{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,13]],"date-time":"2025-11-13T07:20:18Z","timestamp":1763018418991,"version":"build-2065373602"},"reference-count":41,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2021,11,3]],"date-time":"2021-11-03T00:00:00Z","timestamp":1635897600000},"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>In this study, a new modified group iterative scheme for solving the two-dimensional (2D) fractional hyperbolic telegraph differential equation with Dirichlet boundary conditions is obtained from the 2h-spaced standard and rotated Crank\u2013Nicolson FD approximations. The findings of new four-point modified explicit group relaxation method demonstrates the rapid rate of convergence of proposed method as compared to the existing schemes. Numerical tests are performed to test the capability of the group iterative scheme in comparison with the point iterative scheme counterparts. The stability of the derived modified group method is proven by the matrix norm algorithm. The obtained results are tabulated and concluded that exact solutions are exactly symmetric with approximate solutions.<\/jats:p>","DOI":"10.3390\/sym13112078","type":"journal-article","created":{"date-parts":[[2021,11,3]],"date-time":"2021-11-03T21:57:49Z","timestamp":1635976669000},"page":"2078","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["On Unconditionally Stable New Modified Fractional Group Iterative Scheme for the Solution of 2D Time-Fractional Telegraph Model"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8344-4242","authenticated-orcid":false,"given":"Ajmal","family":"Ali","sequence":"first","affiliation":[{"name":"Department of Mathematics, Virtual University of Pakistan, Lahore 54000, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8889-3768","authenticated-orcid":false,"given":"Thabet","family":"Abdeljawad","sequence":"additional","affiliation":[{"name":"Department of Mathematics and General Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia"},{"name":"Department of Medical Research, China Medical University, Taichung 40402, Taiwan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5103-6092","authenticated-orcid":false,"given":"Azhar","family":"Iqbal","sequence":"additional","affiliation":[{"name":"Mathematics and Natural Sciences, Prince Mohammad Bin Fahd University, Al Khobar 31952, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1825-2631","authenticated-orcid":false,"given":"Tayyaba","family":"Akram","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Universiti Sains Malaysia, Gelugor 11800, Malaysia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0491-1528","authenticated-orcid":false,"given":"Muhammad","family":"Abbas","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan"}]}],"member":"1968","published-online":{"date-parts":[[2021,11,3]]},"reference":[{"key":"ref_1","unstructured":"Lin, Y.H., Liu, H., and Liu, X. (2021). Determining a nonlinear hyperbolic system with unknown sources and nonlinearity. arXiv."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"527","DOI":"10.1007\/s00220-020-03889-9","article-title":"Determining a random Schr\u00f6dinger operator: Both potential and source are random","volume":"381","author":"Li","year":"2021","journal-title":"Commun. Math. Phys."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"105005","DOI":"10.1088\/0266-5611\/31\/10\/105005","article-title":"Determining both sound speed and internal source in thermo- and photo-acoustic tomography","volume":"31","author":"Liu","year":"2015","journal-title":"Inverse Probl."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1861","DOI":"10.4310\/CMS.2019.v17.n7.a5","article-title":"Determining a fractional Helmholtz system with unknown source and medium parameter","volume":"17","author":"Cao","year":"2019","journal-title":"Commun. Math. Sci."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"197","DOI":"10.3934\/ipi.2019011","article-title":"Simultaneously recovering potentials and embedded obstacles for anisotropic fractional Schr\u00f6dinger operators","volume":"13","author":"Cao","year":"2019","journal-title":"Inverse Probl. Imaging"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Ali, A., and Ali, N.H.M. (2019, January 25\u201328). Explicit group iterative methods for the solution of two-dimensional time-fractional telegraph equation. Proceedings of the 4th Innovation and Analytics Conference and Exhibition, AIP Conference Proceedings 2138, Sintok Kedah, Malaysia.","DOI":"10.1063\/1.5121043"},{"key":"ref_7","first-page":"1","article-title":"On skewed grid point iterative method for solving 2D hyperbolic telegraph fractional differential equation","volume":"303","author":"Ali","year":"2019","journal-title":"Adv. Differ. Equ."