## Amath 581 Homework Solution

Back to top | Weekly seminars | References

## Class schedule

*Note*: Click on the date to get

**Ibrahim's class notes**. He has also written some supplementray notes:

Wed 1/11 | Idea of distributions. Topological vector spaces. |

Fri 1/13 | Topological vector spaces. Hausdorff property. |

Wed 1/18 | Locally convex spaces. Seminorms. Fréchet spaces. |

Fri 1/20 | LF spaces. |

Wed 1/25 | Distributions. Radon measures. |

Fri 1/27 | Subspaces of distributions. Basic operations on distributions. |

Wed 2/1 | Sheaf structure of distributions. |

Fri 2/3 | Local structure of distributions. |

Wed 2/8 | Convolution. |

Fri 2/10 | Constant coefficient operators. Fundamental solutions. Hypoellipticity. |

Wed 2/15 | Schwartz theorem. Laurent expansion. Analytic hypoellipticity. |

Fri 2/17 | Fourier transform. Liouville's theorem. |

2/20–2/24 | Study break |

Mon 2/27 | Hörmander's characterization of hypoelliptic polynomials. |

Fri 3/2 | Problems in half-space. Cauchy problem. Petrowsky well-posedness. |

Wed 3/7 | Boundary value problems. Lopatinsky-Shapiro condition. |

Fri 3/9 | Strongly hyperbolic and p-parabolic systems. |

Wed 3/14 | Strong hyperbolicity. Inhomogeneous Cauchy problem. |

Fri 3/16 | Well-posedness of a general class of Cauchy problems. Parabolicity. |

Wed 3/21 | Semilinear evolution equations. |

Fri 3/23 | Multiplication in Sobolev spaces. Derivative nonlinearities. |

Wed 3/28 | Elliptic boundary value problems. Gårding inequality. |

Fri 3/30 | Gårding inequality proof. Dirichlet problem. Lax-Milgram lemma. |

Wed 4/4 | Friedrichs inequality. Rellich-Kondrashov compactness. |

Fri 4/6 | Good Friday |

Wed 4/11 | L2-regularity theory. |

Fri 4/13 | Spectral theory. Semigroups. |

## Assignments

## Final project

The final project consists of the student studying an advanced topic, typing up expository notes, and presenting it in class. Here are some ideas for the project:

## Weekly seminars

PDE questions from previous qualifying exams for download.Date | |||

1/16 | Baire's theorem and consequences | Rudin Ch2, Tao | Gantumur |

1/23 | Hahn-Banach theorem | Rudin §3.1-3.7, Tao | Spencer |

1/30, 2/6 | Closed range theorem | Rudin §4.1-4.15 | Ibrahim |

2/13, 2/29 | Fredholm operators | Rudin §4.16-4.25, McLean 2.14-2.17, Tao | Mario |

3/5 | Banach-Alaoglu theorem | Rudin §3.8-3.18, Tao | Andrew |

3/12, 3/19 | Spectral theorem | Rudin Ch13, Jaksic | Sébastien |

3/26 | Hille-Yosida theorem | Rudin §13.34-13.37 | Yang |

4/2 | Navier-Stokes equations | Tao, Clay, Lei-Lin | Gantumur |

## References

## Topics to be covered

## Instructor

Dr. Gantumur Tsogtgerel*Office*: Burnside Hall 1123. Phone: (514) 398-2510.

*Office hours*: Just drop by or make an appointment

## Online resources

PDE Lecture notes by Bruce Driver (UCSD)Xinwei Yu's page (Check the Intermediate PDE Math 527 pages)

John Hunter's teaching page at UC Davis (218B is PDE)

Textbook by Ralph Showalter on Hilbert space methods

Lecture notes by Georg Prokert on elliptic equations

## Prerequisites

MATH 355 or equivalent, MATH 580.## Catalog description

Systems of conservation laws and Riemann invariants. Cauchy- Kowalevskaya theorem, powers series solutions. Distributions and transforms. Weak solutions; introduction to Sobolev spaces with applications. Elliptic equations, Fredholm theory and spectra of elliptic operators. Second order parabolic and hyperbolic equations. Further advanced topics may be included.## Grading

Homework 40%, take-home midterm exam 20%, final project 40%.How to recognize and solve numerically practical problems which may arise in your research. We will solve some serious problems using the full power of MATLAB's built in functions and routines. This class is geared for those who need to get the basics in scientific computing. All major types of PDEs (parabolic, elliptic, and hyperbolic) will be considered in 1D, 2D and 3D in problems ranging from quantum mechanics to fluid flows.

