MAthematics and natural science in the nineteenth century: the classical approaches of poincaré, volterra and hadamard

By the end of the nineteenth Century, the classical triumvirate of elliptic, parabolic, and hyperbolic partial differential equations was well established, and classical mathematics had achieved remarkable success in describing the physical world. To investigate the depth and extent of this success, this chapter examines three areas in which leading applied mathematicians tried to build upon that success in the years 1890–1914: 1

Thomson, Heaviside, and Poincaré on the telegraphist's equation;

2

Hadamard's journey from non-linear problems to ill-posed problems, via Kovalevskaya's Theorem;

3

Volterra on Green's theorem, elasticity theory, and his generalisation of complex function theory to ‘functions of lines'.