{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,5]],"date-time":"2026-08-05T01:17:43Z","timestamp":1785892663014,"version":"3.56.0"},"reference-count":46,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2022,11,18]],"date-time":"2022-11-18T00:00:00Z","timestamp":1668729600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The primary focus of this research study is in the development of an effective hybrid matrix method to solve a class of nonlinear systems of equations of fractional order arising in the modeling of autocatalytic chemical reaction problems. The fractional operator is considered in the sense of Liouville\u2013Caputo. The proposed approach relies on the combination of the quasi-linearization technique and the spectral collocation strategy based on generalized clique bases. The main feature of the hybrid approach is that it converts the governing nonlinear fractional-order systems into a linear algebraic system of equations, which is solved in each iteration. In a weighted L2 norm, we prove the error and convergence analysis of the proposed algorithm. By using various model parameters in the numerical examples, we show the computational efficacy as well as the accuracy of our approach. Comparisons with existing available schemes show the high accuracy and robustness of the designed hybrid matrix collocation technique.<\/jats:p>","DOI":"10.3390\/axioms11110654","type":"journal-article","created":{"date-parts":[[2022,11,21]],"date-time":"2022-11-21T04:33:32Z","timestamp":1669005212000},"page":"654","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":29,"title":["Fractional Clique Collocation Technique for Numerical Simulations of Fractional-Order Brusselator Chemical Model"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6116-4928","authenticated-orcid":false,"given":"Mohammad","family":"Izadi","sequence":"first","affiliation":[{"name":"Department of Applied Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman 76169-14111, Iran"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9277-8092","authenticated-orcid":false,"given":"Hari","family":"Srivastava","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada"},{"name":"Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan"},{"name":"Department of Mathematics and Informatics, Azerbaijan University, 71 Jeyhun Hajibeyli Street, AZ1007 Baku, Azerbaijan"},{"name":"Center for Converging Humanities, Kyung Hee University, 26 Kyungheedae-ro, Dongdaemun-gu, Seoul 02447, Republic of Korea"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2022,11,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1695","DOI":"10.1063\/1.1668896","article-title":"Symmetry breaking instabilities in dissipative systems II","volume":"48","author":"Prigogine","year":"1968","journal-title":"J. Chem. Phys."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Epstein, I.R., and Pojman, J.K. (1998). An Introduction to Nonlinear Chemical Dynamics: Oscillations, Waves, Patterns, and Chaos, Oxford University Press.","DOI":"10.1093\/oso\/9780195096705.001.0001"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"993","DOI":"10.1039\/f19888400993","article-title":"The Brusselator model of oscillatory reactions: Relationships between two-variable and four-variable models with rigorous application of mass conservation and detailed balance","volume":"84","author":"Gray","year":"1988","journal-title":"J. Chem. Soc. Faraday Trans."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Field, R.J., and Gy\u00f6rgyi, L. (1993). Chaos in Chemistry and Biochemistry, World Scientific.","DOI":"10.1142\/1706"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Hilfer, R. (2000). Applications of Fractional Calculus in Physics, World Scientific.","DOI":"10.1142\/9789812817747"},{"key":"ref_6","unstructured":"Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006). Theory and Application of Fractional Differential Equations, North-Holland Mathematics Studies."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2947","DOI":"10.1166\/jctn.2020.9324","article-title":"Numerical investigation of Brusselator chemical model by residual function using Mathematica software","volume":"17","author":"Tong","year":"2020","journal-title":"J. Comput. Theoret. Nanosci."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"4902","DOI":"10.1016\/j.physleta.2008.05.045","article-title":"Stability analysis and limit cycle in fractional system with Brusselator nonlinearities","volume":"372","author":"Gafiychuk","year":"2008","journal-title":"Phys. Lett. A"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"637","DOI":"10.1007\/s10910-008-9362-y","article-title":"Dynamical behaviors of the Brusselator system with impulsive input","volume":"44","author":"Sun","year":"2008","journal-title":"J. Math. Chem."