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.
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This work was supported by the National Basic Research Program of China (Grant No. 2006CB805902), National Natural Science Foundation of China (Grant No. 10531020) and the Research Foundation for Doctor Programme (Grant No. 20050246001)
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Chen, S. The fundamental solution of the Keldysh type operator. Sci. China Ser. A-Math. 52, 1829–1843 (2009). https://doi.org/10.1007/s11425-009-0069-8
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DOI: https://doi.org/10.1007/s11425-009-0069-8