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Ali, A., and Ali, N.H.M. (2019, January 25\u201328). New group fractional damped wave iterative solvers using Mathematica. Proceedings of the International Conference on Mathematical Sciences and Technology, AIP Conference Proceedings 2184, Penang, Malaysia.","DOI":"10.1063\/1.5136466"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"6953","DOI":"10.1016\/j.jcp.2012.06.025","article-title":"New explicit group iterative methods in the solution of two-dimensional hyperbolic equations","volume":"231","author":"Ali","year":"2012","journal-title":"J. Comput. Phys."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"382","DOI":"10.1016\/j.jcp.2015.03.052","article-title":"New explicit group iterative methods in the solution of three-dimensional hyperbolic telegraph equations","volume":"294","author":"Kew","year":"2015","journal-title":"J. Comput. Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1289","DOI":"10.1080\/0020716031000112312","article-title":"The numerical solution of the telegraph equation by the alternating group explicit (AGE) method","volume":"80","author":"Evans","year":"2003","journal-title":"Int. J. Comput. Math."},{"key":"ref_12","first-page":"1272","article-title":"Unconditionally stable difference schemes for a one-space-dimensional linear hyperbolic equation","volume":"187","author":"Gao","year":"2007","journal-title":"Appl. Math. Comput."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"293","DOI":"10.22436\/jnsa.013.05.06","article-title":"Caputo\u2013Katugampola fractional Volterra functional differential equations with a vanishing lag function","volume":"13","author":"Youssef","year":"2020","journal-title":"J. Nonlinear Sci. Appl."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"230","DOI":"10.22436\/jmcs.023.03.06","article-title":"Presence and diversity of positive solutions for a Caputo\u2013type fractional order nonlinear differential equation with an advanced argument","volume":"23","author":"Asaduzzaman","year":"2020","journal-title":"J. Math. Comput. Sci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"58","DOI":"10.22436\/jmcs.023.01.06","article-title":"On approximate solutions for fractional system of differential equations with Caputo\u2013Fabrizio fractional operator","volume":"23","author":"Jassim","year":"2020","journal-title":"J. Math. Comput. Sci."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"325","DOI":"10.22436\/jmcs.022.04.02","article-title":"On Hyers-Ulam-Rassias stability of fractional differential equations with Caputo derivative","volume":"22","year":"2020","journal-title":"J. Math. Comput. Sci."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Lai, J., and Liu, H. (2021). On a novel numerical scheme for Riesz fractional partial differential equations. Mathematics, 9.","DOI":"10.3390\/math9162014"},{"key":"ref_18","first-page":"1","article-title":"Extended cubic B-splines in the numerical solution of time fractional telegraph equation","volume":"365","author":"Akram","year":"2019","journal-title":"Adv. Differ. Equ."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Akram, T., Abbas, M., Iqbal, A., Baleanu, D., and Asad, J.H. (2020). Novel numerical approach based on modified extended cubic B-spline functions for solving non-linear time-fractional telegraph equation. Symmetry, 12.","DOI":"10.3390\/sym12071154"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"684","DOI":"10.1002\/num.1034","article-title":"An unconditionally stable alternating direction implicit scheme for two space dimensional linear hyperbolic equation","volume":"7","author":"Mohanty","year":"2001","journal-title":"Numer. Methods Partial Differ. Equ."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1080\/00207160211918","article-title":"An unconditionally stable {ADI} method for linear hyperbolic equation in three space dimensional","volume":"79","author":"Mohanty","year":"2002","journal-title":"Int. J. Comput. Math."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2061","DOI":"10.1080\/00207160801965271","article-title":"A new unconditionally stable diffierence schemes for the solution of multi-dimensional telegraphic equation","volume":"86","author":"Mohanty","year":"2009","journal-title":"Int. J. Comput. Math."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"897","DOI":"10.1002\/num.20295","article-title":"The combination of collocation finite difference and multigrid methods for solution of the two-dimensional wave equation","volume":"24","author":"Dehghan","year":"2008","journal-title":"Numer. Methods Partial Differ. Equ."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"232","DOI":"10.1002\/num.20341","article-title":"A high order implicit collocation method for the solution of two-dimensional linear hyperbolic equation","volume":"25","author":"Dehghan","year":"2009","journal-title":"Numer. Methods Partial Differ. Equ."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1002\/num.20442","article-title":"Numerical solution of hyperbolic telegraph equation using the Chebyshev tau method","volume":"26","author":"Saadatmandi","year":"2010","journal-title":"Numer. Methods Partial Differ. Equ."