Carleman Estimates with Second Large Parameter for Second Order Operators

2008 ◽  
pp. 135-159 ◽  
Author(s):  
Victor Isakov ◽  
Nanhee Kim
Keyword(s):  
Second Order ◽  
Carleman Estimates ◽  
Large Parameter

Convergent Liouville–Green expansions for second-order linear differential equations, with an application to Bessel functions

1993 ◽  
Vol 440 (1908) ◽  
pp. 37-54 ◽  
Keyword(s):  
Differential Equations ◽  
Bessel Functions ◽  
Second Order ◽  
Linear Differential Equations ◽  
Large Parameter ◽  
Order Linear ◽  
Factorial Series ◽  
Independent Variable ◽  
Linear Differential ◽  
Asymptotic Validity

A class of second-order linear differential equations with a large parameter u is considered. It is shown that Liouville–Green type expansions for solutions can be expressed using factorial series in the parameter, and that such expansions converge for Re ( u ) > 0, uniformly for the independent variable lying in a certain subdomain of the domain of asymptotic validity. The theory is then applied to obtain convergent expansions for modified Bessel functions of large order.

scholarly journals Carleman Estimates with a Second Large Parameter

2000 ◽  
Vol 249 (2) ◽  
pp. 491-514 ◽  
Author(s):  
Matthias M. Eller
Keyword(s):  
Carleman Estimates ◽  
Large Parameter

Uniform approximate solutions of certain nth-order differential equations. I

1963 ◽  
Vol 59 (1) ◽  
pp. 95-110 ◽  
Author(s):  
J. Heading
Keyword(s):  
Differential Equations ◽  
Approximate Solutions ◽  
Second Order ◽  
Large Parameter ◽  
Integral Methods ◽  
History Of ◽  
Transcendental Functions ◽  
Linear Differential ◽  
Image Position ◽  
Complete History

Exact analytical solutions of certain second-order linear differential equations are often employed as approximate solutions of other second-order differential equations when the solutions of this latter equation cannot be expressed in terms of the standard transcendental functions. The classical exposition of this method has been given by Jeffreys (6); approximate solutions of the equation (using Jeffreys's notation)are given in terms of solutions either of the equationor of the equationwhere h is a large parameter. A complete history of this technique is given in the author's recent text An introduction to phase-integral methods (Heading (5)).

scholarly journals Stability of viscous flow between rotating cylinders. III. Integration of a sixth order linear equation

1946 ◽  
Vol 187 (1011) ◽  
pp. 492-504 ◽  
Keyword(s):  
Differential Equation ◽  
Viscous Flow ◽  
Critical Points ◽  
Second Order ◽  
Sixth Order ◽  
Large Parameter ◽  
Rotating Cylinders ◽  
Linear Differential ◽  
High Reynolds ◽  
Second Order Equations

The aim of this paper is to derive the asymptotic integrals, and their transformations through the critical points, of a certain linear differential equation of the sixth order containing a large parameter. This particular equation is of importance in connexion with the question of stability of viscous flow between rotating cylinders. Since, however, similar equations occur in all questions of stability of viscous flow, a development of proper methods of solution of such equations is of very great importance for problems of viscous flow at high Reynolds numbers. The method of finding asymptotic integrals of linear differential equations containing a large parameter is well known; it was developed by Horn (1899), Sehlesinger (1907), Birkhoff (1908) and Fowler & Lock (1922). The main difficulty of the problem consists in the following. The coefficients of the differential equation are expressions like λΦ(x) , where λ is a large parameter, and Φ(x) is a slowly varying function of the independent variable; the function Φ(x) usually vanishes within the range of x under consideration, with the result that the asymptotic expansions become infinite at such critical points, lose their validity round these points and change their form in passing through such points. The main problem of integration consists, thus, in finding the transformations of the asymptotic integrals in passing through critical points. This problem was considered by Jeffreys (1924, 1942), Kramers (1926) and Goldstein (1928, 1932) for certain second-order equations. Langer (1931), using a different method, discussed several cases of second-order equations; a summary of methods used and results obtained was also given by Langer (1934). A case of a fourth-order equation was solved by Meksyn (in Press).

Sign in / Sign up

Export Citation Format

Share Document