IV Prefacetions. Quite often, the interest in modeling centers as much on getting
an appropriate mathematical description of the problem as it does on the
computational methods employed. Nevertheless there is a fruitful interplay
between modeling and con1putational mathematics, as will be evident from
a perusal of the first few chapters of this book.
One can now say that a new inethod arose for theoretical investiga-
tions of rather general and complex processes which allow for mathematical
descriptions or mathematical n1odeling. This is the method of numerical
experi1nent, which makes it possible better to understand real-world phe-
nomena using exploratory devices of computational mathematics. When
investigating some physical process by means of this method, an early step
is the mathematical state1nent of a problem or the well-founded choice of
a tnathematical model capable of describing this process. It is preceded by
a proper choice of physical approximations, that is, to decide which factors
should be taken into account and which may be neglected or omitted in a
general setting. This choice is the privilege of physicists.
What is a 1nathematical modeJ? The group of unknown physical quan-
tities which interest us and the group of available data are closely intercon-
nected. This link may be embodied in algebraic or differential equations. A
proper choice of the mathen1atical model facilitates solving these equations
and providing the subsidiary information on the coefficients of equations as
well as on the initial and boundary data.
Mathematical physics deals with a variety of mathematical models
arising in physics. Equations of mathen1atical physics are mainly partial
differential equations, integral, and integro-differential equations. Usually
these equations reflect the conservation laws of the basic physical quantities
(energy, angular momentum, mass, etc.) and, as a rule, turn out to be
nonlinear.
After writing a system of equations capable of describing the physical
process of interest, it is necessary for the investigator to consider the
resulting mathematical model by methods of the general theory of differ-
ential and integral equations and to make sure that a proble1n has been
completely posed and the available data are sufficient and consistent, to
derive the conditions under which the problem has a unique solution, to
find out whether this solution may be written in explicit form, and whether
particular solutions are possible. Particular solutions are important in giv-
ing the prelin1inaries regarding the nature of the physical process. They
also can serve as "goodness-of-fit'' tests for the desired quality of numerical
methods. At the second stage, one is to construct an approximate (numer-
ical) method and a computer algorithm for solving the problem. The third
stage is computer programming of the algorith1n under such a choice. At