-asserted-by":"publisher","first-page":"63","DOI":"10.1007\/s40314-023-02287-w","volume":"42","author":"MM Alsuyuti","year":"2023","unstructured":"Alsuyuti, M.M., Doha, E.H., Bayoumi, B.I., Ezz-Eldien, S.S.: Robust spectral treatment for time-fractional delay partial differential equations. Comput. Appl. Math. 42, 63\u201373 (2023)","journal-title":"Comput. Appl. Math."},{"key":"10199_CR2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-30726-6","volume-title":"Spectral Methods: Fundamentals in Single Domains","author":"C Canuto","year":"2006","unstructured":"Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods: Fundamentals in Single Domains. Springer, Berlin (2006)"},{"key":"10199_CR3","doi-asserted-by":"publisher","first-page":"529","DOI":"10.1111\/j.1365-246X.1967.tb02303.x","volume":"13","author":"M Caputo","year":"1967","unstructured":"Caputo, M.: Linear models of dissipation whose q is almost frequency independent-II. Geophys. J. R. Astr. Soc 13, 529\u2013539 (1967)","journal-title":"Geophys. J. R. Astr. Soc"},{"key":"10199_CR4","doi-asserted-by":"crossref","unstructured":"Diethelm, K., Freed, A.D.: On the solution of nonlinear fractional-order differential equations used in the modeling of viscoplasticity. In: Scientific Computing in Chemical Engineering II, pp. 217\u2013224 (1999)","DOI":"10.1007\/978-3-642-60185-9_24"},{"issue":"12","key":"10199_CR5","doi-asserted-by":"publisher","first-page":"5662","DOI":"10.1016\/j.apm.2011.05.011","volume":"35","author":"EH Doha","year":"2011","unstructured":"Doha, E.H., Bhrawy, A.H., Ezz-Eldien, S.S.: Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations. Appl. Math. Model. 35(12), 5662\u20135672 (2011)","journal-title":"Appl. Math. Model."},{"key":"10199_CR6","doi-asserted-by":"publisher","DOI":"10.1016\/j.padiff.2024.100717","volume":"10","author":"NA Endrie","year":"2024","unstructured":"Endrie, N.A., Duressa, G.F.: Numerical method for second order singularly perturbed delay differential equations with fractional order in time via fitted computational method. Partial Differ. Equ. Appl. Math. 10, 100717 (2024)","journal-title":"Partial Differ. Equ. Appl. Math."},{"key":"10199_CR7","volume-title":"Partial Differential Equations","author":"LC Evans","year":"1998","unstructured":"Evans, L.C.: Partial Differential Equations. American Mathematical Society, Providence (1998)"},{"key":"10199_CR8","first-page":"63","volume":"321","author":"SS Ezz-Eldien","year":"2018","unstructured":"Ezz-Eldien, S.S.: On solving systems of multi-pantograph equations via spectral tau method. Appl. Math. Comput. 321, 63\u201373 (2018)","journal-title":"Appl. Math. Comput."},{"key":"10199_CR9","doi-asserted-by":"publisher","first-page":"57","DOI":"10.1007\/s11075-018-0535-x","volume":"81","author":"SS Ezz-Eldien","year":"2023","unstructured":"Ezz-Eldien, S.S., Doha, E.H.: Fast and precise sepctral method for solving pantograph type Volterra integro-differential equations. Numer. Alogorithms 81, 57\u201377 (2023)","journal-title":"Numer. Alogorithms"},{"key":"10199_CR10","doi-asserted-by":"publisher","first-page":"84","DOI":"10.1023\/A:1022318402393","volume":"40","author":"W Gander","year":"2000","unstructured":"Gander, W., Gautschi, W.: Adaptive quadrature revisited. BIT Numer. Math. 40, 84\u2013101 (2000)","journal-title":"BIT Numer. Math."},{"issue":"2","key":"10199_CR11","doi-asserted-by":"publisher","first-page":"16","DOI":"10.3390\/math6020016","volume":"6","author":"R Garrappa","year":"2018","unstructured":"Garrappa, R.: Numerical solution of fractional differential equations: a survey and a software tutorial. Mathematics 6(2), 16 (2018)","journal-title":"Mathematics"},{"issue":"9","key":"10199_CR12","first-page":"7504","volume":"4","author":"S Gottlieb","year":"2009","unstructured":"Gottlieb, S., Gottlieb, D.: Spectral methods. Comput. Methods Appl. Mech. Eng. 4(9), 7504 (2009)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"10199_CR13","first-page":"1","volume":"257","author":"R Hans-Gorg","year":"1994","unstructured":"Hans-Gorg, R., Martin, S., Lutz, T.: Robust numerical methods for singularly perturbed differential equations: convection-diffusion-reaction and flow problems. Springer Ser. Comput. Math. 257, 1\u2013604 (1994)","journal-title":"Springer Ser. Comput. Math."},{"key":"10199_CR14","volume":"390","author":"IT Huseynov","year":"2021","unstructured":"Huseynov, I.T., Ahmadova, A., Fernandez, A., Mahmudov, N.I.: Explicit analytical solutions of incommensurate fractional differential equation systems. Appl. Math. Comput. 390, 125590 (2021)","journal-title":"Appl. Math. Comput."