# Math 447: Intro to Partial Differential Equations

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### Title

Introduction to Partial Differential Equations.

(3:3:0)

W (even years)

### Prerequisite

Math 303; or 314 and 334.

### Description

Boundary value problems; transform methods; Fourier series; Bessel functions; Legendre polynomials.

## Desired Learning Outcomes

The main purpose of this course is to teach students how to solve the canonical linear second-order partial differential equations on simple domains. Secondarily, students should be introduced to the theory concerning the validity of such solutions.

### Prerequisites

Current prerequisites ensure that students have had instruction in multivariable calculus and ordinary differential equations.

### Minimal learning outcomes

Primarily, students should be able to use the solution techniques described below. Students should gain a basic understanding of issues concerning solvability and convergence, but the current prerequisites don't guarantee that incoming students will have had any prior exposure to the theory of the convergence of sequences of functions, so expectations in that area are modest.

1. Basic classification of PDEs
• Linearity
• Homogeneity
• Order
• Elliptic, parabolic, or hyperbolic
2. Basic Modeling
• Derivation of the heat equation
• Derivation of the wave equation
3. Basic principles, techniques, and theory
• Principle of superposition
• Method of separation of variables
• Definition of eigenvalues and eigenfunctions corresponding to two-point BVPs
• Basic Sturm-Liouville theory
4. Special eigensystems
• Fourier
• Series representations
• Effect of symmetry and modifications and combinations of functions
• Theorems on pointwise, uniform, and L2 convergence
• Bessel's Inequality and Parseval's Equation
• Integral representations
• Bessel
• Legendre
5. Representation of solutions to the canonical equations on simple domains
• Laplace's equation on rectangles, rectangular strips, quarter-planes, half-planes, disks, and balls
• Wave equation on bounded intervals, half-lines, lines, disks, and balls
• Heat equation on bounded intervals, half-lines, lines, rectangles, disks, and balls

### Textbooks

Possible textbooks for this course include (but are not limited to):

• Richard Haberman, Applied Partial Differential Equations (4th Edition), Prentice Hall, 2003.