Identifying the Type of PDE and Methods for Solving It

[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


please see attachment for PDE

 

[gato_]

Isn't it exactly the same problem you posted a couple of weeks ago here?
https://www.physicsforums.com/showthread.php?t=371542

 

[alokgautam]

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.

looking for you reply

Isn't it exactly the same problem you posted a couple of weeks ago here?
https://www.physicsforums.com/showthread.php?t=371542

 

[gato_]

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

 

[kosovtsov]

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".

 

[alokgautam]

Dear Gato,

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

ALok

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

 

[alokgautam]

Dear kosovtsov,

Thank you very much.

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

with best regards

ALOK

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".

 

[kosovtsov]

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

I do not say that.

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