# Is it parabolic or hyperbolic

1. Feb 3, 2010

### alokgautam

Hello dear friends,

I have this PDE

Can anyone help me in finding what type PDE is it. So that I can try to solve this .

As per my knowledge, I think this is second order and Nonlinear
Is my guess is correct?

Is it parabolic or hyperbolic, or is Elliptic and what method can I use to solve this Pde.
if it is possible please let me know about book for this problem

Last edited: Mar 31, 2010
2. Feb 4, 2010

3. Feb 4, 2010

### alokgautam

Re: Pde

Hi Gota,
thanks for mailing again

yes this is the same problem.
because still i do not know from where shall start to solve this equation.

even i am not able to know to find what type of PDE is it.

parabolic or hyperbolic or elliptic PDEs, because i do not know what is the value of b^2-a*c.

thats why i am not able to find the method for the soluton.

4. Feb 5, 2010

### gato_

Re: Pde

I don't know if the classification applies well for a non linear equation. The linear version of this sure is elliptic, and the tendency of this equation is to "smooth" the resulting surface, but no linear technique will be of help unless the curvatures involved are small. Try $$\Delta z(x,y)=z(x,y)/c$$. Beware with boundary conditions, they are the most tricky part

5. Feb 5, 2010

### kosovtsov

Re: Pde

I think that it is very difficult (if possible) to find the general solution to your PDE. Unfortunately there is not a universal method for this.

Some methods for finding particular solutions of nonlinear PDEs see in

A. D. Polyanin and V. F. Zaitsev, Handbook of Nonlinear Partial Differential Equations, Chapman & Hall/CRC Press, Boca Raton, 2004

These particular solutions, unluckily, rarely meet the physically based boundary conditions.

The most easy way to find particular solution is to suppose that

$$z(x,y) = F(ax+by)$$

which lead to some ODE for function F. This way gives the following particular solution to your PDE in implicit form

$$\pm\int_k^{z(x,y)}\frac{(a^2+b^2)(\xi^2+2C1)}{\sqrt{-(a^2+b^2)(\xi^4+4\xi^2C1+4C1^2-4c^2))}}\,d\xi -ax-by-C2 = 0$$

where a, b, k, C1, C2 are arbitrary constants. Incidentally, the integral above can be evaluated.

One remark - Nevertheless, for some PDEs from your field the general solutions can be found, see, e.g. http://eqworld.ipmnet.ru/eqarchive/view.php?id=271".

Last edited by a moderator: Apr 24, 2017
6. Feb 8, 2010

### alokgautam

Re: Pde

Dear Gato,

Thank you once again.
i will try to apply finite element method to solve this
with best regards

ALok

7. Feb 8, 2010

### alokgautam

Re: Pde

Dear kosovtsov,

Thank you very much.

so there is no numerical method is available to solve this PDE.

with best regards

ALOK

Last edited by a moderator: Apr 24, 2017
8. Feb 11, 2010

### kosovtsov

Re: Pde

I do not say that.

It is clear that you can turn to numerical solution of the PDE if you will.