Partial Differential Equations, Spring 08, Tue, Fri 12:30-2:00  HNS 106

Instructor: Leon Kaganovskiy
Office: HNS 110 or HNS 204 (Physics Computer Lab, 2nd floor)
Office Hours:

Mon: 11-12:30 and 4-6,   Tue :  4-6 (office/computer lab HNS 204),  Wed: 11-12:30 and 4-6,  Thu :  research day,  Fri   :  4:30-6:30 (office/computer lab HNS 204)

I am available at other times by appointment.  

email: lkaganovskiy@ncf.edu - best way to ask a question.    Also if you need me ASAP:  (941) 366-6134;  (941) 961-3896

I will grade the HW myself

Best time:   Tue 3:20-6 (catch me in computer lab at 3:20 or later in the office)  and Fri after 4:30 (or earlier if there is no Nat Sci Seminar)

       

 

Course Goals and Objectives: 

This course will focus on applied partial differential equations and their computational methods.  It is intended for math and science students who apply these techniques in their work.  Topics we will consider include, but are not limited to:

First Order Equations PDEs, constant coefficients, spatially depended velocity of propagation, nonlinear conservation laws, shocks, weak solutions, numerical methods,   Diffusion equation, max principle, heat equation on R and half line, nonlinear diffusion, numerical methods for heat equation, boundary value problems, separation of variables, symmetric BC, inhomogeneous problems, waves in 1D, D’Alambert solution, domains of dependence and influence, conservation of energy, reflection, transmission, finite intervals modes of vibration, higher dimensions, eigenfunctions for the rectangle and disk, Bessel functions, harmonic functions, mean value property, max principle, Dirichlet problem, disk solution, Liouville theorem, Poisson Equation, nonlinear Burgers and Klein-Gordon Equations.   I intend to use J. M. Cooper, Introduction to Partial Differential Equations with Matlab, which will allow us to employ Matlab to investigate computer approaches to solving and visualizing PDE analytically and numerically in most sections of the textbook.

Prerequisites:  Calculus, Ordinary Differential Equations.

Grading Policy: The final grade will be based on tests and problems, as follows:

  1. There will be two exams which will count 35% and 45 % of the final evaluation.

1st exam – 1 week after Spring break, Final – exam week.     Exact dates and times will be announced in class.

  1. Homework assignments and computer projects will count 20% of the course grade. Late homework will be accepted only by special permission of the instructor.

Attendance Policy:

There are no specific attendance credit points, but you are responsible for attending all the classes and keeping abreast of all the material presented in class.

Special Need Students:

Students with the need for special accommodations must work with the Counseling and Wellness Center, which will establish the specific accommodations and communicate them to me.

Academic Dishonesty Policy:

Any suspected instance of plagiarism will be reported to the office of the Provost and handled in accordance with the College’s policy.

 

Books:

J. M. Cooper,  Introduction to Partial Differential Equations with Matlab,  Birkhauser 1998, ISBN 3-8176-3967-5  and  3-7643-3967-5

Extra Reference book (you are not required to buy it):

W. A. Strauss,  Partial Differential Equations an Introduction, John Wiley and Sons Inc, 2008,  ISBN0-471-54868-5

 

Programming Language:  The predominant programming languages used in numerical analysis are Fortran and MATLAB.  In this course we will focus on Matlab. Many numerical analysis programs in Matlab will be provided. For students unacquainted with MATLAB, a short introduction is available from the following sources: Matlab Tutorial Brief,   Matlab Primer (more comprehensive, but longer),   Matlab Intro Plotting,    Matlab Tutorial MIT,  or just Google it.

A student version of MATLAB can be obtained from the company Mathworks, Inc. at a somewhat reduced price (>100$). This student version is essentially the full version, without some special add-on toolboxes. Matlab is available at the second floor Physics Computer Lab. Programs in languages other than Fortran and Matlab are also sometimes acceptable, but no programming assistance will be given in the use of such languages.

 

Topics to be covered and Homework Assigned  (exact due dates will be announced in class).

This course plan may be modified during the semester. Such modifications will be announced in advance during class periods, and the students are responsible for keeping abreast of such changes. The WWW page for the course will also be used to list assignments and other notes, and students are responsible for checking this web page regularly.

Chapter 1:   Preliminaries – skip.    We will review necessary concepts as needed.

Chapter 2:   First Order Equations.

            Constant coefficients, spatially depended velocity of propagation, nonlinear conservation laws, shocks, weak solutions, numerical methods.

            Note that I am going to present an approach which is slightly different from the book and hopefully more understandable.

            HW:  2.1: 1, 3,    Extra Credit:  2.2:  1, 3, 5, 6, 8    2.3:  1-5     2.3: 2       2.5: 2, 3, 5       required:  2.6: 2, 4-8

Chapter 3:   Diffusion

            Diffusion equation, max principle, heat equation on R and half line, nonlinear diffusion, numerical methods for heat equation.

            HW:  3.2:  1d, 2b, 3      3.3:  1, 2, 3a, 4, 9        3.4:  2, 3 (find analytic solution, skip Cd, gamma_d), 5ab (use erf function), 6a      3.5: 1, 2, 5

            Matlab program to illustrate how to include plotting with pause

Chapter 4:   Boundary Value Problems for the Heat Equation

            Separation of Variables, convergence of eigenfunction expansions, symmetric BC, inhomogeneous problems.

            HW  4.2:  4-7  11-14,   4.3:  1, 2, 4, 7, 9      4.4: 1, 2, 4ab, 5, 8a

Chapter 5:   Waves in 1D.

Linearization, IVP and D’Alambert solution, domains of dependence and influence, conservation of energy, inhomogeneous problem, reflection, transmission, finite intervals modes of vibration types, BC, numerical methods for wave equation.

HW:   5.3:  1-5,    5.4: 3-7,      5.5: 1-5, 7-9     5.7: 4-6

Chapter 6:  Fourier Series – mostly covered in Chapter 4, we look only at generalization of Fourier Series.

Chapter 7:  Dispersive Waves and Schrodinger Equation – optional

Chapter 8:  The Heat and Wave Equations in Higher Dimensions.

Diffusion in higher dimensions, BVP for heat equation, eigenfunctions for the rectangle and disk, Bessel functions, asymptotics, wave equation in higher dimension, eigenfunction expansions, nodal curves, energy conservation, inhomogeneous problems.

HW:  8.1:  4, 6, 8     8.2: 1     8.3: 1, 2     8.4: 3-6   8.5: 1, 2      8.9: 1-5, 8

Additional Material on Wave/Heat equation in a spherical ball, Bessel functions, Legendre and Hermite Polynomials. (Strauss Book)

HW (from Strauss book handout):  10.3: 1, 4, 5, 6, 7, 9, 10       10.5: 1, 3, 4, 7, 14-16       10.6: 3, 4, 6

Chapter 9:  Equilibrium

            Harmonic functions, mean value property, max principle, Dirichlet problem, disk solution, Liuville theorem, Poisson Equation.

            HW:  9.1: 4ab, c – extra credit, 6, 8 (ignore sub-harmonic question), 9       9.2: 1, 2, 4-7,     9.3.1