
Date 
Section 
Homework 
Due date 



Hw 1: 

1 

Multiindex notation. Basic definitions. Evans 1.11.3 


2 

Transport equation. Characteristics for first order PDE. Weak solutions. Evans 2.1 
01/23/18 




Hw 2: 

3 
01/23/18 
Characteristics for linear and quasilinear first order PDE. Evans 3.2. 


4 
01/25/18 
Characteristics for quasilinear first order PDE. Inviscid Burgers equation. Evans 3.2. 
01/30/18 




Hw 3: 

5 
01/30/18 
Solution of fully nonlinear PDE of first order. Evans 3.2. 


6 
02/01/18 
Solution of fully nonlinear PDE of first order. Classification of quasilinear second order PDE for two independent variables. Handout 
02/06/18 




Hw 4: 

7 
02/06/18 
Classification of quasilinear second order PDE for n independent variables. Handout 


8 
02/08/18 
Diagonalization of quadratic forms and transforming of second order quasiliner PDE to the canonical form.. Spectral theorem for symmetric matrices. 
02/13/18 




Hw 5: 

9 
02/13/18 
Diagonalization of quadratic forms and transforming of second order quasiliner PDE to the canonical form. Characteristics of second order quasiliner PDE. Using characteristics to bring second order quasiliner PDE to the canonical form in dimension two. 


10 
02/15/18 
The Laplace equation. Radial and fundamental solutions. Dirichlet and Neumann problems. 
02/20/18 




Hw 6: 

11 
02/20/18 
Laplace equation. Necessary condition for wellposedness of Newmann problem. Cauchy problem. Integration by parts. Green's formulas. 


12 
02/22/18 
Green's formulas. Newtonian potential. Singlelayer and doublelayer potentials. Poisson's equation. 
02/27/18 




Hw 7: 

13 
02/27/18 
Dirac delta function. Distributions (generalized functions). Stokes identity. 


14 
03/01/18 
Stokes identity. Green's function of Poisson's equation. Representation formula using Green's function. Method of reflection. Optional reading: Evans pages 2125, 3337 







15 
03/06/18 
Green's function of Poisson's equation in a ball and halfball. Poisson's kernels and Poisson's formulas. 
03/20/18 

16 
03/08/18 
Test 1 in class. No hw due. 







17 
03/13/18 
Spring break 


18 
03/15/18 
Spring break 





Hw 8: 

19 
03/20/18 
Poisson's kernels and Poisson's formulas. Meanvalue formulas. Maximum principle. 


20 
03/22/18 
Mollifiers.Regularity of harmonic functions and estimates on derivatives. Regularity of harmonic functions and estimates on derivatives. Liouville's theorem. 
03/27/18 




Hw 9: 

21 
03/27/18 
Analyticity of harmonic functions. Harnack's inequality. 


22 
03/29/18 
Fundamental solution of heat equation. Solution of initial value problem for homogeneous heat equation. Similarity transformation and derivation of the fundamental solution of heat equation. 
04/03/18 




Hw 10: 

23 
04/03/18 
Fourier transform. Plancherel's theorem. Properties of Fourier transform. 


24 
04/05/18 
Plancherel's theorem. Properties of Fourier transform. Properties of Fourier transform. Solution of the nonhomogeneous heat equation using Fourier transform. 
04/10/18 




Hw 11: 

25 
04/10/18 
Properties of Fourier transform. Solution of the nonhomogeneous heat equation using Fourier transform. 


26 
04/12/18 
Meanvalue property of heat equation. Maximum principle. Separation of variables for heat equation. 







27 
04/17/18 
Test 2 in class. No hw due. 


28 
04/19/18 
Separation of variables for heat equation. Uniquness for heat equation. 
04/24/18 




Hw 12: 

29 
04/24/18 
Uniquness for heat equation. Wave equation. D'Alemberdt formula. Solution of wave equation at halfline. 


30 
04/26/18 
Solution of wave equation on the finite interval. Separation of variables for wave equation on the finite interval. 
05/01/18 




Hw 13: 

31 
05/01/18 
Solution of wave equation in n=3. Kirchhoff's formula. Poisson's formula for n=2. 
05/03/18 

32 
05/03/18 
Nonhomogeneous problems for wave equation. Retarded potential. Boltzmann solution of nonlinear heat equation. 








05/03/18 
FINAL EXAM take home 

