Exact Solution to the Dirichlet Problem for Degenerating on the Boundary Elliptic Equation of Tricomi - Keldysh Type in the Half-Space

Authors: Algazin O.D. Published: 12.10.2016
Published in issue: #5(68)/2016  
DOI: 10.18698/1812-3368-2016-5-4-17

Category: Mathematics and Mechanics | Chapter: Mathematical Physics  
Keywords: Fourier transform, Tricomi equation, Dirichlet problem, approximation to the ide"Шу, self-similar solution, similarйу method, generalized functions of slow growth

In the paper we solve the Dirichlet problem for a multidimensional equation by means of Fourier transform method and similarity method. The problem is a generalization of the Tricomi, Gellerstedt and Keldysh equations in the half-space, the equation is of an elliptic type with the boundary condition on the boundary hyperplane where equation degenerates. We present the solution in the form of an integral with a simple kernel. It is an approximation to the identity and self-similar solution of Tricomi 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 presented as a convolution of this function with the kernel (if a convolution exists).


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