Pub Date : 2020-12-17DOI: 10.1201/9781003072584-11
D. S. Jones
{"title":"Uniform Asymptotic Remainders","authors":"D. S. Jones","doi":"10.1201/9781003072584-11","DOIUrl":"https://doi.org/10.1201/9781003072584-11","url":null,"abstract":"","PeriodicalId":394750,"journal":{"name":"Asymptotic and Computational Analysis","volume":"61 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129855767","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}
Pub Date : 2020-12-17DOI: 10.1201/9781003072584-39
R. Spigler, M. Vianello
{"title":"Approximating Zeros of Solutions of Second-Order Linear ODEs by “Phase Function” Methods","authors":"R. Spigler, M. Vianello","doi":"10.1201/9781003072584-39","DOIUrl":"https://doi.org/10.1201/9781003072584-39","url":null,"abstract":"","PeriodicalId":394750,"journal":{"name":"Asymptotic and Computational Analysis","volume":"61 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133471956","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}
Pub Date : 2020-12-17DOI: 10.1201/9781003072584-36
N. Rudraiah
The effects of Lorentz force and non-uniform temperature gradient on the onset of magnetoconvection in an electrically conducting horizontal Boussinesq fluid layer permeated by a uniform transverse magnetic field are studied analytically using linear stability analysis by specifying constant temperature or constant heat flux at the boundaries. It is shown that when the Chandrasekhar number Q → ∞ the correct asymptotic value of the critical Rayleigh number, Rc, can be obtained from the non-viscous MHD equations using a single-term Galerkin expansion. The criterion for the onset of magnetoconvection is determined using a regular perturbation technique with wave-number as perturbation parameter. The method of matched asymptotics is used to predict explicitly the effect of the Hartmann boundary layer (that exists at the rigid boundary for large values of Q) on the onset of magnetoconvection. It is shown that the effect of the Hartmann boundary layer is to increase the asymptotic value of Rc by an amount proportional to the value of the Hartmann number M. We find that the ratios Rci/Rcl where Rci (i = 1 to 6) are the asymptotic values of RC for different nonlinear temperature profiles, are independent of Q but dependent on thermal depth e. It is also shown that the power law for asymptotic values of Rci depends crucially on the nature of heating and not on the nature of the boundaries.
采用线性稳定性分析方法，在边界处指定恒温或恒热流密度，研究了均匀横向磁场渗透的水平导电Boussinesq流体层中，洛伦兹力和非均匀温度梯度对磁对流发生的影响。结果表明，当钱德拉塞卡数Q→∞时，利用单项伽辽金展开可以得到非粘性MHD方程临界瑞利数Rc的正确渐近值。采用以波数为扰动参数的规则摄动方法，确定了磁对流发生的判据。用匹配渐近法明确地预测了哈特曼边界层(在大Q值时存在于刚性边界处)对磁对流起始的影响。结果表明，哈特曼边界层的作用是使Rc的渐近值增加一个与哈特曼数m成正比的量。我们发现Rci/Rcl的比值，其中Rci (i = 1至6)是不同非线性温度分布下Rc的渐近值。与Q无关，但与热深度e有关。还表明，Rci渐近值的幂律主要取决于加热的性质，而不是边界的性质。
{"title":"Asymptotic Methods in Magnetoconvection","authors":"N. Rudraiah","doi":"10.1201/9781003072584-36","DOIUrl":"https://doi.org/10.1201/9781003072584-36","url":null,"abstract":"The effects of Lorentz force and non-uniform temperature gradient on the onset of magnetoconvection in an electrically conducting horizontal Boussinesq fluid layer permeated by a uniform transverse magnetic field are studied analytically using linear stability analysis by specifying constant temperature or constant heat flux at the boundaries. It is shown that when the Chandrasekhar number Q → ∞ the correct asymptotic value of the critical Rayleigh number, Rc, can be obtained from the non-viscous MHD equations using a single-term Galerkin expansion. The criterion for the onset of magnetoconvection is determined using a regular perturbation technique with wave-number as perturbation parameter. The method of matched asymptotics is used to predict explicitly the effect of the Hartmann boundary layer (that exists at the rigid boundary for large values of Q) on the onset of magnetoconvection. It is shown that the effect of the Hartmann boundary layer is to increase the asymptotic value of Rc by an amount proportional to the value of the Hartmann number M. We find that the ratios Rci/Rcl where Rci (i = 1 to 6) are the asymptotic values of RC for different nonlinear temperature profiles, are independent of Q but dependent on thermal depth e. It is also shown that the power law for asymptotic values of Rci depends crucially on the nature of heating and not on the nature of the boundaries.","PeriodicalId":394750,"journal":{"name":"Asymptotic and Computational Analysis","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125567202","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}
Pub Date : 2020-12-17DOI: 10.1201/9781003072584-22
N. Temme
