{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,6]],"date-time":"2026-01-06T18:49:55Z","timestamp":1767725395944,"version":"build-2238731810"},"reference-count":11,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2017,11,22]],"date-time":"2017-11-22T00:00:00Z","timestamp":1511308800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2017,11,22]],"date-time":"2017-11-22T00:00:00Z","timestamp":1511308800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2018,6]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    This paper begins with a question of existence of a\n                    <jats:italic>regular<\/jats:italic>\n                    integral equation formalism, but different from the existing usual ones, for solving the standard Boussinesq\u2019s equations for\n                    <jats:italic>variable<\/jats:italic>\n                    water depth (or Peregrine\u2019s model). For the question, a\n                    <jats:italic>pseudo<\/jats:italic>\n                    -water depth parameter, suggested by Jang (Commun Nonlinear Sci Numer Simul 43:118\u2013138, 2017), is introduced to alter the standard Boussinesq\u2019s equations into an integral formalism. This enables us to construct a regular (nonlinear) integral equations of\n                    <jats:italic>second<\/jats:italic>\n                    kind (as required), being equivalent to the standard Boussinesq\u2019s equations (of Peregrine\u2019s model). The (constructed) integral equations are, of course, inherently different from the usual integral equation formalisms. For solving them, the successive approximation (or the fixed point iteration) is applied (Jang 2017), whereby a new iterative formula is immediately derived, in this paper, for numerical solutions of the standard Boussinesq\u2019s equations for variable water depth. The formula, semi-analytic and derivative-free, is shown to be useful to observe especially the nonlinear wave phenomena of shallow water waves on a beach. In fact, a numerical experiment is performed on a solitary wave approaching a sloping beach. It shows clearly the main feature of nonlinear wave characteristics, which has reached good agreement with the known (numerical) solutions. Hence, while being theoretical but fundamental in nonlinear computational partial differential equations, the question raised in the study may be solved.\n                  <\/jats:p>","DOI":"10.1007\/s10915-017-0605-6","type":"journal-article","created":{"date-parts":[[2017,11,22]],"date-time":"2017-11-22T10:03:47Z","timestamp":1511345027000},"page":"1721-1756","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["A Regular Integral Equation Formalism for Solving the Standard Boussinesq\u2019s Equations for Variable Water Depth"],"prefix":"10.1007","volume":"75","author":[{"given":"T. S.","family":"Jang","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2017,11,22]]},"reference":[{"key":"605_CR1","doi-asserted-by":"publisher","first-page":"815","DOI":"10.1017\/S0022112067002605","volume":"27","author":"DH Peregrine","year":"1967","unstructured":"Peregrine, D.H.: Long waves on a beach. J. Fluid Mech. 27, 815\u2013827 (1967)","journal-title":"J. Fluid Mech."},{"key":"605_CR2","doi-asserted-by":"publisher","first-page":"33","DOI":"10.1016\/j.cpc.2015.12.013","volume":"201","author":"J Yan","year":"2016","unstructured":"Yan, J., Zhang, Z.: New energy-preserving schemes using Hamiltonian boundary value and Fourier pseudospectral methods for the numerical solution of the \u201cgood\u201d Boussinesq equation. Comput. Phys. Commun. 201, 33\u201342 (2016)","journal-title":"Comput. Phys. 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Eng., American Society of Civil Engineers, Washington, D.C., pp. 345\u2013361 (1970)","DOI":"10.1061\/9780872620285.021"}],"updated-by":[{"DOI":"10.1007\/s10915-019-01094-y","type":"correction","label":"Correction","source":"publisher","updated":{"date-parts":[[2019,11,18]],"date-time":"2019-11-18T00:00:00Z","timestamp":1574035200000}}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10915-017-0605-6\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-017-0605-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-017-0605-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,9,16]],"date-time":"2020-09-16T16:38:07Z","timestamp":1600274287000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10915-017-0605-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,11,22]]},"references-count":11,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2018,6]]}},"alternative-id":["605"],"URL":"https:\/\/doi.org\/10.1007\/s10915-017-0605-6","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,11,22]]},"assertion":[{"value":"2 March 2017","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 November 2017","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 November 2017","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 November 2017","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"18 November 2019","order":5,"name":"change_date","label":"Change Date","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"Correction","order":6,"name":"change_type","label":"Change Type","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The author would like to correct an error in calculation of the Eq. (A.2.8) in Appendix A.2 of the original article. The correct equation is as follows:","order":7,"name":"change_details","label":"Change Details","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"18 November 2019","order":6,"name":"change_date","label":"Change Date","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"Correction","order":6,"name":"change_type","label":"Change Type","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The author would like to correct an error in calculation of the Eq. (A.2.8) in Appendix A.2 of the original article. The correct equation is as follows:","order":7,"name":"change_details","label":"Change Details","group":{"name":"ArticleHistory","label":"Article History"}}]}}