Lame differential equations

Both works [2] and [3] show the separations of the eigenvalue equations for H and H, and H and H, in their respective spheroconal coordinates, into Lame differential equations in the individual elliptical cone angular coordinates. The corresponding solutions are Lam6 spheroconal polynomials included in the classic book of Whittaker and Watson [12]. In practice, the numerical evaluation of such Lame functions was not developed in an efficient manner so that the exact formulation of Ref. [2] did not prosper. Consequently, the analysis of rotations of asymmetric molecules took the route of perturbation theory using the familiar basis of spherical harmonics. 

Reversible bimolecular reactions such asA + B C + D can be solved exactly by the method of separation of variables and the ordinary differential equations in the variable s are Lame equations. This makes the evaluation of the Fourier-type coefficients very difficult since derivative formulas and orthogonality conditions do not seem to exist or at least are not easily used. In addition to this, even if such formulas did exist, it seems unlikely that numerical results could be easily obtained. It does turn out, however, that these reversible bimolecular processes can be solved exactly and conveniently in the equilibrium limit, and this was done by Darvey, Ninham, and Staff.14... 

We also include the differential equations for fhe remaining factor in the Lame functions after removing the singularity factors... 

On substituting these relations into Eq. (14), the stress equilibrium equations can be replaced by a pair of second-order differential equations, the Lame equations ... 

The second method is the Lame approach (Kashani and Young, 2008), which is based on displacement differential equations and is applicable to any cylindrical vessel with any diameter-to-wall-thickness ratio. The Lame method is often referred to as the solution for thick wall cylindrical pressure vessels. Equations for the hoop stress and radial stress in a thick-walled cylinder were developed by Lame in the early nineteenth century (Timoshenko and Goodier, 1969) ... 

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