Abstract

In the article, in the positive domain \(\Omega=\big{\{}(x,y,t):\,x>0,\,y>0,\,t>0\big{\}}\) we consider a degenerate third-order differential equation of the form \(x^{n}y^{m}\,u_{t}=t^{k}y^{m}\,u_{xxx}+t^{k}x^{n}\,u_{yyy}\), \(m,n,k={\textrm{const}}>0\). Nine partial solutions of this equation are expressed through the Campe de Feriet hypergeometric functions \(F_{0;2;2}^{1;0,0}[x,y]\). By generalizing the operator method of J.L. Burchnall and T.W. Chaundy, one-dimensional reciprocal symbolic operators are introduced. Using Burchnall–Chaundy operators, decomposition formulas and integral representations for the Campe de Feriet hypergeometric function \(F_{0;2;2}^{1;0,0}[x,y]\) are obtained.

中文翻译：

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简并三阶微分方程的部分解的积分表示

摘要

文中，在正域\(\Omega=\big{\{}(x,y,t):\,x>0,\,y>0,\,t>0\big{\}} \)我们考虑形式为\(x^{n}y^{m}\,u_{t}=t^{k}y^{m}\,u_{xxx}+ 的简并三阶微分方程t^{k}x^{n}\,u_{yyy}\) , \(m,n,k={\textrm{const}}>0\)。该方程的九个部分解通过 Campe de Feriet 超几何函数\(F_{0;2;2}^{1;0,0}[x,y]\)表示。通过推广JL Burchnall和TW Chaundy的算子方法，引入一维倒数符号算子。使用 Burchnall–Chaundy 算子，获得Campe de Feriet 超几何函数\(F_{0;2;2}^{1;0,0}[x,y]\)的分解公式和积分表示。