{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,12]],"date-time":"2026-06-12T16:57:18Z","timestamp":1781283438733,"version":"3.54.1"},"reference-count":29,"publisher":"Elsevier BV","license":[{"start":{"date-parts":[[2026,12,1]],"date-time":"2026-12-01T00:00:00Z","timestamp":1796083200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"},{"start":{"date-parts":[[2026,12,1]],"date-time":"2026-12-01T00:00:00Z","timestamp":1796083200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/legal\/tdmrep-license"},{"start":{"date-parts":[[2026,12,1]],"date-time":"2026-12-01T00:00:00Z","timestamp":1796083200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-017"},{"start":{"date-parts":[[2026,12,1]],"date-time":"2026-12-01T00:00:00Z","timestamp":1796083200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-037"},{"start":{"date-parts":[[2026,12,1]],"date-time":"2026-12-01T00:00:00Z","timestamp":1796083200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-012"},{"start":{"date-parts":[[2026,12,1]],"date-time":"2026-12-01T00:00:00Z","timestamp":1796083200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-029"},{"start":{"date-parts":[[2026,12,1]],"date-time":"2026-12-01T00:00:00Z","timestamp":1796083200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-004"}],"content-domain":{"domain":["elsevier.com","sciencedirect.com"],"crossmark-restriction":true},"short-container-title":["Journal of Computational and Applied Mathematics"],"published-print":{"date-parts":[[2026,12]]},"DOI":"10.1016\/j.cam.2026.117572","type":"journal-article","created":{"date-parts":[[2026,3,8]],"date-time":"2026-03-08T15:42:30Z","timestamp":1772984550000},"page":"117572","update-policy":"https:\/\/doi.org\/10.1016\/elsevier_cm_policy","source":"Crossref","is-referenced-by-count":1,"special_numbering":"C","title":["Fractional Bernoulli polynomials optimization approach for nonlinear system of time fractional PDEs in (2+1)-dimensional"],"prefix":"10.1016","volume":"488","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2257-1798","authenticated-orcid":false,"given":"Zakieh","family":"Avazzadeh","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6916-1492","authenticated-orcid":false,"given":"Hossein","family":"Hassani","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mohammad Javad","family":"Ebadi","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7556-8942","authenticated-orcid":false,"given":"Praveen","family":"Agarwal","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7206-4455","authenticated-orcid":false,"given":"Roghayeh","family":"Katani","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"78","reference":[{"key":"10.1016\/j.cam.2026.117572_bib0001","series-title":"Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications","volume":"Vol. 198","author":"Podlubny","year":"1999"},{"key":"10.1016\/j.cam.2026.117572_bib0002","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1016\/j.apnum.2022.01.015","article-title":"Two-dimensional Gegenbauer wavelets for the numerical solution of tempered fractional model of the nonlinear Klein-Gordon equation","volume":"174","author":"Rayal","year":"2022","journal-title":"Appl. Numer. Math."},{"key":"10.1016\/j.cam.2026.117572_bib0003","series-title":"Fractional Calculus: Models and Numerical Methods","author":"Baleanu","year":"2012"},{"key":"10.1016\/j.cam.2026.117572_bib0004","doi-asserted-by":"crossref","DOI":"10.1016\/j.cam.2024.115772","article-title":"A numerical method for solutions of tempered fractional differential equations","volume":"443","author":"Bibi","year":"2024","journal-title":"J. Computat. Appl. Math."},{"issue":"5","key":"10.1016\/j.cam.2026.117572_bib0005","doi-asserted-by":"crossref","first-page":"1248","DOI":"10.1016\/j.chaos.2005.10.068","article-title":"Numerical comparison of methods for solving linear differential equations of fractional order","volume":"31","author":"Momani","year":"2007","journal-title":"Chaos Solitons Fractals"},{"key":"10.1016\/j.cam.2026.117572_bib0006","doi-asserted-by":"crossref","DOI":"10.1016\/j.cnsns.2022.106620","article-title":"An efficient hybrid numerical method for multi-term time fractional partial differential equations in fluid mechanics with convergence and error analysis","volume":"114","author":"Joujehi","year":"2022","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"10.1016\/j.cam.2026.117572_bib0007","doi-asserted-by":"crossref","DOI":"10.1016\/j.cam.2022.114647","article-title":"A second order numerical method for the time-fractional Black-Scholes European option pricing model","volume":"418","author":"Kazmi","year":"2023","journal-title":"J. Computat. Appl. Math."