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. Partial differential equations occur in many areas of engineering and technology; electrostatics, heat conduction, magnetism, wave motion, hydrodynamics and aerodynamics all use models that involve partial differential equations. Such equations are difficult to solve, but techniques have been developed for the simpler types. In fact, for all but for the simplest cases, there are a number of numerical methods of solutions of partial differential equations available. To be able to solve simple partial differential equations knowledge of the following is required:
(a) partial integration, (b) first-and second-order partial differentiation – as
explained in Chapter 60, and
(c) the solution of ordinary differential equations – as explained in Chapters 77-82.
