University of Utah
Department of Mathematics

Math 3150-001   Partial Differential Equations for Engineers
T,  M.W.       11:50-12:40     LCB 225

What is PDE? Partial differential equations (PDEs) describe processes in continua that depend on time instances and spatial position. PDEs are, for example, used to describe the vibration of a string or a membrane, waves, propagation of sound or heat, electrostatics, electrodynamics, fluid flow, and elasticity. PDE is a type of differential equation, i.e., a relation involving an unknown function (or functions) of several independent variables and their partial derivatives with respect to those variables.

Course objectives: M3150 is a first course in PDEs intended for students from the sciences and engineering programs. On completion of the course the student should be competent in solving basic linear PDEs using classical solution methods, that is, be able to:

Understand the derivation of the heat, wave and equilibrium (Laplace, Poisson) equations with various boundary conditions;

Understand the classification of PDEs, and their geometric and physical meanings.

Use Fourier series for functions representation;

Use the method of separation of variables;

Use Fourier Transform methods;

Be able to solve boundary value problems (BVPs) of vibrating string and rectangular or circular membranes.

Be able to solve BVPs for the one-dimensional homogeneous and inhomogeneous heat equation with different boundary conditions by using Fourier method.

Be able to solve BVPs for the two-dimensional Laplace and Poission equations in rectangular and circular domains.

Use Maple to visualize PDE solutions.

Textbook:   Partial Differential Equations and Boundary Value Problems, N. Asmar, Prentice Hall 2005; Second Edition
Sections: 1.1, 1.2; 2.1 - 2.4; 3.1 - 2.9; 4.1 - 4.4; 7.1 - 7.4

Student Solutions Manual can be downloaded from the author's website:
http://www.math.missouri.edu/~nakhle/pdebvp/student-manual.pdf

Tutoring is available in T. Benny Rushing Mathematics Center

Syllabus is here

HW 1 Entry quiz

HW2. Section 1.1: Problems 1, 2, 6. Section 2.1: Problems 2, 9, 12.

Tutorial. Maple code for Fourier series expansion
> f := cos((1/2)*x); plot(f, x = -Pi .. Pi);
> a0 := (int(f, x = -Pi .. Pi))/(2*Pi);
> an := (int(f*cos(n*x), x = -Pi .. Pi))/Pi;
> bn := (int(f*sin(n*x), x = -Pi .. Pi))/Pi;
> fs := a0+sum(an*cos(n*x)+bn*sin(n*x), n = 1 .. 5);
> plot({f, fs}, x = -Pi .. 1.4*Pi);
> plot({f-fs}, x = -Pi .. Pi);

HW3 (due Monday September 10)
Section 2.2: Problems 6, 9, 11, 17. Using the Maple code (above), plot partial series for problems 12, 16.

HW4 (due Wednesday September 12)
Section 2.3: Problems 2, 7. 28. Section 2.4: Problems 3, 7. (Find both half-range expansions)

HW5 (due Wednesday September 26)
Section 3.1. Problems 2, 5. Section 3.2. Problem 9 *. Section 3.3. Problems 3, 15 (use # 12), Section 3.4. Problems 15, 18 *.
Problems marked (*) are for extra credit.

HW6 (due Wednesday October 3)
Section 3.5. Problems 4, 12. Section 3.6. Problem 2, 13.

Here is the schedule of tutors in Math Tutoring Center with subjects of tutoring.

HW7 (due Wednesday October 17)
Section 3.7. Problems 2, 11. Section 3.8. Problem 3.

HW8 (due Wednesday October 24)
Section 3.9. Problem 16. Section 4.1 Problem 2, 3.

Tutorial. Maple code for Laplace equation in a rectangle [0, 1]X[[0, 1].
Boundary conditions are: f1=0. g1=0, f2=x^2, g2=y^2

# The basic algorithm
# Solving first problem
> f := x^2; a := 1; b := 1;
> bn := 2*(int(f*sin(Pi*n*x), x = 0 .. a))/a;
> vn := bn*sinh(Pi*n*y/a)*sin(Pi*n*x/a)/sinh(Pi*n*b/a);
> u := sum(vn, n = 1 .. 15);
> plot3d(u, x = 0 .. a, y = 0 .. b);

# Solving a second problem by remaning x <-> y etc.

> eval(u, {a = alpha, b = beta, x = p, y = q}); u2 := eval(%, {alpha = b, beta = a, p = y, q = x});
> plot3d(u+u2, x = 0 .. a, y = 0 .. b, axes = boxed);


# Modified algorithm
> u0 := x*y;
# Recompute boundary conditions:
> f := x*(x-1); a := 1; b := 1;
#Use the basic algorithm:
> bn := 2*(int(f*sin(Pi*n*x), x = 0 .. a))/a;
> vn := bn*sinh(Pi*n*y/a)*sin(Pi*n*x/a)/sinh(Pi*n*b/a);
> u := sum(vn, n = 1 .. 15);
> eval(u, {x = p, y = q}); u2 := eval(%, {p = y, q = x});
> plot3d(u+u2, x = 0 .. a, y = 0 .. b, axes = boxed);
# Collect all three terms
> plot3d(u0+u+u2, x = 0 .. a, y = 0 .. b, axes = boxed);

Extra credit problem: calculate a 2d cloaking annulus from material with conductivity k_c that shields the shperical inclusion with conductivity k_i or radius 1 in a conducting plane with conductivity k_0. (*)

HW 9 Due Wednesday, November 14.
Section 4.4 . Problems 2, 5, 15. Section 4.2. Problem 2. Section 4.3. Problem 1

HW 10 Due Monday, December 3.
Section 7.1 Problem 1. Section 7.2. Problems 19, 53. Section 7.3 Problem 3.