{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,9]],"date-time":"2026-01-09T19:52:50Z","timestamp":1767988370928,"version":"3.49.0"},"reference-count":40,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2023,3,6]],"date-time":"2023-03-06T00:00:00Z","timestamp":1678060800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"University of Oradea"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Fractional calculus, which deals with the concept of fractional derivatives and integrals, has become an important area of research, due to its ability to capture memory effects and non-local behavior in the modeling of real-world phenomena. In this work, we study a new class of fractional Volterra\u2013Fredholm integro-differential equations, involving the Caputo\u2013Katugampola fractional derivative. By applying the Krasnoselskii and Banach fixed-point theorems, we prove the existence and uniqueness of solutions to this problem. The modified Adomian decomposition method is used, to solve the resulting fractional differential equations. This technique rapidly provides convergent successive approximations of the exact solution to the given problem; therefore, we investigate the convergence of approximate solutions, using the modified Adomian decomposition method. Finally, we provide an example, to demonstrate our results. Our findings contribute to the current understanding of fractional integro-differential equations and their solutions, and have the potential to inform future research in this area.<\/jats:p>","DOI":"10.3390\/sym15030662","type":"journal-article","created":{"date-parts":[[2023,3,6]],"date-time":"2023-03-06T06:29:06Z","timestamp":1678084146000},"page":"662","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Symmetrical Solutions for Non-Local Fractional Integro-Differential Equations via Caputo\u2013Katugampola Derivatives"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4392-0742","authenticated-orcid":false,"given":"Khalil S.","family":"Al-Ghafri","sequence":"first","affiliation":[{"name":"University of Technology and Applied Sciences, P.O. Box 14, Ibri 516, Oman"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3123-2360","authenticated-orcid":false,"given":"Awad T.","family":"Alabdala","sequence":"additional","affiliation":[{"name":"Management Department, Universit\u00e9 Fran\u00e7aise d\u2019\u00c9gypte, El Shorouk 11837, Egypt"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4779-7657","authenticated-orcid":false,"given":"Saleh S.","family":"Redhwan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Al-Mahweet University, Al Mahwit, Yemen"},{"name":"Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad 431001, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7251-9608","authenticated-orcid":false,"given":"Omar","family":"Bazighifan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Education, Seiyun University, Seiyun 50512, Yemen"},{"name":"Department of Mathematics, International Telematic University Uninettuno, CorsoVittorio Emanuele II, 39, 00186 Roma, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2959-4212","authenticated-orcid":false,"given":"Ali Hasan","family":"Ali","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Education for Pure Sciences, University of Basrah, Basrah 61001, Iraq"},{"name":"Institute of Mathematics, University of Debrecen, Pf. 400, H-4002 Debrecen, Hungary"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8845-3095","authenticated-orcid":false,"given":"Loredana Florentina","family":"Iambor","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, University of Oradea, 1 University Street, 410087 Oradea, Romania"}]}],"member":"1968","published-online":{"date-parts":[[2023,3,6]]},"reference":[{"key":"ref_1","unstructured":"Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006). Theory and Applications of Fractional Differential Equations, Elsevier."},{"key":"ref_2","unstructured":"Samko, S.G., Kilbas, A.A., and Marichev, O.I. (1993). Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"063502","DOI":"10.1063\/1.4922018","article-title":"Properties of Katugampola fractional derivative with potential application in quantum mechanics","volume":"56","author":"Anderson","year":"2015","journal-title":"J. Math. Phys."},{"key":"ref_4","first-page":"51","article-title":"Existence criteria for Katugampola fractional type impulsive differential equations with inclusions","volume":"2","author":"Janaki","year":"2019","journal-title":"Math. Sci. Model."