{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,22]],"date-time":"2026-06-22T19:34:36Z","timestamp":1782156876113,"version":"3.54.5"},"reference-count":43,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2022,9,18]],"date-time":"2022-09-18T00:00:00Z","timestamp":1663459200000},"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>This article investigates different nonlinear systems of fractional partial differential equations analytically using an attractive modified method known as the Laplace residual power series technique. Based on a combination of the Laplace transformation and the residual power series technique, we achieve analytic and approximation results in rapid convergent series form by employing the notion of the limit, with less time and effort than the residual power series method. Three challenges are evaluated and simulated to validate the suggested method\u2019s practicability, efficiency, and simplicity. The analysis of the acquired findings demonstrates that the method mentioned above is simple, accurate, and appropriate for investigating the solutions to nonlinear applied sciences models.<\/jats:p>","DOI":"10.3390\/sym14091944","type":"journal-article","created":{"date-parts":[[2022,9,20]],"date-time":"2022-09-20T04:28:55Z","timestamp":1663648135000},"page":"1944","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":77,"title":["Fractional Series Solution Construction for Nonlinear Fractional Reaction-Diffusion Brusselator Model Utilizing Laplace Residual Power Series"],"prefix":"10.3390","volume":"14","author":[{"given":"Aisha Abdullah","family":"Alderremy","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, King Khalid University, Abha 61413, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4306-8489","authenticated-orcid":false,"given":"Rasool","family":"Shah","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Abdul Wali Khan University, Mardan P.O. Box 1100, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8548-7078","authenticated-orcid":false,"given":"Naveed","family":"Iqbal","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, University of Ha\u2019il, Ha\u2019il P.O. Box 2440, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8286-8123","authenticated-orcid":false,"given":"Shaban","family":"Aly","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, AL-Azhar University, Assiut 71516, Egypt"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7469-5402","authenticated-orcid":false,"given":"Kamsing","family":"Nonlaopon","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2022,9,18]]},"reference":[{"key":"ref_1","unstructured":"Oldham, K., and Spanier, J. (1974). The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, Academic Press."},{"key":"ref_2","unstructured":"Miller, K.S., and Ross, B. (1993). An Introduction to Fractional Calculus and Fractional Differential Equations, Wiley."},{"key":"ref_3","unstructured":"Podlubny, I. (1999). Fractional Differential Equations, Academic Press."},{"key":"ref_4","unstructured":"Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006). Theory and Applications of Fractional Differential Equations, Elsevier."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"12483","DOI":"10.3934\/math.2022693","article-title":"On the solution of fractional modified Boussinesq and approximate long wave equations with non-singular kernel operators","volume":"7","author":"Botmart","year":"2022","journal-title":"AIMS Math."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1016\/j.jcp.2019.03.008","article-title":"A review of definitions of fractional derivatives and other operators","volume":"388","author":"Teodoro","year":"2019","journal-title":"J. Comput. Phys."},{"key":"ref_7","first-page":"174","article-title":"On the concept of existence and local attractivity of solutions for some quadratic Volterra integral equation of fractional order","volume":"285","author":"Mishra","year":"2016","journal-title":"Appl. Math. Comput."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Pathak, V.K., and Mishra, L.N. (2022). Application of fixed point theorem to solvability for non-linear fractional hadamard functional integral equations. Mathematics, 10.","DOI":"10.3390\/math10142400"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Mukhtar, S., Shah, R., and Noor, S. (2022). The Numerical Investigation of a Fractional-Order Multi-Dimensional Model of Navier-Stokes Equation via Novel Techniques. Symmetry, 14.","DOI":"10.3390\/sym14061102"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Mainardi, F. (2010). Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press.","DOI":"10.1142\/9781848163300"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Klafter, J., Lim, S., and Metzler, R. (2011). Fractional Dynamics in Physics: Recent Advances, World Scientific.","DOI":"10.1142\/9789814340595"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Tarasov, V. (2011). Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer.","DOI":"10.1007\/978-3-642-14003-7"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"18334","DOI":"10.3934\/math.20221010","article-title":"Fractional view analysis of Kersten-Krasil\u2019shchik coupled KdV-mKdV systems with non-singular kernel derivatives","volume":"7","author":"Khan","year":"2022","journal-title":"AIMS Math."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"West, B., Bologna, M., and Grigolini, P. (2003). Physics of Fractal Operators, Springer.","DOI":"10.1007\/978-0-387-21746-8"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"6936","DOI":"10.3934\/math.2022385","article-title":"Analytical investigation of fractional-order Newell-Whitehead-Segel equations via a novel transform","volume":"7","author":"Areshi","year":"2022","journal-title":"AIMS Math."},{"key":"ref_16","unstructured":"Harris, F.E. (2014). Mathematics for Physical Science and Engineering: Symbolic Computing Applications in Maple and Mathematica, Academic Press."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Rashid, S., Khalid, A., Sultana, S., Hammouch, Z., and Alsharif, A.M. (2021). A novel analytical view of time-fractional Korteweg-De Vries equations via a new integral transform. Symmetry, 13.","DOI":"10.3390\/sym13071254"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1002\/num.1690110303","article-title":"Technical note: The numerical solution of the system of 3-D nonlinear elliptic equations with mixed derivatives and variable coefficients using fourth-order difference methods","volume":"11","author":"Mohanty","year":"1995","journal-title":"Numer. Methods Part. Differ. Equ."},{"key":"ref_19","first-page":"53","article-title":"High accuracy numerov type discretization for the solution of one-space dimensional non-linear wave equations with variable coefficients","volume":"3","author":"Mohanty","year":"2011","journal-title":"J. Adv. Res. Sci. Comput."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1007\/s40819-022-01285-6","article-title":"Recent development of Adomian decomposition method for ordinary and partial differential equations","volume":"8","author":"Kumar","year":"2022","journal-title":"Int. J. Appl. Comput. Math."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"8876149","DOI":"10.1155\/2022\/8876149","article-title":"A comparative analysis of the fractional-order coupled Korteweg-De Vries equations with the Mittag-Leffler law","volume":"2022","author":"Aljahdaly","year":"2022","journal-title":"J. Math."},{"key":"ref_22","first-page":"3341754","article-title":"On solutions of fractional-order gas dynamics equation by effective techniques","volume":"2022","author":"Iqbal","year":"2022","journal-title":"J. Funct. Spaces"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1090\/proc\/15174","article-title":"Inverse scattering and soliton solutions of nonlocal reverse-spacetime nonlinear Schr\u00f6dinger equations","volume":"149","author":"Ma","year":"2021","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1007\/s12346-021-00512-7","article-title":"Hetero-B\u00e4cklund transformation, bilinear forms and N solitons for a generalized three-coupled Korteweg-de Vries system","volume":"20","author":"Gao","year":"2021","journal-title":"Qual. Theory Dyn. Syst."},{"key":"ref_25","first-page":"559","article-title":"The sine\u2013cosine method for obtaining solutions with compact and noncompact structures","volume":"159","author":"Wazwaz","year":"2004","journal-title":"Appl. Math. Comput."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Al-Askar, F.M., Mohammed, W.W., Albalahi, A.M., and El-Morshedy, M. (2022). The Impact of the Wiener process on the analytical solutions of the stochastic (2+1)-dimensional breaking soliton equation by using tanh\u2013coth method. Mathematics, 10.","DOI":"10.3390\/math10050817"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1139","DOI":"10.1007\/s10910-021-01237-3","article-title":"Homotopy perturbation method with three expansions","volume":"59","author":"He","year":"2021","journal-title":"J. Math. Chem."