{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,9]],"date-time":"2026-06-09T08:09:16Z","timestamp":1780992556105,"version":"3.54.1"},"reference-count":33,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2020,5,15]],"date-time":"2020-05-15T00:00:00Z","timestamp":1589500800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>This paper presents the algorithms for solving the inverse problems on models with the fractional derivative. The presented algorithm is based on the Real Ant Colony Optimization algorithm. In this paper, the examples of the algorithm application for the inverse heat conduction problem on the model with the fractional derivative of the Caputo type is also presented. Based on those examples, the authors are comparing the proposed algorithm with the iteration method presented in the paper: Zhang, Z. An undetermined coefficient problem for a fractional diffusion equation. Inverse Probl. 2016, 32.<\/jats:p>","DOI":"10.3390\/e22050555","type":"journal-article","created":{"date-parts":[[2020,5,18]],"date-time":"2020-05-18T02:43:42Z","timestamp":1589769822000},"page":"555","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":17,"title":["Comparison of the Probabilistic Ant Colony Optimization Algorithm and Some Iteration Method in Application for Solving the Inverse Problem on Model With the Caputo Type Fractional Derivative"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7255-6951","authenticated-orcid":false,"given":"Rafa\u0142","family":"Brociek","sequence":"first","affiliation":[{"name":"Department of Mathematics Applications and Methods for Artificial Intelligence, Silesian University of Technology, Kaszubska 23, 44-100 Gliwice, Poland"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2268-5023","authenticated-orcid":false,"given":"Agata","family":"Chmielowska","sequence":"additional","affiliation":[{"name":"Department of Mathematics Applications and Methods for Artificial Intelligence, Silesian University of Technology, Kaszubska 23, 44-100 Gliwice, Poland"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9265-5711","authenticated-orcid":false,"given":"Damian","family":"S\u0142ota","sequence":"additional","affiliation":[{"name":"Department of Mathematics Applications and Methods for Artificial Intelligence, Silesian University of Technology, Kaszubska 23, 44-100 Gliwice, Poland"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2020,5,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Mller, J.L., and Siltanen, S. (2012). Linear and Nonlinear Inverse Problems with Practical Applications, Society for Industrial and Applied Mathematics.","DOI":"10.1137\/1.9781611972344"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Vogel, C.R. (2002). Computational Methods for Inverse Problems, Society for Industrial and Applied Mathematics.","DOI":"10.1137\/1.9780898717570"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"4","DOI":"10.1016\/j.jcp.2014.07.019","article-title":"What is a fractional derivative?","volume":"293","author":"Ortigueira","year":"2015","journal-title":"J. Comput. Phys."},{"key":"ref_4","unstructured":"Miller, K., and Ross, B. (1993). An Introduction to the Fractional Calculus and Fractional Differential, Wiley."},{"key":"ref_5","first-page":"338","article-title":"Anomalous heat transfer: Examples, fundamentals, and fractional calculus models","volume":"50","author":"Voller","year":"2018","journal-title":"Adv. Heat Transf."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1016\/j.physb.2014.08.011","article-title":"An inverse problem of parameter estimation for time-fractional heat conduction in a composite medium using carbon-carbon experimental data","volume":"456","author":"Zhuang","year":"2015","journal-title":"Phys. B"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"118440","DOI":"10.1016\/j.ijheatmasstransfer.2019.118440","article-title":"Comparison of mathematical models with fractional derivative for the heat conduction inverse problem based on the measurements of temperature in porous aluminum","volume":"143","author":"Brociek","year":"2019","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Yang, X.J., Hristov, J., Srivastava, H.M., and Ahmad, B. (2014). Modelling fractal waves on shallow water surfaces via local fractional Korteweg-de Vries equation. Abs. Appl. Anal., 278672.","DOI":"10.1155\/2014\/278672"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Hristov, J. (2015). Approximate solutions to time-fractional models by integral-balance approach. Fract. Dyn., 78\u2013109.","DOI":"10.1515\/9783110472097-006"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"4695","DOI":"10.1007\/s00034-018-0794-8","article-title":"A harmonic balance methodology for circuits with fractional and nonlinear elements","volume":"37","author":"Sowa","year":"2018","journal-title":"Circuits Syst. Signal Process."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"649","DOI":"10.2478\/amcs-2018-0050","article-title":"A memory\u2013efficient noninteger\u2013order discrete\u2013time state\u2013space model of a heat transfer process","volume":"28","author":"Mitkowski","year":"2018","journal-title":"Int. J. Appl. Math. Comput. Sci."},{"key":"ref_12","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_13","doi-asserted-by":"crossref","first-page":"276","DOI":"10.1016\/j.physa.2017.04.054","article-title":"A new fractional operator of variable order: Application in the description of anomalous diffusion","volume":"481","author":"Yang","year":"2017","journal-title":"Phys. A: Stat. Mech. Its Appl."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1699","DOI":"10.1007\/s40314-016-0330-z","article-title":"Numerical solution of time-fractional Black\u2013Scholes equation","volume":"36","author":"Koleva","year":"2017","journal-title":"Comput. Appl. Math."