ABSTRACT
At the end of this chapter, you should be able to:
• recognise some important engineering partial differential equations
• solve a partial differential equation by direct partial integration
• solve differential equations by separating the variables
• solve the wave equation ∂2u
∂x2 = 1
c2 ∂2u
∂ t2
• solve the heat conduction equation ∂2u
∂x2 = 1
c2 ∂u
∂ t
• solve Laplace’s equation ∂2u
∂x2 + ∂
∂y2 = 0
A partial differential equation is an equation that contains one or more partial derivatives. Examples include:
(i) a ∂u ∂x
+ b ∂u ∂y
= c
(ii) ∂ 2u
∂x2 = 1
c2 ∂u
∂ t
(known as the heat conduction equation)
(iii) ∂ 2u
∂x2 + ∂
∂y2 = 0
(known as Laplace’s equation)
Equation (i) is a first-order partial differential equation, and equations (ii) and (iii) are second-order partial differential equations since the highest power of the differential is 2.