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Shah, N.A., Alyousef, H.A., El-Tantawy, S.A., Shah, R., and Chung, J.D. (2022). Analytical Investigation of Fractional-Order Korteweg-De-Vries-Type Equations under Atangana-Baleanu-Caputo Operator: Modeling Nonlinear Waves in a Plasma and Fluid. Symmetry, 14.","DOI":"10.3390\/sym14040739"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Ahmad, I., Ahmad, H., Thounthong, P., Chu, Y.M., and Cesarano, C. (2020). Solution of multi-term time-fractional PDE models arising in mathematical biology and physics by local meshless method. Symmetry, 12.","DOI":"10.3390\/sym12071195"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Shah, N.A., Hamed, Y.S., Abualnaja, K.M., Chung, J.D., and Khan, A. (2022). A comparative analysis of fractional-order kaup-kupershmidt equation within different operators. Symmetry, 14.","DOI":"10.3390\/sym14050986"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Kbiri Alaoui, M., Nonlaopon, K., Zidan, A.M., Khan, A., and Shah, R. (2022). Analytical investigation of fractional-order cahn-hilliard and gardner equations using two novel techniques. Mathematics, 10.","DOI":"10.3390\/math10101643"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"109957","DOI":"10.1016\/j.chaos.2020.109957","article-title":"A new attractive analytic approach for solutions of linear and nonlinear neutral fractional pantograph equations","volume":"138","author":"Eriqat","year":"2020","journal-title":"Chaos Solitons Fractals"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1140\/epjp\/s13360-020-01061-9","article-title":"Adapting the Laplace transform to create solitary solutions for the nonlinear time-fractional dispersive PDEs via a new approach","volume":"136","year":"2021","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1069","DOI":"10.1016\/j.aej.2021.07.020","article-title":"A new efficient technique using Laplace transforms and smooth expansions to construct a series solution to the time-fractional Navier-Stokes equations","volume":"61","author":"Burqan","year":"2022","journal-title":"Alex. Eng. J."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"356","DOI":"10.1515\/fca-2020-0017","article-title":"A class of linear non-homogenous higher order matrix fractional differential equations: Analytical solutions and new technique","volume":"23","author":"Oqielat","year":"2020","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"402","DOI":"10.1140\/epjp\/i2019-12731-x","article-title":"Series solutions of non-linear conformable fractional KdV-Burgers equation with some applications","volume":"134","author":"Oqielat","year":"2019","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"2101","DOI":"10.1016\/j.aej.2020.01.023","article-title":"Series solutions for nonlinear time-fractional Schr\u00f6dinger equations: Comparisons between conformable and Caputo derivatives","volume":"59","author":"Oqielat","year":"2020","journal-title":"Alex. Eng. J."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"102500","DOI":"10.1016\/j.rinp.2019.102500","article-title":"Analytical numerical solutions of the fractional multi-pantograph system: Two attractive methods and comparisons","volume":"14","author":"Oqielat","year":"2019","journal-title":"Results Phys."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Shqair, M., El-Ajou, A., and Nairat, M. (2019). Analytical solution for multi-energy groups of neutron diffusion equations by a residual power series method. Mathematics, 7.","DOI":"10.3390\/math7070633"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"1243","DOI":"10.1016\/j.asej.2020.03.016","article-title":"Smooth expansion to solve high-order linear conformable fractional PDEs via residual power series method: Applications to physical and engineering equations","volume":"11","author":"Oqielat","year":"2020","journal-title":"Ain Shams Eng. J."},{"key":"ref_41","unstructured":"Hanna, J., and Rowland, J. (1990). Fourier Series, Transforms, and Boundary Value Problems, Wiley."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"385","DOI":"10.1016\/j.jcp.2014.09.034","article-title":"Construct and predicts solitary pattern solutions for nonlinear time-fractional dispersive partial differential equations","volume":"293","author":"Arqub","year":"2015","journal-title":"J. Comput. Phys."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"103667","DOI":"10.1016\/j.rinp.2020.103667","article-title":"Promoted residual power series technique with Laplace transform to solve some time-fractional problems arising in physics","volume":"19","author":"Alquran","year":"2020","journal-title":"Results Phys."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/9\/1944\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:33:54Z","timestamp":1760142834000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/9\/1944"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,9,18]]},"references-count":43,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2022,9]]}},"alternative-id":["sym14091944"],"URL":"https:\/\/doi.org\/10.3390\/sym14091944","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,9,18]]}}}