{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,19]],"date-time":"2026-02-19T22:20:59Z","timestamp":1771539659145,"version":"3.50.1"},"reference-count":47,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2020,6,4]],"date-time":"2020-06-04T00:00:00Z","timestamp":1591228800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100003086","name":"Basque Government","doi-asserted-by":"publisher","award":["IT1207-19"],"award-info":[{"award-number":["IT1207-19"]}],"id":[{"id":"10.13039\/501100003086","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The newly constructed optimal perturbation iteration procedure with Laplace transform is applied to obtain the new approximate semi-analytical solutions of the fractional type of damped Burgers\u2019 equation. The classical damped Burgers\u2019 equation is remodeled to fractional differential form via the Atangana\u2013Baleanu fractional derivatives described with the help of the Mittag\u2013Leffler function. To display the efficiency of the proposed optimal perturbation iteration technique, an extended example is deeply analyzed.<\/jats:p>","DOI":"10.3390\/sym12060958","type":"journal-article","created":{"date-parts":[[2020,6,5]],"date-time":"2020-06-05T03:32:21Z","timestamp":1591327941000},"page":"958","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":20,"title":["Optimal Perturbation Iteration Method for Solving Fractional Model of Damped Burgers\u2019 Equation"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8884-3680","authenticated-orcid":false,"given":"Sinan","family":"Deniz","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Art and Sciences, Manisa Celal Bayar University, 45140 Manisa, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9983-5742","authenticated-orcid":false,"given":"Ali","family":"Konuralp","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Art and Sciences, Manisa Celal Bayar University, 45140 Manisa, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9320-9433","authenticated-orcid":false,"given":"Mnauel","family":"De la Sen","sequence":"additional","affiliation":[{"name":"Department of Electricity and Electronics, Faculty of Science and Technology, Institute of Research and Development of Processes\u2014IIDP, University of the Basque Country\u2014UPV\/EHU, Bo Sarriena s\/n (48080), Leioa, 48940 Bizkaia, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,6,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"710","DOI":"10.1016\/j.icheatmasstransfer.2008.02.010","article-title":"Application of optimal homotopy asymptotic method for solving nonlinear equations arising in heat transfer","volume":"35","author":"Marinca","year":"2008","journal-title":"Int. Commun. Heat Mass Transf."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"647","DOI":"10.1007\/s40995-016-0039-2","article-title":"Comparative study between optimal homotopy asymptotic method and perturbation-iteration technique for different types of nonlinear equations","volume":"42","author":"Bildik","year":"2018","journal-title":"Iran. J. Sci. Technol. Trans. A Sci."},{"key":"ref_3","first-page":"2898","article-title":"Some solutions of the linear and nonlinear Klein\u2013Gordon equations using the optimal homotopy asymptotic method","volume":"216","author":"Iqbal","year":"2010","journal-title":"Appl. Math. Comput."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"708","DOI":"10.1016\/j.cnsns.2007.09.015","article-title":"Approximate solutions for the Burger and regularized long wave equations by means of the homotopy analysis method","volume":"14","author":"Rashidi","year":"2009","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_5","first-page":"1","article-title":"Solitary wave solutions for the Boussinesq and Fisher equations by the modified simple equation method","volume":"2","author":"Khan","year":"2016","journal-title":"Math. Lett."},{"key":"ref_6","first-page":"14","article-title":"Approximate solutions for a cubic autocatalytic reaction","volume":"7","author":"Saad","year":"2019","journal-title":"Electron. J. Math. Anal. Appl."},{"key":"ref_7","first-page":"332","article-title":"Homotopy perturbation method for the coupled fractional Lotka\u2013Volterra equations","volume":"56","author":"Kadem","year":"2011","journal-title":"Rom. J. Phys."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Deniz, S. (2017). Optimal perturbation iteration method for solving nonlinear heat transfer equations. J. Heat Transf. ASME, 139.","DOI":"10.1115\/1.4036085"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"020046","DOI":"10.1063\/1.4972638","article-title":"Applications of optimal perturbation iteration method for solving nonlinear differential equations","volume":"1798","author":"Deniz","year":"2017","journal-title":"AIP Conf. Proc."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"305","DOI":"10.36045\/bbms\/1503453712","article-title":"A new analytical technique for solving Lane-Emden type equations arising in astrophysics","volume":"24","author":"Deniz","year":"2017","journal-title":"Bull. Belg. Math. Soc. Simon Stevin"},{"key":"ref_11","first-page":"749","article-title":"New analytic approximate solutions to the generalized regularized long wave equations","volume":"55","author":"Bildik","year":"2018","journal-title":"Bull. Korean Math. Soc."