Mathieu functions of general order: connection formulae, base functions and asymptotic formulae. III. The Liouville-Green method and its extensions

1981 ◽  
Vol 301 (1459) ◽  
pp. 99-114 ◽  
Keyword(s):  
Error Estimates ◽  
Error Control ◽  
Control Function ◽  
Mathieu Equation ◽  
Mathieu Functions ◽  
Green Method ◽  
Base Functions ◽  
General Order ◽  
Transcendental Functions ◽  
Connection Formulae

An account is given of the Liouville-Green method for the approximate solution, with error estimates, of linear second-order differential equations, together with certain extensions of the method. The purpose is to make readily available a range of techniques for use in the two final parts of the present series. The topics treated include: ( a ) the construction of approximations in terms of both elementary and higher transcendental functions, ( b ) the relations between approximations of the same solution in terms of different functions, ( c ) the identification of solutions and the estimation of connection coefficients, ( d ) uniform estimation of the error-control function in problems with more than one widely ranging parameter, ( e ) the construction of majorants for approximating functions, the last two being required for the derivation of satisfactory error estimates. There is little in this part that is new, though a method of constructing approximations in terms of Bessel functions is developed specifically for application to the Mathieu equation. Apart from this, some aspects of the presentation are thought to be novel.

Mathieu functions of general order: connection formulae, base functions and asymptotic formulae. IV. The Liouville-Green method applied to the Mathieu equation

1981 ◽  
Vol 301 (1459) ◽  
pp. 115-135 ◽  
Keyword(s):  
Full Range ◽  
Mathieu Equation ◽  
Real Variable ◽  
Mathieu Functions ◽  
Asymptotic Formulae ◽  
Half Strip ◽  
Base Functions ◽  
General Order ◽  
Circular Functions ◽  
Connection Formulae

The methods described in part III and the formulae derived in part II are applied to the construction of a comprehensive set of asymptotic formulae relating to the Mathieu equation y ′ ′ + ( λ + 2 h 2 cos ⁡ 2 z ) y = 0 with real parameters. These comprise formulae both ( a ) for the auxiliary parameters and ( b ), in terms of exponential and circular functions, for the fundamental solution, a function of a complex variable, and the various pairs of real-variable base-functions, all introduced in part II. With the aid of these, together with connection formulae also obtained in part II, approximations can readily be obtained for Mathieu functions of various types, including in particular periodic functions. Formulae for solutions are applicable on the half-strip { z : 0 ⩽ Re z ⩽ 1 2 π , Im z ⩾ 0 } , with the transition point of the differential equation which lies on its frontier removed, or in the case of real-variable solutions of the ordinary or modified equation, on the interval [0,½π] or [0, ∞] respectively, with the same qualification as for the half-strip when this is relevant. The formulae cover the full range of the parameters subject to A ≠ ± 2h 2 . The O -terms providing error estimates are uniformly valid on any subdomain of the independent variable and parameters on which they remain bounded.

Mathieu functions of general order: connection formulae, base functions and asymptotic formulae. I. Introduction

1981 ◽  
Vol 301 (1459) ◽  
pp. 75-79 ◽  
Keyword(s):  
Mathieu Functions ◽  
Asymptotic Formulae ◽  
Mathieu’S Equation ◽  
Actual Error ◽  
Base Functions ◽  
General Order ◽  
Order Of Magnitude ◽  
Introductory Paper ◽  
Transcendental Functions ◽  
Connection Formulae

This is the first, introductory, paper of a series devoted to the derivation of a comprehensive set of approximate formulae for solutions of Mathieu’s equation with real parameters, in terms both of elementary and of higher transcendental functions. Order-of-magnitude error-estimates are obtained; these in every case reflect faithfully the behaviour of the actual error over the appropriate range of parameters and of independent variable. The general scope of the work is outlined in this Introduction, and is compared with that of previous work, in particular that of Langer (1934 b ). There then follows a description of the plan of the work and of the content of the several parts.

