We present the error analysis of class of second order nonlinear hyperbolic interface problem where the spatial and time discretizations are based on finite element method and linearized backward difference scheme respectively. Both semi discrete and fully discrete schemes are analyzed with the assumption that the interface is arbitrary but smooth. Almost optimal convergence rate in (H^1(Omega))-norm is obtained. Examples are given to support the theoretical result.
{"title":"Approximate solution of nonlinear hyperbolic equations with homogeneous jump conditions","authors":"M. O. Adewole","doi":"10.33993/jnaat482-1175","DOIUrl":"https://doi.org/10.33993/jnaat482-1175","url":null,"abstract":"We present the error analysis of class of second order nonlinear hyperbolic interface problem where the spatial and time discretizations are based on finite element method and linearized backward difference scheme respectively. \u0000Both semi discrete and fully discrete schemes are analyzed with the assumption that the interface is arbitrary but smooth. \u0000Almost optimal convergence rate in (H^1(Omega))-norm is obtained. \u0000Examples are given to support the theoretical result.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"127 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122487392","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
E.I. Zolotarev's classical so-called First Problem (ZFP), which was posed to him by P.L. Chebyshev, is to determine, for a given (nin{mathbb N}backslash{1}) and for a given (sin{mathbb R}backslash{0}), the monic polynomial solution (Z^{*}_{n,s}) to the following best approximation problem: Find[min_{a_k}max_{xin[-1,1]}|a_0+a_1 x+dots+a_{n-2}x^{n-2}+(-n s)x^{n-1}+x^n|,]where the (a_k, 0le kle n-2), vary in (mathbb R). It suffices to consider the cases (s>tan^2left(pi/(2n)right)). In 1868 Zolotarev provided a transcendental solution for all (ngeq2) in terms of elliptic functions. An explicit algebraic solution in power form to ZFP, as is suggested by the problem statement, is available only for (2le nle 5.^1) We have now obtained an explicit algebraic solution to ZFP for (6le nle 12) in terms of roots of dedicated polynomials. In this paper, we provide our findings for (6le nle 7) in two alternative fashions, accompanied by concrete examples. The cases (8le nle 12) we treat, due to their bulkiness, in a separate web repository. (^1) Added in proof: But see our recent one-parameter power form solution for (n=6) in [38].
{"title":"Explicit algebraic solution of Zolotarev's First Problem for low-degree polynomials","authors":"H. Rack, Róbert Vajda","doi":"10.33993/jnaat482-1173","DOIUrl":"https://doi.org/10.33993/jnaat482-1173","url":null,"abstract":"E.I. Zolotarev's classical so-called First Problem (ZFP), which was posed to him by P.L. Chebyshev, is to determine, for a given (nin{mathbb N}backslash{1}) and for a given (sin{mathbb R}backslash{0}), the monic polynomial solution (Z^{*}_{n,s}) to the following best approximation problem: Find[min_{a_k}max_{xin[-1,1]}|a_0+a_1 x+dots+a_{n-2}x^{n-2}+(-n s)x^{n-1}+x^n|,]where the (a_k, 0le kle n-2), vary in (mathbb R). It suffices to consider the cases (s>tan^2left(pi/(2n)right)). In 1868 Zolotarev provided a transcendental solution for all (ngeq2) in terms of elliptic functions. An explicit algebraic solution in power form to ZFP, as is suggested by the problem statement, is available only for (2le nle 5.^1) We have now obtained an explicit algebraic solution to ZFP for (6le nle 12) in terms of roots of dedicated polynomials. In this paper, we provide our findings for (6le nle 7) in two alternative fashions, accompanied by concrete examples. The cases (8le nle 12) we treat, due to their bulkiness, in a separate web repository. (^1) Added in proof: But see our recent one-parameter power form solution for (n=6) in [38].","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"64 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128856636","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
An elliptic one-dimensional second order boundary value problem involving discontinuous coefficients, with or without transmission conditions, is considered. For the former case by a direct sum spaces method we show that the eigenvalues are real, geometrically simple and the eigenfunctions are orthogonal. Then the eigenpairs are computed numerically by a local linear finite element method (FEM) and by some global spectral collocation methods. The spectral collocation is based on Chebyshev polynomials (ChC) for problems on bounded intervals respectively on Fourier system (FsC) for periodic problems. The numerical stability in computing eigenvalues is investigated by estimating their (relative) drift with respect to the order of approximation. The accuracy in computing the eigenvectors is addressed by estimating their departure from orthogonality as well as by the asymptotic order of convergence. The discontinuity of coefficients in the problems at hand reduces the exponential order of convergence, usual for any well designed spectral algorithm, to an algebraic one. As expected, the accuracy of ChC outcomes overpasses by far that of FEM outcomes.