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"01001","DOI":"10.1051\/itmconf\/20171301001","article-title":"A practical method for analytical evaluation of approximate solutions of Fisher\u2019s equations","volume":"13","author":"Bildik","year":"2017","journal-title":"ITM Web Conf."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1489","DOI":"10.1002\/num.22214","article-title":"Solving the Burgers\u2019 and regularized long wave equations using the new perturbation iteration technique","volume":"34","author":"Bildik","year":"2018","journal-title":"Numer. Methods Partial Differ. Equ."},{"key":"ref_14","first-page":"488","article-title":"Solving linear and nonlinear fractional diffusion and wave equations by Adomian decomposition","volume":"180","author":"Jafari","year":"2006","journal-title":"Appl. Math. Comput."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"6183","DOI":"10.1016\/j.apm.2012.12.018","article-title":"Variational iteration method for the Burgers\u2019 flow with fractional derivatives\u2014New Lagrange multipliers","volume":"37","author":"Wu","year":"2013","journal-title":"Appl. Math. Model."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"476","DOI":"10.1016\/j.jmaa.2008.04.007","article-title":"The approximate and exact solutions of the space-and time-fractional Burgers equations with initial conditions by variational iteration method","volume":"345","author":"Inc","year":"2008","journal-title":"J. Math. Anal. Appl."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"843","DOI":"10.1016\/j.chaos.2006.05.074","article-title":"Homotopy perturbation method for fractional KdV-Burgers equation","volume":"35","author":"Wang","year":"2008","journal-title":"Chaos Solitons Fractals"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"448","DOI":"10.1108\/09615531111123119","article-title":"He\u2019s homotopy perturbation method for solving the fractional KdV-Burgers-Kuramoto equation","volume":"21","author":"Sezer","year":"2011","journal-title":"Int. J. Numer. Methods Heat Fluid Flow"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1951","DOI":"10.12785\/amis\/070533","article-title":"Approximate analytical solution to time-fractional damped Burger and Cahn-Allen equations","volume":"7","author":"Esen","year":"2013","journal-title":"Appl. Math. Inf. Sci."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Atangana, A., and Baleanu, D. (2016). New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model. arXiv.","DOI":"10.2298\/TSCI160111018A"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"40","DOI":"10.1016\/j.physa.2018.02.014","article-title":"Non-standard finite difference and Chebyshev collocation methods for solving fractional diffusion equation","volume":"500","author":"Agarwal","year":"2018","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"5642","DOI":"10.1002\/mma.4414","article-title":"A new analysis for fractional model of regularized long-wave equation arising in ion acoustic plasma waves","volume":"40","author":"Kumar","year":"2017","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1016\/j.physa.2017.10.002","article-title":"Analysis of regularized long-wave equation associated with a new fractional operator with Mittag\u2013Leffler type kernel","volume":"492","author":"Kumar","year":"2018","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1515\/phys-2017-0005","article-title":"Analysis of a new fractional model for damped Bergers\u2019 equation","volume":"15","author":"Singh","year":"2017","journal-title":"Open Phys."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1016\/j.chaos.2019.02.001","article-title":"A new fractional analysis on the polluted lakes system","volume":"122","author":"Bildik","year":"2019","journal-title":"Chaos Solitons Fractals"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"447","DOI":"10.1016\/j.chaos.2016.02.012","article-title":"Chaos in a simple nonlinear system with Atangana\u2013Baleanu derivatives with fractional order","volume":"89","author":"Atangana","year":"2016","journal-title":"Chaos Solitons Fractals"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"547","DOI":"10.1016\/j.chaos.2016.03.020","article-title":"Chua\u2019s circuit model with Atangana\u2013Baleanu derivative with fractional order","volume":"89","author":"Alkahtani","year":"2016","journal-title":"Chaos Solitons Fractals"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"109555","DOI":"10.1016\/j.chaos.2019.109555","article-title":"A comparative study on solving fractional cubic isothermal auto\u2013catalytic chemical system via new efficient technique","volume":"132","author":"Bildik","year":"2020","journal-title":"Chaos Solitons Fractals"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"552","DOI":"10.1016\/j.chaos.2016.03.026","article-title":"Comparing the Atangana\u2013Baleanu and Caputo\u2013Fabrizio derivative with fractional order: Allen Cahn model","volume":"89","author":"Algahtani","year":"2016","journal-title":"Chaos Solitons Fractals"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"2299","DOI":"10.2298\/TSCI160209103K","article-title":"Solutions of Cattaneo-Hristov model of elastic heat diffusion with Caputo\u2013Fabrizio and Atangana\u2013Baleanu fractional derivatives","volume":"21","author":"Koca","year":"2017","journal-title":"Therm. Sci."