# 12: Partial Differential Equations

• • Contributed by Marcia Levitus

Objectives

• Learn the method of separation of variables to solve simple partial differential equations.
• Understand how to apply the method of separation of variables to two important problems in the physical sciences: The wave equation in one dimension and molecular diffusion.

• 12.1: Introduction to Partial Differential Equations
Many important equations in physical chemistry, engineering, and physics, describe how some physical quantity, such as a temperature or a concentration, varies with position and time.
• 12.2: The Method of Separation of Variables
The separation of variables is a methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation.
• 12.3: The Wave Equation in One Dimension
The wave equation is an important second-order linear partial differential equation that describes waves such as sound waves, light waves and water waves. In this course, we will focus on oscillations in one dimension.
• 12.4: Molecular Diffusion
Molecular diffusion is the thermal motion of molecules at temperatures above absolute zero. The rate of this movement is a function of temperature, viscosity of the fluid and the size and shape of the particles. Diffusion explains the net flux of molecules from a region of higher concentration to one of lower concentration. The term "diffusion" is also generally used to describe the flux of other physical quantities like thermal energy (heat).
• 12.5: Problems