{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,14]],"date-time":"2026-03-14T19:39:58Z","timestamp":1773517198992,"version":"3.50.1"},"reference-count":41,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2024,6,10]],"date-time":"2024-06-10T00:00:00Z","timestamp":1717977600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Princess Nourah bint Abdulrahman University Researchers Supporting Project","award":["PNURSP2024R514"],"award-info":[{"award-number":["PNURSP2024R514"]}]},{"name":"Prince Sultan University","award":["PNURSP2024R514"],"award-info":[{"award-number":["PNURSP2024R514"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Fractional order differential equations often possess inherent symmetries that play a crucial role in governing their dynamics in a variety of scientific fields. In this work, we consider numerical solutions for fractional-order linear delay differential equations. The numerical solution is obtained via the Laplace transform technique. The quadrature approximation of the Bromwich integral provides the foundation for several commonly employed strategies for inverting the Laplace transform. The key factor for quadrature approximation is the contour deformation, and numerous contours have been proposed. However, the highly convergent trapezoidal rule has always been the most common quadrature rule. In this work, the Gauss\u2013Hermite quadrature rule is used as a substitute for the trapezoidal rule. Plotting figures of absolute error and comparing results to other methods from the literature illustrate how effectively the suggested approach works. Functional analysis was used to examine the existence of the solution and the Ulam\u2013Hyers (UH) stability of the considered equation.<\/jats:p>","DOI":"10.3390\/sym16060721","type":"journal-article","created":{"date-parts":[[2024,6,10]],"date-time":"2024-06-10T10:45:56Z","timestamp":1718016356000},"page":"721","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Numerical Solution of the Linear Fractional Delay Differential Equation Using Gauss\u2013Hermite Quadrature"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7928-5624","authenticated-orcid":false,"given":"Salma","family":"Aljawi","sequence":"first","affiliation":[{"name":"Department of Mathematical Sciences, Princess Nourah Bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0009-0008-2035-0880","authenticated-orcid":false,"given":"Sarah","family":"Aljohani","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8217-3308","authenticated-orcid":false,"family":"Kamran","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Islamia College Peshawar, Peshawar 25120, Khyber Pakhtoonkhwa, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0009-0003-1449-1963","authenticated-orcid":false,"given":"Asma","family":"Ahmed","sequence":"additional","affiliation":[{"name":"Department of Computer Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7986-886X","authenticated-orcid":false,"given":"Nabil","family":"Mlaiki","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2024,6,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Alyobi, S., and Jan, R. (2003). Qualitative and quantitative analysis of fractional dynamics of infectious diseases with control measures. Fractal Fract., 7.","DOI":"10.3390\/fractalfract7050400"},{"key":"ref_2","unstructured":"Miller, K.S., and Ross, B. (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons."},{"key":"ref_3","unstructured":"Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006). Theory and Applications of Fractional Differential Equations, Elsevier."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Shah, F.A., Aly, W.H.F., Aksoy, H., Alotaibi, F.M., and Mahariq, I. (2022). Numerical Inverse Laplace Transform Methods for Advection-Diffusion Problems. Symmetry, 14.","DOI":"10.3390\/sym14122544"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1016\/j.chaos.2017.03.022","article-title":"Hyperchaotic behaviour obtained via a nonlocal operator with exponential decay and Mittag-Leffler laws","volume":"102","author":"Atangana","year":"2017","journal-title":"Chaos Solitons Fractals"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Tang, T.Q., Shah, Z., Bonyah, E., Jan, R., Shutaywi, M., and Alreshidi, N. (2022). Modeling and analysis of breast cancer with adverse reactions of chemotherapy treatment through fractional derivative. Computational and Mathematical Methods in Medicine, John Wiley & Sons.","DOI":"10.1155\/2022\/5636844"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2630","DOI":"10.1177\/01423312221085049","article-title":"Analysis of fractional-order dynamics of dengue infection with non-linear incidence functions","volume":"44","author":"Jan","year":"2022","journal-title":"Trans. Inst. Meas. Control."