The spectrum of the simplified magnetohydrodynamic ballooning equation by computation of the titchmarsh-weyl coefficient

Milligan, Gerard
(1989)
The spectrum of the simplified magnetohydrodynamic ballooning equation by computation of the titchmarsh-weyl coefficient.
Master of Science thesis, Dublin City University.

The boundary value problem involving the magnetohydrodynamic ballooning equation,
(1+x2)d2y/dx2 + 2xdy/dx-(X+^’(l+x2)-^(1+x2))y=0, x«R,
with boundary conditions y — > 0 as x — >îoq , arises from a study of the stability of disturbances in a magnetically confined plasma. In order to facilitate the analysis, the equation is transformed to Schrodinger form,
Y"+[X-q(x)]Y=0
q(x) £ R, with the same boundary conditions and the results of Titchmarsh applied. Two cases are examined, The first with S'=0 (modified Legendre equation) and the second with~<*>0. In both cases the Titchmarsh-Weyl coefficients m( x) are constructed and their singularities examined. The singularities yield expressions for the eigenvalues of the problem. In the second case the expressions involve the joining factors of the spheroidal equation, K^(>.) and K ^ f). which join solutions at infinity with solutions about ¿1. For the case '£'>0, the limit of mCX) as X 1-—>0, is shown to be the m(>») coefficient in the case "4,‘= 0.