We discuss asymptotic forms of CJ( x ), L~( x), P~0t,/3) ( x ), the Gegenbauer, Laguerre and Jacobi polynomials. The asymptotic behavior of these classical orthogonal polynomials has been the subject of several investigations. The research usually concentrates on the case that the degree n of the polynomial is the large parameter, and for all classical orthogonal polynomials the asymptotic behavior is well established now. Inside the domain of the zeros of the polynomial the behavior can be described in terms of elementary functions, such as trigonometric functions. In the domain where the transition from oscillatory to monotonic behavior occurs, familiar higher transcendental functions can be used as estimates. For example, the "first" zeros of the Jacobi polynomial can be approximated in terms of the zeros of the J Bessel function. In SzEGO ( 1958) several classical results can be found. For Jacobi polynomials he gives an estimate of
{"title":"Polynomial Asymptotic Estimates of Gegenbauer, Laguerre, and Jacobi Polynomials","authors":"N. Temme","doi":"10.1201/9781003072584-22","DOIUrl":"https://doi.org/10.1201/9781003072584-22","url":null,"abstract":"We discuss asymptotic forms of CJ( x ), L~( x), P~0t,/3) ( x ), the Gegenbauer, Laguerre and Jacobi polynomials. The asymptotic behavior of these classical orthogonal polynomials has been the subject of several investigations. The research usually concentrates on the case that the degree n of the polynomial is the large parameter, and for all classical orthogonal polynomials the asymptotic behavior is well established now. Inside the domain of the zeros of the polynomial the behavior can be described in terms of elementary functions, such as trigonometric functions. In the domain where the transition from oscillatory to monotonic behavior occurs, familiar higher transcendental functions can be used as estimates. For example, the \"first\" zeros of the Jacobi polynomial can be approximated in terms of the zeros of the J Bessel function. In SzEGO ( 1958) several classical results can be found. For Jacobi polynomials he gives an estimate of","PeriodicalId":394750,"journal":{"name":"Asymptotic and Computational Analysis","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128605218","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}
Pub Date : 2020-12-11DOI: 10.1201/9781003072584-35
R. Paris
{"title":"The Asymptotics of Pearcey’s Integral for Complex Variables","authors":"R. Paris","doi":"10.1201/9781003072584-35","DOIUrl":"https://doi.org/10.1201/9781003072584-35","url":null,"abstract":"","PeriodicalId":394750,"journal":{"name":"Asymptotic and Computational Analysis","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117048187","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}
Pub Date : 2020-12-11DOI: 10.1201/9781003072584-26
R. V. Zahar
{"title":"A Generalization of Olver’s Algorithm for Linear Difference Systems","authors":"R. V. Zahar","doi":"10.1201/9781003072584-26","DOIUrl":"https://doi.org/10.1201/9781003072584-26","url":null,"abstract":"","PeriodicalId":394750,"journal":{"name":"Asymptotic and Computational Analysis","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130318727","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":"Graphs as an Aid to Understanding Special Functions","authors":"R. Askey","doi":"10.1201/9781003072584-2","DOIUrl":"https://doi.org/10.1201/9781003072584-2","url":null,"abstract":"","PeriodicalId":394750,"journal":{"name":"Asymptotic and Computational Analysis","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131079501","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}
Pub Date : 2020-12-11DOI: 10.1201/9781003072584-41
A. Wood, R. Paris
{"title":"On Eigenvalues with Exponentially Small Imaginary Part","authors":"A. Wood, R. Paris","doi":"10.1201/9781003072584-41","DOIUrl":"https://doi.org/10.1201/9781003072584-41","url":null,"abstract":"","PeriodicalId":394750,"journal":{"name":"Asymptotic and Computational Analysis","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121598417","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":"Asymptotic Expansions for the Coefficient Functions Associated with Linear Second-Order Differential Equations: The Simple Pole Case","authors":"W. Boyd","doi":"10.1201/9781003072584-4","DOIUrl":"https://doi.org/10.1201/9781003072584-4","url":null,"abstract":"","PeriodicalId":394750,"journal":{"name":"Asymptotic and Computational Analysis","volume":"67 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114356543","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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