mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the qualitative theory of differential equations studies the behavior of differential equations by means other than finding their solutions. It originated from the works of

Henri Poincaré Jules Henri Poincaré (, ; ; 29 April 185417 July 1912) was a French mathematician, Theoretical physics, theoretical physicist, engineer, and philosophy of science, philosopher of science. He is often described as a polymath, and in mathemati ...

and

Aleksandr Lyapunov Aleksandr Mikhailovich Lyapunov (Алекса́ндр Миха́йлович Ляпуно́в, – 3 November 1918) was a Russian mathematician, mechanician and physicist. He was the son of the astronomer Mikhail Lyapunov and the brother of t ...

. There are relatively few differential equations that can be solved explicitly, but using tools from

analysis Analysis (: analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (38 ...

and

topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...

, one can "solve" them in the qualitative sense, obtaining information about their properties. It was used by

Benjamin Kuipers Benjamin Kuipers (born 7 April 1949) is an American computer scientist at the University of Michigan, known for his research in qualitative simulation. Biography Kuipers graduated from Swarthmore College in 1970 with a B.A. in Mathematics. H ...

in the book ''Qualitative reasoning: modeling and simulation with incomplete knowledge'' to demonstrate how the theory of

PDEs In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to how ...

can be applied even in situations where only qualitative knowledge is available.

References

Further reading

Original references