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"324","DOI":"10.1016\/j.enganabound.2009.10.010","article-title":"Combination of meshless local weak and strong {(MLWS)} forms to solve the two dimensional hyperbolic telegraph equation","volume":"34","author":"Dehghan","year":"2010","journal-title":"Eng. Anal. Bound. Elem."},{"key":"ref_27","first-page":"524","article-title":"A finite difference method for fractional partial differential equation","volume":"215","author":"Zhang","year":"2009","journal-title":"Appl. Math. Comput."},{"key":"ref_28","first-page":"1","article-title":"Finite element method for time-space-fractional schrodinger equation","volume":"166","author":"Zhu","year":"2017","journal-title":"Electron. J. Differ. Equ."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"891","DOI":"10.1016\/j.camwa.2011.04.001","article-title":"The boundary element method (BEM) for numerical solution of partial fractional differential equations","volume":"62","author":"Katsikadelis","year":"2011","journal-title":"Comput. Math. Appl."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1080\/00207168508803452","article-title":"Group explicit iterative methods for solving the large linear systems","volume":"17","author":"Evans","year":"1985","journal-title":"Int. J. Comput. Math."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1080\/00207168608803498","article-title":"Explicit group iterative methods for solving elliptic partial differential equations in 3-space dimensions","volume":"18","author":"Evans","year":"1986","journal-title":"Int. J. Comput. Math."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1080\/00207169008803864","article-title":"The explicit block relaxation method as grid smoother in the mutigrid V-cycle scheme","volume":"34","author":"Evans","year":"1990","journal-title":"Int. J. Comput. Math."},{"key":"ref_33","first-page":"383","article-title":"The alternating group explicit iterative method AGE to solve the parabolic and hyperbolic partial differential equations","volume":"2","author":"Evans","year":"1988","journal-title":"Annu. Rev. Numer. Fluid Mech. Heat Transf."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1080\/00207169108803958","article-title":"The four point explicit decoupled group EDG method: A fast Poisson solver","volume":"38","author":"Abdullah","year":"1991","journal-title":"Int. J. Comput. Math."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1080\/10637199508915522","article-title":"Explicit de-coupled group iterative methods anf their parallel implementations","volume":"7","author":"Yousif","year":"1995","journal-title":"Parallel Algorithms Appl."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1080\/00207169508804447","article-title":"Solving the two-dimensional diffusion equation by the four point explicit decoupled EDG iterative method","volume":"58","author":"Ibrahim","year":"1995","journal-title":"Int. J. Comput. Math."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1080\/00207160008805020","article-title":"An efficient four points modified explicit group poisson solver","volume":"76","author":"Othman","year":"2000","journal-title":"Int. J. Comput. Math."},{"key":"ref_38","first-page":"418","article-title":"{MEGSOR} iterative schemes for the solution of {2D} elliptic {PDEs}","volume":"4","author":"Sulaiman","year":"2010","journal-title":"World Acad. Sci. Eng. Technol."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Balasim, A.T., and Ali, N.H.M. (2016, January 24\u201326). The Solution of {2D} Time fractional Diffusion Equation by the Fractional Modified Explicit Group Iterative Method. Proceedings of the International Conference on Mathematics, Engineering and Industrial Applications, AIP Conference Proceedings 1775, Songkhla, Thailand.","DOI":"10.1063\/1.4965134"},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Balasim, A.T., and Ali, N.H.M. (2016, January 24\u201326). Group Iterative Methods for the Solution of Two dimensional Time-fractional Diffusion Equation. Proceedings of the Advances in Industrial and Applied Mathematics, AIP Conference Proceedings 1750, Johor Bahru, Malaysia.","DOI":"10.1063\/1.4954539"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1080\/23311835.2017.1412241","article-title":"New group iterative schemes in the numerical solution of the two-dimensional time fractional advection-diffusion equation","volume":"4","author":"Balasim","year":"2017","journal-title":"Cogent Math."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/11\/2078\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:25:05Z","timestamp":1760167505000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/11\/2078"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,11,3]]},"references-count":41,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2021,11]]}},"alternative-id":["sym13112078"],"URL":"https:\/\/doi.org\/10.3390\/sym13112078","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2021,11,3]]}}}