{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,16]],"date-time":"2026-06-16T04:59:36Z","timestamp":1781585976726,"version":"3.54.5"},"reference-count":26,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2022,5,29]],"date-time":"2022-05-29T00:00:00Z","timestamp":1653782400000},"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>Recently, a lot of attention has been paid to the field of research connected with the wireless sensor network and industrial internet of things. The solutions found by theorists are next used in practice in such area as smart industries, smart devices, smart home, smart transportation and the like. Therefore, there is a need to look for some new techniques for solving the problems described by means of the appropriate equations, including differential equations, integral equations and integro-differential equations. The object of interests of this paper is the method dedicated for solving some integro-differential equations with a retarded (delayed) argument. The proposed procedure is based on the Taylor differential transformation which enables to transform the given integro-differential equation into a respective system of algebraic (nonlinear, very often) equations. The described method is efficient and relatively simple to use, however a high degree of generality and complexity of problems, defined by means of the discussed equations, makes impossible to obtain a general form of their solution and enforces an individual approach to each equation, which, however, does not diminish the benefits associated with its use.<\/jats:p>","DOI":"10.3390\/s22114124","type":"journal-article","created":{"date-parts":[[2022,5,31]],"date-time":"2022-05-31T02:30:06Z","timestamp":1653964206000},"page":"4124","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":21,"title":["Solving the Integral Differential Equations with Delayed Argument by Using the DTM Method"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9291-4891","authenticated-orcid":false,"given":"Edyta","family":"Hetmaniok","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":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4971-5360","authenticated-orcid":false,"given":"Mariusz","family":"Pleszczy\u0144ski","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":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6386-6181","authenticated-orcid":false,"given":"Yasir","family":"Khan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Hafr Al Batin, Hafr Al Batin 31991, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2022,5,29]]},"reference":[{"key":"ref_1","first-page":"153","article-title":"Solution of differential-difference equations by using differential transform method","volume":"174","author":"Arikoglu","year":"2006","journal-title":"Appl. Math. Comput."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"493","DOI":"10.1023\/A:1016386414393","article-title":"A Variant of Newton\u2019s Method for the Computation of Traveling Waves of Bistable Differential-Difference Equations","volume":"14","author":"Elmer","year":"2002","journal-title":"J. Dyn. Differ. Equat."},{"key":"ref_3","unstructured":"Rodr\u00edguez, F., L\u00f3pez, J.C.C., and Castro, M.A. (2021). Models of Delay Differential Equations, MDPI."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Smith, H. (2011). An Introduction to Delay Differential Equations with Applications to the Life Sciences, Springer.","DOI":"10.1007\/978-1-4419-7646-8"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Drozdov, A.D. (1996). Finite Elasticity and Viscoelasticity. A Course in the Nonlinear Mechanics of Solids, World Scientific Publishing.","DOI":"10.1142\/2905"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1016\/0022-5096(72)90027-0","article-title":"A comparison of single-integral non-linear viscoelasticity theories","volume":"20","author":"Smart","year":"1972","journal-title":"J. Mech. Phys. Solids"},{"key":"ref_7","first-page":"249","article-title":"Sur la th\u00e9orie math\u00e9matique des ph\u00e9nom\u00e8nes h\u00e9r\u00e9ditaires","volume":"7","author":"Volterra","year":"1928","journal-title":"J. Math\u00e9matiques Pures Appliqu\u00e9es"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Amin, R., Nazir, S., and Garc\u00eda-Magari\u00f1o, I. (2020). A Collocation Method for Numerical Solution of Nonlinear Delay Integro-Differential Equations for Wireless Sensor Network and Internet of Things. Sensors, 20.","DOI":"10.3390\/s20071962"},{"key":"ref_9","unstructured":"Hetmaniok, E., S\u0142ota, D., Trawi\u0144ski, T., and Witu\u0142a, R. (2017). A novel algorithm for solving the ordinary differential equations. Selected Problems on Experimental Mathematics, Silesian University of Technology Press."