Series expansions of solutions of \(u_{xx}+(2\nu /x)u_ x+\epsilon^ 2u_{tt}=u_ t\)
\textit{P. C. Rosenbloom} and \textit{D. V. Widder} [Trans. Amer. Math. Soc. 92, 220-266 (1959; Zbl 0086.272)] studied series expansions of solutions of the heat equation \(u_{xx}=u_ t\). The paper of Rosenbloom and Widder was the model for further investigations on the generalized heat equation \(u_{xx}+(2\nu /x)u_ x=u_ t\) and the equation \(u_{xx}+\epsilon^ 2u_{tt}=u_ t\) which was studied by the second author [J. Reine Angew. Math. 252, 88-103 (1972; Zbl 0226.35011); Appl. Anal. 10, 197-211 (1980; Zbl 0441.35013)]. In the present paper the authors prove analogous results for the equation (1) \(u_{xx}+(2\nu /x)u_ x+\epsilon^ 2u_{tt}=u_ t.\) Following \textit{J. W. Dettman} [J. Differ. Equations 14, 547-558 (1973; Zbl 0275.34065)] they use an integral transform relating solutions of the generalized heat equation to solutions of the equation (1).
scientific article; zbMATH DE number 3851571
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