Differential Equations and integral equations and Operators
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Differential and integral equations: Here we seek to find functions f
knowing relationships between values of f and its derivatives or
integrals; this includes the study of differential operators and their
applications in mathematics.
As can be seen from their position on the MathMap, these parts of
Analysis are intimately connected with many of the applications of
mathematics, particularly to parts of mathematical physics.
34: Ordinary differential equations are equations
to be solved in which the unknown element is a function, rather than a
number, and in which the known information relates that function to its
derivatives. Few such equations admit an explicit answer, but there is a
wealth of qualitative information describing the solutions and their
dependence on the defining equation. There are many important classes of
differential equations for which detailed information is
available. Applications to engineering and the sciences abound.
Numerical solutions are actively studied.
35: Partial differential equations begin with much
the same formulation as ordinary differential equations, except that the
functions to be found are functions of several variables. Again, one
generally looks for qualitative statements about the solution. For example,
in many cases, solutions exist only if some of the parameters lie in
a specific set (say, the set of integers). Various broad families of
PDE's admit general statements about the behaviour of their solutions.
This area has a long-standing close relationship with the physical sciences,
especially physics, thermodynamics, and quantum mechanics.
45: Integral equations, naturally, seek functions
which satisfy relationships with their integrals. For example, the value
of a function at each time may be related to its average value over all
preceding time. Included in this area are equations mixing integration and
differentiation. Many of the themes from differential equations recur:
qualitative questions, methods of approximation, specific types of
equations of interest, transforms and operators useful for simplifying
49: Calculus of variations and optimization seek
functions or geometric objects which are optimize some objective function.
Certainly this includes a discussion of techniques to find the optima,
such as successive approximations or linear programming. In addition, there
is quite a lot of work establishing the existence of optima and
characterizing them. In many cases, optimal functions or curves can be
expressed as solutions to differential equations. Common applications
include seeking curves and surfaces which are minimal in some sense.
However, the spaces on which the analysis are done may represent configurations
of some physical system, say, so that this field also applies to optimization
problems in economics or control theory for example.
58: Global analysis, or analysis on manifolds,
studies the global nature of differential equations on manifolds, that is,
dynamical systems. In addition to local tools from ordinary differential
equation theory, global techniques include the use of topological spaces of
mappings. In this heading also we find general papers on manifold theory,
including infinite-dimensional manifolds and manifolds with singularities
(hence catastrophe theory), as well as optimization problems (thus
overlapping the Calculus of Variations, above).
[Starting in the year 2000, a new section 37: "Dynamical systems and ergodic theory" will hold some of this material.]
Some of these fields have been increasingly dominated in recent years by
the study of numerical methods in differential equations.
You might want to continue the tour with a trip through numerical analysis.
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Last modified 1999/05/12 by Dave Rusin. Mail: firstname.lastname@example.org