{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,9]],"date-time":"2026-05-09T03:08:55Z","timestamp":1778296135002,"version":"3.51.4"},"reference-count":52,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2023,2,25]],"date-time":"2023-02-25T00:00:00Z","timestamp":1677283200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The time-fractional heat equation governed by nonlocal conditions is solved using a novel method developed in this study, which is based on the spectral tau method. There are two sets of basis functions used. The first set is the set of non-symmetric polynomials, namely, the shifted Chebyshev polynomials of the sixth-kind (CPs6), and the second set is a set of modified shifted CPs6. The approximation of the solution is written as a product of the two chosen basis function sets. For this method, the key concept is to transform the problem governed by the underlying conditions into a set of linear algebraic equations that can be solved by means of an appropriate numerical scheme. The error analysis of the proposed extension is also thoroughly investigated. Finally, a number of examples are shown to illustrate the reliability and accuracy of the suggested tau method.<\/jats:p>","DOI":"10.3390\/sym15030594","type":"journal-article","created":{"date-parts":[[2023,2,27]],"date-time":"2023-02-27T04:05:40Z","timestamp":1677470740000},"page":"594","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":21,"title":["A Tau Approach for Solving Time-Fractional Heat Equation Based on the Shifted Sixth-Kind Chebyshev Polynomials"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9516-884X","authenticated-orcid":false,"given":"Esraa Magdy","family":"Abdelghany","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo 11341, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6102-671X","authenticated-orcid":false,"given":"Waleed Mohamed","family":"Abd-Elhameed","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6833-8903","authenticated-orcid":false,"given":"Galal Mahrous","family":"Moatimid","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo 11341, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0403-8797","authenticated-orcid":false,"given":"Youssri Hassan","family":"Youssri","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt"},{"name":"Faculty of Engineering, Egypt University of Informatics, Knowledge City, New Administrative Capital, Cairo 11865, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1467-640X","authenticated-orcid":false,"given":"Ahmed Gamal","family":"Atta","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo 11341, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,2,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Hesthaven, J., Gottlieb, S., and Gottlieb, D. (2007). Spectral Methods for Time-Dependent Problems, Cambridge University Press.","DOI":"10.1017\/CBO9780511618352"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Shen, J., Tang, T., and Wang, L.L. (2011). Spectral Methods: Algorithms, Analysis and Applications, Springer Science & Business Media.","DOI":"10.1007\/978-3-540-71041-7"},{"key":"ref_3","unstructured":"Boyd, J.P. (2001). Chebyshev and Fourier Spectral Methods, Courier Corporation."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Mason, J.C., and Handscomb, D.C. (2002). Chebyshev Polynomials, CRC Press.","DOI":"10.1201\/9781420036114"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1080\/00207160802036882","article-title":"A new computational method for solution of non-linear Volterra\u2013Fredholm integro-differential equations","volume":"87","author":"Maleknejad","year":"2010","journal-title":"Int. J. Comput. Math."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"2227","DOI":"10.1002\/mma.2969","article-title":"An operational matrix method for solving Lane\u2013Emden equations arising in astrophysics","volume":"37","year":"2014","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"417","DOI":"10.1016\/j.amc.2006.10.008","article-title":"Numerical solution of differential equations by using Chebyshev wavelet operational matrix of integration","volume":"188","author":"Babolian","year":"2007","journal-title":"Appl. Math. Comput."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1007\/s40096-021-00401-9","article-title":"An operational matrix method to solve linear Fredholm\u2013Volterra integro-differential equations with piecewise intervals","volume":"15","year":"2021","journal-title":"Math. Sci."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"\u00d6zt\u00fcrk, Y. (2018). solution for the system of Lane\u2013Emden type equations using Chebyshev polynomials. Mathematics, 6.","DOI":"10.3390\/math6100181"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"218","DOI":"10.1007\/s40314-021-01610-7","article-title":"A spectral collocation matrix method for solving linear Fredholm integro-differential\u2013difference equations","volume":"40","author":"Demir","year":"2021","journal-title":"Comput. Appl. Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"109585","DOI":"10.1016\/j.chaos.2019.109585","article-title":"Shifted-Chebyshev-polynomial-based numerical algorithm for fractional order polymer visco-elastic rotating beam","volume":"132","author":"Wang","year":"2020","journal-title":"Chaos Solitons Fractals"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"315","DOI":"10.1186\/s13662-020-02779-7","article-title":"Numerical solution for the time-fractional Fokker\u2013Planck equation via shifted Chebyshev polynomials of the fourth kind","volume":"2020","author":"Habenom","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_13","first-page":"281","article-title":"On using third and fourth kinds Chebyshev operational matrices for solving Lane-Emden type equations","volume":"60","author":"Doha","year":"2015","journal-title":"Rom. J. Phys."