{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T10:42:05Z","timestamp":1740134525133,"version":"3.37.3"},"reference-count":20,"publisher":"Wiley","license":[{"start":{"date-parts":[[2014,1,1]],"date-time":"2014-01-01T00:00:00Z","timestamp":1388534400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61373003","11271041"],"award-info":[{"award-number":["61373003","11271041"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61373003","11271041"],"award-info":[{"award-number":["61373003","11271041"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2014]]},"abstract":"<jats:p>By using the polynomial expansion in the even order Bernoulli polynomials and using the linear combinations of the shifts of the function<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:mrow><mml:mi>f<\/mml:mi><mml:mrow><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>x<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mrow><mml:mi>x<\/mml:mi><mml:mo>\u2208<\/mml:mo><mml:mi>\u211d<\/mml:mi><\/mml:mrow><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math>to approximate the derivatives of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:mrow><mml:mi>f<\/mml:mi><mml:mrow><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>x<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math>, we propose a family of modified even order Bernoulli-type multiquadric quasi-interpolants which do not require the derivatives of the function approximated at each node and can satisfy any degree polynomial reproduction property. Error estimate indicates that our operators could provide the desired precision by choosing a suitable shape-preserving parameter<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\"><mml:mrow><mml:mi>c<\/mml:mi><\/mml:mrow><\/mml:math>and a nonnegative integer<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\"><mml:mrow><mml:mi>m<\/mml:mi><\/mml:mrow><\/mml:math>. Numerical comparisons show that this technique provides a higher degree of accuracy. Finally, applying our operators to the fitting of discrete solutions of initial value problems, we find that our method has smaller errors than the Runge-Kutta method of order 4 and Wang et al.\u2019s quasi-interpolation scheme.<\/jats:p>","DOI":"10.1155\/2014\/389215","type":"journal-article","created":{"date-parts":[[2014,3,10]],"date-time":"2014-03-10T17:02:06Z","timestamp":1394470926000},"page":"1-14","source":"Crossref","is-referenced-by-count":0,"title":["A Family of Modified Even Order Bernoulli-Type Multiquadric Quasi-Interpolants with Any Degree Polynomial Reproduction Property"],"prefix":"10.1155","volume":"2014","author":[{"given":"Ruifeng","family":"Wu","sequence":"first","affiliation":[{"name":"School of Mathematics, Jilin University, Changchun 130012, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Huilai","family":"Li","sequence":"additional","affiliation":[{"name":"School of Mathematics, Jilin University, Changchun 130012, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tieru","family":"Wu","sequence":"additional","affiliation":[{"name":"School of Mathematics, Jilin University, Changchun 130012, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","reference":[{"key":"1","doi-asserted-by":"publisher","DOI":"10.1029\/JB076i008p01905"},{"key":"2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01890410"},{"key":"3","doi-asserted-by":"publisher","DOI":"10.1007\/BF01891407"},{"key":"4","doi-asserted-by":"publisher","DOI":"10.2307\/2007474"},{"key":"5","doi-asserted-by":"publisher","DOI":"10.1007\/BF01893414"},{"key":"6","doi-asserted-by":"publisher","DOI":"10.1007\/BF01279020"},{"key":"7","doi-asserted-by":"publisher","DOI":"10.1007\/BF02016334"},{"key":"8","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2003.05.014"},{"key":"9","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2008.08.024"},{"key":"10","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2009.06.023"},{"key":"11","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2005.02.027"},{"key":"12","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2006.07.160"},{"issue":"8","key":"13","doi-asserted-by":"crossref","first-page":"1187","DOI":"10.1002\/(SICI)1097-0207(20000720)48:8<1187::AID-NME942>3.0.CO;2-K","volume":"48","year":"2000","journal-title":"International Journal for Numerical Methods in Engineering"},{"key":"14","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2003.12.015"},{"key":"15","doi-asserted-by":"publisher","DOI":"10.1016\/j.enganabound.2004.06.004"},{"key":"16","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-06-01894-1"},{"key":"17","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2004.07.004"},{"key":"21","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142999351042"},{"year":"1960","key":"18"},{"year":"1975","key":"19"}],"container-title":["Journal of Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2014\/389215.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2014\/389215.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2014\/389215.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2017,6,22]],"date-time":"2017-06-22T05:20:14Z","timestamp":1498108814000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.hindawi.com\/journals\/jam\/2014\/389215\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014]]},"references-count":20,"alternative-id":["389215","389215"],"URL":"https:\/\/doi.org\/10.1155\/2014\/389215","relation":{},"ISSN":["1110-757X","1687-0042"],"issn-type":[{"type":"print","value":"1110-757X"},{"type":"electronic","value":"1687-0042"}],"subject":[],"published":{"date-parts":[[2014]]}}}