{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,20]],"date-time":"2026-01-20T01:30:48Z","timestamp":1768872648317,"version":"3.49.0"},"reference-count":42,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2021,5,10]],"date-time":"2021-05-10T00:00:00Z","timestamp":1620604800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2021,5,10]],"date-time":"2021-05-10T00:00:00Z","timestamp":1620604800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2022,1]]},"DOI":"10.1007\/s11075-021-01120-x","type":"journal-article","created":{"date-parts":[[2021,5,10]],"date-time":"2021-05-10T09:03:32Z","timestamp":1620637412000},"page":"431-449","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["A method of fundamental solutions for heat and wave propagation from lateral Cauchy data"],"prefix":"10.1007","volume":"89","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3797-7491","authenticated-orcid":false,"given":"Ihor","family":"Borachok","sequence":"first","affiliation":[]},{"given":"Roman","family":"Chapko","sequence":"additional","affiliation":[]},{"given":"B. Tomas","family":"Johansson","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,5,10]]},"reference":[{"key":"1120_CR1","doi-asserted-by":"publisher","first-page":"413","DOI":"10.1287\/ijoc.8.4.413","volume":"8","author":"J Abate","year":"1996","unstructured":"Abate, J., Choudhury, G.L., Whitt, W.: On the Laguerre method for numerically inverting Laplace transforms. INFORMS J. Comput. 8, 413\u2013427 (1996)","journal-title":"INFORMS J. Comput."},{"key":"1120_CR2","volume-title":"Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables","author":"M Abramowitz","year":"1972","unstructured":"Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover Publications, New York (1972)"},{"key":"1120_CR3","doi-asserted-by":"publisher","first-page":"123004","DOI":"10.1088\/0266-5611\/25\/12\/123004","volume":"25","author":"G Alessandrini","year":"2009","unstructured":"Alessandrini, G., Rondi, L., Rosset, E., Vessella, S.: The stability for the Cauchy problem for elliptic equations. Inverse Probl. 25, 123004 (2009)","journal-title":"Inverse Probl."},{"key":"1120_CR4","doi-asserted-by":"publisher","first-page":"1348","DOI":"10.1016\/j.enganabound.2009.05.007","volume":"33","author":"CJS Alves","year":"2009","unstructured":"Alves, C.J.S.: On the choice of source points in the method of fundamental solutions. Eng. Anal. Bound Elem. 33, 1348\u20131361 (2009)","journal-title":"Eng. Anal. Bound Elem."},{"key":"1120_CR5","doi-asserted-by":"publisher","first-page":"16","DOI":"10.1016\/j.camwa.2018.12.014","volume":"88","author":"CJS Alves","year":"2018","unstructured":"Alves, C.J.S., Martins, N.F.M., Valtchev, S.S.: Domain decomposition methods with fundamental solutions for Helmholtz problems with discontinuous source terms. Comput. Math. Appl. 88, 16\u201332 (2018). (in press)","journal-title":"Comput. Math. Appl."},{"key":"1120_CR6","doi-asserted-by":"publisher","first-page":"885","DOI":"10.1016\/j.aml.2007.07.032","volume":"21","author":"A Amirov","year":"2008","unstructured":"Amirov, A., Yamamoto, M.: A timelike Cauchy problem and an inverse problem for general hyperbolic equations. Appl. Math. Lett. 21, 885\u2013891 (2008)","journal-title":"Appl. Math. Lett."},{"key":"1120_CR7","doi-asserted-by":"publisher","first-page":"971","DOI":"10.3934\/ipi.2015.9.971","volume":"9","author":"E B\u00e9cache","year":"2015","unstructured":"B\u00e9cache, E., Bourgeois, L., Franceschini, L., Dard\u00e9, J.: Application of mixed formulations of quasi-reversibility to solve ill-posed problems for heat and wave equations: the 1D case. Inverse Probl. Imaging 9, 971\u20131002 (2015)","journal-title":"Inverse Probl. Imaging"},{"key":"1120_CR8","volume-title":"Numerical Inversion of the Laplace Transform: Applications to Biology, Economics, Engineering and Physics","author":"R Bellman","year":"1966","unstructured":"Bellman, R., Kalaba, R.E., Lockett, J.A.: Numerical Inversion of the Laplace Transform: Applications to Biology, Economics, Engineering and Physics. American Elsevier Publishing Co., Inc., New York (1966)"},{"key":"1120_CR9","doi-asserted-by":"publisher","first-page":"644","DOI":"10.1137\/0722040","volume":"22","author":"A Bogomolny","year":"1985","unstructured":"Bogomolny, A.: Fundamental solutions method for elliptic boundary value problems. SIAM J. Numer. Anal. 22, 644\u2013669 (1985)","journal-title":"SIAM J. Numer. Anal."