Instructor: Doug Pickrell

Office: Mathematics 703

Office Phone: 621-4767

Email: pickrell@math.arizona.edu

Office Hours: Tu 11-12 (in math tutoring room M203), W 11-12, Th 11-12 and by appointment (I am usually in my office 4:15-5:15 on W, but you should check)

Recommended Text: All of the material for this course is on Wikipedia, and I will be emailing the homework to you, so it is not necessary to buy a book. However, if you prefer a book, some possibilities include Advanced Engineering Mathematics by Peter J. O'Neil (any edition; this is probably most closely aligned with the outline of our course), Mathematical Methods for Scientists and Engineers by McQuarrie, and Advanced Engineering Mathematics by Kreyszig.

About the course: The title of this course is "Advanced Applied Analysis". This does not help much in defining the content of this course, because any idea in advanced mathematics can potentially be applied to something. But this does explain why the aforementioned books are so big - they try to at least mention most of the core ideas that have become associated with applications of analysis. In this course we will concentrate on transform and expansion methods: Laplace transform, series expansions, Fourier series, and Fourier transform. To creat a coherent theme for the course, we will continually apply these methods to partial differential equations: the spring equation, the heat equation, the wave equation, and the potential equation (transform methods are equally applicable to data storage, transmission of information, and interception of information, but these are topics for other courses). We will discuss the derivation of these equations in the first class. Beyond this the plan for the course is to cover significant portions of Chapters 3, 4, 14, 15, 16, 17, 18, and 19 from the book by O'Neil.

Homework: Doing the homework well and on a regular basis is critical in this course. The standard rule of thumb is that you should spend two hours studying outside of class for each hour of classwork. I will email the homework assignments directly to you (It is essential that I have your email address). I will email solutions for most of the problems on the due date (hence late homeworks are not accepted). The problems on the midterms and final exams will mimic those on the homework.

Midterms: There will be two midterms. The dates of each midterm, and the material to be covered on each midterm, will be announced at least one week in advance in class and by email. You must notify me in advance if for some (good) reason you are not able to attend a test; in this event, we will mutually arrange for a makeup in a timely manner (before I send out solutions), or some other method of evaluating your performance in the class.

Final Exam: The final exam for 422-01 is scheduled for Th, Dec 19, 8am. The final exam for 422-02 is scheduled for Tu, Dec 17, 8am.

Grades: I will calculate grades in the following way. I will first compute a total test score, based exclusively upon the midterms and the final exam. Each midterm will count for 25%, and the final exam will count for 50%, toward the total test score. I will use the total test scores to linearly order the class, and attempt to determine clear cutoffs for grades. Your final grade will be at least as high as that determined by this first calculation. I will secondly calculate a total score, based on homework, midterms and the final exam. The homework will count for 20%, each midterm will count for 20%, and the final will count for 40%. I will then linearly order the class, using this second calculation, and attempt to determine clear cutoffs (Generally the second calculation yields higher grades, so it is in your interest to do the homework). If I cannot resolve a borderline grade using either of these two calculations, I will look at the student's test trend and effort on the homework.

Tentative Class Schedule/Due Dates:

Week 1: Derive the heat equation (and Laplace's, or potential, equation) using the divergence theorem from vector calculus. Review of derivation and solution of the spring equation from Math 355. Introduction to Laplace transform.

Weeks 2-3: Laplace transform, application to solving the spring equation my''+cy'+ky=f(t), using convolution (and tables - to be provided)

Week 4: Taylor series solutions of ordinary differential equations (e.g. spring type equations with variable coefficients)

Week 5: More series solutions, special functions (beyond calculus), e.g. Bessel functions,...

Week 6: Review, First midterm (Oct 3).

Weeks 7-8: Fourier series (expanding functions in terms of cosines and sines). This involves applying ideas from Euclidean geometry and linear algebra to function spaces; this is the conceptual core of the course.

Weeks 9-10: Using Fourier series to solve the heat and wave equations on finite intervals. Perhaps some discussion of wavelets.

Week 11: Review, Second midterm (Nov 7 or possibly Nov 14, if we fall behind schedule)

Week 12: Fourier transform on R^n, heat kernel

Week 13 and Week of Thanksgiving: More Fourier transform, eigenvalues and eigenvectors for Laplace operator on a general domain

Week 14: solving heat and wave equations on general domains

Week 15: review

Final Exam

Standard University/Course Policies:

Attendance: Students are expected to attend every scheduled class. • The UA’s policy concerning Class Attendance, Participation, and Administrative Drops is available at: http://catalog.arizona.edu/policy/class-attendance-participation-and-administrative-drop. • The UA’s policy regarding absences for any sincerely held religious belief, observance or practice will be accommodated where reasonable. See: http://policy.arizona.edu/human-resources/religious-accommodation-policy. • Absences pre-approved by the UA Dean of Students (or Dean Designee) will be honored. See: https://deanofstudents.arizona.edu/absences.

Classroom Behavior: To foster a positive learning environment, students and instructors have a shared responsibility. We want a safe, welcoming, and inclusive environment where all of us feel comfortable with each other and where we can challenge ourselves to succeed. To that end, our focus is on the tasks at hand and not on extraneous activities (texting, chatting, reading a newspaper, making phone calls, web surfing).

Communication: It is the student’s responsibility to keep informed of any announcements, syllabus adjustments or policy changes made during scheduled classes, by email.

Students with disabilities: Our goal in this classroom is that learning experiences be as accessible as possible. If you anticipate or experience physical or academic barriers based on disability, please let me know immediately so that we can discuss options. You are also welcome to contact the Disability Resource Center (520-621-3268) to establish reasonable accommodations. For additional information on the Disability Resource Center and reasonable accommodations, please visit http://drc.arizona.edu. If you have reasonable accommodations, please plan to meet with me by appointment or during office hours to discuss accommodations and how my course requirements and activities may impact your ability to fully participate. Please be aware that the accessible table and chairs in this room should remain available for students who find that standard classroom seating is not usable.

Students withdrawing from the course: Must be made in accordance with University policy http://catalog.arizona.edu/policy/grades-and-grading-system#Withdrawal.

Incompletes: Must be made in accordance with University policies, which are available at http://catalog.arizona.edu/policy/grades-and-grading-system#incomplete

University Policies: • The UA Threatening Behavior by Students Policy prohibits threats of physical harm to any member of the University community, including to oneself. See http://policy.arizona.edu/education-and-student-affairs/threatening-behavior-students. • Students are encouraged to share intellectual views and discuss freely the principles and applications of course materials. However, graded work/exercises must be the product of independent effort unless otherwise instructed. Students are expected to adhere to the UA Code of Academic Integrity as described in the UA General Catalog. See: http://deanofstudents.arizona.edu/academic-integrity/students/academic-integrity. • The University is committed to creating and maintaining an environment free of discrimination; see http://policy.arizona.edu/human-resources/nondiscrimination-and-anti-harassment-policy

Note: Information contained in the course syllabus, other than the grade and absence policy, may be subject to change with advance notice, as deemed appropriate by the instructor.