NOTE: This course is a survey of computational methods. The focus is on the implemention of numerical schemes with significant aid from built-in MATLAB functionality such as FFTs, fast matrix solvers, etc. It is not a course in numerical analysis since our coverage of many technical issues is only cursory. A much more comprehensive and detailed treatment of some of the methods covered here is given in AMATH 584, 585, 586.

#### Figure

Dynamics of a repulsive Bose-Einstein condensate trapped in a 3-D lattice potential. The equation was solved using a filtered spectral method in space and 4th-order Runge-Kutta in time. By the end of this course, you should be able to perform this numerical simulation.## Textbook & Notes

There will be no text for this course. I will provide my notes on-line for you to download. I have several texts which will be on reserve at the library to look through the different sections.**Lecture Notes:**.pdf

**Reference Texts on Reserve:**

**1.**R. L. Burden and J. D. Faires,

*Numerical Analysis (Sixth Edition)*. Brooks/Cole, 1997.

**2.**L. N. Trefethen,

*Finite Difference and Spectral Methods.*(freely available).

**3.**L. N. Trefethen,

*Spectral Methods in MATLAB.*SIAM.

**4.**L. N. Trefethen and D. Bau,

*Numerical Linear Algebra.*SIAM.

**5.**J. C. Strikwerda,

*Finite Difference Schemes and Partial Differential Equations.*CRC Press.

**6.**D. R. Durran,

*Numerical Methods for Wave Equations in Geophysical Fluid Dynamics.*Springer.

## Syllabus

**(1) Solution Methods for Differential Equations: (2 weeks)**We will begin with ODE solvers applied to both initial and boundary value problems. Our application will be to finding the eigenstates of a quantum mechanical problem or of an optical waveguide.

**(a)**Initial value problems**(b)**Euler method, 2nd- and 4th-order Runge-Kutta, Adams-Bashford**(c)**Stability and time stepping issues**(d)**Boundary values problems: shooting/collocation/relaxation**(2) Finite Difference Schemes for Partial Differential Equations: (3 weeks)**We will introduce the idea of finite-differencing of differential operators. Our application will be to two problems: vibrating modes of a drum and the evolution of potential vorticity in an advection-diffusion problem of fluid mechanics.

**(a)**Collocation**(b)**Stability and CFL conditions**(c)**Time and space stepping routines**(d)**Tri-diagonal matrix operations**(3) Spectral Methods for Partial Differential Equations: (3 weeks)**Transform methods for PDEs will be introduced with special emphasis given to the Fast-Fourier Transform. We will revisit the potential vorticity in an advection-diffusion problem of fluid mechanics by using these spectral techniques.

**(a)**The Fast-Fourier transform (FFT)**(b)**Chebychev transforms**(c)**Time and space stepping routines**(d)**Numerical filtering algorithms**(4) Finite Element Schemes for Partial Differential Equations: (2 weeks)**For complicated computational domains, the use of a finite element scheme is compulsory. The steady-state flow of a fluid over various airfoils will be considered.

**(a)**Mesh generation**(b)**Advanced matrix operations**(c)**Boundary conditions

## Grading

Your*course grade*will be determined entirely from your homework. There will be no exams.

On or before the due date of each homework, the final homework must be uploaded to SCORELATOR for grading. SCORELATOR will give you up to five chances to get the results correct. The grade for that homework will be based upon the percentage you have exactly right (compared to my master key). The correctness of your codes will determine 100% of your grade.

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