{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,16]],"date-time":"2026-04-16T10:24:59Z","timestamp":1776335099014,"version":"3.51.2"},"reference-count":39,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2022,4,20]],"date-time":"2022-04-20T00:00:00Z","timestamp":1650412800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>In recent times, many different types of systems have been based on fractional derivatives. Thanks to this type of derivatives, it is possible to model certain phenomena in a more precise and desirable way. This article presents a system consisting of a two-dimensional fractional differential equation with the Riemann\u2013Liouville derivative with a numerical algorithm for its solution. The presented algorithm uses the alternating direction implicit method (ADIM). Further, the algorithm for solving the inverse problem consisting of the determination of unknown parameters of the model is also described. For this purpose, the objective function was minimized using the ant algorithm and the Hooke\u2013Jeeves method. Inverse problems with fractional derivatives are important in many engineering applications, such as modeling the phenomenon of anomalous diffusion, designing electrical circuits with a supercapacitor, and application of fractional-order control theory. This paper presents a numerical example illustrating the effectiveness and accuracy of the described methods. The introduction of the example made possible a comparison of the methods of searching for the minimum of the objective function. The presented algorithms can be used as a tool for parameter training in artificial neural networks.<\/jats:p>","DOI":"10.3390\/s22093153","type":"journal-article","created":{"date-parts":[[2022,4,21]],"date-time":"2022-04-21T03:46:11Z","timestamp":1650512771000},"page":"3153","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Computational Methods for Parameter Identification in 2D Fractional System with Riemann\u2013Liouville 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, Faculty of Applied Mathematics, Silesian University of Technology, 44-100 Gliwice, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1667-3328","authenticated-orcid":false,"given":"Agata","family":"Wajda","sequence":"additional","affiliation":[{"name":"Institute for Chemical Processing of Coal, 41-803 Zabrze, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9384-7232","authenticated-orcid":false,"given":"Grazia","family":"Lo Sciuto","sequence":"additional","affiliation":[{"name":"Department of Mechatronics, Silesian University of Technology, Akademicka 10a, 44-100 Gliwice, Poland"},{"name":"Department of Electrical, Electronics and Informatics Engineering, University of Catania, Viale Andrea Doria, 6, 95125 Catania, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"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, Faculty of Applied Mathematics, Silesian University of Technology, 44-100 Gliwice, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2555-9866","authenticated-orcid":false,"given":"Giacomo","family":"Capizzi","sequence":"additional","affiliation":[{"name":"Department of Electrical, Electronics and Informatics Engineering, University of Catania, Viale Andrea Doria, 6, 95125 Catania, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,4,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"28","DOI":"10.1016\/j.fss.2021.04.012","article-title":"The fuzzy fractional SIQR model of computer virus propagation in wireless sensor network using Caputo Atangana-Baleanud erivatives","volume":"429","author":"Dong","year":"2022","journal-title":"Fuzzy Sets Syst."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Viera-Martin, E., Gomez-Aguilar, J.F., Solis-Perez, J.E., Hernandez-Perez, J.A., and Escobar-Jimenez, R.F. (2022). Artificial neural networks: A practical review of applications involving fractional calculus. Eur. Phys. J. Spec. Top., 1\u201337.","DOI":"10.1140\/epjs\/s11734-022-00455-3"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Muresan, C.I., Birs, I.R., Dulf, E.H., Copot, D., and Miclea, L. (2021). A Review of Recent Advances in Fractional-Order Sensing and Filtering Techniques. Sensors, 21.","DOI":"10.3390\/s21175920"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"111408","DOI":"10.1016\/j.bios.2019.111408","article-title":"\u2018Electrical viscosity\u2019 of piezoresistive sensors: Novel signal processing method, assessment of manufacturing quality, and proposal of an industrial standard","volume":"141","author":"Fuss","year":"2019","journal-title":"Biosens. Bioelectron."