{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,13]],"date-time":"2026-01-13T08:26:38Z","timestamp":1768292798213,"version":"3.49.0"},"reference-count":44,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2020,10,9]],"date-time":"2020-10-09T00:00:00Z","timestamp":1602201600000},"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\u2013fractional reaction\u2013diffusion (TFRD) model has broad physical perspectives and theoretical interpretation, and its numerical techniques are of significant conceptual and applied importance. A numerical technique is constructed for the solution of the TFRD model with the non-singular kernel. The Caputo\u2013Fabrizio operator is applied for the discretization of time levels while the extended cubic B-spline (ECBS) function is applied for the space direction. The ECBS function preserves geometrical invariability, convex hull and symmetry property. Unconditional stability and convergence analysis are also proved. The projected numerical method is tested on two numerical examples. The theoretical and numerical results demonstrate that the order of convergence of 2 in time and space directions.<\/jats:p>","DOI":"10.3390\/sym12101653","type":"journal-article","created":{"date-parts":[[2020,10,17]],"date-time":"2020-10-17T07:23:22Z","timestamp":1602919402000},"page":"1653","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":31,"title":["A Numerical Approach of a Time Fractional Reaction\u2013Diffusion Model with a Non-Singular Kernel"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1825-2631","authenticated-orcid":false,"given":"Tayyaba","family":"Akram","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences, Universiti Sains Malaysia, Penang 11800, Malaysia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"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"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8344-4242","authenticated-orcid":false,"given":"Ajmal","family":"Ali","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Virtual University of Pakistan, Lahore 54000, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"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, Saudia Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0286-7244","authenticated-orcid":false,"given":"Dumitru","family":"Baleanu","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Arts and Sciences, Cankaya University, Ankara 06530, Turkey"},{"name":"Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan"},{"name":"Institute of Space-Sciences, 077125 Magurele-Bucharest, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,10,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Murray, J.D. (2003). Mathematical Biology, Springer.","DOI":"10.1007\/b98869"},{"key":"ref_2","unstructured":"Kuramoto, Y. (2003). Chemical Oscillations Waves and Turbulence, Dover Publications, Inc."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Wilhelmsson, H., and Lazzaro, E. (2001). Reaction\u2013Diffusion Problems in the Physics of hot Plasmas, Institute of Physics Publishing.","DOI":"10.1887\/0750306157"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Hundsdorfer, W., and Verwer, J.G. (2003). Numerical Solution of Time Dependent Advection-Diffusion-Reaction Equations, Springer.","DOI":"10.1007\/978-3-662-09017-6"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1202","DOI":"10.1063\/1.466650","article-title":"Spiral waves in a surface reaction: Model calculations","volume":"100","author":"Bar","year":"1994","journal-title":"J. Chem. Phys."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"468","DOI":"10.1016\/S0378-4371(00)00386-1","article-title":"Fractional calculus and continuous-time finance. II: The waiting-time distribution","volume":"287","author":"Mainardi","year":"2000","journal-title":"Physica A"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1403","DOI":"10.1029\/2000WR900031","article-title":"Application of a fractional advection-dispersion equation","volume":"36","author":"Benson","year":"2000","journal-title":"Water Resour. Res."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0370-1573(00)00070-3","article-title":"The random walks guide to anomalous diffusion: A fractional dynamics approach","volume":"339","author":"Metzler","year":"2000","journal-title":"Phys. Rep."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"5315","DOI":"10.1007\/s40314-018-0633-3","article-title":"Jacobi collocation scheme for variable-order fractional reaction sub-diffusion equation","volume":"37","author":"Hafez","year":"2018","journal-title":"Comput. Appl. Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"4376","DOI":"10.1007\/s40314-018-0579-5","article-title":"A class of efficient difference method for time fractional reaction-diffusion equation","volume":"37","author":"Zhang","year":"2018","journal-title":"Comput. Appl. Math."},{"key":"ref_11","first-page":"09","article-title":"A numerical approach for a class of time-fractional reaction-diffusion equation through exponential B-spline method","volume":"39","author":"Kanth","year":"2019","journal-title":"Comput. Appl. Math."},{"key":"ref_12","first-page":"73","article-title":"A new Definition of Fractional Derivative without Singular Kernel","volume":"1","author":"Caputo","year":"2015","journal-title":"Prog. Fract. Differ. Appl."},{"key":"ref_13","first-page":"1","article-title":"Extension of the resistance inductance, capacitance electrical circuit of fractional derivative without singular kernel","volume":"7","author":"Atangana","year":"2015","journal-title":"Adv. Mech. Eng."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"467","DOI":"10.1016\/j.physa.2015.12.066","article-title":"Modeling diffusive transport with a fractional derivative without singular kernel","volume":"447","year":"2016","journal-title":"Physica A"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"6289","DOI":"10.3390\/e17096289","article-title":"Modeling of a mass-spring-damper system by fractional derivative with and without a singular kernel","volume":"17","author":"Gomez","year":"2015","journal-title":"Entropy"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"948","DOI":"10.1016\/j.amc.2015.10.021","article-title":"On the new fractional derivative and application to nonlinear Fisher\u2019s reaction-diffusion equation","volume":"273","author":"Atangana","year":"2016","journal-title":"Appl. Math. Comput."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"833","DOI":"10.2298\/TSCI16S3833Y","article-title":"Some new applications for heat and fluid flows via fractional derivatives without singular kernel","volume":"20","author":"Yang","year":"2016","journal-title":"Therm. Sci."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"3847","DOI":"10.1016\/j.cnsns.2010.02.007","article-title":"On the solutions of time-fractional reaction-diffusion equations","volume":"15","author":"Rida","year":"2010","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_19","first-page":"28p","article-title":"Comparing numerical methods for solving time-fractional reaction-diffusion equations","volume":"2012","author":"Turut","year":"2012","journal-title":"Math. Anal."},{"key":"ref_20","first-page":"5p","article-title":"A domain decomposition method for time fractional reaction-diffusion equation","volume":"2014","author":"Gong","year":"2014","journal-title":"Sci. World J."},{"key":"ref_21","first-page":"13p","article-title":"A new approach and solution technique to solve time fractional non-linear reaction-diffusion equation","volume":"2015","author":"Sungu","year":"2015","journal-title":"Math. Prob. Eng."},{"key":"ref_22","first-page":"820162","article-title":"Solving the Caputo fractional reaction-diffusion equation on GPU","volume":"2014","author":"Liu","year":"2014","journal-title":"Discret. Dyn. Nat. Soc."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1007\/s12190-014-0764-7","article-title":"An H1-Galerkin mixed finite element method for time fractional reaction-diffusion equation","volume":"47","author":"Liu","year":"2015","journal-title":"J. Appl. Math. Comput."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1186\/s13662-016-0929-9","article-title":"An efficient parallel algorithm for Caputo fractional reaction-diffusion equation with implicit finite difference method","volume":"1","author":"Wang","year":"2016","journal-title":"Adv. Differ. Equ."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"402","DOI":"10.1140\/epjp\/i2018-12200-2","article-title":"Convergence analysis of tau scheme for the fractional reaction-diffusion equation","volume":"133","author":"Rashidinia","year":"2018","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"414","DOI":"10.1515\/phys-2015-0047","article-title":"Numerical solutions of the reaction diffusion system by using exponential cubic B-spline collocation algorithms","volume":"13","author":"Ersoy","year":"2015","journal-title":"Open Phys."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"493","DOI":"10.1016\/j.jcp.2017.03.006","article-title":"Numerical solution of the time fractional reaction-diffusion equation with a moving boundary","volume":"338","author":"Zheng","year":"2017","journal-title":"J. Comput. Phys."