},{"key":"10.1016\/j.cam.2026.117572_bib0008","doi-asserted-by":"crossref","DOI":"10.1016\/j.cnsns.2022.106608","article-title":"Galerkin operational approach for multi-dimensions fractional differential equations","volume":"114","author":"Alsuyuti","year":"2022","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"10.1016\/j.cam.2026.117572_bib0009","doi-asserted-by":"crossref","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":"10.1016\/j.cam.2026.117572_bib0010","doi-asserted-by":"crossref","first-page":"349","DOI":"10.1016\/j.apnum.2020.10.018","article-title":"A numerical method based on fractional-order generalized Taylor wavelets for solving distributed-order fractional partial differential equations","volume":"160","author":"Yuttanan","year":"2021","journal-title":"Appl. Numer. Math."},{"issue":"30","key":"10.1016\/j.cam.2026.117572_bib0011","first-page":"29","article-title":"On an operational matrix method based on generalized Bernoulli polynomials of level m","volume":"55","author":"Quintana","year":"2018","journal-title":"Calcolo"},{"key":"10.1016\/j.cam.2026.117572_bib0012","series-title":"New U-Bernoulli, U-Euler and U-Genocchi Polynomials and Their Matrices","volume":"Vol. 15","author":"Ramirez","year":"2023"},{"key":"10.1016\/j.cam.2026.117572_bib0013","doi-asserted-by":"crossref","first-page":"377","DOI":"10.22436\/jmcs.032.04.07","article-title":"On F-Frobenius-Euler polynomials and their matrix approach","volume":"32","author":"Urielesa","year":"2024","journal-title":"J. Math. Comput. Sci."},{"key":"10.1016\/j.cam.2026.117572_bib0014","doi-asserted-by":"crossref","first-page":"291","DOI":"10.22436\/jmcs.035.03.03","article-title":"Properties and applications of Bell polynomials of two variables","volume":"35","author":"Zayeda","year":"2024","journal-title":"J. Math. Comput. Sci."},{"issue":"4","key":"10.1016\/j.cam.2026.117572_bib0015","doi-asserted-by":"crossref","first-page":"2391","DOI":"10.1016\/j.aej.2020.02.035","article-title":"Efficient sustainable algorithm for numerical solutions of systems of fractional order differential equations by Haar wavelet collocation method","volume":"59","author":"Abdeljawad","year":"2020","journal-title":"Alex. Eng. J."},{"key":"10.1016\/j.cam.2026.117572_bib0016","doi-asserted-by":"crossref","first-page":"36","DOI":"10.1016\/j.amc.2017.12.010","article-title":"Numerical simulation for coupled systems of nonlinear fractional order integro-differential equations via wavelets method","volume":"324","author":"Wang","year":"2018","journal-title":"Appl. Math. Computat."},{"key":"10.1016\/j.cam.2026.117572_bib0017","doi-asserted-by":"crossref","first-page":"475","DOI":"10.1016\/j.amc.2014.11.079","article-title":"Numerical algorithm to solve system of nonlinear fractional differential equations based on wavelets method and the error analysis","volume":"251","author":"Chen","year":"2015","journal-title":"Appl. Math. Computat."},{"issue":"3","key":"10.1016\/j.cam.2026.117572_bib0018","doi-asserted-by":"crossref","first-page":"1351","DOI":"10.1016\/j.amc.2004.03.014","article-title":"Numerical solutions for systems of fractional differential equations by the decomposition method","volume":"162","author":"Momani","year":"2005","journal-title":"Appl. Math. Computat."},{"issue":"1","key":"10.1016\/j.cam.2026.117572_bib0019","doi-asserted-by":"crossref","first-page":"371","DOI":"10.1007\/s10957-023-02346-6","article-title":"Generalized Bernoulli-Laguerre polynomials: applications in coupled nonlinear system of variable-order fractional","volume":"200","author":"Hassani","year":"2024","journal-title":"PDEs J. Optim. Theory Appl."},{"issue":"4","key":"10.1016\/j.cam.2026.117572_bib0020","doi-asserted-by":"crossref","first-page":"300","DOI":"10.1007\/s40314-020-01362-w","article-title":"Solving nonlinear systems of fractional-order partial differential equations using an optimization technique based on generalized polynomials","volume":"39","author":"Hassani","year":"2020","journal-title":"Computat. Appl. Math."