},{"key":"10199_CR15","first-page":"457","volume":"130","author":"MK Kadalbajoo","year":"2002","unstructured":"Kadalbajoo, M.K., Patidar, K.C.: A survey of numerical techniques for solving singularly perturbed ordinary differential equations. Appl. Math. Comput. 130, 457\u2013510 (2002)","journal-title":"Appl. Math. Comput."},{"issue":"3","key":"10199_CR16","doi-asserted-by":"publisher","first-page":"383","DOI":"10.2478\/s13540-012-0028-x","volume":"15","author":"C Li","year":"2012","unstructured":"Li, C., Zeng, F., Liu, F.: Spectral approximation to the fractional integral and derivative. Fract. Calc. Appl. Anal. 15(3), 383\u2013406 (2012)","journal-title":"Fract. Calc. Appl. Anal."},{"key":"10199_CR17","volume-title":"An Introduction to Nonlinear Partial Differential Equations","author":"JD Logan","year":"1994","unstructured":"Logan, J.D.: An Introduction to Nonlinear Partial Differential Equations. Wiley, New York (1994)"},{"issue":"4","key":"10199_CR18","first-page":"936","volume":"10","author":"S Motsa","year":"2012","unstructured":"Motsa, S.: A new piecewise-quasilinearization method for solving chaotic systems of initial value problems. Cent. Eur. J. Phys. 10(4), 936\u2013946 (2012)","journal-title":"Cent. Eur. J. Phys."},{"key":"10199_CR19","doi-asserted-by":"publisher","first-page":"265","DOI":"10.1007\/s11071-012-0712-8","volume":"72","author":"SS Motsa","year":"2013","unstructured":"Motsa, S.S., Dlamini, P., Khumalo, M.: A new multistage spectral relaxation method for solving chaotic initial value systems. Nonlinear Dyn. 72, 265\u2013283 (2013)","journal-title":"Nonlinear Dyn."},{"key":"10199_CR20","volume-title":"Fractional Calculus for Hydrology, Soil Science and Geomechanics","author":"S Ninghu","year":"2020","unstructured":"Ninghu, S.: Fractional Calculus for Hydrology, Soil Science and Geomechanics. Taylor & Francis, New York (2020)"},{"key":"10199_CR21","unstructured":"Nnakenyi, C.A., Weideman, J.A.C., Hale, N.: Spectral methods for fractional differential equations. AIMS Master Thesis (2015)"},{"key":"10199_CR22","doi-asserted-by":"publisher","first-page":"3","DOI":"10.5098\/hmt.13.19","volume":"13","author":"SD Oloniiju","year":"2019","unstructured":"Oloniiju, S.D., Goqo, S.P., Sibanda, P.: A Chebyshev spectral method for heat and mass transfer in MHD nanofluid flow with space fractional constitutive model. Front. Heat and Mass Transf. 13, 3\u201314 (2019)","journal-title":"Front. Heat and Mass Transf."},{"issue":"5","key":"10199_CR23","doi-asserted-by":"publisher","first-page":"959","DOI":"10.11650\/tjm\/210501","volume":"25","author":"SD Oloniiju","year":"2021","unstructured":"Oloniiju, S.D., Goqo, S.P., Sibanda, P.: A pseudo-spectral method for time distributed order two-sided space fractional differential equations. Taiwan. J. Math. 25(5), 959\u2013979 (2021)","journal-title":"Taiwan. J. Math."},{"issue":"3","key":"10199_CR24","first-page":"950","volume":"4","author":"SD Oloniiju","year":"2024","unstructured":"Oloniiju, S.D., Mukwevho, N., Tijani, Y.O., Otegbeye, O.: Chebyshev pseudospectral method for fractional differential equations in non-overlapping partitioned domains. Appl. Math. 4(3), 950\u2013974 (2024)","journal-title":"Appl. Math."},{"key":"10199_CR25","doi-asserted-by":"crossref","unstructured":"Otegbeye, O.: On paired decoupled quasi-linearization methods for solving nonlinear systems of differential equations that model boundary layer fluid flow problems. Ph.D. thesis (2018)","DOI":"10.1063\/1.5042190"},{"key":"10199_CR26","doi-asserted-by":"crossref","unstructured":"Owolabi, K., Atangana, A.: Numerical Methods for Fractional Differentiation. Springer, Berlin (2019)","DOI":"10.1007\/978-981-15-0098-5"},{"key":"10199_CR27","volume-title":"Adaptive Numerical Solution of PDEs","author":"D Peter","year":"2012","unstructured":"Peter, D., Martin, W.: Adaptive Numerical Solution of PDEs. De Gruyter, Berlin (2012)"},{"key":"10199_CR28","volume-title":"Fractional Differential Equations","author":"I Podlubny","year":"1999","unstructured":"Podlubny, I.: Fractional Differential Equations. B Academic Press, San Diego. (1999)"},{"key":"10199_CR29","doi-asserted-by":"publisher","first-page":"585","DOI":"10.1016\/j.aej.2023.05.055","volume":"74","author":"S Qureshi","year":"2023","unstructured":"Qureshi, S., Akanbi, M.A., Shaikh, A.A., Wusu, A.A., Ogunlaran, O.M., Mahmoud, W., Osman, M.S.: A new adaptive nonlinear numerical method for singular and stiff differential problems. Alex. Eng. J. 74, 585\u2013597 (2023)","journal-title":"Alex. Eng. J."