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"52","DOI":"10.1177\/1077546318769163","article-title":"Towards fractional sensors","volume":"25","author":"Lopes","year":"2019","journal-title":"J. Vib. Control"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Oprz\u0119dkiewicz, K., Mitkowski, W., and Ros\u00f3\u0142, M. (2021). Fractional Order Model of the Two Dimensional Heat Transfer Process. Energies, 14.","DOI":"10.3390\/en14196371"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"616","DOI":"10.1080\/10407782.2019.1608770","article-title":"A new LRBFCM-GBEM modeling algorithm for general solution of time fractional-order dual phase lag bioheat transfer problems in functionally graded tissues","volume":"75","author":"Fahmy","year":"2019","journal-title":"Numer. Heat Transf. Part A Appl."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"2385","DOI":"10.1016\/j.apm.2014.10.053","article-title":"The numerical method for the moving boundary problem with space-fractional derivative in drug release devices","volume":"39","author":"Gao","year":"2015","journal-title":"Appl. Math. Model."},{"key":"ref_9","first-page":"3214","article-title":"Numerical solution of the one phase 1D fractional Stefan problem using the front fixing method","volume":"38","author":"Klimek","year":"2014","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"676","DOI":"10.1007\/s10439-018-02177-x","article-title":"Modeling Heat Transfer in Tumors: A Review of Thermal Therapies","volume":"47","author":"Andreozzi","year":"2018","journal-title":"Ann. Biomed. Eng."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Chen, D., Zhang, J., and Li, Z. (2022). A Novel Fixed-Time Trajectory Tracking Strategy of Unmanned Surface Vessel Based on the Fractional Sliding Mode Control Method. Electronics, 11.","DOI":"10.3390\/electronics11050726"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1109\/TIV.2019.2904415","article-title":"Electric Vehicle Power Propulsion System Control Based on Time-Varying Fractional Calculus: Implementation and Experimental Results","volume":"4","author":"Khooban","year":"2019","journal-title":"IEEE Trans. Intell. Veh."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"568","DOI":"10.1007\/978-3-030-63128-4_44","article-title":"Numerical Method for the One Phase 1D Fractional Stefan Problem Supported by an Artificial Neural Network","volume":"1288","year":"2021","journal-title":"Adv. Intell. Syst. Comput."},{"key":"ref_14","first-page":"125863","article-title":"A computational algorithm for the numerical solution of fractional order delay differential equations","volume":"402","author":"Amin","year":"2021","journal-title":"Appl. Math. Comput."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1367","DOI":"10.1016\/j.camwa.2018.11.033","article-title":"Space\u2013time finite element method for the multi-term time\u2013space fractional diffusion equation on a two-dimensional domain","volume":"78","author":"Bu","year":"2019","journal-title":"Comput. Math. Appl."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Concezzi, M., and Spigler, R. (2020). An ADI Method for the Numerical Solution of 3D Fractional Reaction-Diffusion Equations. Fractal Fract., 4.","DOI":"10.3390\/fractalfract4040057"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Moura Neto, F.D., and da Silva Neto, A.J. (2013). An Introduction to Inverse Problems with Applications, Springer.","DOI":"10.1007\/978-3-642-32557-1"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Yuldashev, T.K., and Kadirkulov, B.J. (2021). Inverse Problem for a Partial Differential Equation with Gerasimov\u2013Caputo-Type Operator and Degeneration. Fractal Fract., 5.","DOI":"10.3390\/fractalfract5020058"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Kinash, N., and Janno, J. (2019). An Inverse Problem for a Generalized Fractional Derivative with an Application in Reconstruction of Time- and Space-Dependent Sources in Fractional Diffusion and Wave Equations. Mathematics, 7.","DOI":"10.3390\/math7121138"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Brociek, R., Chmielowska, A., and S\u0142ota, D. (2020). 