Mathieu functions of general order: connection formulae, base functions and asymptotic formulae. II. Connection formulae and base functions

1981 ◽  
Vol 301 (1459) ◽  
pp. 81-97 ◽  
Keyword(s):  
Mathieu Equation ◽  
Real Variable ◽  
Stability Diagram ◽  
Mathieu Function ◽  
Mathieu Functions ◽  
Stability And Instability ◽  
Base Functions ◽  
General Order ◽  
Parameter Ranges ◽  
Connection Formulae

Connection formulae are examined which relate a solution y(z) of the Mathieu equation y" + (A + 2h 2 cos 2z) y = 0 with the solutions y ( ± z ± nπ) generated from it by the symmetry group of the equation. The treatment is exact, and is made first in the context of more general periodic differential equations; the results are then specialized to the Mathieu equation, a function of the third kind, characterized by its asymptotic behaviour as z → ∞i, being taken as fundamental. Two parameter ranges are then distinguished, corresponding to the regions of the stability diagram (a) where the solutions are always unstable and ( b ) where subregions of stability and instability alternate. Auxiliary parameters are defined in the two cases, and pairs of real-variable base-functions are constructed, appropriate to the ordinary Mathieu equation and to two types of modified equation. These pairs satisfy criteria introduced by Miller (1950). Comprehensive formulae are derived, relating these base-functions to standard types of Mathieu function, and special attention is given to periodic solutions.

Mathieu functions of general order: connection formulae, base functions and asymptotic formulae. V. Approximations in terms of higher transcendental functions

1981 ◽  
Vol 301 (1459) ◽  
pp. 137-162 ◽  
Keyword(s):  
Exceptional Point ◽  
Parabolic Cylinder ◽  
Maximum Relative Error ◽  
Mathieu Functions ◽  
Airy Functions ◽  
Half Strip ◽  
Base Functions ◽  
General Order ◽  
Transcendental Functions ◽  
Function Approximations

The methods of part III are further applied to the construction of approximations for the fundamental solution and base functions of part II in terms of higher transcendental functions. The domain of validity is now the complete half-strip {z; 0 ≤ Re z ≤ ½π, Im z ≥. 0} without exceptional point. Relative remainder estimates are again uniformly valid provided they are bounded. Specifically, approximations are obtained in terms of: ( a ) Airy functions, applicable if A ≠ ± 2h 2 ; ( b ) parabolic cylinder functions, applicable if |A ≥ 4h 2 , including A = ± 2h 2 ; ( c ) Bessel functions, applicable if |A| ≥ 4h 2 ; these formulae have maximum relative error A - 3/2 h 2 O (l) on the half-strip, even if h is arbitrarily small, provided only that A -1 is bounded. This is significantly better when A/h 2 is large than the corresponding estimate, A -½ 0(1), for the Airy function approximations. Certain more refined estimates for the auxiliary parameters introduced in part II are also obtained.

A new method of solving Oseen’s equations and its application to the flow past an inclined elliptic cylinder

1954 ◽  
Vol 224 (1157) ◽  
pp. 141-160 ◽  
Keyword(s):  
Reynolds Number ◽  
Elliptic Cylinder ◽  
Cylindrical Body ◽  
Thickness Ratio ◽  
Successive Approximations ◽  
Angle Of Incidence ◽  
Mathieu Functions ◽  
Flow Past ◽  
Transcendental Functions ◽  
Lift And Drag

In this paper is developed a general method of solving Oseen’s linearized equations for a two-dimensional steady flow of a viscous fluid past an arbitrary cylindrical body. The method is based on the fact that the velocity in the neighbourhood of the cylinder can be generally expressed in terms of a pair of analytic functions, the determination of which from the appropriate boundary condition can be effected by successive approximations in powers of the Reynolds number, R . The method enables one to obtain the velocity distribution near the cylinder and the lift and drag acting on it in the form of power series in R , without recourse to manipulation of higher transcendental functions such as Bessel and Mathieu functions for circular and elliptic cylinders, respectively. As an example of the application of the method, the uniform flow past an elliptic cylinder at an arbitrary angle of incidence is considered. Analytical expressions for the lift and drag coefficients are obtained, which are correct to the order of R , the lowest order terms being O ( R -1 ) and numerical calculations are carried out for the thickness ratio t = 0, 0.1, 0.5, 1 and the Reynolds number R = 0.1, 1. It is found that drag increases slightly with increase of either thickness ratio or angle of incidence, and that lift decreases with increase of thickness ratio while, as a function of the angle of incidence, it has a maximum at about 45°.