},{"key":"ref_5","first-page":"53","article-title":"Analytic study on fractional implicit differential equations with impulses via Katugampola fractional Derivative","volume":"6","author":"Janaki","year":"2018","journal-title":"Int. J. Math. Appl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1","DOI":"10.22436\/mns.04.01.01","article-title":"Dynamics and stability results for impulsive type integro-differential equations with generalized fractional derivative","volume":"4","author":"Vivek","year":"2019","journal-title":"Math. Nat. Sci."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"9","DOI":"10.5890\/JVTSD.2018.03.002","article-title":"Theory and analysis of impulsive type pantograph equations with Katugampola fractioanl derivative","volume":"2","author":"Vivek","year":"2018","journal-title":"J. Vabration Test. Syst. Dyn."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"2111","DOI":"10.1109\/TVT.2022.3211661","article-title":"Channel Prediction Using Ordinary Differential Equations for MIMO Systems","volume":"72","author":"Wang","year":"2023","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1016\/j.apm.2022.12.025","article-title":"A magnetic field coupling fractional step lattice Boltzmann model for the complex interfacial behavior in magnetic multiphase flows","volume":"117","author":"Li","year":"2023","journal-title":"Appl. Math. Model."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"108418","DOI":"10.1016\/j.aml.2022.108418","article-title":"Existence of solutions for the (p, q)-Laplacian equation with nonlocal Choquard reaction","volume":"135","author":"Xie","year":"2023","journal-title":"Appl. Math. Lett."},{"key":"ref_11","first-page":"1049","article-title":"Approximate solutions and existence of solution for a Caputo nonlocal fractional volterra fredholm integro-differential equation","volume":"33","author":"Abood","year":"2020","journal-title":"Int. J. Appl. Math."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"10","DOI":"10.1186\/1687-1847-2014-10","article-title":"On Caputo modification of the Hadamard fractional derivatives","volume":"2014","author":"Gambo","year":"2014","journal-title":"Adv. Differ. Equ."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Hilfer, R. (2008). Threefold introduction to fractional derivatives. Anomalous Transp. Found. Appl., 17\u201373.","DOI":"10.1002\/9783527622979.ch2"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"501","DOI":"10.1016\/0022-247X(88)90170-9","article-title":"A review of the decomposition method in applied mathematics","volume":"135","author":"Adomian","year":"1988","journal-title":"J. Math. Anal. Appl."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"348","DOI":"10.3934\/Math.2017.2.348","article-title":"Large deviations for stochastic fractional integrodifferential equations","volume":"2","author":"Suvinthra","year":"2018","journal-title":"AIMS Math."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Almarri, B., Ali, A.H., Al-Ghafri, K.S., Almutairi, A., Bazighifan, O., and Awrejcewicz, J. (2022). Symmetric and Non-Oscillatory Characteristics of the Neutral Differential Equations Solutions Related to p-Laplacian Operators. Symmetry, 14.","DOI":"10.3390\/sym14030566"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Almarri, B., Ali, A.H., Lopes, A.M., and Bazighifan, O. (2022). Nonlinear Differential Equations with Distributed Delay: Some New Oscillatory Solutions. Mathematics, 10.","DOI":"10.3390\/math10060995"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Almarri, B., Janaki, S., Ganesan, V., Ali, A.H., Nonlaopon, K., and Bazighifan, O. (2022). Novel Oscillation Theorems and Symmetric Properties of Nonlinear Delay Differential Equations of Fourth-Order with a Middle Term. Symmetry, 14.","DOI":"10.3390\/sym14030585"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Bazighifan, O., Ali, A.H., Mofarreh, F., and Raffoul, Y.N. (2022). Extended Approach to the Asymptotic Behavior and Symmetric Solutions of Advanced Differential Equations. Symmetry, 14.","DOI":"10.3390\/sym14040686"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Khan, F.S., Khalid, M., Al-Moneef, A.A., Ali, A.H., and Bazighifan, O. (2022). Freelance Model with Atangana\u2013Baleanu Caputo Fractional Derivative. Symmetry, 14.","DOI":"10.3390\/sym14112424"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Arshad, U., Sultana, M., Ali, A.H., Bazighifan, O., Al-moneef, A.A., and Nonlaopon, K. (2022). Numerical Solutions of Fractional-Order Electrical RLC Circuit Equations via Three Numerical Techniques. Mathematics, 10.","DOI":"10.3390\/math10173071"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Sultana, M., Arshad, U., Ali, A.H., Bazighifan, O., Al-Moneef, A.A., and Nonlaopon, K. (2022). New Efficient Computations with Symmetrical and Dynamic Analysis for Solving Higher-Order Fractional Partial Differential Equations. Symmetry, 14.","DOI":"10.3390\/sym14081653"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Bazighifan, O., and Kumam, P. (2020). Oscillation Theorems for Advanced Differential Equations with p-Laplacian Like Operators. Mathematics, 8.","DOI":"10.3390\/math8050821"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Bazighifan, O., Alotaibi, H., and Mousa, A.A.A. (2021). Neutral Delay Differential Equations: Oscillation Conditions for the Solutions. Symmetry, 13.","DOI":"10.3390\/sym13010101"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1016\/0022-247X(83)90090-2","article-title":"Inversion of nonlinear stochastic operators","volume":"91","author":"Adomian","year":"1983","journal-title":"J. Math. Anal. Appl."