9: Partial Differential Equations
Differential equations containing partial derivatives with two or more independent variables are called partial differential equations (pdes). These equations are of fundamental scientific interest but are substantially more difficult to solve, both analytically and computationally, than odes. In this chapter, we begin by deriving two fundamental pdes: the diffusion equation and the wave equation, and show how to solve them with prescribed boundary conditions using the technique of separation of variables. We then discuss solutions of the two-dimensional Laplace equation in Cartesian and polar coordinates, and finish with a lengthy discussion of the Schrödinger equation, a partial differential equation fundamental to both physics and chemistry.
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- 9.1: Derivation of the Diffusion Equation
- To derive the diffusion equation in one spacial dimension, we imagine a still liquid in a long pipe of constant cross sectional area. A small quantity of dye is placed in a cross section of the pipe and allowed to diffuse up and down the pipe. The dye diffuses from regions of higher concentration to regions of lower concentration.