# MAT 413 An Introduction to Partial Differential Equations

This is a sample syllabus only. Ask your instructor for the official syllabus for your course.

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### Course Description

Solutions to partial differential equations by separation of variables and Fourier series. Applications to heat flow and diffusion, wave motion, and potentials. Some discussion of existence and uniqueness of solutions.

MAT 413 meets for three hours of lecture per week.

### Prerequisites

MAT 311 with a grade of "C" or better is required; MAT 213 is recommended.

### Objectives

After completing MAT 413 the student should be able to

• demonstrate knowledge and understanding of the concepts and techniques used to solve basic problems of heat flow, wave motion, and potentials
• find the Fourier series for a given function
• determine the nature of the convergence of the Fourier series of a given function
• apply operations (addition, scalar multiplication, integration, differentiation) to Fourier series to derive other results
• find the Fourier integral for a given function
• use the technique of separation of variables to solve boundary value problems for the heat equation, the wave equation, and the potential equation in various domains
• derive the D'Alembert solution for the wave equation and use it to determine properties of particular boundary value problems
• solve Sturm-Liouville problems by using more general eigenfunctions
• use Bessel functions to solve the heat problem in a cylinder and the wave problem in a disk

### Expected outcomes

Students should be able to demonstrate through written assignments, tests, and/or oral presentations, that they have achieved the objectives of MAT 413.

### Method of Evaluating Outcomes

Evaluations are based on homework, class participation, short tests and scheduled examinations covering students' understanding of topics covered in MAT 413.

### Text

Boundary Value Problems (2nd ed.), by David L. Powers.

#### Topics to be Covered

• Fourier Series and Integrals
• Periodic functions and Fourier series
• Arbitrary period and half-range expansions
• Convergence of Fourier series
• Uniform convergence
• Operations on Fourier series
• Fourier integral
• The Heat Equation
• Derivation and boundary conditions
• Examples: Fixed end temperatures, Insulated bar, Different boundary conditions, Convection
• Sturm-Liouville problems
• Expansion in series of eigenfunctions
• Generalities on the heat conduction problem
• Semi-infinite rod
• Infinite rod
• The Wave Equation
• The vibrating string problem and its solution
• D'Alembert's solution
• Generalities on the one-dimensional wave equation
• Wave equation in unbounded regions
• The Potential Equation
• Potential equation
• Potential in a rectangle, a slot, a disk
• Classification of partial differential equations
• Existence and Uniqueness of Solutions, Well-posed problems
• Problems in Several Dimensions
• Two-dimensional wave equation
• Two-dimensional heat equation
• Problems in polar coordinates
• Bessel's equation
• Other topics as time allows
• Spherical coordinates and Legendre polynomials
• Laplace transform

Students' grades are based on homework, class participation, short tests, and scheduled examinations covering students' understanding of the topics covered in MAT 413. The instructor determines the relative weights of these factors.

### Attendance Requirements

Attendance policy is set by the instructor.

### Policy on Due Dates and Make-Up Work

Due dates and policy regarding make-up work are set by the instructor.

### Schedule of Examinations

The instructor sets all test dates except the date of the final exam. The final exam is given at the date and time announced in the Schedule of Classes.