{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,9]],"date-time":"2026-04-09T17:08:16Z","timestamp":1775754496104,"version":"3.50.1"},"reference-count":29,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2025,1,25]],"date-time":"2025-01-25T00:00:00Z","timestamp":1737763200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Center for Research and Development in Mathematics and Applications (CIDMA) through the Portuguese Foundation for Science and Technology (FCT\u2014Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","award":["UIDB\/04106\/2020"],"award-info":[{"award-number":["UIDB\/04106\/2020"]}]},{"name":"Center for Research and Development in Mathematics and Applications (CIDMA) through the Portuguese Foundation for Science and Technology (FCT\u2014Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","award":["UIDP\/04106\/2020"],"award-info":[{"award-number":["UIDP\/04106\/2020"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This study introduces a novel approach for investigating the solvability of boundary value problems for differential equations that incorporate both ordinary and fractional derivatives, specifically within the context of non-autonomous variable order. Unlike traditional methods in the literature, which often rely on generalized intervals and piecewise constant functions, we propose a new fractional operator better suited for this problem. We analyze the existence and uniqueness of solutions, establishing the conditions necessary for these properties to hold using the Krasnoselskii fixed-point theorem and Banach\u2019s contraction principle. Our study also addresses the Ulam\u2013Hyers stability of the proposed problems, examining how variations in boundary conditions influence the solution dynamics. To support our theoretical framework, we provide numerical examples that not only validate our findings but also demonstrate the practical applicability of these mixed derivative equations across various scientific domains. Additionally, concepts such as symmetry may offer further insights into the behavior of solutions. This research contributes to a deeper understanding of complex differential equations and their implications in real-world scenarios.<\/jats:p>","DOI":"10.3390\/sym17020184","type":"journal-article","created":{"date-parts":[[2025,1,27]],"date-time":"2025-01-27T04:59:10Z","timestamp":1737953950000},"page":"184","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Solvability of Boundary Value Problems for Differential Equations Combining Ordinary and Fractional Derivatives of Non-Autonomous Variable Order"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4342-5231","authenticated-orcid":false,"given":"Mohammed Said","family":"Souid","sequence":"first","affiliation":[{"name":"Department of Economic Sciences, University of Tiaret, Tiaret 14035, Algeria"}]},{"given":"Amar","family":"Benkerrouche","sequence":"additional","affiliation":[{"name":"Faculty of Exact Science and Computer Science, University of Djelfa, P.O. Box 3117, Djelfa 17000, Algeria"}]},{"given":"Souad","family":"Guedim","sequence":"additional","affiliation":[{"name":"Laboratory of Applied Mathematics, Kasdi Merbah University, Ouargla 30000, Algeria"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0984-0159","authenticated-orcid":false,"given":"Sandra","family":"Pinelas","sequence":"additional","affiliation":[{"name":"Departamento de Ciencias Exatas e Engenharia, Academia Militar, Av. Conde Castro Guimaraes, 2720-113 Amadora, Portugal"},{"name":"Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5834-7053","authenticated-orcid":false,"given":"Abdelkader","family":"Amara","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Kasdi Merbah University, Ouargla 30000, Algeria"}]}],"member":"1968","published-online":{"date-parts":[[2025,1,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"52","DOI":"10.1016\/j.physa.2017.12.007","article-title":"Analytical and numerical solutions of nonlinear alcoholism model via variable-order fractional differential equations","volume":"494","year":"2018","journal-title":"Phys. A"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"236","DOI":"10.1007\/s40314-022-01934-y","article-title":"Efficient alternating direction implicit numerical approaches for multi-dimensional distributed-order fractional integro differential problems","volume":"41","author":"Guo","year":"2022","journal-title":"Comp. Appl. Math."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"226","DOI":"10.22436\/jmcs.030.03.04","article-title":"Existence of a weak solution for a nonlinear parabolic problem with fractional derivates","volume":"30","author":"Caucha","year":"2023","journal-title":"J. Math. Computer Sci."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1140\/epjst\/e2011-01390-6","article-title":"A comparative study of constant-order and variable-order fractional models in characterizing memory property of systems","volume":"193","author":"Sun","year":"2011","journal-title":"Eur. Phys. J. Spec. Top."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1016\/j.cnsns.2015.10.027","article-title":"Caputo derivatives of fractional variable order Numerical approximations","volume":"35","author":"Tavares","year":"2016","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"5375","DOI":"10.1007\/s40314-018-0639-x","article-title":"Two new fractional derivatives of variable order with non-singular kernel and fractional differential equation","volume":"37","author":"Sousa","year":"2018","journal-title":"Comput. Appl. Math."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1016\/j.aml.2017.08.020","article-title":"An efficient numberical method for variable order fractional functional differential equation","volume":"76","author":"Yang","year":"2018","journal-title":"Appl. Math. Lett."