12: Fourier Solutions of Partial Differential Equations

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Contributed by William F. Trench
Andrew G. Cowles Distinguished Professor Emeritus (Mathamatics) at Trinity University

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In this chapter, we use the series discussed in Chapter 11 to solve partial differential equations that arise in problems of mathematical physics.

12.1: The Heat Equation
This section deals with the partial differential equation $u_t = a^2u_{xx}$, which arises in problems of conduction of heat
- 12.1.1: The Heat Equation (Exercises)
12.2: The Wave Equation
This section deals with the partial differential equation $u_{tt} = a^2u_{xx}$, which arises in the problem of the vibrating string.
- 12.2.1: The Wave Equation (Exercises)
12.3: Laplace's Equation in Rectangular Coordinates
This section deals with the partial differential equation $u_{xx} + u_{yy} = 0$, which arises in steady state problems of heat conduction and potential theory.
- 12.3.1: Laplace's Equation in Rectangular Coordinates (Exercises)
12.4: Laplace's Equation in Polar Coordinates
This section deals with the partial differential equation $u_{rr} + 1/r u_r + 1 r^2 u_{θθ} = 0$, which is the equivalent to the equation studied in Section 1.3 when the independent variables are polar coordinates.
- 12.4.1: Laplace's Equation in Polar Coordinates (Exercises)