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1016\/0096-3003(81)90002-3","article-title":"Numerical solution of differential equations in the deterministie limit of stochastic theory","volume":"8","author":"Adomian","year":"1981","journal-title":"Appl. Math. Comput."},{"key":"ref_27","first-page":"73","article-title":"A review of the Adomian decomposition method and its applications to fractional differential equations","volume":"3","author":"Duan","year":"2012","journal-title":"Commun. Frac. Calc."},{"key":"ref_28","first-page":"87","article-title":"Solution of fractional integro-differential equations by Adomian decomposition method","volume":"4","author":"Mittal","year":"2008","journal-title":"Int. J. Appl. Math. Mech."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"480","DOI":"10.1016\/0022-247X(87)90199-5","article-title":"On the Adomian decomposition method and comparisons with Picard\u2019s method","volume":"128","author":"Rach","year":"1987","journal-title":"J. Math. Anal. Appl."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1016\/S0096-3003(98)10024-3","article-title":"A reliable modification of Adomian decomposition method","volume":"102","author":"Wazwaz","year":"1999","journal-title":"Appl. Math. Comput."},{"key":"ref_31","first-page":"1","article-title":"Modification on Adomian decomposition method for solving fractional Riccati differential equation","volume":"4","author":"Ismail","year":"2017","journal-title":"Int. Adv. Res. J. Sci. Technol."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"12","DOI":"10.53006\/rna.974148","article-title":"Caputo-Katugampola-type implicit fractional differential equation with anti-periodic boundary conditions","volume":"5","author":"Redhwan","year":"2022","journal-title":"Results Nonlinear Anal."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"3714","DOI":"10.3934\/math.2020240","article-title":"Implicit fractional differential equation with anti-periodic boundary condition involving Caputo-Katugampola type","volume":"5","author":"Redhwan","year":"2020","journal-title":"Aims Math"},{"key":"ref_34","unstructured":"Redhwan, S.S., Shaikh, S.L., and Abdo, M.S. (2020). Theory of Nonlinear Caputo-Katugampola Fractional Differential Equations. arXiv."},{"key":"ref_35","first-page":"497","article-title":"Analytical and approximate solutions for generalized fractional quadratic integral equation","volume":"26","author":"Abood","year":"2021","journal-title":"Nonlinear Funct. Anal. Appl."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"2877","DOI":"10.1007\/s12190-021-01647-1","article-title":"A New Fifth-Order Iterative Method Free from Second Derivative for Solving Nonlinear Equations","volume":"68","author":"Ali","year":"2022","journal-title":"J. Appl. Math. Comput."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"860","DOI":"10.1016\/j.amc.2011.03.062","article-title":"New approach to a generalized fractional integral","volume":"3","author":"Katugampola","year":"2011","journal-title":"Appl. Math. Comput."},{"key":"ref_38","first-page":"1","article-title":"A new approach to generalized fractional derivatives","volume":"6","author":"Katugampola","year":"2014","journal-title":"Bull. Math. Anal. Appl."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Almeida, R. (2017). A Gronwall inequality for a general Caputo fractional operator. arXiv.","DOI":"10.7153\/mia-2017-20-70"},{"key":"ref_40","unstructured":"Smart, D.R. (1980). Fixed Point Theorems, Cambridge University Press."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/3\/662\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:48:57Z","timestamp":1760122137000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/3\/662"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,3,6]]},"references-count":40,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2023,3]]}},"alternative-id":["sym15030662"],"URL":"https:\/\/doi.org\/10.3390\/sym15030662","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,3,6]]}}}