Math 463/513 Homework Assignments (Prof. Lushnikov),  Spring 2018

Date below is filled with dates for sections I (Prof. Lushnikov)  finished lecturing on. Evans means L.C. Evans textbook with recommended sections for extra readings.

Solutions of hw 1ญญญ

Solutions of hw 2ญญญ

Solutions of hw 3ญญญ

Solutions of hw 4ญญญ

Solutions of hw 5ญญญ

 

 

 

 

 

Date

Section

Homework

Due date

 

 

 

Hw 1:

  

1

01/16/18

Multiindex notation. Basic definitions.   Evans 1.1-1.3

   

  

2

01/18/18

Transport equation. Characteristics for first order PDE. Weak solutions. Evans 2.1 

 Problems 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.

Problems 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

Problems 3    

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.

Problems 4

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.

Problems 5 

02/20/18

 

 

 

Hw 6:    

 

11

02/20/18

Laplace equation. Necessary condition for well-posedness of Newmann problem.  Cauchy problem. Integration by parts. Green's formulas.   

 

 

12

02/22/18

Green's formulas. Newtonian potential. Single-layer and double-layer potentials. Poisson's equation.  

Problems 6 

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 21-25, 33-37 

 

 

 

 

  

 

 

15

03/06/18

Green's function of Poisson's equation in a ball and half-ball.  Poisson's kernels and Poisson's formulas.  

 

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.  Mean-value formulas. Maximum principle. 

 

 

20

03/22/18

Mollifiers.Regularity of harmonic functions and estimates on derivatives. 

 

 

 

 

Hw 9:       

 

21

03/27/18

 Regularity of harmonic functions and estimates on derivatives. Liouville's theorem. 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.  

 

 

 

 

Hw 10:   

 

23

04/03/18

Similarity transformation and derivation of the fundamental solutionof heat equation. Fourier transform. Plancherel's theorem. Properties of Fourier transform. 

 

 

24

04/05/18

Plancherel's theorem. Properties of Fourier transform.  

 

 

 

 

 Hw 11:   

 

25

04/10/18

Properties of Fourier transform. Solution of the nonhomogeneous heat equation using Fourier transform.  

 

 

26

04/12/18

Mean-value property of heat equation. Maximum principle. Separation of variables for heat equation.   

 

 

 

 

 

 

 

27

04/17/18

Separation of variables for heat equation. Uniquness for heat equation. 

 

28

04/19/18

Test 2 in class. No hw due.   

 

 

 

 

  

Hw 12:  

 

29

04/24/18

Uniquness for heat equation.   Wave equation. D'Alemberdt formula. Solution of wave equation at half-line. 

 

 

30

04/26/18

Sotuion of wave equation on the finite interval. Separation of variables for wave equation on the fintie interval.  

 

 

 

 

Hw 13:  

 

31

05/01/18

Solution of wave equation in n=3. Kirchhoff's formula. 

Poisson's formula for n=2.

 

32

05/03/18

Nonhomogeneous problems for wave equation. Retarded potential.  Boltzmann solution of nonlinear heat equation. 

 

 

 

  

  

  

  

 

TBA 

FINAL EXAM  TBA