University of Utah
Department of Mathematics
Math 3150-001
Partial
Differential Equations for Engineers
T, M.W. 10:45 - 11:35 am WEB L110
Instructor:
Andrej Cherkaev
Office: JWB 225 ph: 581-6822
email:cherk@math.utah.edu
Office hours: F, 10-12 and by appointment
About the subject:
Partial differential equations (PDE) are 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. PDEs are
used to formulate, and thus aid the solution of problems involving functions of
several variables.
PDEs are for example used to describe the propagation of sound or heat,
electrostatics, electrodynamics, Fuid flow, and elasticity. These seemingly dis-
tinct physical phenomena can be formalized identically (in terms of PDEs),
which shows that they are governed by the same underlying dynamic. PDEs
fnd their generalization in stochastic partial dierential equations. Just as orinary differential equations often model dynamical systems, partial dierential
equations often model multidimensional systems.
Wikipedia: Partial differential equation
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
Home work
HW1. Section 1.2: Problems 5, 11, 16. Section 2.1: Problems 2, 9, 12, 14,
19.
HW2. Section 2.2: Problems 5, 11, 15. Section 2.3: Problems 4, 17. (You
may use Maple)
HW3. Section 2.3: Problems 2, 9, 17. Section 2.4: Problems 4, 16.
HW4. Section 3.3: Problems 1, 9, 11. Section 3.4: Problem 2.
HW5. Section 3.5: Problems 2, 4, 12. Section 3.6: Problems 2, 5, Section 3.7: Problems 1, 2.
HW 6. Section 3.8: Problem 4. (Graph the solution). Section 3.9: Problems 3, 7
Example: Maple code for Fourier series calculations
# Example of computing Fourier series (Ch. 2)
# 1. Set the periodic function f and period p
f := x - exp(x);
p := 1;
# 2. Compute Fourier coefficients:
a0 := (int(f, x = -p .. p))/(2 p);
an := (int(f*cos(n*Pi*x/p), x = -p .. p))/p;
bn := (int(f*sin(n*Pi*x/p), x = -p .. p))/p;
# 3. Compute the partial sum fn (Here, n=15)
fn := a0+sum(an*cos(n*Pi*x/p)+bn*sin(n*Pi*x/p), n = 1 .. 15);
# 4. Plot the function, the partial sum, and their difference
plot({f, fn}, x = -p .. p);
plot(f-fn, x = -p .. p);
# 5. Repeat for any function and any period.
Hints for Bessel functions (Maple)
Plot of Bessel functions
plot(BesselJ(0, x), x=0 .. 30);
plot(BesselJ(1, x), x=0 .. 30);
First zeros of Bessel Function:
z1 := evalf(BesselJZeros(0, 1));
z2 := evalf(BesselJZeros(0, 2));
z3 := evalf(BesselJZeros(0, 3));
etc.
Alternatively, one can compute them at once:
> for n to 10 do w[n] := evalf(BesselJZeros(0, n)) end do;
2.404825558
5.520078110
8.653727913
11.79153444
14.93091771
18.07106397
21.21163663
24.35247153
27.49347913
30.63460647
Computing coefficients of Bessel Series. f is an arbitrary function: Here I put a=1
f := 100-r;
Notice, to perform numerical intergation in Maple I put the upper limit in the form a=1.0 (floating point)
for n to 10 do A[n] := 2*(int(f*BesselJ(0, w[n]*r)*r, r = 0 .. 1.0))/BesselJ(1, w[n]) end do;
0.4072812946
-0.02337240050
0.01441795517
-0.004036123910
0.003507320274
-0.001467399340
0.001418570623
-0.0007181654672
The summarion of Bessel Series, Example c=2
u := sum(A[k]*BesselJ(0, w[k]*r)*cos(2*w[k]*t), k = 1 .. 10);
Plotting
plot3d(u, r=0 .. 1, t=0 .. 2, axes=boxed);
with(plots):
animate(plot, [u, r=0 .. 1], t=0 .. 3);
HW 8
4.4 Problems 2, 4, 12, 15.
Create a cloaking device.
Create a cloaking device.
HW 9 The last one!
7.1 Problem 5
7.2 Problems 2, 3
7.3 Problem 5