},{"key":"ref_10","unstructured":"Damasevicius, R., and Vasiljeviene, G. (2018, January 4\u20136). Application of the Taylor transformation to the systems of ordinary differential equations. Proceedings of the Information and Software Technologies, ICIST 2018, Communications in Computer and Information Science, Vilnius, Lithuania."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Hetmaniok, E., and Pleszczy\u0144ski, M. (2022). Comparison of the Selected Methods Used for Solving the Ordinary Differential Equations and Their Systems. Mathematics, 10.","DOI":"10.3390\/math10030306"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1016\/j.jksus.2010.08.015","article-title":"Differential transform method for special systems of integral equations","volume":"24","author":"Biazar","year":"2012","journal-title":"J. King Saud Univ. Sci."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1016\/j.jksus.2010.07.013","article-title":"Differential transform method for solving singularly perturbed Volterra integral equations","volume":"23","author":"Momani","year":"2011","journal-title":"J. King Saud Univ. Sci."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1144","DOI":"10.1016\/j.mcm.2007.12.022","article-title":"Differential transform method for solving Volterra integral equation with separable kernels","volume":"48","author":"Odibat","year":"2008","journal-title":"Math. Comput. Model."},{"key":"ref_15","first-page":"708","article-title":"Differential transform method for solving the linear and nonlinear Klein\u2013Gordon equation","volume":"185","author":"Aruna","year":"2009","journal-title":"Comput. Phys. Commun."},{"key":"ref_16","first-page":"166","article-title":"Reduced differential transform method to solve two and three dimensional second order hyperbolic telegraph equations","volume":"29","author":"Aruna","year":"2017","journal-title":"J. King Saud Univ. Eng. Sci."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"6896","DOI":"10.1016\/j.physleta.2008.10.008","article-title":"Differential transform method for solving linear and non-linear systems of partial differential equations","volume":"372","author":"Srivastava","year":"2008","journal-title":"Phys. Lett. A"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2621","DOI":"10.1016\/j.camwa.2011.03.007","article-title":"Differential transform method for the system of two-dimensional nonlinear Volterra integro-differential equations","volume":"61","author":"Tari","year":"2011","journal-title":"Comput. Math. Appl."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"585","DOI":"10.4236\/am.2015.63053","article-title":"Differential Transform Method for Some Delay Differential Equations","volume":"6","author":"Liu","year":"2015","journal-title":"Appl. Math."},{"key":"ref_20","first-page":"521","article-title":"A Comparison Between Two Approaches to Solve Functional Differential Equations: DTM and DJM","volume":"5","author":"Sharma","year":"2017","journal-title":"Int. J. Math. Appl."},{"key":"ref_21","unstructured":"Wolfram, S. (2017). An Elementary Introduction to the Wolfram Language, Wolfram Media, Inc.. [2nd ed.]."},{"key":"ref_22","unstructured":"Wolfram, S. (2003). The Mathematica Book, Wolfram Media, Inc.. [5th ed.]."},{"key":"ref_23","first-page":"63","article-title":"The Differential Transform Method for solving the model describing biological species living together","volume":"7","author":"Tari","year":"2012","journal-title":"Iran. J. Math. Sci. Inform."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"58","DOI":"10.1007\/BF02414092","article-title":"Sur le probl\u00e8me biologique h\u00e9r\u00e9ditaiare de deux espec\u00e8s d\u00e9vorante et d\u00e9vor\u00e9e","volume":"9","author":"Brelot","year":"1931","journal-title":"Ann. Mat. Pura Appl."},{"key":"ref_25","first-page":"1145","article-title":"Solution of boundary value problems for integro-differential equations by using differential transform method","volume":"168","author":"Arikoglu","year":"2005","journal-title":"Appl. Math. Comput."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"2411","DOI":"10.1016\/j.camwa.2008.05.017","article-title":"Solutions of integral and integro-differential equation systems by using differential transform method","volume":"56","author":"Arikoglu","year":"2008","journal-title":"Comput. Math. Appl."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/11\/4124\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:20:52Z","timestamp":1760138452000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/11\/4124"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,5,29]]},"references-count":26,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2022,6]]}},"alternative-id":["s22114124"],"URL":"https:\/\/doi.org\/10.3390\/s22114124","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,5,29]]}}}