COURSE SYLLABUS Course Number: MATH7440 Course Title: PARTIAL DIFFERENTIAL EQUATIONS I Credit Hours: 3 Prerequisites: Departmental approval. Corequisite: Objectives: To present the fundamentals of the theory of partial differential equations, to prepare students for independent research in the field, and to enable students to apply the theory in other fields. Course Content: The following is a sample course outline. Actual topics may vary according to the students' backgrounds and interests. Prototypes of (second-order linear) partial differential equations (elliptic, parabolic, hyperbolic equations), review of elementary solution methods, well-posedness of boundary and initial-boundary value problems [8 days] Laplace's and Poisson's equations, harmonic and subharmonic functions, maximum principles, existence and uniqueness results for the Dirichlet and other boundary-value problems [12 days] General second-order linear elliptic partial differential equations, Schauder estimates, H\"older spaces, existence, uniqueness, and regularity results, the Fredholm alternative [12 days] Higher-order linear elliptic partial differential equations, weak solutions, Sobolev spaces, existence, uniqueness, and regularity results, Hilbert space methods, spectrum and resolvent, eigenfunction expansions [12 days] Possible Textbooks: M.Renardy & R.C.Rogers, An Introduction to Partial Differential Equations, Springer, New York, 1993 F.John, Partial Differential Equations, Springer, New York, 1982 Sample Grading and Evaluation Procedures A graduate student is expected to creatively engage the mathematical material of the course. The student will be given problems to solve; these problems may include the derivation of proofs to theorems. These solutions may be presented in class on the blackboard or be written up to be handed in to the instructor. Extended projects may also be assigned. Grade Calculation Presentation of solutions to problems/theorems, homework: 40% Midterm exam or midterm project: 25% Final Exam or culminating project: 35% There may be variations in these procedures depending on the individual instructors and the nature of the specific material. Sample Statement Re: Accommodations Students who need accommodations are asked to arrange a meeting during office hours the first week of classes, or as soon as possible if accommodations are needed immediately. If you have a conflict with my office hours, an alternate time can, be arranged. To set up this meeting, please contact me by E-mail. Bring a Copy of your Accommodation Memo and an Instructor Verification Form to the meeting. If you do not have an Accommodation Memo but need accommodations, make an appointment with The Program for Students with Disabilities, 1244 Haley Center, 844-2096 (V/TT). (Note: Instructor office room, office hours and email address will be made available on the course syllabus and on the first day of class.) JUSTIFICATION FOR GRADUATE CREDIT This course is part of a modified semester version of a 600-level sequence under the quarter system. Under the quarter system it was specifically designed as a graduate course. It was approved as part of our graduate program and has been a traditional part of our graduate program offerings. Outside of modernization, the standard of the course remains at the same graduate level that the department has maintained in the past. Due to the semester changes the number of hours for the original material has decreased. The loss of material will not affect the depth of instruction of the remaining topics. The topics selected for the semester course from the quarter course are those critical to the understanding of the subject. The course demands considerable mathematics background and a degree of mathematical maturity traditionally found at the graduate level. The 7000-level course will inculcate the same analytical skills and depth of understanding previously demanded by the comparable 600-level quarter course. In order to successfully complete the course the student will have to demonstrate an ability to creatively examine and apply the mathematics