APMA 4200 Final Exam Study Guide
April 11, 2021
This is a study guide for the first exam. The first part is a list of definitions, facts,
and theorems to know for the exam and the second is a list of suggested problems from
Haberman’s text.
On the exam you will be given a list of formula’s including the definitions of Fourier
series, Fourier sine/cosine series, the Rayleigh quotient as well as any necessary trig iden-
tities.
Some things to know
• Heat, Laplace, Wave equation in 3 and 4 variables.
• Method of separation of variables for problems with 3 or 4 variables on bounded
domains with regular boundary conditions.
• Polar, cylindrical and spherical coordinates.
• Bessel’s equation.
• Statement of theorems for Helmholtz equation ∆φ` λφ “ 0 in 2 and 3 dimensions.
• 2 and 3 dimensional Rayleigh quotient for the Helmholtz equation.
• How to replace a problem with nonhomogeneous boundary conditions with one that
has homogeneous boundary conditions.
• Solving a problem with a nonhomogeneous term or nonhomogeneous boundary con-
ditions using eigenfunction expansion.
• Definition of Fourier transform and inverse Fourier transform.
• Basic properties of the Fourier transform.
• Convolution theorem.
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• How to solve a PDE using Fourier transforms.
• Defining equation for a Green’s function.
• How to find Green’s functions for ODEs.
• The Fredholm alternative for a Sturm Lioville operator.
Suggested Problems
Chapter 7
7.3.1, 7.3.4, 7.5.2, 7.5.7, 7.5.8 7.6.1, 7.6.2, 7.7.3, 7.7.9 7.9.1, 7.9.2
Chapter 8
8.2.2, 8.3.1, 8.4.1, 8.4.4 8.6.1, 8.6.2, 8.6.7
Chapter 9
9.3.6, 9.3.11, 9.3.14, 9.4.5, 9.4.7, 9.5.3, 9.5.4, 9.5.6
Chapter 10
10.3.8, 10.3.11, 10.4.4, 10.4.6, 10.4.7ab, 10.4.9
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