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1138","DOI":"10.1016\/j.camwa.2008.02.015","article-title":"Implicit finite difference approximation for time fractional diffusion equations","volume":"56","author":"Murio","year":"2008","journal-title":"Comput. Math. Appl."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1155","DOI":"10.1016\/j.ejor.2006.06.046","article-title":"Ant colony optimization for continuous domains","volume":"185","author":"Socha","year":"2008","journal-title":"Eur. J. Oper. Res."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1080\/10407790.2014.915680","article-title":"Experimental veryfication of selected artificial intelligence algorithms used for solving the inverse Stefan problem","volume":"66","author":"Hetmaniok","year":"2014","journal-title":"Numer. Heat Transf. Part B-Fundam."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"015011","DOI":"10.1088\/0266-5611\/32\/1\/015011","article-title":"An undetermined coefficient problem for a fractional diffusion equation","volume":"32","author":"Zhang","year":"2016","journal-title":"Inverse Probl."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"625","DOI":"10.2298\/TSCI121124216Y","article-title":"Fractal heat conduction problem solved by local fractional variation iteration method","volume":"17","author":"Baleanu","year":"2013","journal-title":"Therm. Sci."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Brociek, R., S\u0142ota, D., Kr\u00f3l, M., Matula, G., and Kwa\u015bny, W. (2017). Modeling of heat distribution in porous aluminum using fractional differential equation. Fractal Fract., 1.","DOI":"10.3390\/fractalfract1010017"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"624","DOI":"10.1108\/HFF-08-2015-0326","article-title":"Computations of anomalous phase change","volume":"26","author":"Voller","year":"2016","journal-title":"Int. J. Numer. Methods Heat Fluid Flow"},{"key":"ref_22","first-page":"171","article-title":"Application of Real Ant Colony Optimization algorithm to solve space and time fractional heat conduction inverse problem","volume":"46","author":"Brociek","year":"2017","journal-title":"Inf. Technol. Control"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.swevo.2016.12.005","article-title":"A survey of swarm intelligence for dynamic optimization: Algorithms and applications","volume":"33","author":"Mavrovouniotis","year":"2017","journal-title":"Swarm Evol. Comput."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1016\/S0020-0190(01)00258-7","article-title":"ACO algorithms with guaranteed convergence to the optimal solution","volume":"82","author":"Gutjahr","year":"2002","journal-title":"Inf. Process. Lett."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"243","DOI":"10.1016\/j.tcs.2005.05.020","article-title":"Ant colony optimization theory: A survey","volume":"344","author":"Dorigo","year":"2005","journal-title":"Theor. Comput. Sci."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Panigrahi, B.K., Shi, Y., and Lim, M.-H. (2011). Ant colony optimization: Principle, convergence and application. Handbook of Swarm Intelligence. Adaptation, Learning, and Optimization, Springer.","DOI":"10.1007\/978-3-642-17390-5"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"649","DOI":"10.1108\/03321641311297089","article-title":"Research on convergence of ACO subset algorithms","volume":"32","author":"Chen","year":"2013","journal-title":"COMPEL: Int. J. Comput. Math. Electr. Electron. Eng."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1080\/1068276031000074288","article-title":"Inverse design of airfoils based on a novel formulation of the ant colony optimization method","volume":"11","author":"Fainekos","year":"2003","journal-title":"Inverse Probl. Eng."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"705","DOI":"10.1080\/17415970802083276","article-title":"Fuzzy ant colony optimization for estimating chlorophyll concentration profile in offshore sea water","volume":"16","author":"Carvalho","year":"2008","journal-title":"Inverse Probl. Sci. Eng."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1043","DOI":"10.1080\/17415977.2012.691881","article-title":"Estimation of space-dependent single scattering albedo in a radiative transfer problem using differential evolution","volume":"20","author":"Lobato","year":"2012","journal-title":"Inverse Probl. Sci. Eng."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1657","DOI":"10.1007\/s00231-016-1923-1","article-title":"Identification of the heat transfer coefficient in the two-dimensional model of binary alloy solidification","volume":"53","author":"Hetmaniok","year":"2017","journal-title":"Heat Mass Transf."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1432","DOI":"10.1080\/17415977.2017.1411911","article-title":"Inverse identification of elastic properties of composite materials using hybrid GA-ACO-PSO algorithm","volume":"26","author":"Tam","year":"2018","journal-title":"Inverse Probl. Sci. Eng."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Kaipio, J., and Somersalo, E. (2005). Statistical and Computational Inverse Problems, Springer.","DOI":"10.1007\/b138659"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/5\/555\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:29:21Z","timestamp":1760174961000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/5\/555"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,5,15]]},"references-count":33,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2020,5]]}},"alternative-id":["e22050555"],"URL":"https:\/\/doi.org\/10.3390\/e22050555","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,5,15]]}}}