The fundamental solution of the Keldysh type operator

Abstract

In this paper we discuss the fundamental solution of the Keldysh type operator \( L_\alpha u \triangleq \frac{{\partial ^2 u}} {{\partial x^2 }} + y\frac{{\partial ^2 u}} {{\partial y^2 }} + \alpha \frac{{\partial u}} {{\partial y}} \), which is a basic mixed type operator different from the Tricomi operator. The fundamental solution of the Keldysh type operator with \( \alpha > - \frac{1} {2} \) is obtained. It is shown that the fundamental solution for such an operator generally has stronger singularity than that for the Tricomi operator. Particularly, the fundamental solution of the Keldysh type operator with \( \alpha < \frac{1} {2} \) has to be defined by using the finite part of divergent integrals in the theory of distributions.