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The article is devoted to a numerical investigation of the Boussinesq -- L$\grave{o}$ve mathematical model. Algorithm for finding of the numerical solution to the Cauchy -- Dirichlet problem for the Boussinesq -- L$\grave{o}$ve equation modeling longitudinal oscillations in a thin elastic rod with regard to transverse inertia was obtained on the basis of a phase space method and by using a  finite differences method. This problem can be reduced to the Cauchy problem for the Sobolev type equation of the second order, which is not solvable for arbitrary initial values. The constructed algorithm includes the additional check if initial data belongs to the phase space. The algorithm is implemented as a program in Matlab. The results of numerical experiments are obtained both in  regular and degenerate cases. The graphs of obtained solutions are presented in each case.

Numerical Investigation of One Sobolev Type Mathematical Model