},{"key":"ref_10","first-page":"38","article-title":"Stability and a numerical solution of fractional-order Brusselator chemical reaction system","volume":"8","author":"Yuan","year":"2017","journal-title":"J. Fract. Calc. Appl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"414","DOI":"10.1016\/j.physleta.2006.11.038","article-title":"Does the fractional Brusselator with efficient dimension less than 1 have a limit cycle?","volume":"363","author":"Wang","year":"2007","journal-title":"Phys. Lett. A"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"102","DOI":"10.1186\/1687-1847-2013-102","article-title":"Nonstandard finite difference schemes for a fractional-order Brusselator system","volume":"2013","author":"Ongun","year":"2013","journal-title":"Adv. Differ. Equ."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"521","DOI":"10.1080\/10236198.2016.1257005","article-title":"Numerical treatment for nonlinear Brusselator chemical model","volume":"23","author":"Zafar","year":"2017","journal-title":"J. Differ. Equ. Appl."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"496323","DOI":"10.1155\/2014\/496323","article-title":"Variational iteration method for a fractional-order Brusselator system","volume":"2014","author":"Jafari","year":"2014","journal-title":"Abstr. Appl. Anal."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"450235","DOI":"10.1155\/2015\/450235","article-title":"Approximate analytical solutions of the fractional-order Brusselator system using the polynomial least squares method","volume":"2015","author":"Bota","year":"2015","journal-title":"Adv. Math. Phys."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"257484","DOI":"10.1155\/2014\/257484","article-title":"Numerical solutions of the nonlinear fractional-order Brusselator system by Bernstein polynomials","volume":"2014","author":"Khan","year":"2014","journal-title":"Sci. World J."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"012001","DOI":"10.1088\/1742-6596\/693\/1\/012001","article-title":"Legendre wavelet operational matrix of fractional derivative through wavelet-polynomial transformation and its applications in solving fractional order Brusselator system","volume":"693","author":"Chang","year":"2016","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"165","DOI":"10.19113\/sdufbed.24679","article-title":"Numerical solutions of fractional order autocatalytic chemical reaction model","volume":"21","author":"Ongun","year":"2017","journal-title":"S\u00fcleyman Demirel \u00dcnivers. Fen Bilim. Enstit\u00fcs\u00fc Dergisi"},{"key":"ref_19","first-page":"1","article-title":"A novel three-step iterative approach for oscillatory chemical reactions of fractional Brusselator model","volume":"204","author":"Asv","year":"2022","journal-title":"Int. J. Model. Simul."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"11187","DOI":"10.1016\/j.chaos.2021.111087","article-title":"Fractal-fractional Brusselator chemical reaction","volume":"150","author":"Saad","year":"2021","journal-title":"Chaos Solit. Fract."},{"key":"ref_21","first-page":"113","article-title":"A meshless method for solving the 2D Brusselator reaction-diffusion system","volume":"101","author":"Mohammadi","year":"2014","journal-title":"Comput. Model. Eng. Sci."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Tlidi, M., Gandica, Y., Sonnino, G., Averlant, E., and Panajotov, K. (2016). Self-Replicating spots in the Brusselator model and extreme events in the one-dimensional case with delay. Entropy, 18.","DOI":"10.3390\/e18030064"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Alfifi, H.Y. (2021). Feedback control for a diffusive and delayed Brusselator model: Semi-analytical solutions. Symmetry, 13.","DOI":"10.3390\/sym13040725"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"140","DOI":"10.1007\/s40819-019-0727-7","article-title":"A highly accurate time\u2013space pseudospectral approximation and stability analysis of two dimensional Brusselator model for chemical systems","volume":"5","author":"Mittal","year":"2019","journal-title":"Int. J. Appl. Comput. Math."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1186\/s13661-022-01598-x","article-title":"Time accurate solution to Benjamin-Bona-Mahony Burgers equation via Taylor-Boubaker series scheme","volume":"2022","author":"Izadi","year":"2022","journal-title":"Bound. Value Probl."},{"key":"ref_26","first-page":"914","article-title":"Error analysis and Kronecker implementation of Chebyshev spectral collocation method for solving linear PDEs","volume":"10","author":"Razavi","year":"2022","journal-title":"Comput. Methods Differ. Equ."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Izadi, M., Srivastava, H.M., and Adel, W. (2022). An effective approximation algorithm for second-order singular functional differential equations. Axioms, 11.","DOI":"10.3390\/axioms11030133"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"100146","DOI":"10.1016\/j.rinam.2021.100146","article-title":"Bernoulli wavelet method for numerical solution of anomalous infiltration and diffusion modeling by nonlinear fractional differential equations of variable order","volume":"10","author":"Chouhan","year":"2021","journal-title":"Results Appl. Math."},{"key":"ref_29","first-page":"126123","article-title":"An efficient approximation technique applied to a non-linear Lane-Emden pantograph delay differential model","volume":"401","author":"Izadi","year":"2021","journal-title":"Appl. Math. Comput."