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

2016/03/18 by Oleg D. Algazin, Algazin, Oleg D.
Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.AP #math.MP

paper · pdf · doi:10.48550/arxiv.1603.05760

arxiv created 2016/03/18 · arxiv updated 2016/03/21

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).

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