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INITIAL AND BOUNDARY VALUES PROBLEMS FOR SEMI-INFINITE CUSPIDAL BARS

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{\it G. Pruidze}

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{\it A.Razmadze Mathematical Institute}\\
{\it of the Georgian Academy of Sciences}\\
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\par 
By B.-W. Schulze and his followers the Mellin pseudo-differential operator theory
has been created. This theory is suitable to investigate 
partial differential Equations on manifolds with singularities [1].
In particular, the application of this theory to the investigation of 
initial boundary value problems for
elastic bodies with non-smooth boundaries on the base of three-, two- and one-dimensional
mathematical models has a good chance of success. In order to illustrate this
in the present paper using the Mellin transform in (0,0) approximation of a theory
of cuspidal bars [2] the boundary value and initial boundary value problems
have been solved in an explicit form if the area of the cross-section is the 
following function
$$
h_0x^\alpha,\;\;h_0,\;\alpha=const>0,\;\;x\ge 0.
$$
In statical case the result coincides with known resalts [2] obtained by 
direct integration. In dynamical case it is not possible to apply the latter
approach. But using the Mellin transform,  in case of special density, we menage to solve 
initial boundary value problem under usual initial conditions and admissible
Dirichlet data indicated in [2].

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{\bf R e f e r e n c e s}

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\par
[1] Schulze B.-W., Pseodo-Differential Operators on Manifolds with Singularities,
North-Holland, Amsteram, 1991.
\par
[2] Jaiani G.V., On a Model of a Bar with Variable Thickness, BULLETIN of TICMI, 
2, 1998, 36-40.  (electronic version: http://www.viam.hepi. 
edu.ge/others/TICMI).
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