Applied Engineering Analysis

Chapter 9Applications of Partial Differential Equations in Mechanical Engineering Analysis

Chapter Learning Objectives

• Learn the physical meaning of partial derivatives of functions.
• Learn that there are different order of partial derivatives describing the rate of changes of functions representing real physical quantities.
• Learn the two commonly used techniques for solving partial differential equations by (1) Integral transform methods that include the Laplace transform for physical problems covering half-space, and the Fourier transform method for problems that cover the entire space; (2) the separation of variables technique.
• Learn the use of the separation of variables technique to solve partial differential equations relating to heat conduction in solids and vibration of solids in multidimensional systems.

9.1 Introduction

Partial differential equations such as that shown in Equation (2.5) are the equations that involve partial derivatives described in Section 2.2.5. A partial derivative represents the rate of change of a function (a physical quantity in engineering analysis) with respect to one of several variables that the function is associated with.

The independent variables in partial derivatives can be (1) spatial variables represented by (x,y,z) in a rectangular coordinate system or (r,θ,z) in a cylindrical polar coordinate system and (2) temporal variables represented by time t.

Partial differential equations can be categorized as “boundary-value problems” ...