{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:04:49Z","timestamp":1760148289676,"version":"build-2065373602"},"reference-count":39,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2023,4,17]],"date-time":"2023-04-17T00:00:00Z","timestamp":1681689600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This work focuses on solving and analyzing two-point fuzzy boundary value problems in the form of fractional ordinary differential equations (FFOBVPs) using a new version of the approximation analytical approach. FFOBVPs are useful in describing complex scientific phenomena that include heritable characteristics and uncertainty, and obtaining exact or close analytical solutions for these equations can be challenging, especially in the case of nonlinear problems. To address these difficulties, the optimal homotopy asymptotic method (OHAM) was studied and extended in a new form to solve FFOBVPs. The OHAM is known for its ability to solve both linear and nonlinear fractional models and provides a straightforward methodology that uses multiple convergence control parameters to optimally manage the convergence of approximate series solutions. The new form of the OHAM presented in this work incorporates the concepts of fuzzy sets theory and some fractional calculus principles to include fuzzy analysis in the method. The steps of fuzzification and defuzzification are used to transform the fuzzy problem into a crisp problem that can be solved using the OHAM. The method is demonstrated by solving and analyzing linear and nonlinear FFOBVPs at different values of fractional derivatives. The results obtained using the new form of the fuzzy OHAM are analyzed and compared to those found in the literature to demonstrate the method\u2019s efficiency and high accuracy in the fuzzy domain. Overall, this work presents a feasible and efficient approach for solving FFOBVPs using a new form of the OHAM with fuzzy analysis.<\/jats:p>","DOI":"10.3390\/axioms12040387","type":"journal-article","created":{"date-parts":[[2023,4,17]],"date-time":"2023-04-17T05:51:41Z","timestamp":1681710701000},"page":"387","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Application of the Optimal Homotopy Asymptotic Approach for Solving Two-Point Fuzzy Ordinary Differential Equations of Fractional Order Arising in Physics"],"prefix":"10.3390","volume":"12","author":[{"given":"Ali Fareed","family":"Jameel","sequence":"first","affiliation":[{"name":"Faculty of Education and Arts, Sohar University, Sohar 3111, Oman"},{"name":"Institute of Strategic Industrial Decision Modelling (ISIDM), School of Quantitative Sciences (SQS), Universiti Utara Malaysia (UUM), Sintok 06010, Malaysia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Dulfikar","family":"Jawad Hashim","sequence":"additional","affiliation":[{"name":"Computer Engineering Technique, Mazaya University College, Nasiriyah 64001, Thi-Qar, Iraq"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nidal","family":"Anakira","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science and Technology, Irbid National University, Irbid P.O. Box 2600, Jordan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7614-4727","authenticated-orcid":false,"given":"Osama","family":"Ababneh","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Zarqa University, Zarqa 13110, Jordan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8404-3897","authenticated-orcid":false,"given":"Ahmad","family":"Qazza","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Zarqa University, Zarqa 13110, Jordan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5374-0916","authenticated-orcid":false,"given":"Abedel-Karrem","family":"Alomari","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Yarmouk University, Irbid 21163, Jordan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Khamis S.","family":"Al Kalbani","sequence":"additional","affiliation":[{"name":"Faculty of Education and Arts, Sohar University, Sohar 3111, Oman"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,4,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"577","DOI":"10.1007\/s11192-013-1032-6","article-title":"On development of fractional calculus during the last fifty years","volume":"98","author":"Machado","year":"2014","journal-title":"Scientometrics"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"20200050","DOI":"10.1098\/rsta.2020.0050","article-title":"Advanced materials modelling via fractional calculus: Challenges and perspectives","volume":"378","author":"Failla","year":"2020","journal-title":"Philos. Trans. R. Soc. A Math. Phys. Eng. Sci."},{"key":"ref_3","first-page":"1021","article-title":"Applications of fractional calculus","volume":"4","author":"Dalir","year":"2010","journal-title":"Appl. Math. Sci."