Tips for Solving a 1D Second Order ODE in an Open Channel Experiment

[Beniguette]

Hi all,

My maths are very rusty and I would need some advice. I have some experimental results obtained in an open channel and got depth-averaged velocity u(y) at different cross-sectional locations y. I tried different models but there is one I don't know how to tackle. The following one dimensional second order ODE describes the evolution of u(y):

$$a^{2}-b^{2}u^{2}+c^{2} \frac{du}{dy}+e^{2}y\frac{du}{dy}+f^{2}y^{2}\frac{d^{2}u}{dy^{2}}+r^{2}y\frac{d^{2}u}{dy^{2}}$$

I tried the dsolve function in Matlab, even putting some numbers in the constants without success. I don't really need an analytical solution (is it likely there would be any?). I have the experimental results and want to adjust part of the constant r to see how some parameters affect the results. Boundary conditions are known from experiments. Do I need to discretise that equation to get a solution? How should I do that?

I really welcome any advice or suggestion.

Regards,

Ben

 

[scorpion990]

Have you tried using the series method? Except for the u^2 term, the rest is pretty easy to treat in this manner. You might get some useful information that way.

 

[HallsofIvy]

No. Series solutions depend on the fact that you can add solutions of a linear equation to create a new solution. That isn't true for non-linear equations so series solutions don't work for non-linear equations.

 

[Beniguette]

Thank you for your interest scorpion990 and HallsolIvy. I have read someone mentioning he used a Newton iteration method to solve the same type of ODE in a similar context. He starts with an assumed profile of U(y) and an initial guess for the unknown parameter r. In each iteration the value of r at the previous time step is used as a known constant. He states that the profile evolves to a final form that is independant from the original profile. Does that make sense to anyone?

 

[matematikawan]

Since you don't mind numerical solution, try making the substitution v=du/dy. You then solve the system of differential equation.


$$\frac{du}{dy}=v$$ and

$$\frac{dv}{dy}=\frac{-a^2+b^2u^2-c^2v+c^2yv}{f^2y^2+r^2y}$$

Please refer to the thread: Laser rate equation(ODE) simulation problem
if you want to try solving with matlab.

 

[Beniguette]

I've just moved house and had no access to the Internet so I'm just getting your post. Thanks a lot for the suggestion matematikawan, that seems like an interesting read. I shall give it a go.