A Comment on the Approximate Calculation of Eigenvalues for Certain Second-Order Linear Differential Equations

1962 ◽  
Vol 66 (623) ◽  
pp. 722-722
Author(s):  
William Squire
Keyword(s):  
Approximate Calculation ◽  
Quadratic Equation ◽  
Second Order ◽  
Interesting Aspect ◽  
Approximate Evaluation ◽  
Order Linear ◽  
Combined Use ◽  
Linear Differential ◽  
Lowest Eigenvalue ◽  
Variational Expression

In a note by Goodey on the combined use of the WKB solution and Rayleigh's principle to estimate the lowest eigenvalue of a second-order linear differential equation, a numerical error concealed an interesting aspect of the result. The exact value λ = 2·062 corresponds to 0·6564π; and not 0·654π as stated. The approximate value 0·6559π is therefore lower than the exact value.The possibility of approximate evaluation of the integral in a variational expression affecting the direction of approach to the exact value was pointed out by the author in a recent paper on the application of quadrature by differentiation. The application of the method described there to the example considered by Goodey may be of some interest as it gives the eigenvalue to within 1/2 per cent by the solution of a quadratic equation.

The Approximate Calculation of Eigenvalues for Certain Second-Order Linear Differential Equations

1961 ◽  
Vol 65 (605) ◽  
pp. 360-360 ◽  
Author(s):  
W. J. Goodey
Keyword(s):  
Approximate Solution ◽  
Differential Equations ◽  
Simple Formula ◽  
Approximate Calculation ◽  
Technical Note ◽  
Second Order ◽  
Numerical Results ◽  
Linear Differential Equations ◽  
Order Linear ◽  
Linear Differential

In a recent technical note, Squire discussed the approximate solution of certain second-order linear differential equations by the method attributed variously to Riccati, Madelung, Wentzel, Kramers and Brillouin (the W.K.B. method), and others. The problem of eigenvalues, frequently met with in this type of equation, does not, however, appear to have received much attention by this method, and in this note a simple formula is developed which appears to give excellent numerical results in many cases.

A property of the asymptotic series for a class of Titchmarsh–Weyl m-functions

1986 ◽  
Vol 102 (3-4) ◽  
pp. 253-257 ◽  
Author(s):  
B. J. Harris
Keyword(s):  
Differential Equation ◽  
Asymptotic Expansion ◽  
Linear Differential Equation ◽  
Asymptotic Series ◽  
Second Order ◽  
Order Linear Differential Equation ◽  
Order Linear ◽  
The Real ◽  
Linear Differential ◽  
Image Position

SynopsisIn an earlier paper [6] we showed that if q ϵ CN[0, ε) for some ε > 0, then the Titchmarsh–Weyl m(λ) function associated with the second order linear differential equationhas the asymptotic expansionas |A| →∞ in a sector of the form 0 < δ < arg λ < π – δ.We show that if the real valued function q admits the expansionin a neighbourhood of 0, then

A mechanical method for the solution of second-order linear differential equations

1931 ◽  
Vol 27 (4) ◽  
pp. 546-552 ◽  
Author(s):  
E. C. Bullard ◽  
P. B. Moon
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Order Differential Equation ◽  
Second Order ◽  
Linear Differential Equations ◽  
Mechanical Method ◽  
Order Linear ◽  
Second Order Differential Equation ◽  
Linear Differential

A mechanical method of integrating a second-order differential equation, with any boundary conditions, is described and its applications are discussed.

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