Temur Jangveladze
abstract:
The present monograph is concerned with
the investigation and numerical solution of the initial-boundary
value problems for some nonlinear partial differential and parabolic
type integro-differential models. The models are based on the
well-known system of Maxwell equations which describes the process
of propagation of an electromagnetic field into a medium. The
existence, uniqueness and asymptotic behavior of solutions, as time
tends to infinity, for some types of initial-boundary value problems
are studied. The examples of one-dimensional nonlinear systems and
their analytical solutions are given which show that those systems
do not, in general, have global solutions. Consequently, the case of
a blow-up solution is observed. Linear stability of the stationary
solution of the initial-boundary value problem for one nonlinear
system is proved. The possibility of occurrence of the Hopf-type
bifurcation is established. Semi-discrete and finite difference
approximations are discussed. The splitting-up scheme with respect
to physical processes for one-dimensional case as well as additive
Rothe-type semi-discrete schemes for multi-dimensional cases are
investigated. The stability and convergence properties for those
schemes are studied. Algorithms for finding approximate solutions
are constructed. Results of numerical experiments with tables and
graphical illustrations are given. Their analysis is carried out.
Mathematics Subject Classification: 5K05, 35R09, 65M06, 65M12, 35K55, 35Q61, 35B32, 35B35, 35B40, 35B44, 65Y05
Key words and phrases: Nonlinear parabolic integro-differential equations, nonlinear partial differential systems, Maxwell equations, unique solvability, asymptotic behavior, blow-up, Hopf-type bifurcation, semi-discrete and discrete models, splitting-up schemes, stability, convergence