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APM346-2015S => APM346--Home Assignments => HA3 => Topic started by: Victor Ivrii on February 05, 2015, 07:27:12 PM

Title: HA3 problem 2
Post by: Victor Ivrii on February 05, 2015, 07:27:12 PM

Consider heat equation with a convection term
\begin{equation}
u_t+\underbracket{v u_x}_{\text{convection term}} =ku_{xx}.
\label{eq-HA3.4}
\end{equation}

a.  Using change of variables $u(x,t)=U(x-vt,t)$ reduce it to ordinary heat equation and using (1)-(2) of http://www.math.toronto.edu/courses/apm346h1/20151/HA3.html (http://www.math.toronto.edu/courses/apm346h1/20151/HA3.html) for a latter write a formula for solution $u (x,t)$.
b.  Can we use the method of continuation to solve IBVP with Dirichlet or Neumann boundary condition at $x>0$ for (\ref{eq-HA3.4}) on $\{x>0,t>0\}$? Justify your answer.

Title: Re: HA3 problem 2
Post by: Yang Liu on February 06, 2015, 02:35:06 AM

Attached ;D

Title: Re: HA3 problem 2
Post by: Victor Ivrii on February 08, 2015, 05:47:12 AM

a Idea correct but it will be $(x-vt)$ rather than $(x+vt)$