Abstract

The Cauchy problem for an iterated generalized two-axially symmetric equation of hyperbolic type is investigated. Unlike traditional methods, the generalized Erdélyi–Kober operator of fractional order is applied to solve the problem. The application of this operator allows us to reduce the equations with the lowest term and with the singular Bessel operator acting in one of the variables to an equation without the Bessel operator acting on this variable and without the lowest term. The proposed approach makes it possible to construct an explicit formula for solving the formulated problem. These formulas express solutions of the Cauchy problem in terms of the sum of the initial functions under the action of powers of the Bessel operator. They allow us to directly see the character of the dependence of the solution on the initial functions.

中文翻译：

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关于双曲型广义广义两轴对称方程的柯西问题

摘要

研究了双曲型广义广义两轴对称方程的柯西问题。与传统方法不同，应用分数阶广义Erdélyi-Kober算子来解决该问题。该运算符的应用使我们能够简化具有最低项的方程，并且将奇异的Bessel运算符作用于变量中的方程成为方程，而没有Bessel运算符作用于该变量且没有最低项。所提出的方法使得有可能构造用于解决所提出的问题的明确公式。这些公式用贝塞尔算子的力量在初始函数的总和上表达柯西问题的解。它们使我们可以直接看到解决方案对初始函数的依赖性的特征。