{"title":"Solving a non-local linear differential equation model of the Newtonian-type","authors":"Wen-Xiu Ma","doi":"10.1007/s12043-024-02765-8","DOIUrl":null,"url":null,"abstract":"<div><p>Motivated by recent studies on non-local integrable models, we consider a non-local inhomogeneous linear differential equation model of Newtonian type: </p><div><div><span>$$\\begin{aligned} \\hspace{42pt}x''(t)=\\lambda x(t)+\\mu x(-t) +f(t),\\ t\\in {\\mathbb {R}}, \\end{aligned}$$</span></div></div><p>where <span>\\(\\lambda \\)</span> and <span>\\(\\mu \\)</span> are real constants and <i>f</i> is continuous. Through decomposing functions into their even and odd parts, we transform the non-local model into a local model, and then with the classical ODE technique, solve the resulting local model under the even and odd constraints. The general solution involving two arbitrary constants is presented in nine cases of the coefficients.\n</p></div>","PeriodicalId":743,"journal":{"name":"Pramana","volume":"98 2","pages":""},"PeriodicalIF":1.9000,"publicationDate":"2024-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Pramana","FirstCategoryId":"4","ListUrlMain":"https://link.springer.com/article/10.1007/s12043-024-02765-8","RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"PHYSICS, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
Abstract
Motivated by recent studies on non-local integrable models, we consider a non-local inhomogeneous linear differential equation model of Newtonian type:
where \(\lambda \) and \(\mu \) are real constants and f is continuous. Through decomposing functions into their even and odd parts, we transform the non-local model into a local model, and then with the classical ODE technique, solve the resulting local model under the even and odd constraints. The general solution involving two arbitrary constants is presented in nine cases of the coefficients.
期刊介绍:
Pramana - Journal of Physics is a monthly research journal in English published by the Indian Academy of Sciences in collaboration with Indian National Science Academy and Indian Physics Association. The journal publishes refereed papers covering current research in Physics, both original contributions - research papers, brief reports or rapid communications - and invited reviews. Pramana also publishes special issues devoted to advances in specific areas of Physics and proceedings of select high quality conferences.