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1007\/978-3-030-12232-4_5","article-title":"Numerical Solution of space-time fractional reaction-diffusion equations via the Caputo and Riesz derivatives","volume":"39","author":"Owelabi","year":"2019","journal-title":"Math. Appl. Eng. Model Soc. Issues"},{"key":"ref_29","first-page":"3401","article-title":"B-spline wavelet operational method for numerical solution of time-space fractional partial differential equations","volume":"15","author":"Zeynab","year":"2017","journal-title":"Int. J. Wavelets Multiresolut. Inf. Process."},{"key":"ref_30","unstructured":"Pandey, P., Kumar, S., and G\u00f6mez-Aguilar, J.F. (2019). Numerical Solution of the Time Fractional reaction-advection- diffusion Equation in Porous Media. J. Appl Comput. Mech., 7."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"060017","DOI":"10.1063\/1.5136449","article-title":"An extended cubic B-spline collocation scheme for time fractional sub-diffusion equation","volume":"2184","author":"Akram","year":"2019","journal-title":"AIP Conf. Proc."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"030004","DOI":"10.1063\/1.5121041","article-title":"Numerical solution of fractional cable equation via extended cubic B-spline","volume":"2138","author":"Akram","year":"2019","journal-title":"AIP Conf. Proc."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"365","DOI":"10.1186\/s13662-019-2296-9","article-title":"Extended cubic B-splines in the numerical solution of time fractional telegraph equation","volume":"2019","author":"Akram","year":"2019","journal-title":"Adv. Differ. Equ."},{"key":"ref_34","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_35","doi-asserted-by":"crossref","first-page":"85","DOI":"10.22436\/jmcs.022.01.08","article-title":"A numerical study on time fractional Fisher equation using an extended cubic B-spline approximation","volume":"22","author":"Akram","year":"2021","journal-title":"J. Math. Comput. Sci."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"2201","DOI":"10.1016\/j.aej.2020.01.048","article-title":"An efficient numerical technique for solving time fractional Burgers equation","volume":"59","author":"Akram","year":"2020","journal-title":"Alex Eng. J."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Akram, T., Abbas, M., Riaz, M.B., Ismail, A.I., and Ali, N.M. (2020). Development and analysis of new approximation of extended cubic B-spline to the non-linear time fractional Klein-Gordon equation. Fractals, in press.","DOI":"10.1142\/S0218348X20400393"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"378","DOI":"10.1186\/s13662-019-2318-7","article-title":"A numerical algorithm based on modified extended B-spline functions for solving time fractional diffusion wave equation involving reaction and damping terms","volume":"2019","author":"Khalid","year":"2019","journal-title":"Adv. Differ. Equ."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Khalid, N., Abbas, M., Iqbal, M.K., Singh, J., and Ismail, A.I. (2020). A computational approach for solving time fractional differential equation via spline functions. Alex Eng. J., in press.","DOI":"10.1016\/j.aej.2020.06.007"},{"key":"ref_40","first-page":"87","article-title":"Properties of a New Fractional Derivative without Singular Kernel","volume":"1","author":"Losada","year":"2015","journal-title":"Prog. Fract. Differ. Appl."},{"key":"ref_41","first-page":"576","article-title":"An extension of the cubic uniform B-spline curves","volume":"15","author":"Han","year":"2003","journal-title":"Comput. Aided Des. Comput. Graph."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1016\/0021-9045(68)90025-7","article-title":"On error bounds for spline interpolation","volume":"1","author":"Hall","year":"1968","journal-title":"J. Approx. Theory"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"452","DOI":"10.1016\/0021-9045(68)90033-6","article-title":"On the convergence of odd degree spline interpolation","volume":"1","author":"Boor","year":"1968","journal-title":"J. Approx. Theory"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"28","DOI":"10.1016\/j.amc.2016.01.049","article-title":"Numerical solution of hyperbolic telegraph equation by cubic B-spline collocation method","volume":"281","author":"Sharifi","year":"2016","journal-title":"Appl. Math. Comput."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/10\/1653\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:18:03Z","timestamp":1760177883000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/10\/1653"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,10,9]]},"references-count":44,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2020,10]]}},"alternative-id":["sym12101653"],"URL":"https:\/\/doi.org\/10.3390\/sym12101653","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,10,9]]}}}