Page 11 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

1

CHAPTER 1

Partial Differential Equations

in Engineering

1.1 INTRODUCTORY COMMENTS
This book covers the material presented in a course in applied mathematics that is required
for first-year graduate students in the departments of Chemical and Mechanical Engineering
at Tulane University. A great deal of material is presented, covering boundary value problems,
complex variables, and Fourier transforms. Therefore the depth of coverage is not as extensive
as in many books. Our intent in the course is to introduce students to methods of solving
linear partial differential equations. Subsequent courses such as conduction, solid mechanics,
and fracture mechanics then provide necessary depth.
The reader will note some similarity to the three books, Fourier Series and Boundary
Value Problems, Complex Variables and Applications,and Operational Mathematics, originally by
R. V. Churchill. The first of these has been recently updated by James Ward Brown. The
current author greatly admires these works, and studied them during his own tenure as a
graduate student. The present book is more concise and leaves out some of the proofs in an
attempt to present more material in a way that is still useful and is acceptable for engineering
students.
First we review a few concepts about differential equations in general.

1.2 FUNDAMENTAL CONCEPTS
An ordinary differential equation expresses a dependent variable, say u, as a function of one
independent variable, say x, and its derivatives. The order of the differential equation is given
by the order of the highest derivative of the dependent variable. A boundary value problem
consists of a differential equation that is defined for a given range of the independent variable
(domain) along with conditions on the boundary of the domain. In order for the boundary value
problem to have a unique solution the number of boundary conditions must equal the order of
the differential equation. If the differential equation and the boundary conditions contain only
terms of first degree in u and its derivatives the problem is linear. Otherwise it is nonlinear.