History of mathematicians
In this document we give some information of mathematicians which
work or names are used in the course "Numerical methods for
differential equations" at the TU Delft.
Modern applied mathematics started in the 17 and 18 century with people
like
Simon Stevin (1548-1620),
René Descartes (1596-1650),
Isaac Newton (1642-1727) and
Leonhard Euler (1707-1783).
Numerical aspects were used in analysis in a natural way; the name
numerical
mathematics was unknown. Numerical methods developed by Newton, Euler
and later
Carl Friedrich Gauss (1777-1855)
play an important role in present day numerical mathematics.
November 1947 can be seen as the
birthday of modern numerical analysis.
The Taylor polynomial
(
Brook Taylor (1685-1731))
is used to analyse the error in various numerical approximations.
The order symbol of Landau (
Edmund Georg Hermann Landau (1877-1938))
is used to give a short notation of the
approximation errors.
To approximate a funtion where only a limited number of function values
are given we use a Lagrange polynomial
(
Joseph-Louis Lagrange (1736-1813)). To do this is in a smooth way
cubic splines can be used. Schoenberg
(
Isaac Jacob Schoenberg (1903-1990)) initiated work on splines.
Birkhoff
(
Garrett Birkhoff (1911-1996)) was quick to recommend the use of cubic splines
for the representation of smooth curves.
The Taylor polynomial
(
Brook Taylor (1685-1731))
is used to analyse the error in the
approximation of a derivative by a finite difference formula.
Richardson' s extrapolation
(
Lewis Fry Richardson (1881-1953))
is used to obtain an error estimate or a more
accurate formula.
Some of the considered boundary value problems have a
nonlinear differential equation. Such problems are solved by
the Picard
(
Charles Émile Picard (1856-1941)),
or the Newton-Raphson method
(Isaac Newton (1642-1727)
and
Joseph Raphson (1648-1715)).
The value of definite integrals can be computed by numerical integration
methods. Below some of these methods are mentioned.
In order to prove that an initial value problem is well posed it is
necessary that the function is Lipschitz continuous
(
Rudolf Otto Sigismund Lipschitz (1832-1903)).
Several numerical integration methods for initial value problems
are given and analysed as there are
In boundary value problems a differential equation is given together
with appropriate boundary conditions, in order to make the solution
unique. There are various boundary conditions possible.
We consider a heat equation, where the required solution describes
the temperature (T). To derive the differential equation equation
the law of
Jean Baptiste Joseph Fourier (1768-1830)
is used, which the heat flux with the first derivative of the
temperature.
As boundary conditions one can prescribe the temperature (called a
Dirichlet condition
Johann Peter Gustav Lejeune Dirichlet (1805-1859))
or one can prescribe the flux, the first derivative of the
temperature (called a Neumann condition
Carl Gottfried Neumann (1832-1925))
or one can prescribe a combination of the temperature and the flux
(called a Robbins condition).
In order to measure convergence some norms are used. We use the scaled Euclid
norm
(
Euclid (365 BC-300 BC ).
In this chapter the methods for initial value problems and boundary
value problems are combined.
The time derivative in the heat equation
can be discretized by the Euler Forward, Euler
Backward or the Trapeziodal Rule. For this equation the Trapeziodal
Rule is called the Crank-Nicolson method
(
John Crank
(1916-2006) and
Phyllis
Nicolson
(1917-1968)
)
Biographies index
Contact information:
Kees Vuik
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