{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:41:05Z","timestamp":1760060465108,"version":"build-2065373602"},"reference-count":28,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2025,8,29]],"date-time":"2025-08-29T00:00:00Z","timestamp":1756425600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>We consider a two-step numerical approach for solving parabolic initial boundary value problems in 3D simply connected smooth regions. The method uses the Laplace transform in time, reducing the problem to a set of independent stationary boundary value problems for the Helmholtz equation with complex parameters. The inverse Laplace transform is computed using a sinc quadrature along a suitably chosen contour in the complex plane. We show that due to a symmetry of the quadrature nodes, the number of stationary problems can be decreased by almost a factor of two. The influence of the integration contour parameters on the approximation error is also researched. Stationary problems are numerically solved using a boundary integral equation approach applying the Nystr\u00f6m method, based on the quadratures for smooth surface integrals. Numerical experiments support the expectations.<\/jats:p>","DOI":"10.3390\/axioms14090666","type":"journal-article","created":{"date-parts":[[2025,8,29]],"date-time":"2025-08-29T16:42:21Z","timestamp":1756485741000},"page":"666","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On the Combination of the Laplace Transform and Integral Equation Method to Solve the 3D Parabolic Initial Boundary Value Problem"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9918-8407","authenticated-orcid":false,"given":"Roman","family":"Chapko","sequence":"first","affiliation":[{"name":"Department of Computational Mathematics, Ivan Franko National University of Lviv, Universytetska Str. 1, 79000 Lviv, Ukraine"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0006-9975-7936","authenticated-orcid":false,"given":"Svyatoslav","family":"Lavryk","sequence":"additional","affiliation":[{"name":"Department of Computational Mathematics, Ivan Franko National University of Lviv, Universytetska Str. 1, 79000 Lviv, Ukraine"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,8,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Kress, R. (2014). Linear Integral Equations, Springer. [3rd ed.].","DOI":"10.1007\/978-1-4614-9593-2"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Costabel, M., and Sayas, F.J. (2017). Time-dependent problems with the boundary integral equation method. Encyclopedia of Computational Mechanics, John Wiley & Sons. [2nd ed.].","DOI":"10.1002\/9781119176817.ecm2022"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"455","DOI":"10.1007\/BF01385870","article-title":"Time discretization of parabolic boundary integral equations","volume":"63","author":"Lubich","year":"1992","journal-title":"Numer. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1216\/jiea\/1181075987","article-title":"Rothe\u2019s method for the heat equation and boundary integral equations","volume":"9","author":"Chapko","year":"1997","journal-title":"J. Integr. Equat. Appl."},{"key":"ref_5","first-page":"55","article-title":"On the numerical solution of initial boundary value problems by the Laguerre transformation and boundary integral equations","volume":"Volume 2","author":"Agarwal","year":"2000","journal-title":"Integral and Integrodifferential Equations: Theory, Methods and Applications"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1007\/s10665-016-9858-6","article-title":"Numerical solution of the Dirichlet initial boundary value problem for the heat equation in exterior 3-dimensional domains using integral equations","volume":"103","author":"Chapko","year":"2017","journal-title":"J. Eng. Math."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1007\/s00211-005-0645-y","article-title":"Numerical solution of a heat diffusion problem by boundary elements methods using the Laplace transform","volume":"102","author":"Hohage","year":"2005","journal-title":"Numer. Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1080\/10407788808913648","article-title":"Application of hybrid Laplace transform\/finite-difference method to transient heat conduction problems","volume":"14","author":"Chen","year":"1988","journal-title":"Numer. Heat Transf."