By Yehuda Pinchover and Jacob Rubinstein
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Additional resources for An introduction to partial differential equations
Hamilton investigated the propagation of light. He sought to derive the rules governing this propagation 26 First-order equations from a purely geometric theory, akin to Euclidean geometry. Hamilton was well aware of the wave theory of light, which was proposed by the Dutch physicist Christian Huygens (1629–1695) and advanced early in the nineteenth century by the English scientist Thomas Young (1773–1829) and the French physicist Augustin Fresnel (1788–1827). ) Pierre de Fermat (1601–1665). Fermat proposed a uniﬁed principle, according to which light rays travel from a point A to a point B in an orbit that takes the least amount of time.
1) where F is a given function of 2n + 1 variables. First-order equations appear in a variety of physical and engineering processes, such as the transport of material in a ﬂuid ﬂow and propagation of wavefronts in optics. Nevertheless they appear less frequently than second-order equations. For simplicity we shall limit the presentation in this chapter to functions in two variables. The reason for this is not just to simplify the algebra. As we shall soon observe, the solution method is based on the geometrical interpretation of u as a surface in an (n + 1)-dimensional space.
They do not explicitly include the variable t), it follows that the equations for the characteristics can be written simply as the ﬁrst-order ODE b(x, y) dy = . 4 The current example will be useful for us in Chapter 3, where we shall need to solve linear equations of the form a(x, y)u x + b(x, y)u y = 0. 22) 32 First-order equations The equations for the characteristic curves dx dy du = a, = b, =0 dt dt dt imply at once that the solution u is constant on the characteristics that are determined by dy b(x, y) = .
An introduction to partial differential equations by Yehuda Pinchover and Jacob Rubinstein