{"title":"New method for solving non-homogeneous periodic second-order difference equation and some applications","authors":"R. Ben Taher, M. Lassri, M. Rachidi","doi":"10.1007/s40065-023-00422-3","DOIUrl":null,"url":null,"abstract":"<div><p>In the present study, we are interested in solving the nonhomogeneous second-order linear difference equation with periodic coefficients of period <span>\\( p\\ge 2\\)</span>, by bringing two new approaches enabling us to provide both analytic and combinatorial solutions to this family of equations. First, we get around the problem by converting this kind of equations to an equivalent family of nonhomogeneous linear difference equations of order <i>p</i> with constant coefficients. Second, we propose new expressions of the solutions of this family of equations, using our techniques of calculating the powers of product of companion matrices and some properties of generalized Fibonacci sequences. The study of the special case <span>\\( p=2 \\)</span> is provided. And to enhance the effectiveness of our approaches, some numerical examples are discussed.</p></div>","PeriodicalId":54135,"journal":{"name":"Arabian Journal of Mathematics","volume":"12 3","pages":"647 - 665"},"PeriodicalIF":0.9000,"publicationDate":"2023-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40065-023-00422-3.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Arabian Journal of Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://link.springer.com/article/10.1007/s40065-023-00422-3","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
In the present study, we are interested in solving the nonhomogeneous second-order linear difference equation with periodic coefficients of period \( p\ge 2\), by bringing two new approaches enabling us to provide both analytic and combinatorial solutions to this family of equations. First, we get around the problem by converting this kind of equations to an equivalent family of nonhomogeneous linear difference equations of order p with constant coefficients. Second, we propose new expressions of the solutions of this family of equations, using our techniques of calculating the powers of product of companion matrices and some properties of generalized Fibonacci sequences. The study of the special case \( p=2 \) is provided. And to enhance the effectiveness of our approaches, some numerical examples are discussed.
期刊介绍：
The Arabian Journal of Mathematics is a quarterly, peer-reviewed open access journal published under the SpringerOpen brand, covering all mainstream branches of pure and applied mathematics.
Owned by King Fahd University of Petroleum and Minerals, AJM publishes carefully refereed research papers in all main-stream branches of pure and applied mathematics. Survey papers may be submitted for publication by invitation only.To be published in AJM, a paper should be a significant contribution to the mathematics literature, well-written, and of interest to a wide audience. All manuscripts will undergo a strict refereeing process; acceptance for publication is based on two positive reviews from experts in the field.Submission of a manuscript acknowledges that the manuscript is original and is not, in whole or in part, published or submitted for publication elsewhere. A copyright agreement is required before the publication of the paper.Manuscripts must be written in English. It is the author''s responsibility to make sure her/his manuscript is written in clear, unambiguous and grammatically correct language.
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