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I am having trouble trying to find example or figure out how to solve systems of PDEs of this form.

$$ u_{t} + au_{x} = f(u,v,x) $$

$$ v_{t} + bv_{x} = g(u,v,x) $$

For cases where $a \geq 0$ and $b \geq 0$ and constant with $f(u,v,x)$ and $g(u,v,x)$ being linear.

For example:

$$ u_{t} + au_{x} = k(u - bv),$$ $$ v_{t} + bv_{x} = k(bv - u),$$

where $k > 0$, $b \geq 0$ and $c \geq 0$. Ideally I would like to have $b = b(x)$. Im using these equations to test a finite volume PDE code I am developing as part of my PhD work.

My question is do these equations have analytical solutions? If so how do I find them?

Zack Taylor
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  • It may not be easy to solve the general case for arbitrary $f$, $g$ analytically, as well as the example. However, some cases can be solved in closed form using the method of characteristics, for instance the case $a=b=\text{cst}$ with linear $f$, $g$. – EditPiAf Dec 04 '20 at 23:44

1 Answers1

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No explicit solution formulas in general, but the theory does suggest a way to test your code. (The existence theory is on page 466 of Courant and Hilbert, Methods of Mathematical Physics Vol 2.) The first equation gives the derivative of $u$ in the direction $(a,1)$, that is, $$ \frac{d}{dt}u(x_0+at,t) = ku(x_0+at,t)-kbv(x_0+at,t) $$ and a similar result using the second equation in the direction $(b,1)$.

So integrate $$ ku(x_0+at,t)-kbv(x_0+at,t) $$ numerically, using the output of your code, from $t = 0$ to $t = T$ and you ought to get $$ u(x_0+aT,T)-u(x_0,0)$$ pretty nearly, as a test of your code that does not require doing any numerical derivatives.

Bob Terrell
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