},{"key":"1120_CR10","doi-asserted-by":"publisher","first-page":"1550","DOI":"10.1080\/17415977.2015.1130042","volume":"24","author":"I Borachok","year":"2016","unstructured":"Borachok, I., Chapko, R., Johansson, B.T.: Numerical solution of a Cauchy problem for Laplace equation in 3-dimensional domains by integral equations. Inverse Probl Sci. Eng. 24, 1550\u20131568 (2016)","journal-title":"Inverse Probl Sci. Eng."},{"key":"1120_CR11","doi-asserted-by":"publisher","first-page":"711","DOI":"10.1515\/jiip-2015-0053","volume":"24","author":"I Borachok","year":"2016","unstructured":"Borachok, I., Chapko, R., Johansson, B.T.: Numerical solution of an elliptic 3-dimensional Cauchy problem by the alternating method and boundary integral equations. J. Inverse Ill-Posed Probl. 24, 711\u2013725 (2016)","journal-title":"J. Inverse Ill-Posed Probl."},{"key":"1120_CR12","doi-asserted-by":"publisher","first-page":"264","DOI":"10.1016\/j.trmi.2017.04.004","volume":"171","author":"T Buchukuri","year":"2017","unstructured":"Buchukuri, T., Chkadua, O., Natroshvili, D.: Method of fundamental solutions for mixed and crack type problems in the classical theory of elasticity. Trans. A. Razmadze Math. Inst. 171, 264\u2013292 (2017)","journal-title":"Trans. A. Razmadze Math. Inst."},{"key":"1120_CR13","doi-asserted-by":"publisher","first-page":"104","DOI":"10.1016\/j.apnum.2018.03.004","volume":"129","author":"R Chapko","year":"2018","unstructured":"Chapko, R., Johansson, B.T.: A boundary integral equation method for numerical solution of parabolic and hyperbolic Cauchy problems. Appl. Numer. Math. 129, 104\u2013119 (2018)","journal-title":"Appl. Numer. Math."},{"key":"1120_CR14","doi-asserted-by":"publisher","first-page":"25","DOI":"10.3934\/ipi.2012.6.25","volume":"6","author":"R Chapko","year":"2012","unstructured":"Chapko, R., Johansson, B.T.: On the numerical solution of a Cauchy problem for the Laplace equation via a direct integral equation approach. Inverse Probl. Imaging 6, 25\u201336 (2012)","journal-title":"Inverse Probl. Imaging"},{"key":"1120_CR15","doi-asserted-by":"publisher","first-page":"102385","DOI":"10.1016\/j.wavemoti.2019.102385","volume":"91","author":"R Chapko","year":"2019","unstructured":"Chapko, R., Johansson, B.T., Muzychuk, Y., Hlova, A.: Wave propagation from lateral Cauchy data using a boundary element method. Wave Motion 91, 102385 (2019)","journal-title":"Wave Motion"},{"key":"1120_CR16","first-page":"55","volume":"2","author":"R Chapko","year":"2000","unstructured":"Chapko, R., Kress, R.: On the numerical solution of initial boundary value problems by the Laguerre transformation and boundary integral equations. Integr. Integrodifferential Equ. Theory Methods Appl. Ser. Math. Anal. Appl. 2, 55\u201369 (2000)","journal-title":"Integr. Integrodifferential Equ. Theory Methods Appl. Ser. Math. Anal. Appl."},{"key":"1120_CR17","doi-asserted-by":"publisher","first-page":"075009","DOI":"10.1088\/1361-6420\/ab8f85","volume":"36","author":"B Chen","year":"2020","unstructured":"Chen, B., Guo, Y., Ma, F., Sun, Y.: Numerical schemes to reconstruct three-dimensional time-dependent point sources of acoustic waves. Inverse Probl. 36, 075009 (2020)","journal-title":"Inverse Probl."},{"key":"1120_CR18","first-page":"21","volume":"16","author":"CS Chen","year":"2009","unstructured":"Chen, C.S., Reutskiy, S.Y., Rozov, V.Y.: The method of the fundamental solutions and its modifications for electromagnetic field problems. Assist. Mech. Eng. Sci. 16, 21\u201333 (2009)","journal-title":"Assist. Mech. Eng. Sci."},{"key":"1120_CR19","doi-asserted-by":"publisher","first-page":"118","DOI":"10.1016\/j.enganabound.2020.08.013","volume":"120","author":"AHD Cheng","year":"2020","unstructured":"Cheng, A.H.D., Hong, Y.: An overview of the method of fundamental solutions\u2014solvability, uniqueness, convergence, and stability. Eng. Anal. Bound. Elem. 120, 118\u2013152 (2020)","journal-title":"Eng. Anal. Bound. Elem."