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.

1. Introduction

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.

2. Interpolation

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.

3. Numerical differentiation

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.

4. Numerical methods for nonlinear equations

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)).

5. Numerical integration

The value of definite integrals can be computed by numerical integration methods. Below some of these methods are mentioned.

6. Numerical methods for initial value problems

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

7. Finite differences for boundary value problems

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 ).

8. The instationary heat equation

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

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Kees Vuik

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