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"100007","DOI":"10.1016\/j.csfx.2019.100007","article-title":"Analysis of 3D IS-LM macroeconomic system model within the scope of fractional calculus","volume":"2","author":"Bonyah","year":"2019","journal-title":"Chaos Solitons Fractals X"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"046118","DOI":"10.1103\/PhysRevE.66.046118","article-title":"Fractional langevin model of memory in financial markets","volume":"66","author":"Picozzi","year":"2002","journal-title":"Phys. Rev. E"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"You, X., Li, S., Kang, L., and Cheng, L. (2023). A Study of the Non-Linear Seepage Problem in Porous Media via the Homotopy Analysis Method. Energies, 16.","DOI":"10.3390\/en16052175"},{"key":"ref_7","first-page":"186","article-title":"Approximate solution of the fuzzy fractional Bagley-Torvik equation by the RBF collocation method","volume":"6","author":"Esmaeilbeigi","year":"2018","journal-title":"Comput. Methods Differ. Equ."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Chakraverty, S., Tapaswini, S., and Behera, D. (2016). Fuzzy Arbitrary Order System: Fuzzy Fractional Differential Equations and Applications, John Wiley.","DOI":"10.1002\/9781119004233"},{"key":"ref_9","first-page":"252","article-title":"Linear fractional fuzzy differential equations with Caputo derivative","volume":"7","author":"Abdollahi","year":"2019","journal-title":"Comput. Methods Differ. Equ."},{"key":"ref_10","first-page":"5427","article-title":"Exact solution of some linear fuzzy fractional differential equation using Laplace transform method","volume":"13","author":"Das","year":"2017","journal-title":"Glob. J. Pure Appl. Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"163550","DOI":"10.1016\/j.ijleo.2019.163550","article-title":"Method for finding highly dispersive optical solitons of nonlinear differential equations","volume":"206","author":"Kudryashov","year":"2020","journal-title":"Optik"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"203875","DOI":"10.1155\/2013\/203875","article-title":"The analytical solution of some fractional ordinary differential equations by the Sumudu transform method","volume":"2013","author":"Bulut","year":"2013","journal-title":"Abstr. Appl. Anal."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"075201","DOI":"10.1088\/1402-4896\/ab8b27","article-title":"Abundant new analytical and approximate solutions to the generalized schamel equation","volume":"95","author":"Ghanbari","year":"2020","journal-title":"Phys. Scr."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1007\/s40010-020-00723-8","article-title":"An analytical solution of linear\/nonlinear fractional-order partial differential equations and with new existence and uniqueness conditions","volume":"92","author":"Verma","year":"2022","journal-title":"Proc. Natl. Acad. Sci. India Sect. A Phys. Sci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1100","DOI":"10.2478\/s13540-014-0216-y","article-title":"On the operational solutions of fuzzy fractional differential equations","volume":"17","year":"2014","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"454969","DOI":"10.1155\/2013\/454969","article-title":"Solving fuzzy fractional differential equations using Zadeh\u2019s extension principle","volume":"2013","author":"Ahmad","year":"2013","journal-title":"Sci. World J."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Ahmadian, A., Senu, N., Salahshour, S., and Suleiman, M. (2015, January 19\u201320). On a numerical solution for fuzzy fractional differential equation using an operational matrix method. Proceedings of the 2015 International Symposium on Mathematical Sciences and Computing Research (iSMSC), Ipoh, Malaysia.","DOI":"10.1109\/ISMSC.2015.7594093"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"104","DOI":"10.1186\/1687-1847-2013-104","article-title":"A Jacobi operational matrix for solving a fuzzy linear fractional differential equation","volume":"2013","author":"Ahmadian","year":"2013","journal-title":"Adv. Differ. Equ."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"12","DOI":"10.1016\/j.cnsns.2012.06.008","article-title":"Modified fractional Euler method for solving fuzzy fractional initial value problem","volume":"18","author":"Mazandarani","year":"2013","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"44","DOI":"10.1016\/j.cnsns.2017.03.012","article-title":"Uncertain viscoelastic models with fractional order: A new spectral tau method to study the numerical simulations of the solution","volume":"53","author":"Ahmadian","year":"2017","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2691","DOI":"10.3233\/IFS-151547","article-title":"Fuzzy fractional initial value problem","volume":"28","author":"Prakash","year":"2015","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1186\/1687-1812-2013-13","article-title":"The variational iteration method for fuzzy fractional differential equations with uncertainty","volume":"2013","author":"Khodadadi","year":"2013","journal-title":"Fixed Point Theory Appl."