Ngô Quốc Anh

April 23, 2009

Đề thi OLP toàn quốc 2009

Filed under: Các Bài Tập Nhỏ, Linh Tinh — Ngô Quốc Anh @ 22:05

Kỳ thi được tổ chức tại Đại học Quảng Bình từ 17/4 đến 19/4. Sau đây xin giới thiệu đề thi môn Đại số và Giải tích năm nay:

April 13, 2009

Green’s function for viscous Burgers type equation

Filed under: Các Bài Tập Nhỏ, Linh Tinh, Nghiên Cứu Khoa Học, PDEs — Tags: — Ngô Quốc Anh @ 12:08

We now find Green function for the following equation u_t+au_x-u_{xx}=0. In this topic, I will not consider any boundary value condition. My approach is the Fourier transform. From the equation u_t+au_x-u_{xx}=0 we apply Fourier transform to the both sides with respect to x. We then obtain

\displaystyle {\widehat u_t}\left( {\xi ,t} \right) + a\left( {i\xi } \right)\widehat u\left( {\xi ,t} \right) - {\left( {i\xi } \right)^2}\widehat u\left( {\xi ,t} \right) = 0

or equivalently,

\displaystyle {\widehat u_t}\left( {\xi ,t} \right) = - \left( {ai\xi + {\xi ^2}} \right)\widehat u\left( {\xi ,t} \right).

The above identity can be regarded as a first order ODE with respect to t with constant coefficients, solving this ODE we obtain

\displaystyle\widehat u\left( {\xi ,t} \right) = {e^{ - \left( {ai\xi + {\xi ^2}} \right)t}}\widehat u\left( {\xi ,0} \right).

Taking inverse Fourier transform gives

\displaystyle u\left( {x,t} \right) = \frac{1}{2\pi} \int_\mathbb{R} {{e^{ix \cdot \xi }}{e^{ - \left( {ai\xi + {\xi ^2}} \right)t}}\widehat u\left( {\xi ,0} \right)d\xi }.

Note that

\displaystyle \widehat u\left( {\xi ,0} \right) = \int_\mathbb{R} {{e^{ - iy \cdot \xi }}u\left( {y,0} \right)dy}.

Therefore,

\displaystyle \begin{gathered} u\left( {x,t} \right) = \frac{1} {{2\pi }}\int_\mathbb{R} {{e^{ix\cdot\xi }}{e^{ - \left( {ai\xi + {\xi ^2}} \right)t}}\left( {\int_\mathbb{R} {{e^{ - iy\cdot\xi }}u\left( {y,0} \right)dy} } \right)d\xi } \hfill \\ \quad \qquad = \int_\mathbb{R} {\left( {\frac{1} {{2\pi }}\int_\mathbb{R} {{e^{ix\cdot\xi }}{e^{ - \left( {ai\xi + {\xi ^2}} \right)t}}{e^{ - iy\cdot\xi }}d\xi } } \right)u\left( {y,0} \right)dy} n \hfill \\ \quad \qquad = \int_\mathbb{R} {\left( {\frac{1} {{2\pi }}\int_\mathbb{R} {{e^{i\left( {x - y} \right)\cdot\xi }}{e^{ - \left( {ai\xi + {\xi ^2}} \right)t}}d\xi } } \right)u\left( {y,0} \right)dy} . \hfill \\ \end{gathered}

Thus, the Green function for equation u_t+au_x-u_{xx}=0 is the following

\displaystyle\mathbb{G}(x,t; y, \tau) = \frac{1} {{2\pi }}\int_\mathbb{R} {{e^{i (x-y) \cdot \xi }}{e^{ - \left( {ai\xi + {\xi ^2}} \right)(t-\tau)}}d\xi }.

Clearly, the above Green function has the following properties

  • \displaystyle\mathbb{G}(x,t; y, t)=\delta(x-y).
  • \displaystyle{\mathbb{G}_t}(x,t; y, \tau) + a{\mathbb{G}_x}(x,t; y, \tau) - {\mathbb{G}_{xx}}(x,t; y, \tau) = 0.

I now propose the following question: if we impose a boundary condition, then how can we find an appropriate Green function for such problem?

April 6, 2009

The QE – Harvard University, Department of Mathematics

Filed under: Đề Thi — Ngô Quốc Anh @ 9:45

The qualifying exam in mathematics is designed to measure the breadth of a student’s knowledge in mathematics. The exam may identify those areas in which a student’s knowledge is weak. Passing the exam is an indication that a student is ready to begin more specialized study leading to research work.

The exam is given at the very start of each semester. A student may take the exam as often as (s)he likes. There is absolutely no stigma attached to `failing’ the exam. `Failing’ it may well provide more useful information than `passing’ it. `Passing’ the exam early is mainly an indication that a student has been an undergraduate at a university with a broad undergraduate program in mathematics. It is not a good predictor of the quality of the eventual PhD thesis.

Students are strongly encouraged to first take the exam no later than their second semester. Before passing the qualifying exam, students should take three beginning 200 level (or 100 level) math courses each semester. In a semester in which they are teaching they need only take two such courses. After passing the qualifying exam students are usually excused from grades in any math courses they take. Students are expected to pass the qualifying exam by the end of their second year.

The exam consists of three three hour papers on three consecutive days. Each paper typically has 6 questions covering a broad range of mathematics. The questions aim to test your ability to solve concrete problems by identifying and applying important theorems. They should not require great ingenuity. In any given year the exam may not cover every topic on the syllabus, but it should cover a broadly representative set of quals/topics and over time all quals/topics should be examined.

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