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Modify the derivation of the conservation laws to justify that the 1-D diffusion with source/sink admits the following form $u_t(x,t)=k u_{xx}(x,t)+f(x,t)$. $(u(x,t)$ represents the concentration of certain particles on R at spatial point x and time t. The flux $F(x,t)$ is given by the Fick's law, $F=-k u_x$. In addition, $f(x,t)$ is a source/ sink that shows the amount of particles added into/ taken away from the real line in the unit amount of time.

I know that to derive conservation laws, let M(t) be the total amount of particles in [a,b] at time t, then $M(t)= \int_a^b u(x,t) dx$

On the other hand, $\frac{dM}{dt}=flux in-flux out=F(a,t)-F(b,t)$

Combine these two, we have \begin{align} \\ \int_a^b u(x,t) dx=F(a,t)-F(b,t) \\ =-\int_a^b F_x(x,t) dx \\ \int_a^b u_t(x,t)+F_x(x,t) dx=0 \\ = u_t(x,t)+F_x(x,t)=0 \end{align}

I am not sure how to modify that for the question. A hint is appreciated.

spruce
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1 Answers1

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Here, the balance equation over $[a, b]$ reads $$ \frac{\text d M}{\text d t} = F(a,t) - F(b,t) + S(t) $$ where the integrated variable is $M = \int_{a}^{b} u(x, t)\, \text d x$, the flux is given by Fick's law $F(x, t) = -k u_x(x, t)$, and the source is $S(t) = \int_a^b f(x, t)\, \text d x$. Therefore, $$ \int_a^b u_t(x, t)\, \text d x = k\int_a^b u_{xx}(x, t)\, \text d x + \int_a^b f(x, t)\, \text d x $$ i.e. $u_t = k u_{xx} + f$.

EditPiAf
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