{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,8,2]],"date-time":"2025-08-02T18:12:08Z","timestamp":1754158328569,"version":"3.41.2"},"reference-count":20,"publisher":"Emerald","issue":"7","license":[{"start":{"date-parts":[[2014,8,26]],"date-time":"2014-08-26T00:00:00Z","timestamp":1409011200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.emerald.com\/insight\/site-policies"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2014,8,26]]},"abstract":"<jats:sec>\n               <jats:title content-type=\"abstract-heading\">Purpose<\/jats:title>\n               <jats:p> \u2013 The purpose of this paper is to address a novel method for solving parabolic partial differential equations (PDEs) in general, wherein the heat conduction equation constitutes an important particular case. The new method, appropriately named the Improved Transversal Method of Lines (ITMOL), is inspired in the Transversal Method of Lines (TMOL), with strong insight from the method of separation of variables. <\/jats:p>\n            <\/jats:sec>\n            <jats:sec>\n               <jats:title content-type=\"abstract-heading\">Design\/methodology\/approach<\/jats:title>\n               <jats:p> \u2013 The essence of ITMOL revolves around an exponential variation of the dependent variable in the parabolic PDE for the evaluation of the time derivative. As will be demonstrated later, this key step is responsible for improving the accuracy of ITMOL over its predecessor TMOL. Throughout the paper, the theoretical properties of ITMOL, such as consistency, stability, convergence and accuracy are analyzed in depth. In addition, ITMOL has proven to be unconditionally stable in the Fourier sense. <\/jats:p>\n            <\/jats:sec>\n            <jats:sec>\n               <jats:title content-type=\"abstract-heading\">Findings<\/jats:title>\n               <jats:p> \u2013 In a case study, the 1-D heat conduction equation for a large plate with symmetric Dirichlet boundary conditions is transformed into a nonlinear ordinary differential equation by means of ITMOL. The numerical solution of the resulting differential equation is straightforward and brings forth a nearly zero truncation error over the entire time domain, which is practically nonexistent. <\/jats:p>\n            <\/jats:sec>\n            <jats:sec>\n               <jats:title content-type=\"abstract-heading\">Originality\/value<\/jats:title>\n               <jats:p> \u2013 Accurate levels of the analytical\/numerical solution of the 1-D heat conduction equation by ITMOL are easily established in the entire time domain.<\/jats:p>\n            <\/jats:sec>","DOI":"10.1108\/hff-01-2013-0030","type":"journal-article","created":{"date-parts":[[2014,11,10]],"date-time":"2014-11-10T07:47:32Z","timestamp":1415605652000},"page":"1519-1536","source":"Crossref","is-referenced-by-count":3,"title":["Accurate analytical\/numerical solution of the heat conduction equation"],"prefix":"10.1108","volume":"24","author":[{"given":"Antonio","family":"Campo","sequence":"first","affiliation":[]},{"given":"Abraham","family":"J. Salazar","sequence":"additional","affiliation":[]},{"given":"Diego","family":"J. Celentano","sequence":"additional","affiliation":[]},{"given":"Marcos","family":"Raydan","sequence":"additional","affiliation":[]}],"member":"140","reference":[{"key":"key2020122922365103900_b1","unstructured":"Arpaci, V.\n                (1966), Conduction Heat Transfer, Addison-Wesley, Reading, MA."},{"key":"key2020122922365103900_b2","doi-asserted-by":"crossref","unstructured":"Cannon, J.R.\n                (1984), The One-Dimensional Heat Equation, Addison-Wesley, Reading, MA.","DOI":"10.1017\/CBO9781139086967"},{"key":"key2020122922365103900_b3","unstructured":"Carslaw, H.S.\n                and \n                  Jaeger, J.C.\n                (1959), Conduction of Heat in Solids, 2nd ed., Clarendon Press, Oxford University Press, London."},{"key":"key2020122922365103900_b4","doi-asserted-by":"crossref","unstructured":"Crank, J.\n               , \n                  Nicolson, P.\n                and \n                  Hartree, D.R.\n                (1947), \u201cA practical method for numerical evaluation of solutions of partial differential equations of the heat conduction type\u201d, Proceedings of the Cambridge Philosophical Society, Vol. 43, pp. 50-67.","DOI":"10.1017\/S0305004100023197"},{"key":"key2020122922365103900_b5","doi-asserted-by":"crossref","unstructured":"Dahlquist, G.G.\n                (1963), \u201cA special stability problem for linear multistep methods\u201d, BIT, Vol. 3, No. 1, pp. 27-43.","DOI":"10.1007\/BF01963532"},{"key":"key2020122922365103900_b6","doi-asserted-by":"crossref","unstructured":"Duffy, D.J.