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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 the problems.
- 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.

Last modified 1999/05/12 by Dave Rusin. Mail: feedback@math-atlas.org