},{"key":"10199_CR30","first-page":"1143","volume":"163","author":"JI Ramos","year":"2005","unstructured":"Ramos, J.I.: Linearization techniques for singularly-perturbed initial-value problems of ordinary differential equations. Appl. Math. Comput. 163, 1143\u20131163 (2005)","journal-title":"Appl. Math. Comput."},{"issue":"1","key":"10199_CR31","doi-asserted-by":"publisher","first-page":"15","DOI":"10.1115\/1.3101682","volume":"50","author":"YA Rossikhin","year":"1997","unstructured":"Rossikhin, Y.A., Shitikova, M.V.: Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids. Appl. Mech. Rev. 50(1), 15\u201367 (1997)","journal-title":"Appl. Mech. Rev."},{"key":"10199_CR32","doi-asserted-by":"publisher","first-page":"57","DOI":"10.1016\/j.wavemoti.2019.01.014","volume":"88","author":"FM Samuel","year":"2019","unstructured":"Samuel, F.M., Motsa, S.S.: Solving hyperbolic partial differential equations using a highly accurate multidomain bivariate spectral collocation method. Wave Motion 88, 57\u201372 (2019)","journal-title":"Wave Motion"},{"issue":"3","key":"10199_CR33","first-page":"10575","volume":"219","author":"KK Sharma","year":"2013","unstructured":"Sharma, K.K., Rai, R., Patidar, K.C.: A review on singularly perturbed differential equation with turning points and interior layers. Appl. Math. Comput. 219(3), 10575\u201310609 (2013)","journal-title":"Appl. Math. Comput."},{"key":"10199_CR34","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-540-71041-7","volume-title":"Spectral Methods: Algorithms, Analysis and Applications","author":"J Shen","year":"2011","unstructured":"Shen, J., Tang, T., Wang, L.-L.: Spectral Methods: Algorithms, Analysis and Applications, vol. 41. Springer, Berlin (2011)"},{"key":"10199_CR35","first-page":"76","volume":"5","author":"B Sikora","year":"2023","unstructured":"Sikora, B.: Remarks on the caputo fractional derivative. Matematyka Informatyka Na Uczelniach Technicznych (MINUT) 5, 76\u201384 (2023)","journal-title":"Matematyka Informatyka Na Uczelniach Technicznych (MINUT)"},{"issue":"4","key":"10199_CR36","doi-asserted-by":"publisher","first-page":"464","DOI":"10.1017\/S1446181110000830","volume":"51","author":"NH Sweilam","year":"2010","unstructured":"Sweilam, N.H., Khader, M.M.: Efficient chebyshev spectral methods for solving multi-term fractional orders differential equations. ANZIAM J. 51(4), 464\u2013475 (2010)","journal-title":"ANZIAM J."},{"key":"10199_CR37","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898719598","volume-title":"Spectral Methods in MATLAB","author":"LN Trefethen","year":"2000","unstructured":"Trefethen, L.N.: Spectral Methods in MATLAB. Society for Industrial and Applied Mathematics, Providence (2000)"},{"key":"10199_CR38","doi-asserted-by":"publisher","first-page":"241","DOI":"10.1016\/j.jcp.2013.09.041","volume":"257","author":"Q Xu","year":"2014","unstructured":"Xu, Q., Hesthaven, J.S.: Stable multi-domain spectral penalty methods for fractional partial differential equations. J. Comput. Phys. 257, 241\u2013258 (2014)","journal-title":"J. Comput. Phys."},{"issue":"1","key":"10199_CR39","doi-asserted-by":"publisher","first-page":"A40","DOI":"10.1137\/130933216","volume":"36","author":"M Zayernouri","year":"2014","unstructured":"Zayernouri, M., Em Karniadakis, G.: Fractional spectral collocation method. SIAM J. Sci. Comput. 36(1), A40\u2013A62 (2014)","journal-title":"SIAM J. Sci. Comput."}],"container-title":["Journal of Nonlinear Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-025-10199-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00332-025-10199-8\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-025-10199-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,9,10]],"date-time":"2025-09-10T16:57:59Z","timestamp":1757523479000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00332-025-10199-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,8,14]]},"references-count":39,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2025,10]]}},"alternative-id":["10199"],"URL":"https:\/\/doi.org\/10.1007\/s00332-025-10199-8","relation":{},"ISSN":["0938-8974","1432-1467"],"issn-type":[{"value":"0938-8974","type":"print"},{"value":"1432-1467","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,8,14]]},"assertion":[{"value":"24 October 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"28 July 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 August 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that there are no conflict of interest. The findings and conclusions expressed in this article are solely those of the authors.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"100"}}