Numerical Analysis and the Dirichlet Problem

Abstract

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Approximate solution of axially symmetric problems

“…[1][2][3][4][5]). If one wishes a matrix with a prescribed distribution of characteristic roots, however, it is natural to resort to the similarity transformation A = CRC -1 where the characteristic roots of R are known.…”

“…A number of physical problems associated with such diverse fields as electrostatic field theory, heat and ideal fluid flow, and stress concentration theory [1][2][3] reduce, under the assumption of axial symmetry, to the study on a bounded, simply connected region R of the elliptic differential equation u~ + u~ + k In general, I~ is an arbitrary, but fixed, real constant; but since the special case ,~ = 0 has been treated rather completely in recent years (see, for example, [4,5] and the references therein), the discussion p]aces particular emphasis on the more difficult cases when k ~ 0. If R is bounded away from the x-axis, so that the coefficient of u~ is bounded on R, then Dirichlet problems associated with (1.1) have been explored extensively both theoretically and numerically (see, for example, [1,[6][7][8]).…”