General Solution for 2nd Order PDE: Is it Possible?

Click For Summary
SUMMARY

A general solution for second-order partial differential equations (PDEs) can be achieved through specific transformations. The discussion highlights the use of Maple software for calculating solutions, although the results may be complex. By applying suitable transformations, one can eliminate mixed and linear terms to convert the PDE into canonical forms such as parabolic, hyperbolic, or elliptic equations. This simplification allows for more straightforward solutions to the original PDE.

PREREQUISITES
  • Understanding of second-order partial differential equations (PDEs)
  • Familiarity with canonical forms of PDEs (parabolic, hyperbolic, elliptic)
  • Experience with Maple software for computational mathematics
  • Knowledge of transformation techniques in differential equations
NEXT STEPS
  • Research transformation techniques for simplifying second-order PDEs
  • Learn how to use Maple for solving complex PDEs
  • Study canonical forms of PDEs in detail
  • Explore specific examples of parabolic, hyperbolic, and elliptic equations
USEFUL FOR

Mathematicians, physicists, and engineers dealing with differential equations, as well as students and researchers looking to deepen their understanding of PDE solutions and transformations.

Jhenrique
Messages
676
Reaction score
4
Hellow everybody!

A simple question: exist a general formulation, a solution general, for a PDE of order 2 like:
## au_{xx}(x,y)+2bu_{xy}(x,y)+cu_{yy}(x,y)+du_x(x,y)+eu_y(x,y)+fu(x,y)=g(x,y) ##
?

The maple is able to calculate the solution, however, is a *monstrous* solution!
 
Last edited:
Physics news on Phys.org
With a suitable transformation you can change it to one of the canonical representations of pdes (parabolic,hyperbolic and elliptic) and solve it.
 
MathematicalPhysicist said:
With a suitable transformation you can change it to one of the canonical representations of pdes (parabolic,hyperbolic and elliptic) and solve it.

This transformation consists in eliminate the mixed and linear terms? Resulting an equation of kind:

## Au_{xx}(x,y)+Cu_{yy}(x,y)+Fu(x,y)=g(x,y) ##

?
 

Similar threads

Graduate Separation of variables possible in this problem?
  • · Replies 3 ·
Replies
3
Views
2K
Graduate Generic Solution of a Coupled System of 2nd Order PDEs
  • · Replies 2 ·
Replies
2
Views
2K
Graduate Reduction of 2nd order PDE to a first order equations system
  • · Replies 2 ·
Replies
2
Views
4K
Graduate About ellipticity and a proof that a system of PDEs is elliptic
  • · Replies 0 ·
Replies
0
Views
3K
Undergrad 2nd order ODE's, variation of parameters and the notorious constraint
  • · Replies 5 ·
Replies
5
Views
2K
Is this the correct general solution of the given PDE?
  • · Replies 12 ·
Replies
12
Views
2K
PDE and the separation of variables
  • · Replies 11 ·
Replies
11
Views
2K
Graduate 2nd Order PDE Using Similarity Method
  • · Replies 5 ·
Replies
5
Views
2K
Graduate Solving Linear Second Order PDE with Mixed Derivative Term?
  • · Replies 10 ·
Replies
10
Views
9K
Graduate Conceptual Solution of a First Order PDE
  • · Replies 3 ·
Replies
3
Views
2K