Math
3150-001 Partial Differential Equations for
Engineers
Spring 2015.
MWF WEB L110, 10:45 -11:35
|
Instructor: Andrej Cherkaev |
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 distinct 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 di erential equations. Just as orinary differential equations often model dynamical systems, partial di erential equations often model multidimensional systems.
Wikipedia: Partial differential equation
Course learning objectives
Basic topics:
Students will become knowledgable about partial differential equations (PDEs) and how they can serve as models for physical processes such as mechanical vibrations, transport phenomena including diffusion, heat transfer, and electrostatics. Students will be able to derive heat and wave equations in 2D and 3D using the divergence theorem.
Students will master how solutions of PDEs is determined by conditions at the boundary of the spatial domain and initial conditions at time zero.
Students will be able to understand and use inner product spaces and the property of orthogonality of functions to determine Fourier coefficients, and solution of PDEs using separation of variables. Students will master the method of separation of variables to solve the heat and wave equation under a variety of boundary conditions. Students will be familiar with the use of Fourier series for representation of functions, and the conditions for series convergence.
Students will be able to solve for the electric potential in an area or volume region by specifying the charge distribution on the boundary of the region (i.e., boundary conditions) and use separation of variables to obtain the solution. Students will be able to derive basic properties of these electric potentials, including points of minimum/maximum potentials, and use Stokes' theorem to determine work done moving charges in a closed path through the potential.
Students will also master the use of the Fourier transform and integral convolution to solve the heat equation on the real line using the heat kernel.
Problem solving fluency:
In addition to topical
content, students will also improve their problem solving
skills. Students will practise reading and interpreting problem
objectives, selecting and executing appropriate methods to
achieve objectives, and finally, be able to interpret and
communicate results.
Week 1: Subject of PDE; 12.1-2: Heat/diffusion equation, conduction/transport in 1D
Week 2-3: 12.3-5: Boundary conditions, Equilibrium temperature, Derivation of heat equation in 2-3D using the divergence theorem.
Week 4: 4.6, 4.10, Orthogonal Vectors, Inner product, inner product space, Inner product on a function space, orthogonal projection onto a subspace with orthogonal basis.
Week 5-7: 13.1-5: Fourier coefficients, solving 1D heat equation with zero-endpoint temperatures, 1D Heat equation with insulated ends, periodic ends, Laplace equation in a rectangle and disk, mean value theorem, maximum condition, uniqueness, net-zero boundary flux via divergence theorem
Week 8-9: 14.1-4: Fourier series, Convergence theorem, Sine and Cosine series, Term-by-term differentiation
Week 10-11: 15.1-5: Derivation of wave equation in 1D, Boundary conditions, Solution with fixed ends, Vibrating rectangular membrane
Week 12-13-14: 16.1-4: Heat equation on infinite 1D domain, Fourier transform pairs, Transforming the heat equation, Heat kernel.
Week 15: Slack
time and review.
Grading: The grade is based on the homework
(30%), two midterm exams (20% + 20%), and the final exam
(30%). The problems in the exams are based on the homework
problems.
Handouts with proofs of the divergence theorem (a.k.a Gauss
theorm or Ostrogradsky-Gauss theorem) 1
and
2
About the Sturm-Liouville theory
http://en.wikipedia.org/wiki/Sturm%E2%80%93Liouville_theory,
http://www.math.iitb.ac.in/~siva/ma41707/ode7.pdf
Convergence of Fourier series
http://en.wikipedia.org/wiki/Convergence_of_Fourier_series
Home work: set 1
12.2 ## 3, 5, 8.
12.3 ## 1, 2.
12.4 ## 1 (a), (b), (c), (e), (g).
Home work: set 2
13.2 # 2,
13.3 ## 1(a), 1(c), 1(d), 1(f).
Home work: set 3
13.3 ## 2(a), 2(b), 2(c),
13.3 ## 3(a), 3(b), 3(d),
13.3 #6.
First Midterm (chapters 12, 13) will be at Wednesday,
February 18
Home work: set 4
13.5 ## 1(g), 3(a), 3(b), 5 (a), 5(b).
Home work: set 5
Home work: set 6
Example of Maple program for computing and plotting the
Fourier series for f=x^2, 0<x<1, and
computing the derivative
f:=x^2;
L:=1;
an := 2*(int(f*cos(Pi*n*x/L), x = 0 .. L));
a0 := (int(f, x = 0 .. 1));
ff := a0+sum(an*cos(Pi*n*x/L), n = 1 .. 10);
plot({f, ff}, x = 0 .. 1);
plot(f-ff, x = 0 .. 1);
dff := diff(ff, x);
df:= diff(f, x);
plot({df, dff}, x = 0 .. 1);
Second Midterm (chapters 14, 15) will be at Wednesday, March
25.
15.6 # 3,
16.2 ##1,2
Due Friday, April 10.
Due Monday,, April 27.