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"419","DOI":"10.1002\/jnm.1660070606","article-title":"A Laplace domain finite element method (LDFEM) applied to diffusion and propagation problems in electrical engineering","volume":"7","author":"Cai","year":"1994","journal-title":"Int. J. Numer. Model. Electron. Netw. Devices Fields"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Costabel, M. (1994). Developments in boundary element methods for time-dependent problems. Problems and Methods in Mathematical Physics, Vieweg+Teubner Verlag.","DOI":"10.1007\/978-3-322-85161-1_2"},{"key":"ref_11","unstructured":"Brebbia, C., Telles, J., and Wrobel, L. (2012). Boundary Element Techniques: Theory and Applications in Engineering, Springer."},{"key":"ref_12","unstructured":"Cohen, A.M. (2007). Numerical Methods for Laplace Transform Inversion, Springer."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1016\/j.apnum.2004.06.015","article-title":"On the numerical inversion of the Laplace transform of certain holomorphic mappings","volume":"51","author":"Palencia","year":"2004","journal-title":"Appl. Numer. Math."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1007\/s11075-014-9895-z","article-title":"An improved Talbot method for numerical Laplace transform inversion","volume":"68","author":"Dingfelder","year":"2015","journal-title":"Numer. Algorithms"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1341","DOI":"10.1090\/S0025-5718-07-01945-X","article-title":"Parabolic and hyperbolic contours for computing the Bromwich integral","volume":"76","author":"Weideman","year":"2007","journal-title":"Math. Comput."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1093\/imamat\/23.1.97","article-title":"The accurate numerical inversion of Laplace transforms","volume":"23","author":"Talbot","year":"1979","journal-title":"IMA J. Appl. Math."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"339","DOI":"10.1007\/s11075-012-9625-3","article-title":"Review of inverse Laplace transform algorithms for Laplace-space numerical approaches","volume":"63","author":"Kuhlman","year":"2013","journal-title":"Numer. Algorithms"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0021-9991(79)90025-1","article-title":"Numerical inversion of the Laplace transform: A survey and comparison of methods","volume":"33","author":"Davies","year":"1979","journal-title":"J. Comput. Phys."},{"key":"ref_19","unstructured":"Davies, B. (2012). Integral Transforms and Their Applications, Springer."},{"key":"ref_20","unstructured":"Wienert, L. (1990). Die Numerische Approximation von Randintegraloperatoren f\u00fcr die Helmholtzgleichung im R3. [Ph.D. Thesis, University of G\u00f6ttingen]."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Bellman, R.E., and Roth, R.S. (1984). The Laplace Transform, World Scientific.","DOI":"10.1142\/0107"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"985","DOI":"10.1007\/s11075-022-01368-x","article-title":"Fully numerical Laplace transform methods","volume":"92","author":"Weideman","year":"2023","journal-title":"Numer. Algorithms"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"419","DOI":"10.1016\/j.acha.2012.11.005","article-title":"Algorithms for unequally spaced fast Laplace transforms","volume":"35","author":"Andersson","year":"2013","journal-title":"Appl. Comput. Harmon. Anal."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1052","DOI":"10.1109\/61.772353","article-title":"Rational approximation of frequency domain responses by vector fitting","volume":"14","author":"Gustavsen","year":"1999","journal-title":"IEEE Trans. Power Deliv."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"477","DOI":"10.1007\/s10543-009-0234-z","article-title":"Quadrature formulas for the Laplace and Mellin transforms","volume":"49","author":"Campos","year":"2009","journal-title":"BIT Numer. Math."},{"key":"ref_26","unstructured":"Churchill, R.V. (1972). Operational Mathematics, McGraw-Hill. [3rd ed.]."},{"key":"ref_27","unstructured":"Colton, D., and Kress, R. (1983). Integral Equation Methods in Scattering Theory, John Wiley & Sons."},{"key":"ref_28","unstructured":"Abramowitz, M., and Stegun, I. (1972). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/9\/666\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:35:22Z","timestamp":1760034922000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/9\/666"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,8,29]]},"references-count":28,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2025,9]]}},"alternative-id":["axioms14090666"],"URL":"https:\/\/doi.org\/10.3390\/axioms14090666","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,8,29]]}}}