# 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?