{"title":"Analytic vs. numerical solutions to a Sturm-Liouville transmission eigenproblem","authors":"C. Gheorghiu, Bertin Zinsou","doi":"10.33993/jnaat482-1201","DOIUrl":"https://doi.org/10.33993/jnaat482-1201","url":null,"abstract":"An elliptic one-dimensional second order boundary value problem involving discontinuous coefficients, with or without transmission conditions, is considered. For the former case by a direct sum spaces method we show that the eigenvalues are real, geometrically simple and the eigenfunctions are orthogonal. \u0000Then the eigenpairs are computed numerically by a local linear finite element method (FEM) and by some global spectral collocation methods. \u0000The spectral collocation is based on Chebyshev polynomials (ChC) for problems on bounded intervals respectively on Fourier system (FsC) for periodic problems. \u0000The numerical stability in computing eigenvalues is investigated by estimating their (relative) drift with respect to the order of approximation. The accuracy in computing the eigenvectors is addressed by estimating their departure from orthogonality as well as by the asymptotic order of convergence. The discontinuity of coefficients in the problems at hand reduces the exponential order of convergence, usual for any well designed spectral algorithm, to an algebraic one. \u0000As expected, the accuracy of ChC outcomes overpasses by far that of FEM outcomes.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"90 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131929733","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Using the weakly Picard operators technique and the contraction principle, we study the convergence of the iterates of some modified Bernstein type operators.
利用弱Picard算子技术和收缩原理，研究了一类修正Bernstein型算子的迭代收敛性。
{"title":"Iterates of a modified Bernstein type operator","authors":"Teodora Cătinaş","doi":"10.33993/jnaat482-1205","DOIUrl":"https://doi.org/10.33993/jnaat482-1205","url":null,"abstract":"Using the weakly Picard operators technique and the contraction principle, we study the convergence of the iterates of some modified Bernstein type operators.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125439985","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The aim of this paper is twofold. Firstly, we consider the unique solvability of absolute value equations (AVE), (Ax-Bvert xvert =b), when the condition (Vert A^{-1}Vert
本文的目的是双重的。首先，我们考虑了当条件(Vert A^{-1}Vert
{"title":"On the unique solvability and numerical study of absolute value equations","authors":"Achache Mohamed","doi":"10.33993/jnaat482-1182","DOIUrl":"https://doi.org/10.33993/jnaat482-1182","url":null,"abstract":"The aim of this paper is twofold. Firstly, we consider the unique solvability of absolute value equations (AVE), (Ax-Bvert xvert =b), when the condition (Vert A^{-1}Vert <frac{1}{leftVert BrightVert }) holds. This is a generalization of an earlier result by Mangasarian and Meyer for the special case where (B=I). \u0000Secondly, a generalized Newton method for solving the AVE is proposed. We show under the condition (Vert A^{-1}Vert <frac{1}{4Vert BVert }), that the algorithm converges linearly global to the unique solution of the AVE. \u0000Numerical results are reported to show the efficiency of the proposed method and to compare with an available method.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"74 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128213195","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Professor Ion Păvăloiu at his 80'th anniversary","authors":"E. Catinas","doi":"10.33993/jnaat482-1211","DOIUrl":"https://doi.org/10.33993/jnaat482-1211","url":null,"abstract":"","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"56 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116098155","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper considers the construction of second derivative general linear methods (SD-GLM) from hybrid LMM and their transformation to NordsieckGLM. How the Runge-Kutta starters for the methods can be derived are given. The representation of the methods in Nordsieck form has the advantage of easy implementation in variable stepsize.