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. Entropy, 22.","DOI":"10.3390\/e22050555"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"53040","DOI":"10.1109\/ACCESS.2019.2912200","article-title":"Review of deep learning algorithms and architectures","volume":"7","author":"Shrestha","year":"2019","journal-title":"IEEE Access"},{"key":"ref_22","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_23","doi-asserted-by":"crossref","first-page":"20120146","DOI":"10.1098\/rsta.2012.0146","article-title":"Modelling heat transfer in heterogeneous media using fractional calculus","volume":"371","author":"Sierociuk","year":"2013","journal-title":"Philos. Trans. R. Soc. A"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"032115","DOI":"10.1103\/PhysRevE.103.032115","article-title":"Anomalous diffusion and Noether\u2019s second theorem","volume":"103","author":"Bagiolli","year":"2021","journal-title":"Phys. Rev. E"},{"key":"ref_25","unstructured":"Podlubny, I. (1999). Fractional Differential Equations, Academic Press."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1703","DOI":"10.1090\/S0025-5718-2015-02917-2","article-title":"A class of second order difference approximations for solving space fractional diffusion equations","volume":"84","author":"Tian","year":"2015","journal-title":"Math. Comput."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Brociek, R., Wajda, A., and S\u0142ota, D. (2021). Inverse problem for a two-dimensional anomalous diffusion equation with a fractional derivative of the Riemann\u2013Liouville type. Energies, 14.","DOI":"10.3390\/en14113082"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Barrett, R., Berry, M., Chan, T.F., Demmel, J., Donato, J., Dongarra, J., Eijkhout, V., Pozo, R., Romine, C., and der Vorst, H.V. (1994). Templates for the Solution of Linear System: Building Blocks for Iterative Methods, SIAM.","DOI":"10.1137\/1.9781611971538"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"631","DOI":"10.1137\/0913035","article-title":"Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems","volume":"13","year":"1992","journal-title":"SIAM J. Sci. Stat. Comput."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1016\/j.apnum.2020.05.016","article-title":"Efficient numerical methods for the nonlinear two-sided space-fractional diffusion equation with variable coefficients","volume":"157","author":"Yang","year":"2020","journal-title":"Appl. Numer. Math."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"035003","DOI":"10.1088\/0266-5611\/31\/3\/035003","article-title":"A tutorial on inverse problems for anomalous diffusion processes","volume":"13","author":"Jin","year":"2015","journal-title":"Inverse Probl."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Mohammad-Djafari, A. (2021). Regularization, Bayesian Inference, and Machine Learning Methods for Inverse Problems. Entropy, 23.","DOI":"10.20944\/preprints202111.0092.v1"},{"key":"ref_33","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_34","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1016\/j.swevo.2017.07.004","article-title":"Multimodal continuous ant colony optimization for multisensor remote sensing image registration with local search","volume":"47","author":"Wu","year":"2019","journal-title":"Swarm Evol. Comput."},{"key":"ref_35","first-page":"369","article-title":"Application of real ant colony optimization algorithm to solve space fractional heat conduction inverse problem","volume":"639","author":"Brociek","year":"2016","journal-title":"Commun. Comput. Inf. Sci."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"212","DOI":"10.1145\/321062.321069","article-title":"\u201cDirect Search\u201d Solution of Numerical and Statistical Problems","volume":"8","author":"Hook","year":"1961","journal-title":"J. ACM"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"799","DOI":"10.1007\/s42452-019-0808-6","article-title":"Structural health monitoring based on the hybrid ant colony algorithm by using Hooke\u2013Jeeves pattern search","volume":"1","author":"Shakya","year":"2019","journal-title":"SN Appl. Sci."},{"key":"ref_38","first-page":"1","article-title":"Inverse problem in space fractional advection diffusion equation","volume":"7","author":"Marinho","year":"2020","journal-title":"Proceeding Ser. Braz. Soc. Comput. Appl. Math."},{"key":"ref_39","unstructured":"\u00d6zi\u015fik, M., and Orlande, H. (2000). Inverse Heat Transfer: Fundamentals and Applications, Taylor & Francis."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/9\/3153\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:57:35Z","timestamp":1760137055000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/9\/3153"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,4,20]]},"references-count":39,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2022,5]]}},"alternative-id":["s22093153"],"URL":"https:\/\/doi.org\/10.3390\/s22093153","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,4,20]]}}}