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Functions of relaxed controls

Mathematical control theory problems involving solutions of certain partial differential equations, nonadditive set functions, or other functionals - approximation and existence theorems

FUNCTIONAL ANALYSIS↗

Solution of the Boltzmann kinetic equation for the relaxation of a gas mixture

The temporal behavior is considered of a quiescent mixture of gases of different temperatures with spatially uniform distribution. The process of heating a cold gas by a hot gas is treated on the basis of the Boltzmann kinetic equation. The mixture is assumed to be composed of absolutely hard smooth spheres, and the initial distribution functions for each gas is taken to the Maxwellian. With such a choice of initial distribution functions, it is shown that the solution of the Boltzmann kinetic equation depends only on the velocity modulus and the time.

Rykov, V. A.↗

Analysis of embedded shock waves calculated by relaxation methods.

The requirements for uniqueness of the calculated jump conditions across embedded shock waves are investigated for type-dependent difference systems used in transonic flow studies. A mathematical analysis shows that sufficient conditions are (1) the equations should be differenced in conservative form and (2) a special difference operator should be used when switching from a hyperbolic to an elliptic operator. The latter results in a consistency condition on the integral equations, rather than the differential, at these points. Calculated jump conditions for several embedded and detached shock waves are analyzed in the physical and hodograph planes. Comparisons are made with previous results, a time-dependent calculation, and data.

Murman, E. M.↗