Exact solution to the Dirichlet problem for degenerating on the boundary elliptic equation of Tricomi-Keldysh type in the half-space

TL;DR

This paper provides an exact integral solution to a class of elliptic equations degenerating on the boundary, generalizing classical equations like Tricomi and Keldysh, using Fourier and similarity methods.

Contribution

It introduces a new integral formula for the Dirichlet problem of a generalized Tricomi-Keldysh type equation in the half-space, extending classical solutions.

Findings

Solution expressed as an integral with an approximation to the identity kernel

Contains a Poisson's formula for the Laplace equation as a special case

Allows convolution representation for boundary data of slow growth

Abstract

In the paper by means of Fourier transform method and similarity method we solve the Dirichlet problem for a multidimensional equation wich is a generalization of the Tricomi, Gellerstedt and Keldysh equations in the half-space, in which equation have elliptic type, with the boundary condition on the boundary hyperplane where equation degenerates.The solution is presented in the form of an integral with a simple kernel which is an approximation to the identity and self-similar solution of Tricomi-Keldysh type equation . In particular, this formula contains a Poisson's formula, which gives the solution of the Dirichlet problem for the Laplace equation for the half-space. If the given boundary value is a generalized function of slow growth, the solution of the Dirichlet problem can be written as a convolution of this function with the kernel (if a convolution exists).