},{"key":"ref_23","first-page":"111","article-title":"Approximate solution of fuzzy fractional differential equations","volume":"9","author":"Panahi","year":"2017","journal-title":"Int. J. Ind. Math."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1597","DOI":"10.18576\/amis\/110606","article-title":"Second order fuzzy fractional differential equations under Caputo\u2019s H-differentiability","volume":"11","author":"Hasan","year":"2017","journal-title":"Appl. Math. Inf. Sci."},{"key":"ref_25","first-page":"875","article-title":"Solution and analysis of the fuzzy Volterra integral equations via homotopy analysis method","volume":"127","author":"Ali","year":"2021","journal-title":"Comput. Model. Eng. Sci."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1007\/s40324-015-0061-x","article-title":"Solving fuzzy fractional differential equations by a generalized differential transform method","volume":"73","author":"Rivaz","year":"2016","journal-title":"SeMA J."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"81","DOI":"10.32732\/jmo.2018.10.2.81","article-title":"New implementation of residual power series for solving fuzzy fractional Riccati equation","volume":"10","author":"Alshorman","year":"2018","journal-title":"J. Model. Optim."},{"key":"ref_28","first-page":"170","article-title":"Solving nonlinear fuzzy fractional IVPs using fractional residual power series algorithm","volume":"2019","author":"Alaroud","year":"2019","journal-title":"IACM"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"4931","DOI":"10.1016\/j.aej.2021.09.060","article-title":"Optimal homotopy asymptotic method for solving several models of first order fuzzy fractional IVPs","volume":"61","author":"Dulfikar","year":"2022","journal-title":"Alex. Eng. J."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Mousa, M.M., and Alsharari, F. (2021). Convergence and error estimation of a new formulation of homotopy perturbation method for classes of nonlinear integral\/integro-differential equations. Mathematics, 9.","DOI":"10.3390\/math9182244"},{"key":"ref_31","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_32","doi-asserted-by":"crossref","first-page":"527","DOI":"10.13189\/ms.2020.080505","article-title":"Comparison for the Approximate Solution of the Second-Order Fuzzy Nonlinear Differential Equation with Fuzzy Initial Conditions","volume":"8","author":"Ali","year":"2020","journal-title":"Math. Stat."},{"key":"ref_33","first-page":"6497","article-title":"Numerical solution of n-th order fuzzy initial value problems by six stages Range Kutta method of order five","volume":"9","author":"Ali","year":"2019","journal-title":"Int. J. Electr. Comput. Eng."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Yuldashev, T.K., and Karimov, E.T. (2020). Inverse problem for a mixed type integro-differential equation with fractional order Caputo operators and spectral parameters. Axioms, 9.","DOI":"10.3390\/axioms9040121"},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Radi, D., Sorini, L., and Stefanini, L. (2020). On the numerical solution of ordinary, interval and fuzzy differential equations by use of F-transform. Axioms, 9.","DOI":"10.3390\/axioms9010015"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"167","DOI":"10.18576\/pfda\/090112","article-title":"Approximate Solutions of Fuzzy Fractional Differential Equations via Homotopy Analysis Method","volume":"9","author":"Dulfikar","year":"2023","journal-title":"Fract. Differ. Appl."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"15","DOI":"10.2478\/amns.2020.2.00011","article-title":"Fractional calculus involving (p, q)-mathieu type series","volume":"5","author":"Kaur","year":"2020","journal-title":"Appl. Math. Nonlinear Sci."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"879","DOI":"10.1016\/j.fss.2005.10.015","article-title":"Median alpha-levels of a fuzzy number","volume":"157","author":"Bodjanova","year":"2002","journal-title":"Fuzzy Sets Syst."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Ismail, M., Saeed, U., Alzabut, J., and Rehman, M. (2019). Approximate solutions for fractional boundary value problems via green-CAS wavelet method. Mathematics, 7.","DOI":"10.3390\/math7121164"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/4\/387\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:17:24Z","timestamp":1760123844000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/4\/387"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,4,17]]},"references-count":39,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2023,4]]}},"alternative-id":["axioms12040387"],"URL":"https:\/\/doi.org\/10.3390\/axioms12040387","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2023,4,17]]}}}