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"66","DOI":"10.1016\/j.amc.2019.01.030","article-title":"Numerical solutions of integral and integro-differential equations using Chebyshev polynomials of the third kind","volume":"351","author":"Sakran","year":"2019","journal-title":"Appl. Math. Comput."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"5145","DOI":"10.1016\/j.aej.2021.10.036","article-title":"Numerical solution method for multi-term variable order fractional differential equations by shifted Chebyshev polynomials of the third kind","volume":"61","author":"Dincel","year":"2022","journal-title":"Alex. Eng. J."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"328","DOI":"10.1016\/S0196-8858(02)00017-9","article-title":"An integral formula for generalized Gegenbauer polynomials and Jacobi polynomials","volume":"29","author":"Xu","year":"2002","journal-title":"Adv. Appl. Math."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Masjed\u2013Jamei, M. (2006). Some New Classes of Orthogonal Polynomials and Special functions: A Symmetric Generalization of Sturm-Liouville Problems and Its Consequences. [Ph.D Thesis, Department of Mathematics, University of Kassel].","DOI":"10.1080\/10652460701510949"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1515\/ijnsns-2018-0118","article-title":"Sixth-kind Chebyshev spectral approach for solving fractional differential equations","volume":"20","author":"Youssri","year":"2019","journal-title":"Int. J. Nonlinear Sci. Numer. Simul."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2897","DOI":"10.1007\/s40314-017-0488-z","article-title":"Fifth-kind orthonormal Chebyshev polynomial solutions for fractional differential equations","volume":"37","author":"Youssri","year":"2018","journal-title":"Comput. Appl. Math."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"112908","DOI":"10.1016\/j.cam.2020.112908","article-title":"Numerical solution of variable order fractional nonlinear quadratic integro-differential equations based on the sixth-kind Chebyshev collocation method","volume":"377","author":"Babaei","year":"2020","journal-title":"J. Comput. Appl. Math."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"966","DOI":"10.1080\/00207160.2021.1940977","article-title":"A new efficient algorithm based on fifth-kind Chebyshev polynomials for solving multi-term variable-order time-fractional diffusion-wave equation","volume":"99","author":"Sadri","year":"2022","journal-title":"Int. J. Comput. Math."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"101412","DOI":"10.1016\/j.jocs.2021.101412","article-title":"Solving fractional optimal control problems with inequality constraints by a new kind of Chebyshev wavelets method","volume":"54","author":"Xu","year":"2021","journal-title":"J. Comput. Sci."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"9087359","DOI":"10.1155\/2022\/9087359","article-title":"A new efficient method for solving system of weakly singular fractional integro-differential equations by shifted sixth-kind Chebyshev polynomials","volume":"2022","author":"Yaghoubi","year":"2022","journal-title":"J. Math."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1016\/j.cnsns.2018.04.019","article-title":"A new collection of real world applications of fractional calculus in science and engineering","volume":"64","author":"Sun","year":"2018","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"753601","DOI":"10.1155\/S0161171203301486","article-title":"Recent applications of fractional calculus to science and engineering","volume":"2003","author":"Debnath","year":"2003","journal-title":"Int. J. Math. Math. Sci."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Atanackovi\u0107, T., Pilipovi\u0107, S., Stankovi\u0107, B., and Zorica, D. (2014). Fractional Calculus with Applications in Mechanics: Vibrations and Diffusion Processes, Wiley.","DOI":"10.1002\/9781118577530"},{"key":"ref_27","unstructured":"Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006). Theory and Applications of Fractional Differential Equations, Elsevier."},{"key":"ref_28","unstructured":"Podlubny, I. (1997). The Laplace transform method for linear differential equations of the fractional order. arXiv."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"256","DOI":"10.1016\/j.jcp.2017.12.044","article-title":"A fast linearized conservative finite element method for the strongly coupled nonlinear fractional Schr\u00f6dinger equations","volume":"358","author":"Li","year":"2018","journal-title":"J. Comput. Phys."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"957","DOI":"10.1007\/s10915-017-0388-9","article-title":"Fast iterative method with a second-order implicit difference scheme for time-space fractional convection\u2013diffusion equation","volume":"72","author":"Gu","year":"2017","journal-title":"J. Sci. Comput."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1016\/j.cam.2018.02.018","article-title":"Application Jacobi spectral method for solving the time-fractional differential equation","volume":"339","author":"Sazmand","year":"2018","journal-title":"J. Comput. Appl. Math."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1515\/IJNSNS.2005.6.2.209","article-title":"Search for variational principles in electrodynamics by Lagrange method","volume":"6","author":"Hao","year":"2005","journal-title":"Int. J. Nonlinear Sci. Numer. Simul."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1515\/IJNSNS.2002.3.2.107","article-title":"An approximate solution for one-dimensional weakly nonlinear oscillations","volume":"3","author":"Marinca","year":"2002","journal-title":"Int. J. Nonlinear Sci. Numer. Simul."