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For elliptic in the half-space and parabolic degenerating on the boundaryequation of Keldysh type we construct by similarity method the self-similarsolution, which is the approximation to the identity in the class of integrablefunctions. This solution is the fundamental solution of the Dirichlet problem,i.e. the solution of the Dirichlet problem with the Dirac delta-function in theboundary condition. Solution of the Dirichlet problem with an arbitraryfunction in the boundary condition can be written as the convolution of thefunction with the fundamental solution of the Dirichlet problem, if aconvolution exists. For a bounded and piecewise continuous boundary functionconvolution exists and is written in the form of an integral, which gives theclassical solution of the Dirichlet problem, and is a generalization of thePoisson integral for the Laplace equation. If the boundary function is ageneralized function, the convolution is a generalized solution of theDirichlet problem.

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