Similarity Method in Constructing Fundamental Solution of the Dirichlet Problem for Equation of Keldysh Type in Half-Space

Authors: Algazin O.D. Published: 26.01.2018
Published in issue: #1(76)/2018  
DOI: 10.18698/1812-3368-2018-1-4-15

Category: Mathematics | Chapter: Differential Equations, Dynamic Systems, and Optimal Control  
Keywords: equation of Keldysh type, Dirichlet problem, similarity method, self-similar solution, approximate identity, generalized functions

The purpose of this research was to use the similarity method for the equation of Keldysh type, which is elliptic in half-space and degenerating on the boundary. As a result, we found a self-similar solution, which is the approximate identity in the class of integrable functions. It is a fundamental solution of the Dirichlet problem, i. e. the solution of the Dirichlet problem with the Dirac δ-function in the boundary condition. The 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 the convolution exists. For a bounded and piecewise continuous boundary function the 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


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