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Fourier Transform Applied to Partial Differential Equations

On the previous page on the Fourier Transform applied to differential equations. we looked at the solution to ordinary differential equations. On this page, we'll examine using the Fourier Transform to solve partial differential equations (known as PDEs), which are essentially multi-variable functions within differential equations of two or more variables.

As an example of solving Partial Differential Equations, we will take a look at the classic problem of heat flow on an infinite rod. That is, essentially we are interested in the temperature of the rod; we'll call the temperature as a function of position (x) and time (t) by G(x, t). Side Note: solving heat flow on a circular ring is actually the original motivation for Fourier Series and Fourier Transforms by Fourier!

[Equation 1]

A couple of things to note in equation [1]:

  • The operator

    src="/img/74/partial634.gif" />

    represents the partial derivative with respect to time. That is, the derivative is taken with respect to t while treating x as a constant. The same goes for the partial derivative with respect to x (t is held constant).

  • To simplify, we will use the subscript notation for partial derivatives, as in the second line of Equation [1]. That is, g_x represents the partial derivative of g with respect to x

  • The constant c is a positive number and is a measure of how quickly the heat diffuses from one part of the rod to another.

    We will introduce an initial condition and look to solve for the heat function g(x,t) for all values of t >0. That is, suppose we know the temperature distribution on the rod for all x at t =0:

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