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"367","DOI":"10.1140\/epjp\/s13360-022-02591-0","article-title":"Modeling the dynamics of tumor\u2013immune cells interactions via fractional calculus","volume":"137","author":"Tang","year":"2022","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_9","first-page":"321","article-title":"Analyses of an antiviral immune response model with time delays","volume":"6","author":"Cooke","year":"1998","journal-title":"Can. Appl. Math. Q."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"332","DOI":"10.1007\/s002850050194","article-title":"Interaction of maturation delay and nonlinear birth in population and epidemic models","volume":"39","author":"Cooke","year":"1999","journal-title":"J. Math. Biol."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1016\/S0025-5564(99)00055-3","article-title":"A model of HIV-1 pathogenesis that includes an intracellular delay","volume":"163","author":"Nelson","year":"2000","journal-title":"Math. Biosci."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1016\/S0025-5564(98)10028-7","article-title":"Delay equation analysis of human respiratory stability","volume":"152","author":"Vielle","year":"1998","journal-title":"Math. Biosci."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1016\/j.cam.2012.06.034","article-title":"Analysis and numerical methods for fractional differential equations with delay","volume":"252","author":"Morgado","year":"2013","journal-title":"J. Comput. Appl. Math."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"120","DOI":"10.1016\/j.jtusci.2013.07.002","article-title":"A numerical method based on finite difference for solving fractional delay differential equations","volume":"7","author":"Moghaddam","year":"2013","journal-title":"J. Taibah Univ. Sci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s40314-019-0951-0","article-title":"A new numerical method for solving fractional delay differential equations","volume":"38","author":"Jhinga","year":"2019","journal-title":"Comput. Appl. Math."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"471","DOI":"10.1080\/00207160.2015.1114610","article-title":"Numerical solution of fractional delay differential equation by shifted Jacobi polynomials","volume":"94","author":"Muthukumar","year":"2017","journal-title":"Int. J. Comput. Math."},{"key":"ref_17","first-page":"1","article-title":"A Comparative Study on Solution Methods for Fractional order Delay Differential Equations and its Applications","volume":"2","author":"Chishti","year":"2023","journal-title":"Math. Sci. Appl."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1007\/s40096-022-00468-y","article-title":"A new Chelyshkov matrix method to solve linear and nonlinear fractional delay differential equations with error analysis","volume":"17","author":"Izadi","year":"2023","journal-title":"Math. Sci."},{"key":"ref_19","first-page":"104","article-title":"Reduce Differential Transform Method for Analytical Approximation of Fractional Delay Differential Equation","volume":"1","author":"Naseem","year":"2022","journal-title":"Int. J. Emerg. Multidiscip. Math."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"246","DOI":"10.1016\/j.cnsns.2016.12.027","article-title":"A differential transformation approach for solving functional differential equations with multiple delays","volume":"48","author":"Rebenda","year":"2017","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"6930","DOI":"10.1038\/s41598-024-56944-z","article-title":"Numerical simulation of a fractional stochastic delay differential equations using spectral scheme: A comprehensive stability analysis","volume":"14","author":"Li","year":"2024","journal-title":"Sci. Rep."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"7321","DOI":"10.3934\/math.2024355","article-title":"Analysis of a nonlinear problem involving discrete and proportional delay with application to Houseflies model","volume":"9","author":"Shah","year":"2024","journal-title":"Aims Math."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"6464","DOI":"10.1002\/mma.6390","article-title":"On qualitative theory of fractional order delay evolution equation via the prior estimate method","volume":"43","author":"Sher","year":"2020","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"951","DOI":"10.1016\/j.cjph.2023.03.014","article-title":"Stability analysis of fractional-order differential equations with multiple delays: The 1<\u03b1<2 case","volume":"89","author":"Yao","year":"2024","journal-title":"Chin. J. Phys."