12a show that the function u0 x t 1 x2 4t e 4t is

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Unformatted text preview: elop techniques that can be used to solve a wide variety of partial diﬀerential equations. In this chapter, we will give two important simple examples of partial differential equations, the heat equation and the wave equation, and we will show how to solve them by the techniques of “separation of variables” and Fourier analysis. Higher dimensional examples will be given in the following chapter. We will see that just as in the case of ordinary diﬀerential equations, there is an important dichotomy between linear and nonlinear equations. The techniques of separation of variables and Fourier analysis are eﬀective only for linear partial diﬀerential equations. Nonlinear partial diﬀerential equations are far more 81 diﬃcult to solve, and form a key topic of contemporary mathematical research.1 Our ﬁrst example is the equation governing propagation of heat in a bar of length L. We imagine that the bar is located along the x-axis and we let u(x, t) = temperature of the bar at the point x at time t. Heat in a small segment of a homogeneous bar is proportional to temperature, the constant of propor...
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This document was uploaded on 01/12/2014.

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