PDE Solving Continuity Equation

[shen07]

Hi, I am trying to find the exact solution of the Continuity Equation. Any Idea how can i start solving it, i need it for some calculation in Image Processing.

$$\pd{C}{t}+\pd{UC}{x}+\pd{VC}{y}=0$$

Where $U$ and $V$ is velocity in $X$ and $Y$ direction. The initial condition is as follows.$$C(x,y,0)=\left\{ \begin{array}{cc}
1\enspace \text{if}\;\sqrt{(x-0.5)^2+(y-0.5)^2}\le0.2\\

0\enspace\enspace \text{otherwise} \end{array} \right.
$$

 

[chisigma]

Hi, I am trying to find the exact solution of the Continuity Equation. Any Idea how can i start solving it, i need it for some calculation in Image Processing.

$$\pd{C}{t}+\pd{UC}{x}+\pd{VC}{y}=0$$

Where $U$ and $V$ is velocity in $X$ and $Y$ direction. The initial condition is as follows.$$C(x,y,0)=\left\{ \begin{array}{cc}
1\enspace \text{if}\;\sqrt{(x-0.5)^2+(y-0.5)^2}\le0.2\\

0\enspace\enspace \text{otherwise} \end{array} \right.
$$

If You assume that u and v are constants, then the PDE becomes...

$\displaystyle c_{t} + u\ c_{x} + v\ c_{y} = 0\ (1)$

... which is linear... in this case the solution is of the type...

$\displaystyle c(x,y,t) = c(x - u\ t, y - v\ t)\ (2)$

... and in this particular case You have a circular spot of radious .04 centered at t=0 in (.5,.5) and traveling with speed u in the x direction and with speed v in the y direction...

Kind regards

$\chi$ $\sigma$