Exceptional cauchy problems and regular singular points

Abstract

The classical Euler-Poisson-Darboux problem is called exceptional if For such a choice of k, the associated ordinary differential equation problem obtained by replacing Δ and Ø(×) by constants has a regular singular point at t = 0. One of the roots of the indicial equation is a positive integer while the other is zero and the construction of a solution of this associated problem corresponds to using this smaller zero root. Solution functions obtained can be used to connect a solution (non-unique) of the original exceptional Cauchy problem to a non-exceptional one. In this paper, this method will be applied to two examples of exceptional Cauchy problems.

AMS(MOS) Subject classifications: Primary 47A50

Secondary 47E05, 47D10

Keywords:

• Euler-Poisson-Darboux equation
• exceptional case
• Banach space
• strongly continuous group
• regular singular point
• connecting relation

Additional information

Notes on contributors

L. R. Bragg

R. P. Gilbert