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The dynamic response of an elastic beam attached to a spring rigid bar of two degrees of freedom is studied in this paper. The partial differential equation (PDE) describing the beam and the ordinary differential equations (ODEs) concerning the rigid bar are transformed into a homogeneous first order system of equations. The eigenfrequencies and mode shapes of the beam are determined and the problem is solved by the matrix exponential method. A numerical experiment based on true-life values of the parameters is presented. The results are verified by Finite Element (FE) code modeling. An optimization problem with respect to the system parameters is solved. [ABSTRACT FROM AUTHOR]

In this paper we will present an effective numerical method for the Black-Scholes equation with transaction costs for the limiting price u(s, t;a). The technique combines the Rothe method with a two-grid (coarse-fine) algorithm for computation of numerical solutions to initial boundary-values problems to this equation. Numerical experiments for comparison the accuracy ant the computational cost of the method with other known numerical schemes are discussed. [ABSTRACT FROM AUTHOR]