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I am very new to the theory of Partial Differential Equations (PDEs) and I am having trouble understanding the following excerpt. The author is gently introducing PDEs through the heat equation, and at some point he briefly describes the characteristics of Second Order Linear PDEs. The heat equation is noted:

$$ \frac{\partial u}{\partial \tau} = \frac{\partial^2 u}{\partial x^2}$$

where $x$ represents distance and $\tau$ time. The excerpt reads as follows:

We can think of the characteristics of a second order linear equation as curves along which information can propagate, or as curves across which discontinuities in the second derivatives of $u$ can occur. Suppose that $u(x,\tau)$ satisfies the general second order linear equation:

$$ \begin{align} a(x,\tau)\frac{\partial^2 u}{\partial x^2} & + b(x,\tau)\frac{\partial^2 u}{\partial x \partial \tau} + c(x,\tau)\frac{\partial^2 u}{\partial \tau^2} \\[6pt] & + d(x,\tau)\frac{\partial u}{\partial x} + e(x,\tau)\frac{\partial u}{\partial \tau} + f(x,\tau)u + g(x,\tau) = 0 \end{align}$$

The idea is to see whether the derivative terms can be written in terms of directional derivatives, so that the equation is partly like an ordinary differential equation along curves with these vectors as tangents. These curves are the characteristics. If we write them as $x=x(\xi)$, $\tau=\tau(\xi)$, where $\xi$ is a parameter along the curves, then $x(\xi)$ and $\tau(\xi)$ satisfy:

$$ a(x,\tau)\left(\frac{d\tau}{d\xi}\right)^2 - b(x,\tau)\frac{d\tau}{d\xi}\frac{dx}{d\xi} + c(x,\tau)\left(\frac{dx}{d\xi}\right)^2 = 0$$

My questions are the following:

  • I basically do not understand the two paragraphs above; in particular I am not sure what the author mean by "characteristics". I am also not familiar with directional derivatives, although I have looked them up in Wikipedia. Could someone explain what the author mean?
  • I do not see how the author derives the 2nd equation from the 1st one. Could someone explain the derivation?

Reference

Wilmott, P. (1995). The Mathematics of Financial Derivatives. University of Cambridge, 1st Edition.

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    $\begingroup$ A characteristic is a path where the solution to the PDE can be written as a solution to an ODE. The simplest example is in constant-velocity transport: for $u_t+u_x=0$, the material is "transported" to the right at a speed of $1$, meaning that the solution is constant along paths of the form $x=x_0+t$. You derive the equation for the PDE along the characteristic by plugging the expression for it into the PDE and using the chain rule. $\endgroup$ – Ian Jun 2 '17 at 14:33
  • $\begingroup$ The heat equation here seems completely unrelated to the excerpt. Don't let that part confuse you. The author is discussing general 2nd order linear PDEs. $\endgroup$ – Merkh Jun 2 '17 at 14:34
  • $\begingroup$ @Merkh yes I know, I just wanted to explain the general notation used in the text but in this case $u(x,\tau)$ is simply a generic function. $\endgroup$ – Morris Fletcher Jun 2 '17 at 16:25

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