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"365","DOI":"10.1186\/s13662-021-03520-8","article-title":"Boundary value problem for nonlinear fractional differential equations of variable order via Kuratowski MNC technique","volume":"2021","author":"Benkerrouche","year":"2021","journal-title":"Adv. Differ. Equations"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"7967880","DOI":"10.1155\/2021\/7967880","article-title":"Existence and stability of a Caputo variable-order boundary value problem","volume":"2021","author":"Benkerrouche","year":"2021","journal-title":"J. Math."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Benkerrouche, A., Souid, M.S., Etemad, S., Hakem, A., Agarwal, P., Rezapour, S., Ntouyas, S.K., and Tariboon, J. (2021). Qualitative study on solutions of a Hadamard variable order boundary problem via the Ulam-Hyers-Rassias stability. Fractal Fract., 5.","DOI":"10.3390\/fractalfract5030108"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"3187","DOI":"10.1002\/mma.8306","article-title":"On the boundary value problems of Hadamard fractional differential equations of variable order","volume":"46","author":"Benkerrouche","year":"2022","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Benkerrouche, A., Souid, M.S., Stamov, G., and Stamova, I. (2022). Multiterm impulsive Caputo-Hadamard type differential equations of fractional variable order. Axioms, 11.","DOI":"10.3390\/axioms11110634"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Hristova, S., Benkerrouche, A., Souid, M.S., and Hakem, A. (2021). Boundary value problems of Hadamard fractional differential equations of variable order. Symmetry, 13.","DOI":"10.3390\/sym13050896"},{"key":"ref_14","first-page":"435","article-title":"Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation","volume":"2","author":"Lin","year":"2009","journal-title":"Appl. Math. Comput."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1106","DOI":"10.1016\/j.mcm.2011.09.034","article-title":"Solution existence for non-autonomous variable-order fractional differential equations","volume":"55","author":"Razminia","year":"2012","journal-title":"Math. Comput. Model."},{"key":"ref_16","first-page":"910","article-title":"Symmetry analysis and exact solutions of fractional differential equations with mixed derivatives","volume":"14","author":"Ahmed","year":"2022","journal-title":"Symmetry"},{"key":"ref_17","first-page":"1481","article-title":"Symmetrical approach to solving fractional boundary value problems with variable-order derivatives","volume":"12","author":"Wu","year":"2020","journal-title":"Symmetry"},{"key":"ref_18","first-page":"2058","article-title":"On the boundary value problems of fractional differential equations involving mixed order derivatives with symmetry properties","volume":"13","author":"Zhao","year":"2021","journal-title":"Symmetry"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1007\/BF01911126","article-title":"Fractional integration and differentiation of variable order","volume":"21","author":"Samko","year":"1995","journal-title":"Anal. Math."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1080\/10652469308819027","article-title":"Integration and differentiation to a variable fractional order","volume":"1","author":"Samko","year":"1993","journal-title":"Integr. Transforms Spec. Funct."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"470","DOI":"10.1016\/j.sigpro.2010.04.006","article-title":"Variable-order fractional derivatives and their numerical approximations","volume":"91","author":"Valerio","year":"2011","journal-title":"Signal Process"},{"key":"ref_22","first-page":"1","article-title":"Existence of solutions for two point boundary value problems with singular differential equations of variable order","volume":"245","author":"Zhang","year":"2013","journal-title":"Electron. J. Differ. Equations"},{"key":"ref_23","first-page":"22","article-title":"The existeness and uniqueness result of solutions to initial value problems of nonlinear di usion equations involving with the conformable variable","volume":"9","author":"Zhang","year":"2019","journal-title":"Azerbaijan J. Math."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Zhang, S., and Hu, L. (2019). Unique existence result of approximate solution to initial value problem for fractional differential equation of variable order involving the derivative arguments on the half-axis. Mathematics, 7.","DOI":"10.3390\/math7030286"},{"key":"ref_25","first-page":"93","article-title":"Approximate solutions to initial value problem for differential equation of variable order","volume":"9","author":"Zhang","year":"2018","journal-title":"J. Fract. Calc. Appl."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1095","DOI":"10.1016\/j.camwa.2009.05.010","article-title":"Existence of fractional neutral functional differential equations","volume":"59","author":"Agarwal","year":"2010","journal-title":"Comput. Math. Appl."},{"key":"ref_27","first-page":"82","article-title":"Existence and uniqueness result of solutions to initial value problems of fractional differential equations of variable-order","volume":"4","author":"Zhang","year":"2013","journal-title":"J. Fract. Calc. Appl."},{"key":"ref_28","unstructured":"Lions, J.L. (1969). Quelques M\u00e9thodes de R\u00e9solution des Probl\u00e9mes Aux Limites Non Lin\u00e9aires, Dunod."},{"key":"ref_29","first-page":"103","article-title":"Ulam stabilities of ordinary differential equations in a Banach space","volume":"26","author":"Rus","year":"2010","journal-title":"Carpathian J. Math."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/2\/184\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,8]],"date-time":"2025-10-08T10:35:57Z","timestamp":1759919757000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/2\/184"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,1,25]]},"references-count":29,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2025,2]]}},"alternative-id":["sym17020184"],"URL":"https:\/\/doi.org\/10.3390\/sym17020184","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,1,25]]}}}