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"110","DOI":"10.1016\/j.matcom.2005.05.001","article-title":"An explicit and numerical solutions of the fractional KdV equation","volume":"70","author":"Momani","year":"2005","journal-title":"Math. Comput. Simul."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"5233","DOI":"10.1016\/j.apm.2012.10.045","article-title":"Numerical solutions to initial and boundary value problems for linear fractional partial differential equations","volume":"37","author":"Khan","year":"2013","journal-title":"Appl. Math. Model."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"100099","DOI":"10.1016\/j.padiff.2021.100099","article-title":"Numerical solution of two-dimensional fractional-order partial differential equations using hybrid functions","volume":"4","author":"Postavaru","year":"2021","journal-title":"Partial. Differ. Equ. Appl. Math."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"262","DOI":"10.1002\/num.20247","article-title":"Variational iteration method for solving the space-and time-fractional KdV equation","volume":"24","author":"Momani","year":"2008","journal-title":"Numer. Methods Partial Differ. Equ."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"15138","DOI":"10.3934\/math.2022830","article-title":"Spectral tau solution of the linearized time-fractional KdV-Type equations","volume":"7","author":"Youssri","year":"2022","journal-title":"AIMS Math."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"110753","DOI":"10.1016\/j.chaos.2021.110753","article-title":"A robust numerical scheme for a time-fractional Black\u2013Scholes partial differential equation describing stock exchange dynamics","volume":"145","author":"Nuugulu","year":"2021","journal-title":"Chaos Solitons Fractals"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"671","DOI":"10.1016\/j.apm.2015.06.014","article-title":"The M\u00fcntz-Legendre Tau method for fractional differential equations","volume":"40","author":"Mokhtary","year":"2016","journal-title":"Appl. Math. Model."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"240","DOI":"10.3389\/fphy.2019.00240","article-title":"Jacobi spectral Galerkin method for distributed-order fractional Rayleigh\u2013Stokes problem for a generalized second grade fluid","volume":"7","author":"Hafez","year":"2020","journal-title":"Front. Phys."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"101811","DOI":"10.1016\/j.asej.2022.101811","article-title":"Convective-radiative thermal investigation of a porous dovetail fin using spectral collocation method","volume":"14","author":"Weera","year":"2023","journal-title":"Ain Shams Eng. J."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1007\/s40819-018-0597-4","article-title":"Generalized Fibonacci operational collocation approach for fractional initial value problems","volume":"5","author":"Atta","year":"2019","journal-title":"Int. J. Appl. Comput. Math."},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Atta, A.G., Abd-Elhameed, W.M., Moatimid, G.M., and Youssri, Y.H. (2022). Advanced shifted sixth-kind Chebyshev tau approach for solving linear one-dimensional hyperbolic telegraph type problem. Math. Sci.","DOI":"10.1007\/s40096-022-00460-6"},{"key":"ref_45","first-page":"6833404","article-title":"Newfangled linearization formula of certain Nonsymmetric Jacobi polynomials: Numerical treatment of nonlinear Fisher\u2019s equation","volume":"2023","author":"Ali","year":"2023","journal-title":"J. Funct. Spaces"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1186\/s13662-017-1123-4","article-title":"A new operational matrix of Caputo fractional derivatives of Fermat polynomials: An application for solving the Bagley\u2013Torvik equation","volume":"2017","author":"Youssri","year":"2017","journal-title":"Adv. Differ. Equ."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"655","DOI":"10.1590\/S1807-03022011000300010","article-title":"Operational Tau approximation for a general class of fractional integro-differential equations","volume":"30","author":"Vanani","year":"2011","journal-title":"Comput. Appl. Math."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"1850118","DOI":"10.1142\/S0219876218501189","article-title":"Fully Legendre spectral Galerkin algorithm for solving linear one-dimensional telegraph type equation","volume":"16","author":"Doha","year":"2019","journal-title":"Int. J. Comput. Methods"},{"key":"ref_49","unstructured":"Podlubny, I. (1999). Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Academic Press."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"1281","DOI":"10.1016\/j.apnum.2011.08.007","article-title":"Implicit difference approximation for the time fractional heat equation with the nonlocal condition","volume":"61","author":"Karatay","year":"2011","journal-title":"Appl. Numer. Math."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"501","DOI":"10.1007\/s40324-021-00245-2","article-title":"A fast collocation algorithm for solving the time fractional heat equation","volume":"78","year":"2021","journal-title":"SeMA J."},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"Atta, A., Abd-Elhameed, W., Moatimid, G., and Youssri, Y. (2022). A fast Galerkin approach for solving the fractional Rayleigh\u2013Stokes problem via sixth-kind Chebyshev polynomials. Mathematics, 10.","DOI":"10.3390\/math10111843"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/3\/594\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:42:25Z","timestamp":1760121745000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/3\/594"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,2,25]]},"references-count":52,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2023,3]]}},"alternative-id":["sym15030594"],"URL":"https:\/\/doi.org\/10.3390\/sym15030594","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,2,25]]}}}