\n                (2005), Finite Difference Methods in Financial Engineering: A Partial Differential Equation Approach, John Wiley, New York, NY.","DOI":"10.1002\/9781118673447"},{"key":"key2020122922365103900_b7","unstructured":"Fourier, J.B.J.\n                (1822), Th\u00e9orie Analytique de la Chaleur, Firmin Didot P\u00e8re et Fils, Paris."},{"key":"key2020122922365103900_b8","unstructured":"Lambert, J.D.\n                (1987), \u201cDevelopments in stability theory for ordinary differential equations\u201d, in \n                  Iserles, A.\n                and \n                  Powell, M.J.D.\n                (Eds), The State of the Art in Numerical Analysis, Clarendon Press, Oxford, pp. 409-431."},{"key":"key2020122922365103900_b9","unstructured":"Liskovets, O.A.\n                (1965), \u201cThe method of lines (Review)\u201d, Differentsial\u2019nye Uravneniya, Vol. 1 No. 12, pp. 1662-1678, English translation in Differential Equations, Vol. 1 No. 12, pp. 1308-1323 (in Russian)."},{"key":"key2020122922365103900_b10","unstructured":"Luikov, A.V.\n                (1968), Analytical Heat Diffusion Theory, Academic Press, London."},{"key":"key2020122922365103900_b11","unstructured":"Ozisik, M.N.\n                (1993), Heat Conduction, 2nd ed., Wiley-Interscience, New York, NY."},{"key":"key2020122922365103900_b12","unstructured":"Ozisik, M.N.\n                (1994), Finite Difference Methods in Heat Transfer, CRC Press, Boca Raton, FL."},{"key":"key2020122922365103900_b13","doi-asserted-by":"crossref","unstructured":"Polyanin, A.D.\n                (2002), Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall\/CRC Press, Boca Raton, FL.","DOI":"10.1201\/9781420035322"},{"key":"key2020122922365103900_b14","unstructured":"Protter, M.H.\n                and \n                  Weinberger, H.F.\n                (1967), Maximum Principles in Differential Equations, Prentice-Hall, Englewood Cliffs, NJ."},{"key":"key2020122922365103900_b15","unstructured":"Rektorys, K.\n                (1982), The Method of Discretization in Time and Partial Differential Equations, D. Reidel Publishing Co, Dordrecht."},{"key":"key2020122922365103900_b16","unstructured":"Risken, H.\n                (1966), The Fokker-Planck Equation: Methods of Solutions and Applications, 2nd ed., Springer-Verlag, Berlin."},{"key":"key2020122922365103900_b17","doi-asserted-by":"crossref","unstructured":"Rothe, E.\n                (1930), \u201cZweidimensionale parabolische Randwertaufgaben als Grenzfall eindimensionaler Randwertaufgabe\u201d, Math. Anal., Vol. 102 No. 5, pp. 650-670.","DOI":"10.1007\/BF01782368"},{"key":"key2020122922365103900_b18","doi-asserted-by":"crossref","unstructured":"Rubel, L.A.\n                (1992), \u201cClosed-form solutions of some partial differential equations via quasi-solutions\u201d, Illinois Journal of Mathematics, Vol. 36 No. 1, pp. 116-135.","DOI":"10.1215\/ijm\/1255987610"},{"key":"key2020122922365103900_b19","doi-asserted-by":"crossref","unstructured":"Salazar, A.J.\n               , \n                  Campo, A.\n                and \n                  Morrone, B.\n                (1998), \u201cApproximate solutions for unsteady heat conduction in large slabs with uniform surface heat flux\u201d, Journal of Computational Physics, Vol. 144 No. 2, pp. 402-422.","DOI":"10.1006\/jcph.1998.5999"},{"key":"key2020122922365103900_b20","unstructured":"Shimakura, N.\n                (1992), Partial Differential Operators of Elliptic Type (Transactions of Mathematical Monographs), Vol. 99, American Mathematical Society, Providence, RI."}],"container-title":["International Journal of Numerical Methods for Heat &amp; Fluid Flow"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.emeraldinsight.com\/doi\/full-xml\/10.1108\/HFF-01-2013-0030","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.emerald.com\/insight\/content\/doi\/10.1108\/HFF-01-2013-0030\/full\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.emerald.com\/insight\/content\/doi\/10.1108\/HFF-01-2013-0030\/full\/html","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,7,25]],"date-time":"2025-07-25T00:51:56Z","timestamp":1753404716000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.emerald.com\/hff\/article\/24\/7\/1519-1536\/94841"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,8,26]]},"references-count":20,"journal-issue":{"issue":"7","published-print":{"date-parts":[[2014,8,26]]}},"alternative-id":["10.1108\/HFF-01-2013-0030"],"URL":"https:\/\/doi.org\/10.1108\/hff-01-2013-0030","relation":{},"ISSN":["0961-5539"],"issn-type":[{"type":"print","value":"0961-5539"}],"subject":[],"published":{"date-parts":[[2014,8,26]]}}}