# Solving a system of differential equation including non-linearity

I am trying to solve three equations which are of the form shown below numerically:

$\frac{\partial v}{\partial t}=f(v,y)$

$\frac{\partial}{\partial t}\left(vy\right)=f(v,y,z)$

$\frac{\partial}{\partial t}\left(vy^{2}\right)+\frac{\partial}{\partial t}\left(z\right)=f(v,y,z)$

where $v$, $y$ and $z$ are all functions of time ($t$) and space ($x$).

I know the initial values of the variables. I am desperate to know which numerical method to use to solve them. I was thinking of explicit Euler but I think that would not be possible due to the non-linearity cause by the $y^2$ term. I will be coding in Matlab. Would I need to linearize the equation and how?

Thanks very much.

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I will use $v'$, $y'$, $z'$ for the derivatives with respect to time, and $f$ for $f(v,y,z)$. We would like to reformulate this system to give us explicit expressions for these derivatives.

The first equation remains unchanged: $$v' = f.$$

Apply the product rule in the second equation and eliminate the $v'$ term using the first equation and rearrange terms: $$y' = \frac{f}{v} - \frac{fy}{v}.$$

Apply the product rule in the third equation, eliminate $v'$, $y'$ using the first two equations and rearrange: $$z' = -f -fy^2 +2f^2y.$$

These equations can now be solved by one of many methods. Explicit Euler is a poor choice; it has stability issues and large error. I suggest using one of MatLab's built in solvers. You can find the details of the implementation on their website.

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Thanks very much. Just to check something I have done previously, if I was to solve the first two equations (without any $z$), without applying the product rule, using explicit Euler, could I just solve the first equation for $v$ and the second equation for $vy$ and then divide $vy$ by the old $v$ to get the new $y$? – Hooman May 14 '12 at 8:34
Yeah, that sounds like it should work. It's equivalent to solving: $v' = f(v,Y/v)$ and $Y' = f(v,Y/v)$ with $Y = vy$. – in_wolframAlpha_we_trust May 14 '12 at 8:41
Thanks again. Another concern I had was that the third equation is non-linear and hence it would need to be linearized first somehow. I was told to look into methods such as predictor-corrector and so treat the $y^2$ in left hand side of the third equation as:$y^{n+1}y^{n}$ to find $y$ at an intermediate time step and then correct its result to find the new $y$. I am a bit confused by that really. – Hooman May 15 '12 at 8:05
The built in solvers I linked to can handle the non-linearity without a problem. If you prefer a less "black-box" approach to numerical analysis you can have a look at this page en.wikipedia.org/wiki/… which explains a very simple predictor-corrector method. There isn't really space to explain the entire field of numerical analysis in this comment box, so I recommend you pick up "A First Course in Numerical Analysis" by A. Iserles. – in_wolframAlpha_we_trust May 15 '12 at 11:47
Also, you've asked six questions and had four of them answered. I suggest that you accept some of those answers if they have helped you. To accept an answer click on the check mark beneath the upvote/downvote arrows next to the answer. – in_wolframAlpha_we_trust May 15 '12 at 11:50