Ngô Quốc Anh

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?

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