{"title":"Second derivative General Linear Method in Nordsieck form","authors":"R. I. Okuonghae, M. Ikhile","doi":"10.33993/jnaat481-1140","DOIUrl":"https://doi.org/10.33993/jnaat481-1140","url":null,"abstract":"This paper considers the construction of second derivative general linear methods (SD-GLM) from hybrid LMM and their transformation to NordsieckGLM. How the Runge-Kutta starters for the methods can be derived are given. The representation of the methods in Nordsieck form has the advantage of easy implementation in variable stepsize. \u0000 ","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121203805","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A rigorous convergence analysis is presented for arbitrary order cardinal spline subdivision with general integer arity, for which the binary case, with arity two, is a well-studied subject. Explicit geometric convergence rates are derived, and particular attention is devoted to the rendering of cardinal spline graphs and parametric curves.
{"title":"Geometric convergence rates for cardinal spline subdivision with general integer arity","authors":"J. De Villiers, Mpafereleni Rejoyce Gavhi-Molefe","doi":"10.33993/jnaat481-1164","DOIUrl":"https://doi.org/10.33993/jnaat481-1164","url":null,"abstract":"A rigorous convergence analysis is presented for arbitrary order cardinal spline subdivision with general integer arity, for which the binary case, with arity two, is a well-studied subject. Explicit geometric convergence rates are derived, and particular attention is devoted to the rendering of cardinal spline graphs and parametric curves.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133144653","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
An integral formula for the Goldbach partitions requires uniform convergence of a complex exponential sum. The dependence of the coefficients of the series is found to be bounded by that of cusp forms. Norms may be defined for these forms on a fundamental domain of a modular group. The relation with the integral formula is found to be sufficient to establish the consistency of the interchange of the integral and the sum, which must remain valid as the even integer $N$ tends to infinity.
{"title":"Goldbach partitions and norms of cusp forms","authors":"S. Davis","doi":"10.33993/jnaat481-1152","DOIUrl":"https://doi.org/10.33993/jnaat481-1152","url":null,"abstract":"An integral formula for the Goldbach partitions requires uniform convergence of a complex exponential sum. The dependence of the coefficients of the series is found to be bounded by that of cusp forms. Norms may be defined for these forms on a fundamental domain of a modular group. The relation with the integral formula is found to be sufficient to establish the consistency of the interchange of the integral and the sum, which must remain valid as the even integer $N$ tends to infinity.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133180751","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Some variants of the numerical Picard iterations method are presented to solve an IVP for an ordinary differential system. The term "numerical" emphasizes that a numerical solution is computed. The method consists in replacing the right hand side of the differential system by Lagrange interpolation polynomials followed by successive approximations. In the case when the number of interpolation point is fixed a convergence result is given. Finally some numerical experiments are reported.
{"title":"On the numerical Picard iterations with collocations for the initial value problem","authors":"E. Scheiber","doi":"10.33993/jnaat481-1146","DOIUrl":"https://doi.org/10.33993/jnaat481-1146","url":null,"abstract":"Some variants of the numerical Picard iterations method are presented to solve an IVP for an ordinary differential system. The term \"numerical\" emphasizes that a numerical solution is computed. The method consists in replacing the right hand side of the differential system by Lagrange interpolation polynomials followed by successive approximations. In the case when the number of interpolation point is fixed a convergence result is given. Finally some numerical experiments are reported.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117127399","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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