chapter 12, Partial Differential Equations

Excerpt

Historical Comments: In 1727, John Bernoulli (1667–1748) treated the vibrating string problem by imagining the string to be a thin thread having a number of equally spaced weights placed along it. However, because his governing equation was not time-dependent, it was not truly a partial differential equation. The French mathematician Jean Le Rond d'Alembert (1717–1783) derived the one-dimensional wave equation as we know it today by letting the number of weights in Bernoulli's model become infinite while at the same time allowing the space between them to go to zero. His famous solution, widely known as d'Alembert's solution, appeared around 1746, six years before the separation of variables technique was introduced by Daniel Bernoulli (1700–1782). Leonhard Euler (1707–1783), D. Bernoulli, and Joseph Louis Lagrange (1736–1813) all solved the wave equation in the mid-1700's by the method of separating the variables, also called the Bernoulli product method.

Laplace's equation arose in the study of gravitational attraction by Pierre-Simon Laplace (1749–1827) and Adrien-Marie Legendre (1752–1833), both of whom were professors of mathematics at the Ecole Militaire in France. In fact, it was Legendre's famous 1782 study of gravitational attraction of spheroids that introduced what are now called Legendre polynomials (see Chapter 2).

The first major step toward developing a general method of solution of partial differential equations began in the early 1800's when Joseph Fourier (1768–1830) made his famous study of the heat equation. He is best known today for his celebrated 1822 book Théorie analytique de la chaleur. The chief contribution of Fourier was the idea that almost any function can be represented in a sine series.

Mathematicians in the nineteenth century worked vigorously on problems associated with Laplace's equation, trying to extend the work of Legendre and Laplace. Despite this great effort by well-known mathematicians, very little was known about the general properties of the solutions of Laplace's equation until 1828 when George Green ( 1793–1841 ), a self-taught British mathematician whose main interest was in electricity and magnetism, and the Russian mathematician Michel Ostrogradsky (1801–1861) independently studied properties of a class of solutions known as harmonic functions. In the second half of the nineteenth century, mathematicians began to make progress on problems concerning the existence of solutions to partial differential equations.