{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:46:49Z","timestamp":1760244409415,"version":"build-2065373602"},"reference-count":40,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2022,11,26]],"date-time":"2022-11-26T00:00:00Z","timestamp":1669420800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100003593","name":"Brazilian Agency CNPq","doi-asserted-by":"publisher","award":["306364\/2018-1","001"],"award-info":[{"award-number":["306364\/2018-1","001"]}],"id":[{"id":"10.13039\/501100003593","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Brazilian Agency CAPES","award":["306364\/2018-1","001"],"award-info":[{"award-number":["306364\/2018-1","001"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this work, a large class of integro-differential equations, arising from the description of heat transfer problems, is considered, particularly the nonlinear equations. We propose a procedure for constructing their solution in a very simple and reliable way in which the only needed tool is the same one employed to solve a linear second-order ordinary differential equation, subject to Robin boundary conditions. Proofs of the convergence, existence, and uniqueness are presented. Some special cases are simulated to illustrate the proposed tools.<\/jats:p>","DOI":"10.3390\/axioms11120672","type":"journal-article","created":{"date-parts":[[2022,11,28]],"date-time":"2022-11-28T06:04:36Z","timestamp":1669615476000},"page":"672","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Procedure for Constructing the Solution of a Nonlinear Fredholm Integro-Differential Equation of Second Order"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2947-8142","authenticated-orcid":false,"given":"Rog\u00e9rio Martins","family":"Saldanha da Gama","sequence":"first","affiliation":[{"name":"Mechanical Engineering Department, Rio de Janeiro State University, S\u00e3o Fco Xavier Street 524, Rio de Janeiro 20550-013, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Rog\u00e9rio Pazetto","family":"Saldanha da Gama","sequence":"additional","affiliation":[{"name":"Mechanical Engineering Department, Rio de Janeiro State University, S\u00e3o Fco Xavier Street 524, Rio de Janeiro 20550-013, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,11,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"527","DOI":"10.1080\/00207160701227848","article-title":"A Taylor polynomial approach for solving high-order linear Fredholm integro-differential equations in the most general form","volume":"84","author":"Sezer","year":"2007","journal-title":"Int. J. Comput. Math."},{"key":"ref_2","first-page":"799","article-title":"Numerical solution of linear Fredholm integral equation by using hybrid Taylor and block-pulse functions","volume":"149","author":"Maleknejad","year":"2004","journal-title":"Appl. Math. Comput."},{"key":"ref_3","first-page":"15","article-title":"Numerical solution of integral equations system of the second kind by block-pulse functions","volume":"166","author":"Maleknejad","year":"2005","journal-title":"Appl. Math. Comput."},{"key":"ref_4","first-page":"60","article-title":"Numerical solution of integro-differential equations by using CAS wavelet operational matrix of integration","volume":"194","author":"Danfu","year":"2007","journal-title":"Appl. Math. Comput."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"96184","DOI":"10.1155\/MPE\/2006\/96184","article-title":"Semiorthogonal spline wavelets approximation for Fredholm integro-differential equations","volume":"2006","author":"Lakestani","year":"2006","journal-title":"Math. Probl. Eng."},{"key":"ref_6","first-page":"27","article-title":"Wavelet methods for the numerical solution of Fredholm integro-differential equations","volume":"11","author":"Behiry","year":"2002","journal-title":"Int. J. Appl. Math."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1005","DOI":"10.1016\/j.apm.2005.02.003","article-title":"Numerical piecewise approximate solution of Fredholm integro-differential equations by the Tau method","volume":"29","author":"Hosseini","year":"2005","journal-title":"Appl. Math. Model."},{"key":"ref_8","first-page":"674364","article-title":"Numerical solution of nonlinear Fredholm integro-differential equations using Spectral Homotopy Analysis method","volume":"2013","author":"Atabakan","year":"2013","journal-title":"Math. Probl. Eng."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"960289","DOI":"10.1155\/2012\/960289","article-title":"On spectral homotopy analysis method for solving linear Volterra and Fredholm integro-differential equations","volume":"2012","author":"Atabakan","year":"2012","journal-title":"Abstr. Appl. Anal."},{"key":"ref_10","first-page":"334","article-title":"Legendre polynomial solutions of high-order linear Fredholm integro-differential equations","volume":"210","author":"Sezer","year":"2009","journal-title":"Appl. Math. Comput."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1080\/00207160701405436","article-title":"Chebyshev finite difference method for Fredholm integro-differential equation","volume":"85","author":"Dehghan","year":"2008","journal-title":"Int. J. Comput. Math."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"258","DOI":"10.22436\/jmcs.05.04.02","article-title":"Solution of Fredholm Integro-differential Equations System by Modified Decomposition Method","volume":"5","author":"Rabbani","year":"2012","journal-title":"J. Math. Comput. Sci."},{"key":"ref_13","first-page":"1691","article-title":"An operational approach with Pade approximant for the numerical solution of non-linear Fredholm integro-differential equations","volume":"19","author":"Khani","year":"2011","journal-title":"Sharif Univ. Technol. Sci. Iran."},{"key":"ref_14","first-page":"657","article-title":"A method for the numerical solution of the integro-differential equations","volume":"188","author":"Darania","year":"2007","journal-title":"Appl. Math. Comput."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2367","DOI":"10.1016\/j.cam.2011.11.022","article-title":"The numerical solution of the non-linear integro-differential equations based on the meshless method","volume":"236","author":"Dehghan","year":"2012","journal-title":"J. Comput. Appl. Math."