Identifying the Type of PDE and Methods for Solving It

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Discussion Overview

The discussion revolves around identifying the type of a nonlinear partial differential equation (PDE) and exploring potential methods for solving it. Participants express uncertainty regarding the classification of the PDE as parabolic, hyperbolic, or elliptic, and discuss various approaches to find solutions, including numerical methods and specific techniques for nonlinear equations.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant suggests that the PDE is second order and nonlinear, but is unsure about its classification as parabolic, hyperbolic, or elliptic due to the unknown value of \(b^2 - ac\).
  • Another participant questions whether classification applies well to nonlinear equations, noting that the linear version is elliptic and that boundary conditions are particularly challenging.
  • A different participant mentions the difficulty of finding a general solution for the PDE and states that there is no universal method for solving nonlinear PDEs.
  • One proposed method for finding particular solutions involves assuming a specific form for \(z(x,y)\) that leads to an ordinary differential equation (ODE).
  • Another participant expresses intent to apply the finite element method to solve the PDE.
  • There is a discussion about the availability of numerical methods, with one participant asserting that numerical solutions can be pursued, while another expresses doubt about their applicability.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the classification of the PDE or the availability of numerical methods. Multiple competing views remain regarding the methods for solving the PDE and the challenges associated with nonlinear equations.

Contextual Notes

Participants highlight limitations related to the classification of nonlinear PDEs and the complexity of boundary conditions, which remain unresolved in the discussion.

alokgautam
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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
 
Last edited:
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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



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


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
 


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:


Dear Gato,

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

ALok


gato_ said:
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
 


Dear kosovtsov,

Thank you very much.

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

with best regards

ALOK


kosovtsov said:
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:


alokgautam said:
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.
 

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