Construction by similarity method of the fundamental solution of the Dirichlet problem for Keldysh type equation in the half-space

TL;DR

This paper constructs the fundamental solution for the Dirichlet problem of Keldysh type equations in a half-space using similarity methods, enabling solutions for various boundary conditions through convolution.

Contribution

It introduces a similarity-based method to explicitly construct the fundamental solution for Keldysh type equations in the half-space, extending classical potential theory.

Findings

Constructed the fundamental solution as a self-similar solution.

Expressed the general solution via convolution with the fundamental solution.

Extended the solution framework to generalized boundary functions.

Abstract

For elliptic in the half-space and parabolic degenerating on the boundary equation of Keldysh type we construct by similarity method the self-similar solution, which is the approximation to the identity in the class of integrable functions. This solution is the fundamental solution of the Dirichlet problem, i.e. the solution of the Dirichlet problem with the Dirac delta-function in the boundary condition. Solution of the Dirichlet problem with an arbitrary function in the boundary condition can be written as the convolution of the function with the fundamental solution of the Dirichlet problem, if a convolution exists. For a bounded and piecewise continuous boundary function convolution exists and is written in the form of an integral, which gives the classical solution of the Dirichlet problem, and is a generalization of the Poisson integral for the Laplace equation. If the boundary function is a generalized function, the convolution is a generalized solution of the Dirichlet problem.