},{"key":"ref_30","first-page":"127319","article-title":"Bessel-quasilinearization technique to solve the fractional-order HIV-1 infection of CD4+ T-cells considering the impact of antiviral drug treatment","volume":"431","author":"Izadi","year":"2022","journal-title":"Appl. Math. Comput."},{"key":"ref_31","first-page":"1501","article-title":"Some parametric and argument variations of the operators of fractional calculus and related special functions and integral transformations","volume":"22","author":"Srivastava","year":"2021","journal-title":"J. Nonlinear Convex Anal."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"135","DOI":"10.55579\/jaec.202153.340","article-title":"An introductory overview of fractional-calculus operators based upon the Fox-Wright and related higher transcendental functions","volume":"5","author":"Srivastava","year":"2021","journal-title":"J. Adv. Eng. Comput."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1016\/0012-365X(94)90163-5","article-title":"Clique polynomials and independent set polynomials of graphs","volume":"125","author":"Hoede","year":"1994","journal-title":"Discrete Math."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"435","DOI":"10.1007\/s40324-020-00225-y","article-title":"A new approach for the numerical solution for the non-linear Klein\u2013Gordon equation","volume":"77","author":"Kumbinarasaiah","year":"2020","journal-title":"SeMA J."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"4563","DOI":"10.1016\/j.aej.2021.03.026","article-title":"Numerical solutions of time-fractional Klein-Gordon equations by clique polynomials","volume":"60","author":"Ganji","year":"2021","journal-title":"Alexandria Eng. J."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Adel, W., and Kumbinarasaiah, S. (2022). A new clique polynomial approach for fractional partial differential equations. Int. J. Nonlinear Sci. Numer. Simul.","DOI":"10.1515\/ijnsns-2021-0258"},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Heydari, M.H., and Razzaghi, M. (2021). Highly accurate solutions for space-time fractional Schr\u00f6dinger equations with non-smooth continuous solution using the hybrid clique functions. Math. Sci.","DOI":"10.1007\/s40096-021-00437-x"},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Stewart, G.W. (1996). Afternotes on Numerical Analysis, SIAM.","DOI":"10.1137\/1.9781611971491"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"4156","DOI":"10.1002\/mma.7020","article-title":"A fractional-order generalized Taylor wavelet method for nonlinear fractional delay and nonlinear fractional pantograph differential equations","volume":"44","author":"Yuttanan","year":"2021","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_40","first-page":"739","article-title":"An approximation technique for first Painlev\u00e9 equation","volume":"11","author":"Izadi","year":"2021","journal-title":"TWMS J. Appl. Eng. Math."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Izadi, M., Y\u00fczbas\u0131, \u015e., and Noeiaghdam, S. (2021). Approximating solutions of non-linear Troesch\u2019s problem via an efficient quasi-linearization Bessel approach. Mathematics, 9.","DOI":"10.3390\/math9161841"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"8753","DOI":"10.1002\/mma.6542","article-title":"Generalized wavelet quasi-linearization method for solving population growth model of fractional order","volume":"43","author":"Srivastava","year":"2020","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Izadi, M., and Srivastava, H.M. (2021). Generalized Bessel quasilinearlization technique applied to Bratu and Lane-Emden type equations of arbitrary order. Fractal Fract., 5.","DOI":"10.3390\/fractalfract5040179"},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Izadi, M., Y\u00fczba\u015f\u0131, \u015e., and Adel, W. (2022). A new Chelyshkov matrix method to solve linear and nonlinear fractional delay differential equations with error analysis. Math. Sci.","DOI":"10.1007\/s40096-022-00468-y"},{"key":"ref_45","first-page":"2021031","article-title":"A novel matrix technique for multi-order pantograph differential equations of fractional order","volume":"477","author":"Izadi","year":"2021","journal-title":"Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci."},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Izadi, M., Y\u00fczba\u015f\u0131, \u015e., and Cattani, C. (2021). Approximating solutions to fractional-order Bagley-Torvik equation via generalized Bessel polynomial on large domains. Ricerche Mat.","DOI":"10.1007\/s11587-021-00650-9"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/11\/654\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:21:13Z","timestamp":1760145673000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/11\/654"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,11,18]]},"references-count":46,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2022,11]]}},"alternative-id":["axioms11110654"],"URL":"https:\/\/doi.org\/10.3390\/axioms11110654","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,11,18]]}}}