},{"key":"1120_CR20","volume-title":"Numerical Methods for Laplace Transform Inversion","author":"AM Cohen","year":"2007","unstructured":"Cohen, A.M.: Numerical Methods for Laplace Transform Inversion. Springer, Berlin (2007)"},{"key":"1120_CR21","unstructured":"Eld\u00e9n, L.: Numerical solution of the sideways heat equation. In: Engl, H., Rundell, W. (eds.) Inverse Problems in Diffusion Processes, pp 130\u2013150. SIAM, Philadelphia (1995)"},{"key":"1120_CR22","doi-asserted-by":"publisher","first-page":"69","DOI":"10.1023\/A:1018981221740","volume":"9","author":"G Fairweather","year":"1998","unstructured":"Fairweather, G., Karageorghis, A.: The method of fundamental solutions for elliptic boundary value problems. Adv. Comput. Math. 9, 69\u201395 (1998)","journal-title":"Adv. Comput. Math."},{"key":"1120_CR23","unstructured":"Golberg, M.A., Chen, C.S.: The method of fundamental solutions for potential Helmholtz and diffusion problems. In: Golberg, M.A. (ed.) Boundary Integral Methods: Numerical and Mathematical Aspects, pp 103\u2013176. WIT Press\/Comput. Mech. Publ, Boston (1999)"},{"key":"1120_CR24","unstructured":"Golberg, M.A., Chen, C.S., Muleshkov, A.S.: The method of fundamental solutions for time-dependent problems. In: Brebia, C.S., Chen C.A., Pepper, D.W. (eds.) Boundary Element Technology XIII, pp 377\u2013386. WIT Press, Southampton (1999)"},{"key":"1120_CR25","first-page":"185","volume":"11","author":"MH Gu","year":"2009","unstructured":"Gu, M.H., Young, D.L., Fan, C.M.: The method of fundamental solutions for one-dimensional wave equations. CMC Comput. Mater. Continua 11, 185\u2013208 (2009)","journal-title":"CMC Comput. Mater. Continua"},{"key":"1120_CR26","doi-asserted-by":"publisher","first-page":"586","DOI":"10.51400\/2709-6998.2200","volume":"19","author":"MH Gu","year":"2011","unstructured":"Gu, M.H., Fan, C.M., Young, D.L.: The method of fundamental solutions for the multi-dimensional wave equations. J. Mar. Sci. Technol. 19, 586\u2013595 (2011)","journal-title":"J. Mar. Sci. Technol."},{"key":"1120_CR27","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-51658-5","volume-title":"Inverse Problems for Partial Differential Equations","author":"V Isakov","year":"2017","unstructured":"Isakov, V.: Inverse Problems for Partial Differential Equations, 3rd edn. Springer-Verlag, Cham (2017)","edition":"3rd edn."},{"key":"1120_CR28","doi-asserted-by":"publisher","first-page":"697","DOI":"10.1016\/j.enganabound.2007.11.012","volume":"32","author":"BT Johansson","year":"2008","unstructured":"Johansson, B.T., Lesnic, D.: A method of fundamental solutions for transient heat conduction. Eng. Anal. Bound. Elem. 32, 697\u2013703 (2008)","journal-title":"Eng. Anal. Bound. Elem."},{"key":"1120_CR29","first-page":"309","volume":"19","author":"A Karageorghis","year":"2011","unstructured":"Karageorghis, A., Lesnic, D., Marin, L.: A survey of applications of the MFS to inverse problems. Inv. Pr. Sci. Engn. 19, 309\u2013336 (2011)","journal-title":"Inv. Pr. Sci. Engn."},{"key":"1120_CR30","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1016\/0096-3003(79)90021-3","volume":"5","author":"J Keilson","year":"1979","unstructured":"Keilson, J., Nunn, W.R.: Laguerre transformation as a tool for the numerical solution of integral equations of convolution type. Appl. Math. Comput. 5, 313\u2013359 (1979)","journal-title":"Appl. Math. Comput."},{"key":"1120_CR31","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4419-8474-6","volume-title":"An Introduction to the Mathematical Theory of Inverse Problems","author":"A Kirsch","year":"2011","unstructured":"Kirsch, A.: An Introduction to the Mathematical Theory of Inverse Problems. Springer, Berlin (2011)"},{"key":"1120_CR32","doi-asserted-by":"publisher","first-page":"559","DOI":"10.1002\/mma.1670150805","volume":"15","author":"M Klibanov","year":"1992","unstructured":"Klibanov, M., Rakesh: Numerical solution of a time-like Cauchy problem for the wave equation. Math Methods Appl. Sci 15, 559\u2013570 (1992)","journal-title":"Math Methods Appl. Sci"},{"key":"1120_CR33","doi-asserted-by":"publisher","first-page":"199","DOI":"10.1016\/0041-5553(64)90092-8","volume":"4","author":"VD Kupradze","year":"1964","unstructured":"Kupradze, V.D.: A method for the approximate solution of limiting problems in mathematical physics. USSR Comput. Maths. Math. Phys. 4, 199\u2013205 (1964)","journal-title":"USSR Comput. Maths. Math. Phys."