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1007\/s11242-006-0010-3","article-title":"Thermal Analysis of Natural Convection Porous Fins","volume":"67","author":"Kiwan","year":"2007","journal-title":"Transp. Porous Media"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1046","DOI":"10.1016\/j.ijthermalsci.2006.11.013","article-title":"Effect of radiative losses on the heat transfer from porous fins","volume":"46","author":"Kiwan","year":"2007","journal-title":"Int. J. Therm. Sci."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"729","DOI":"10.1016\/j.ijheatmasstransfer.2018.12.156","article-title":"Comprehensive investigation of solid and porous fins influence on natural convection in an inclined rectangular enclosure","volume":"133","author":"Asl","year":"2019","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1406","DOI":"10.1108\/HFF-12-2011-0264","article-title":"Natural convection and radiation in porous fins","volume":"23","author":"Darvishi","year":"2013","journal-title":"Int. J. Numer. Methods Heat Fluid Flow"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"638","DOI":"10.1016\/j.icheatmasstransfer.2010.12.024","article-title":"Thermal analysis of natural convection and radiation in porous fins","volume":"38","author":"Gorla","year":"2011","journal-title":"Int. Commun. Heat Mass Transf."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"956","DOI":"10.1016\/j.proeng.2013.09.172","article-title":"Thermal analysis of porous pin fin used for electronic cooling. International Conference on Design and Manufacturing","volume":"64","author":"Bhanja","year":"2013","journal-title":"IConDM Procedia Eng."},{"key":"ref_22","unstructured":"Sparrow, E.M., and Cess, R.D. (1978). Radiation Heat Transfer, Hemisphere Publishing Corporation."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"433","DOI":"10.1016\/0020-7462(95)00012-D","article-title":"An alternative mathematical modelling for coupled conduction\/radiation energy transfer phenomenon in a system of gray bodies surrounded by a vacuum","volume":"30","author":"Gama","year":"1995","journal-title":"Int. J. Non-Linear Mech."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1006\/jcph.1996.0215","article-title":"Numerical simulation of the (nonlinear) conduction\/radiation heat transfer process in a nonconvex and black cylindrical body","volume":"128","author":"Gama","year":"1996","journal-title":"J. Comput. Phys."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"310","DOI":"10.1016\/0021-9991(92)90210-P","article-title":"Simulation of the steady-state energy transfer in rigid bodies, with convective\/radiative boundary conditions employing a minimum principle","volume":"99","author":"Gama","year":"1992","journal-title":"J. Comput. Phys."},{"key":"ref_26","unstructured":"Holman, J.P. (1996). Heat Transfer, McGraw-Hill."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1615\/JPorMedia.2022040580","article-title":"Solution construction for the nonlinear heat transfer problem in a cylindrical porous fin","volume":"25","year":"2022","journal-title":"J. Porous Media"},{"key":"ref_28","unstructured":"Kreyszig, E. (1978). Introductory Functional Analysis with Applications, John Wiley & Sons."},{"key":"ref_29","unstructured":"Taylor, A.E. (1958). Introduction to Functional Analysis, Wiley."},{"key":"ref_30","unstructured":"Sagan, H. (1992). Introduction to the Calculus of Variations, Dover."},{"key":"ref_31","unstructured":"Berger, M.S. (1977). Nonlinearity & Functional Analysis: Lectures on Nonlinear Problems in Mathematical Analysis, Academic Press."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1016\/S0362-546X(00)00186-3","article-title":"The control of boundary value problems for quasilinear impulsive integro-differential equations","volume":"48","author":"Akhmetov","year":"2002","journal-title":"Nonlinear Anal. Theory Methods Appl."},{"key":"ref_33","first-page":"169","article-title":"A note on the stability and boundedness of solutions to non-linear differential systems of second order","volume":"24","year":"2017","journal-title":"J. Assoc. Arab. Univ. Basic Appl. Sci."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1216\/rmj.2021.51.17","article-title":"Nonlinear Hadamard fractional boundary value problems with different orders","volume":"51","author":"Abuasbeh","year":"2021","journal-title":"Rocky Mt. J. Math."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"4278","DOI":"10.1002\/mma.8039","article-title":"Dynamics of the worm transmission in wireless sensor network in the framework of fractional derivatives","volume":"45","author":"Achar","year":"2022","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Makhlouf, A.B., and Baleanu, D. (2022). Finite time stability of fractional order systems of neutral type. Fractal Fract., 6.","DOI":"10.3390\/fractalfract6060289"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1007\/s13398-014-0175-5","article-title":"Continuability and boundedness of multi-delay functional integro-differential equations of the second order","volume":"109","author":"Graef","year":"2015","journal-title":"Rev. R Acad. Cienc. Exactas F\u00eds. Nat. Ser. A Math. RACSAM"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1105","DOI":"10.1070\/IM2014v078n06ABEH002722","article-title":"Asymptotic limit of an integro-differential equation modelling complex systems","volume":"78","author":"Ferrara","year":"2014","journal-title":"Izv. Math."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1036","DOI":"10.1007\/s10957-012-0021-x","article-title":"Controllability of a nonholonomic macroeconomic system","volume":"154","author":"Ferrara","year":"2012","journal-title":"J. Optim. Theory Appl."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1016\/0093-6413(93)90080-8","article-title":"On the energy balance for continuous mixtures","volume":"20","author":"Sampaio","year":"1993","journal-title":"Mech. Res. Commun."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/12\/672\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:27:14Z","timestamp":1760146034000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/12\/672"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,11,26]]},"references-count":40,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2022,12]]}},"alternative-id":["axioms11120672"],"URL":"https:\/\/doi.org\/10.3390\/axioms11120672","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2022,11,26]]}}}