{"title":"A numerical method for solving the Cauchy problem for ODEs using a system of polynomials generated by a system of modified Laguerre polynomials","authors":"G. Akniyev, R. Gadzhimirzaev","doi":"10.31029/demr.12.2","DOIUrl":null,"url":null,"abstract":"In this paper, we consider a numerical realization of an iterative method for solving the Cauchy problem for ordinary differential equations, based on representing the solution in the form of a Fourier series by the system of polynomials $\\{L_{1,n}(x;b)\\}_{n=0}^\\infty$, orthonormal with respect to the Sobolev-type inner product $$ \\langle f,g\\rangle=f(0)g(0)+\\int_{0}^\\infty f'(x)g'(x)\\rho(x;b)dx $$ and generated by the system of modified Laguerre polynomials $\\{L_{n}(x;b)\\}_{n=0}^\\infty$, where $b>0$. In the approximate calculation of the Fourier coefficients of the desired solution, the Gauss -- Laguerre quadrature formula is used.","PeriodicalId":431345,"journal":{"name":"Daghestan Electronic Mathematical Reports","volume":"69 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Daghestan Electronic Mathematical Reports","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.31029/demr.12.2","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper, we consider a numerical realization of an iterative method for solving the Cauchy problem for ordinary differential equations, based on representing the solution in the form of a Fourier series by the system of polynomials $\{L_{1,n}(x;b)\}_{n=0}^\infty$, orthonormal with respect to the Sobolev-type inner product $$ \langle f,g\rangle=f(0)g(0)+\int_{0}^\infty f'(x)g'(x)\rho(x;b)dx $$ and generated by the system of modified Laguerre polynomials $\{L_{n}(x;b)\}_{n=0}^\infty$, where $b>0$. In the approximate calculation of the Fourier coefficients of the desired solution, the Gauss -- Laguerre quadrature formula is used.