Volume 5, Issue 4

The numerical approximation of the Laplace equation with inhomogeneous mixed boundary conditions in 2D with lowest-order Raviart-Thomas mixed finite elements is realized in three flexible and short MATLAB programs. It is the aim of this paper to derive, document, illustrate, and validate the three MATLAB implementations EBmfem, LMmfem, and CRmfem for further use and modification in education and research. A posteriori error control with a reliable and efficient averaging technique is included to monitor the discretization error. Therein, emphasis is on the correct treatment of mixed boundary conditions. Numerical examples illustrate some applications of the provided software and the quality of the error estimation.

A new algorithm for Sturm|Liouville problems with matrix coefficients is proposed which possesses the convergence rate of a geometric progression with a denominator depending inversely proportional on the order number of eigenvalues. The asymptotic behavior of the distance between neighboring eigenvalues if the order number tends to infinity is investigated too. Numerical examples confirming the theoretical results are given.

We present analytical computational tools that permit us to identify, in an automatic way, conservation laws in optimal control. The central result we use is the famous Noether's theorem, a classical theory developed by Emmy Noether in 1918, in the context of the calculus of variations and mathematical physics, which was extended recently to the more general context of optimal control. We show how a Computer Algebra System can be very helpful in finding the symmetries and corresponding conservation laws in optimal control theory, thus making useful in practice the theoretical results recently obtained in the literature. A Maple implementation is provided and several illustrative examples are given.

In this paper we consider the system of nonlinear stochastic differential equations. The solution of these equations is approximated by finite series of definite integrals where every term of the series can be evaluated. To construct the approximation, the linearization technique and the technique for noncommuting matrices are used.

For one-dimensional and multidimensional semilinear transport equations of quite a general form with given initial data and boundary conditions the exact difference schemes (EDSs) are constructed. In the case of constant coe±cients, such numerical methods can be created on rectangular grids, while in the case of variable coefficients - on moving grids only. The questions of developing difference schemes of arbitrary order for quasi-linear transport equations with a nonlinear right-hand side are discussed. In this paper, the EDSs are constructed also for certain classes of linear and quasilinear parabolic equations, for convection-diffusion problems with a small parameter, as well as inhomogeneous wave equations with constant coe±cients.