},{"key":"1120_CR34","doi-asserted-by":"publisher","first-page":"3512","DOI":"10.1109\/20.489553","volume":"31","author":"JD Lavers","year":"1995","unstructured":"Lavers, J.D., Wang, J.: On the determination of the locations for the virtual sources in the method of fundamental solutions for eddy current problems. IEEE T. Magn. 31, 3512\u20133514 (1995)","journal-title":"IEEE T. Magn."},{"key":"1120_CR35","doi-asserted-by":"publisher","first-page":"267","DOI":"10.1016\/j.compstruc.2004.10.005","volume":"83","author":"D Lesnic","year":"2005","unstructured":"Lesnic, D., Marin, L.: The method of fundamental solutions for Cauchy problem associated with two-dimensional Helmholtz equations. Comput. Struct. 83, 267\u2013278 (2005)","journal-title":"Comput. Struct."},{"key":"1120_CR36","doi-asserted-by":"publisher","first-page":"355","DOI":"10.1016\/j.cam.2008.09.027","volume":"228","author":"ZC Li","year":"2009","unstructured":"Li, Z.C.: The method of fundamental solutions for annular shaped domains. J. Comput. Appl. Math. 228, 355\u2013372 (2009)","journal-title":"J. Comput. Appl. Math."},{"key":"1120_CR37","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-65217-2","volume-title":"Non-homogeneous boundary value problems and applications","author":"J-L Lions","year":"1972","unstructured":"Lions, J.-L., Magenes, E.: Non-homogeneous boundary value problems and applications. Springer-Verlag, New York-Heidelberg I (1972)"},{"key":"1120_CR38","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1145\/3306346.3323002","volume":"38","author":"C Schreck","year":"2019","unstructured":"Schreck, C., Hafner, C., Wojtan, C.: Fundamental solutions for water wave animation. ACM Trans. Graph. (TOG) 38, 1\u201314 (2019)","journal-title":"ACM Trans. Graph. (TOG)"},{"key":"1120_CR39","doi-asserted-by":"publisher","first-page":"1903","DOI":"10.1016\/j.jcp.2008.11.018","volume":"228","author":"T Shigeta","year":"2008","unstructured":"Shigeta, T., Young, D.L.: Method of fundamental solutions with optimal regularization techniques for the Cauchy problem of the Laplace equation with singular points. J. Comput. Phys. 228, 1903\u20131915 (2008)","journal-title":"J. Comput. Phys."},{"key":"1120_CR40","doi-asserted-by":"publisher","first-page":"1399","DOI":"10.1090\/S0025-5718-09-02191-7","volume":"78","author":"YG Smyrlis","year":"2009","unstructured":"Smyrlis, Y.G.: Applicability and applications of the method of fundamental solutions. Math. Comput. 78, 1399\u20131434 (2009)","journal-title":"Math. Comput."},{"key":"1120_CR41","first-page":"185","volume":"4","author":"D Tataru","year":"1998","unstructured":"Tataru, D.: On the regularity of boundary traces for the wave equation. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 4, 185\u2013206 (1998)","journal-title":"Ann. Scuola Norm. Sup. Pisa Cl. Sci."},{"key":"1120_CR42","doi-asserted-by":"publisher","first-page":"1411","DOI":"10.1016\/j.enganabound.2009.05.008","volume":"33","author":"DL Young","year":"2009","unstructured":"Young, D.L., Gu, M.H., Fan, C.M.: The time-marching method of fundamental solutions for wave equations. Eng. Anal. Bound Elem. 33, 1411\u20131425 (2009)","journal-title":"Eng. Anal. Bound Elem."}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-021-01120-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11075-021-01120-x\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-021-01120-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,11,3]],"date-time":"2023-11-03T14:04:13Z","timestamp":1699020253000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11075-021-01120-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,5,10]]},"references-count":42,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2022,1]]}},"alternative-id":["1120"],"URL":"https:\/\/doi.org\/10.1007\/s11075-021-01120-x","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,5,10]]},"assertion":[{"value":"27 January 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 April 2021","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 May 2021","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}