},{"issue":"4","key":"10.1016\/j.cam.2026.117572_bib0021","doi-asserted-by":"crossref","first-page":"2401","DOI":"10.1007\/s00366-022-01627-4","article-title":"Optimal solution of a general class of nonlinear system of fractional partial differential equations using hybrid functions","volume":"39","author":"Hassani","year":"2023","journal-title":"Eng. Comput."},{"issue":"2","key":"10.1016\/j.cam.2026.117572_bib0022","doi-asserted-by":"crossref","first-page":"1344","DOI":"10.1002\/num.22582","article-title":"Numerical implementation of nonlinear system of fractional Volterra integral-differential equations by Legendre wavelet method and error estimation","volume":"37","author":"Shen","year":"2021","journal-title":"Numer. Methods Partial Differ. Equ."},{"issue":"5","key":"10.1016\/j.cam.2026.117572_bib0023","doi-asserted-by":"crossref","first-page":"1890","DOI":"10.1002\/num.22383","article-title":"Numerical scheme for solving system of fractional partial differential equations with Volterra-type integral term through two-dimensional block-pulse functions","volume":"35","author":"Xie","year":"2019","journal-title":"Numer. Methods Partial Differ. Equ."},{"key":"10.1016\/j.cam.2026.117572_bib0024","doi-asserted-by":"crossref","first-page":"130","DOI":"10.1140\/epjp\/i2017-11407-y","article-title":"A new optimization method for a class of time fractional convection-diffusion-wave equations with variable coefficients","volume":"132","author":"Dahaghin","year":"2017","journal-title":"Eur. Phys. J. Plus"},{"key":"10.1016\/j.cam.2026.117572_bib0025","first-page":"235","article-title":"A computational method for solving variable-order fractional nonlinear diffusion-wave equation","volume":"352","author":"Heydari","year":"2019","journal-title":"Appl. Math. Comput."},{"key":"10.1016\/j.cam.2026.117572_bib0026","doi-asserted-by":"crossref","first-page":"44","DOI":"10.1016\/j.cam.2014.07.018","article-title":"Bernoulli polynomials for the numerical solution of some classes of linear and nonlinear integral equations","volume":"275","author":"Bazm","year":"2015","journal-title":"J. Comput. Appl. Math."},{"issue":"7","key":"10.1016\/j.cam.2026.117572_bib0027","doi-asserted-by":"crossref","first-page":"2280","DOI":"10.1016\/j.camwa.2017.12.003","article-title":"Application of bernoulli matrix method for solving two-dimensional hyperbolic telegraph equations with dirichlet boundary conditions","volume":"75","author":"Singh","year":"2018","journal-title":"Comput. Math. Appl."},{"issue":"21\u201322","key":"10.1016\/j.cam.2026.117572_bib0028","doi-asserted-by":"crossref","first-page":"8886","DOI":"10.1016\/j.apm.2016.05.018","article-title":"Numerical solution of the static beam problem by bernoulli collocation method","volume":"40","author":"Ren","year":"2016","journal-title":"Appl. Math. Model."},{"key":"10.1016\/j.cam.2026.117572_bib0029","doi-asserted-by":"crossref","first-page":"569","DOI":"10.1016\/j.jmaa.2019.03.071","article-title":"Identities for Bernoulli polynomials related to multiple Tornheim zeta functions","volume":"476","author":"Dilcher","year":"2019","journal-title":"J. Math. Anal. Appl."}],"container-title":["Journal of Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0377042726002207?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0377042726002207?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2026,6,12]],"date-time":"2026-06-12T16:11:26Z","timestamp":1781280686000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S0377042726002207"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,12]]},"references-count":29,"alternative-id":["S0377042726002207"],"URL":"https:\/\/doi.org\/10.1016\/j.cam.2026.117572","relation":{},"ISSN":["0377-0427"],"issn-type":[{"value":"0377-0427","type":"print"}],"subject":[],"published":{"date-parts":[[2026,12]]},"assertion":[{"value":"Elsevier","name":"publisher","label":"This article is maintained by"},{"value":"Fractional Bernoulli polynomials optimization approach for nonlinear system of time fractional PDEs in -dimensional","name":"articletitle","label":"Article Title"},{"value":"Journal of Computational and Applied Mathematics","name":"journaltitle","label":"Journal Title"},{"value":"https:\/\/doi.org\/10.1016\/j.cam.2026.117572","name":"articlelink","label":"CrossRef DOI link to publisher maintained version"},{"value":"article","name":"content_type","label":"Content Type"},{"value":"\u00a9 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.","name":"copyright","label":"Copyright"}],"article-number":"117572"}}