The Second Solution of Mathieu's Differential Equation

1928 ◽  
Vol 24 (2) ◽  
pp. 223-230 ◽  
Author(s):  
S. Goldstein
Keyword(s):  
Differential Equation ◽  
Numerical Computation ◽  
Mathieu Equation ◽  
Mathieu Functions ◽  
Similar Construction

The Mathieu functions of period π and 2π have recently been constructed by the help of analysis similar to that developed by Laplace, Kelvin, Darwin and Hough to find the free tides symmetrical about the axis of a rotating globe. The purpose of this note is to show that a similar construction can be carried out for the second solution of the Mathieu equation, when one solution is periodic in π or 2π, by the help of analysis similar to that used for forced tides. The construction is effected in a form suitable for numerical computation.

XXII.—Researches into the Characteristic Numbers of the Mathieu Equation—(Third Paper)

1928 ◽  
Vol 47 ◽  
pp. 294-301 ◽  
Author(s):  
E. L. Ince
Keyword(s):  
Bessel Functions ◽  
Mathieu Equation ◽  
Mathematical Physics ◽  
Mathieu Functions ◽  
Characteristic Numbers ◽  
Asymptotic Developments ◽  
The Way

The importance in Mathematical Physics of the Bessel functions, whose order is half an odd integer, suggests that the corresponding Mathieu functions may be worthy of a closer attention than they have yet received. At the very least it is expedient to pave the way for their computation. In the second paper bearing the above title, asymptotic developments of the characteristic numbers which correspond to these functions were given; it is here proposed, in the first place, to take the more direct line of approach.

IV.—Researches into the Characteristic Numbers of the Mathieu Equation

1927 ◽  
Vol 46 ◽  
pp. 20-29 ◽  
Author(s):  
E. L. Ince
Keyword(s):  
Fourier Series ◽  
Periodic Solutions ◽  
Mathieu Equation ◽  
Mathieu Functions ◽  
Characteristic Numbers ◽  
Image Position

The characteristic numbers of the Mathieu equationare those values of a for which, when q is given, the equation admits of a solution of period π or 2π. The periodic solutions, or Mathieu functions, may be developed as a Fourier-series convergent for all values of q,multiplied by one or other of the factors

Perturbation Solution for Secondary Bifurcation in the Quadratically-Damped Mathieu Equation

2001 ◽  
Author(s):  
Deepak V. Ramani ◽  
Richard H. Rand ◽  
William L. Keith
Keyword(s):  
Perturbation Method ◽  
Limit Cycles ◽  
Bifurcation Point ◽  
Mathieu Equation ◽  
Bifurcation Curve ◽  
Mathieu Functions ◽  
Bifurcation Of Limit Cycles ◽  
Saddle Node Bifurcation ◽  
Secondary Bifurcation ◽  
Transition Curves

Abstract This paper concerns the quadratically-damped Mathieu equation:x..+(δ+ϵcos⁡t)x+x.|x.|=0. Numerical integration shows the existence of a secondary-bifurcation in which a pair of limit cycles come together and disappear (a saddle-node bifurcation of limit cycles). In δ–ϵ parameter space, this secondary bifurcation appears as a curve which emanates from one of the transition curves of the linear Mathieu equation for ϵ ≈ 1.5. The bifurcation point along with an approximation for the bifurcation curve is obtained by a perturbation method which uses Mathieu functions rather than the usual sines and cosines.

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