TITLE:
Some Remarks to Numerical Solutions of the Equations of Mathematical Physics
AUTHORS:
Ludmila Petrova
KEYWORDS:
Two Systems of Reference; Nonintegrable Manifolds and Integrable Structures; Solutions of Two Types; Discrete Transitions; Observable Formations
JOURNAL NAME:
American Journal of Computational Mathematics,
Vol.3 No.3,
August
15,
2013
ABSTRACT: The equations of mathematical physics, which describe some actual
processes, are defined on manifolds (tangent, a companying or others) that are
not integrable. The derivatives on such manifolds turn out to be inconsistent, i.e.
they don’t form a differential. Therefore, the solutions to equations obtained
in numerical modelling the derivatives on such manifolds are not functions.
They will depend on the commutator made up by noncommutative mixed derivatives,
and this fact relates to inconsistence of derivatives. (As it will be shown,
such solutions have a physical meaning). The exact
solutions (functions) to the equations of mathematical physics are obtained
only in the case when the integrable structures are realized. So called
generalized solutions are solutions on integrable structures. They are
functions (depend only on variables) but are defined only on integrable
structure, and, hence, the derivatives of functions or the functions themselves
have discontinuities in the direction normal to integrable structure. In
numerical simulation of the derivatives of differential equations, one cannot
obtain such generalized solutions by continuous way, since this is connected
with going from initial nonintegrable manifold to integrable structures. In
numerical solving the equations of mathematical physics, it is possible to
obtain exact solutions to differential equations only with the help of
additional methods. The analysis of the solutions to differential equations
with the help of skew-symmetric forms [1,2] can give certain recommendations
for numerical solving the differential equations.