Partial Differential Equations
Revised: March 2024
Course Description
Solution techniques, modeling and analysis of partial differential equations (PDEs)
and boundary value problems. Techniques include separation of variables methods for
second-order linear PDEs, Fourier series, Transform methods, Bessel functions, and
Sturm-Liouville Theory. Additional topics may include Green’s functions, numerical
methods, and method of characteristics. Prerequisites: Math 256 and Math 320.
Student Learning Objectives
By the end of the course students will be able to:
- formulate partial differential equations (PDEs) and boundary value problems (BVPs)
via modeling;
- apply analytic techniques to solve and examine both PDEs and BVPs;
- discuss the importance of eigenvalues and eigenfunctions, both mathematically and
physically;
- construct the decomposition of a function via eigenfunction expansion;
- discuss the importance of integral operators and how they are utilized to simplify
linear PDEs;
- interpret the solutions to PDEs and BVPs in a physically relevant fashion.
Text
Boundary Value Problems and Partial Differential Equations, 6th edition, Powers.
Grading Procedure
Grading procedures and factors influencing course grade are left to the discretion
of individual instructors, subject to general university policy.
Attendance Policy
Attendance policy is left to the discretion of individual instructors, subject to
general university policy.
Course Outline
- Chapter 0: Ordinary Differential Equations (2 weeks)
Review, Boundary Value Problems, Eigenvalues, Eigenfunctions, Sturm-Liouville Theory
- Chapter 1: Fourier Series and Integrals (2 weeks)
Fourier Series, Fourier Integrals, Fourier Transform
- Chapter 2: The Heat Equation (4 weeks)
Derivation and Analysis, Steady States, Separation of Variables, Variants, Integral
Solutions for Semi-Infinite/Infinite Domains
- Chapter 3: The Wave Equation (1 week)
Derivation and Analysis, Separation of Variables, Vibrating Beam, d’Alembert’s Solution
- Chapter 4: The Potential Equation (2.5 weeks)
Derivation and Analysis, Separation of Variables, Splitting BCs, Potential Equation
on a disk, Helmholtz Equation, Poisson Equation in rectangular and circular domains
- Chapter 5: Higher Dimensions and Other Coordinates (1.5 weeks)
Heat and Wave Equations in higher dimensions (rectangular coordinates), Polar Coordinates,
Bessel Functions, Vibrations on a Circular Membrane, Spherical Coordinates and Legendre
Polynomials
- Chapter 6: Laplace Transform (0.5 week)
Integral Transforms, Convolution, Application to PDEs
- Additional Topics as time allows, based on student and instructor interest.