ACTIVITIES  PERCENTAGES 

Assignments  50% 
Midterm exam  20% 
Final exam  30% 
Lectures: 3x / week, 1 hour / session
This course provides students with the basic analytical and computational tools of linear partial differential equations (PDEs) for practical applications in science engineering, including heat/diffusion, wave, and Poisson equations.
Analytics emphasize the viewpoint of linear algebra and the analogy with finite matrix problems including operator adjoints and eigenproblems, series solutions, Green's functions, and separation of variables.
Numerics focus on finitedifference and finiteelement techniques to reduce PDEs to matrix problems, including stability and convergence analysis and implicit/explicit timestepping.
There is no required text for this course, though the following book is recommended (emphasized more the numerical part of the course).
Strang, Gilbert. Computational Science and Engineering. Wellesley, MA: WellesleyCambridge Press, 2007. ISBN: 9780961408817. More information, including online chapters, can be found on Prof. Strang's CSE Web site.
Late problem sets are not accepted, however the lowest problem set score will be dropped.
ACTIVITIES  PERCENTAGES 

Assignments  50% 
Midterm exam  20% 
Final exam  30% 
SES #  TOPICS  KEY DATES 

L1  Overview of linear PDEs and analogies with matrix algebra  Problem set 1 out 
L2  Poisson's equation and eigenfunctions in 1d: Fourier sine series  
L3  Finitedifference methods and accuracy  
L4  Discrete vs. continuous Laplacians: symmetry and dot products 
Problem set 1 due Problem set 2 out 
L5  Hilbert spaces and adjoints for differential operators  
L6  Algebraic properties of the 1d Laplacian: consequences for Poisson, heat, and wave equations  
L7  Laplacians in higher dimensions, and general Dirichlet and Neumann boundary conditions 
Problem set 2 due Problem set 3 out 
L8  Separation of variables, in time and space  
L9  Separation of variables in cylindrical geometries: Bessel functions  
L10  Multidimensional finite differences and Kronecker products 
Problem set 3 due Problem set 4 out 
L11  Rayleigh quotients, the minmax theorem, and estimating the first few eigenfunctions  
L12  Green's functions and inverse operators  
L13 
Green's function of the 1d Laplacian Reciprocity 
Problem set 4 due Problem set 5 out 
L14  Delta functions and distributions I  
L15 
Delta functions and distributions II Green's functions via delta functions 

L16  Green's function of the 3d Laplacian 
Problem set 5 due Problem set 6 out 
L17  The method of images, interfaces, and surface integral equations  
L18  Green's functions in inhomogeneous media: Integral equations and Born approximations 
Problem set 6 due Problem set 7 out 
L19 
Dipole sources and approximations Overview of timedependent problems 

L20  Timestepping and stability: Definitions, Lax equivalence  
L21  Von Neumann analysis and the heat equation  
L22 
Explicit and implicit timestepping, and CrankNicolson schemes Wave equations in firstorder form 

L23 
Algebraic properties of wave equations and unitary time evolution Conservation of energy in a stretched string 

Midterm exam  
L24 
Wave equations in higher dimensions D'Alembert's solution and planewaves 
Problem set 7 due Problem set 8 out 
L25  Staggered discretizations of wave equations  
L26 
Wave propagation examples Phase and group velocity via Fourier analysis 
Problem set 8 due Problem set 9 out 
L27 
Group velocity dispersion Waveguides with hard walls 

L28  Reflection and refraction, evanescent waves, and dispersion relations 
Problem set 9 due Problem set 10 out 
L29  Waveguide eigenproblems  
L30  Maxwell's equations  
L31  Numerical simulation of Maxwell's equations: computational electromagnetism  
L32  Wave equations in frequency domain: Helmoltz equations and Green's functions 
Problem set 10 due 
L33  Perfectly matched layers (PML)  
L34 
PML in the time domain Finite element methods: introduction 

L35  Galerkin methods  
L36  Tent functions and recovering finitedifference methods from the Galerkin approach  
L37  Symmetry and linear PDEs  
Final exam 