# MAT 413 An Introduction to Partial Differential Equations

Instructor:
Office:
Office hours:
Phone:
Email:

### 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.

3 units credit.

### Prerequisites

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

### Text

Texts are chosen by the instructor. For example:

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

### Course Requirements, Tentative Schedule of Class Meetings and Topics, Readings, Assignments and Due dates, Exams

A schedule of class meetings, topics, assignments, due dates, exam dates, etc. will be provided by instructor. See your class syllabus.

Here is an example course outline, based on the above text.

• 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

The final exam is given at the date and time announced in the Schedule of Classes.

### Learning Objectives

After completing MAT 413 the student will

• 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

### Computers and Calculators, Computer Literacy

Most instructors encourage the use of machines, calculators computers, phones etc., for analyzing data. The use of machines may be restricted during examinations or at certain other times. Ask your instructor for the policy in your class.

Students are not expected to be programmers or to know any particular computer language before starting this class. Some instructors may expect students to be able to access information on the internet, or to use calculators, or to learn to use particular software with instruction. Basic skill in algebra and the use of mathematical symbols, order of operations etc., and the willingness to read and follow instruction manuals and help files will suffice.

Students' grades are based on homework, class participation, short tests, and scheduled examinations covering students' understanding of the topics covered in this course. The instructor determines the relative weights of these factors and the grading scale. See the syllabus for your particular class.

### Location of Class Meetings

Classes meet on the dates and room announced in the official Schedule of Classes. This is a traditional, face-to-face class.

### Attendance Requirements

Attendance policy is set by the instructor.

### Policy on Due Dates, Make-Up Work, Missed Exams, and Extra-Credit Assignments

Due dates and policy regarding make-up work and missed exams are set by the instructor. Instructors may, or may not, choose to offer extra credit assignments. If extra credit assignments are offered they will be available to all students.

The mathematics department does not tolerate cheating. Students who have questions or concerns about academic integrity should ask their professors or the counselors in the Student Development Office, or refer to the University Catalog for more information. (Look in the index under "academic integrity".)

### Accomodations for Students with Disabilities

Cal State Dominguez Hills adheres to all applicable federal, state, and local laws, regulations, and guidelines with respect to providing reasonable accommodations for students with temporary and permanent disabilities. If you have a disability that may adversely affect your work in this class, I encourage you to register with Disabled Student Services (DSS) and to talk with me about how we best can help you. All disclosures of disabilities will be kept strictly confidential. Please note: you must register with DSS to arrange an no accommodation. For information call (310) 243-3660 or send an email message to dss@csudh.edu or visit the DSS website http://www4.csudh.edu/dss/contact-us/index or visit their office WH D-180

### Behavioral Expectations

We all are adults so behavior rarely is an issue. Just follow the Golden Rule: "do unto others as you would have them do unto you" then everything will be fine.

The university must maintain a classroom environment that is suitable for learning, so anyone who insists on disrupting that environment will be expelled from the class.

Revision history:

Prepared by J. Barab 2/4/00. Revised 7/7/01, 7/25/06, 1/10/15 (G. Jennings).