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Kamal, R., Alzahrani, S.M., and Alzahrani, T. (2023). A Hybrid Local Radial Basis Function Method for the Numerical Modeling of Mixed Diffusion and Wave-Diffusion Equations of Fractional Order Using Caputo\u2019s Derivatives. Fractal Fract., 7.","DOI":"10.3390\/fractalfract7050381"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"4640467","DOI":"10.1155\/2021\/4640467","article-title":"On the numerical approximation of three-dimensional time fractional convection-diffusion equations","volume":"2021","author":"Kamran","year":"2021","journal-title":"Math. Probl. Eng."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"831","DOI":"10.5802\/crmath.98","article-title":"A transform based local RBF method for 2D linear PDE with Caputo\u2013Fabrizio derivative","volume":"358","author":"Kamran","year":"2020","journal-title":"Comptes Rendus. Math\u00e9matique"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1016\/j.apnum.2004.06.015","article-title":"On the numerical inversion of the Laplace transform of certain holomorphic mappings","volume":"51","author":"Palencia","year":"2004","journal-title":"Appl. Numer. Math."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1214","DOI":"10.3390\/sym15061214","article-title":"On the Approximation of Fractional-Order Differential Equations Using Laplace Transform and Weeks Method","volume":"15","author":"Kamran","year":"2023","journal-title":"Symmetry"},{"key":"ref_30","first-page":"379","article-title":"On the solution of the delay differential equation via Laplace transform","volume":"11","author":"Cimen","year":"2020","journal-title":"Commun. Math. Appl."},{"key":"ref_31","first-page":"1","article-title":"Initial value problems for Caputo fractional equations with singular nonlinearities","volume":"2019","author":"Webb","year":"2019","journal-title":"Electron. J. Differ. Equ."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Agarwal, R.P., Meehan, M., and O\u2019regan, D. (2001). Fixed Point Theory and Applications, Cambridge University Press.","DOI":"10.1017\/CBO9780511543005"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1093\/imamat\/23.1.97","article-title":"The accurate numerical inversion of Laplace transforms","volume":"23","author":"Talbot","year":"1979","journal-title":"Ima J. Appl. Math."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"385","DOI":"10.1137\/130932132","article-title":"The exponentially convergent trapezoidal rule","volume":"56","author":"Trefethen","year":"2014","journal-title":"Siam Rev."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1341","DOI":"10.1090\/S0025-5718-07-01945-X","article-title":"Parabolic and hyperbolic contours for computing the Bromwich integral","volume":"76","author":"Weideman","year":"2007","journal-title":"Math. Comput."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"6542787","DOI":"10.1155\/2023\/6542787","article-title":"Computational approach for differential equations with local and nonlocal fractional-order differential operators","volume":"2023","author":"Kamran","year":"2023","journal-title":"J. Math."},{"key":"ref_37","first-page":"2743","article-title":"On the Approximation of Fractal-Fractional Differential Equations Using Numerical Inverse Laplace Transform Methods","volume":"135","author":"Kamran","year":"2023","journal-title":"Cmes-Comput. Model. Eng. Sci."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"2200","DOI":"10.1137\/18M1196273","article-title":"Gauss\u2013Hermite quadrature for the Bromwich integral","volume":"57","author":"Weideman","year":"2019","journal-title":"Siam J. Numer. Anal."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"272","DOI":"10.1093\/comjnl\/3.4.272","article-title":"Convergence properties of Gaussian quadrature formulae","volume":"3","author":"Barrett","year":"1961","journal-title":"Comput. J."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1080\/00036817108839015","article-title":"Estimation of errors in the numerical quadrature of analytic functions","volume":"1","author":"Takahasi","year":"1971","journal-title":"Appl. Anal."},{"key":"ref_41","first-page":"256071","article-title":"A numerical method for delayed fractional-order differential equations","volume":"2013","author":"Wang","year":"2013","journal-title":"J. Appl. Math."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/6\/721\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:56:20Z","timestamp":1760108180000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/6\/721"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,6,10]]},"references-count":41,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2024,6]]}},"alternative-id":["sym16060721"],"URL":"https:\/\/doi.org\/10.3390\/sym16060721","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,6,10]]}}}