{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,13]],"date-time":"2026-01-13T23:10:43Z","timestamp":1768345843893,"version":"3.49.0"},"reference-count":64,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2023,10,25]],"date-time":"2023-10-25T00:00:00Z","timestamp":1698192000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001659","name":"German Research Foundation","doi-asserted-by":"publisher","award":["DFG\u2013FOR5409"],"award-info":[{"award-number":["DFG\u2013FOR5409"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Nonlinear coupled reaction\u2013diffusion (NCRD) systems have played a crucial role in the emergence of spatiotemporal patterns across various scientific and engineering domains. The NCRD systems considered in this study encompass various models, such as linear, Gray\u2013Scott, Brusselator, isothermal chemical, and Schnakenberg, with the aim of capturing the spatiotemporal patterns they generate. These models cover a diverse range of intricate spatiotemporal patterns found in nature, including spots, spot replication, stripes, hexagons, and more. A mixed-type modal discontinuous Galerkin approach is employed for solving one- and two-dimensional NCRD systems. This approach introduces a mathematical formulation to handle the occurrence of second-order derivatives in diffusion terms. For spatial discretization, hierarchical modal basis functions premised on orthogonal scaled Legendre polynomials are used. Moreover, a novel reaction term treatment is proposed for the NCRD systems, demonstrating an intrinsic feature of the new DG scheme and preventing erroneous solutions due to extremely nonlinear reaction terms. The proposed approach reduces the NCRD systems into a framework of ordinary differential equations in time, which are addressed by an explicit third-order TVD Runge\u2013Kutta algorithm. The spatiotemporal patterns generated with the present approach are comparable to those found in the literature. This approach can readily be expanded to handle large multi-dimensional problems that appear as model equations in developed biological and chemical applications.<\/jats:p>","DOI":"10.3390\/axioms12111004","type":"journal-article","created":{"date-parts":[[2023,10,25]],"date-time":"2023-10-25T09:54:08Z","timestamp":1698227648000},"page":"1004","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["On the Spatiotemporal Pattern Formation in Nonlinear Coupled Reaction\u2013Diffusion Systems"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6669-5296","authenticated-orcid":false,"given":"Satyvir","family":"Singh","sequence":"first","affiliation":[{"name":"Applied and Computational Mathematics, RWTH Aachen University, 52062 Aachen, Germany"},{"name":"Department of Mathematics, Graphic Era Deemed to be University, Dehradun 248002, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0186-0526","authenticated-orcid":false,"given":"Ahmed Hussein","family":"Msmali","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, Jazan University, Jazan 45142, Saudi Arabia"},{"name":"School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW 2500, Australia"}]}],"member":"1968","published-online":{"date-parts":[[2023,10,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Murray, J.D. 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