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1716","DOI":"10.1002\/num.22219","article-title":"Chaos in a nonlinear Bloch system with Atangana\u2013Baleanu fractional derivatives","volume":"34","year":"2018","journal-title":"Numer. Methods Partial Differ. Equ."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"100","DOI":"10.1140\/epjp\/i2018-11949-4","article-title":"Modelling the spread of Ebola virus with Atangana\u2013Baleanu fractional operators","volume":"133","author":"Koca","year":"2018","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1140\/epjp\/i2018-12051-9","article-title":"Solutions of partial differential equations using the fractional operator involving Mittag\u2013Leffler kernel","volume":"133","author":"Yavuz","year":"2018","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"063109","DOI":"10.1063\/1.5026284","article-title":"New fractional derivatives with non-singular kernel applied to the Burgers equation","volume":"28","author":"Saad","year":"2018","journal-title":"Chaos Interdiscip. J. Nonlinear Sci."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1502","DOI":"10.1002\/num.22195","article-title":"Numerical approximation of Riemann-Liouville definition of fractional derivative: From Riemann-Liouville to Atangana-Baleanu","volume":"34","author":"Atangana","year":"2018","journal-title":"Numer. Methods Partial Differ. Equ."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"540","DOI":"10.1140\/epjp\/i2017-11809-9","article-title":"On the applications of nanofluids to enhance the performance of solar collectors: A comparative analysis of Atangana\u2013Baleanu and Caputo\u2013Fabrizio fractional models","volume":"132","author":"Sheikh","year":"2017","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"444","DOI":"10.1140\/epjp\/i2017-11717-0","article-title":"New numerical approximation of fractional derivative with non-local and non-singular kernel: Application to chaotic models","volume":"132","author":"Toufik","year":"2017","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"2222","DOI":"10.1016\/j.nonrwa.2007.08.001","article-title":"Kummer function solutions of damped Burgers equations with time-dependent viscosity by exact linearization","volume":"9","author":"Vaganan","year":"2008","journal-title":"Nonlinear Anal. Real World Appl."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"L723","DOI":"10.1088\/0305-4470\/26\/16\/003","article-title":"Approximate solution of the damped Burgers equation","volume":"26","author":"Malfliet","year":"1993","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"4839","DOI":"10.1016\/j.cam.2011.01.002","article-title":"Solving optimal control problems for the unsteady Burgers equation in COMSOL Multiphysics","volume":"235","year":"2011","journal-title":"J. Comput. Appl. Math."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1016\/j.jksus.2016.09.001","article-title":"Optimal perturbation iteration method for Bratu\u2013type problems","volume":"30","author":"Deniz","year":"2018","journal-title":"J. King Saud Univ. Sci."},{"key":"ref_42","first-page":"327","article-title":"Semi-analytical investigation of modified Boussinesq-Burger equations","volume":"22","author":"Deniz","year":"2020","journal-title":"J. Bal\u0131kesir Univ. Inst. Sci. Technol."},{"key":"ref_43","first-page":"35","article-title":"Modification of coupled Drinfel\u2019d-Sokolov-Wilson Equation and approximate solutions by optimal perturbation iteration method","volume":"20","author":"Deniz","year":"2020","journal-title":"Afyon Kocatepe Univ. J. Sci. Eng."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1140\/epjp\/i2017-11344-9","article-title":"A new efficient method for solving delay differential equations and a comparison with other methods","volume":"132","author":"Bildik","year":"2017","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_45","first-page":"503","article-title":"New approximate solutions to the nonlinear Klein-Gordon equations using perturbation iteration techniques","volume":"13","author":"Bildik","year":"2020","journal-title":"Discret. Contin. Dyn. Syst. S"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"122769","DOI":"10.1016\/j.physa.2019.122769","article-title":"A new analysis of a partial differential equation arising in biology and population genetics via semi analytical techniques","volume":"542","author":"Agarwal","year":"2020","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_47","unstructured":"Nayfeh, A.H. (2008). Perturbation Methods, John Wiley & Sons."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/6\/958\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:35:43Z","timestamp":1760175343000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/6\/958"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,6,4]]},"references-count":47,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2020,6]]}},"alternative-id":["sym12060958"],"URL":"https:\/\/doi.org\/10.3390\/sym12060958","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,6,4]]}}}