Euler-Poisson-Darboux方程的一般形式及变形法的应用
The general form of the Euler--Poisson--Darboux equation and application of transmutation method
摘要：
本文通过广义平移和球平均算子，对参数$k$的所有取值，包括在特殊的奇负值之前，得到了具有Bessel算子的一般形式的Euler-Poisson-Darboux方程的Cauchy问题的紧积分形式的解表示.我们使用汉克尔变换的方法来证明结果在一个统一的方式。在附加条件下，我们证明了分配解也是经典解。证明了连通广义球平均的一个变形性质，强调了用变形方法求解含Bessel算子的微分方程的重要性。本文还包含一个简短的历史介绍微分方程与贝塞尔算子和一个相当详细的参考清单的专著和论文的数学理论和应用这一类的微分方程。
英文摘要：
In the paper we find solution representations in the compact integral form to the Cauchy problem for a general form of the Euler--Poisson--Darboux equation with Bessel operators via generalized translation and spherical mean operators for all values of the parameter $k$, including also not studying before exceptional odd negative values. We use a Hankel transform method to prove results in a unified way. Under additional conditions we prove that a distributional solution is a classical one too. A transmutation property for connected generalized spherical mean is proved and importance of applying transmutation methods for differential equations with Bessel operators is emphasized. The paper also contains a short historical introduction on differential equations with Bessel operators and a rather detailed reference list of monographs and papers on mathematical theory and applications of this class of differential equations.
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