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Canonical Forms of Second Order PDEs

Any second order partial differential equation (PDE) in two variables can be transformed into canonical form via a suitable coordinate transformation. This reduces the PDE to one of three types: hyperbolic, parabolic, or elliptic. The document demonstrates transforming a sample PDE into canonical form by applying a coordinate transformation and substituting the transformed partial derivatives. It is shown that the type of PDE (hyperbolic, parabolic, elliptic) is preserved under such a coordinate transformation.

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152 views1 page

Canonical Forms of Second Order PDEs

Any second order partial differential equation (PDE) in two variables can be transformed into canonical form via a suitable coordinate transformation. This reduces the PDE to one of three types: hyperbolic, parabolic, or elliptic. The document demonstrates transforming a sample PDE into canonical form by applying a coordinate transformation and substituting the transformed partial derivatives. It is shown that the type of PDE (hyperbolic, parabolic, elliptic) is preserved under such a coordinate transformation.

Uploaded by

shashank
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Canonical forms

Any second order PDE in two variables can be reduced to canonical


form of hyperbolic, parabolic and elliptic PDE’s under suitable trans-
formation. Let us convert the PDE Equation 2.1.1 using the coordi-
nate transformation _(x, y), _(x, y). Also take w(_, _) = u(x(_, _), y(_, _))
ux = w_ _x + w_ _x
uy = w_ _y + w_ _y
Similarly compute uxx, uyy and uxy in terms of w__, w__ and w__. After
substitution of these terms into the Equation 2.1.1 gives
a(_, _)w__ + 2 b(_, _)w__ + c(_, _)w__ = (w_,w_,w, _, _)
where
a(_, _) = A_x
2 + 2B _x _y + C_y
2
b(_, _) = A_x _x + B (_x _y + _y _x) + C _y _y
c(_, _) = A_x
2 + 2B _x_y + C _y
2
We need a justification that the form of the PDE remains invariant
even after the coordinate transformation i.e., hyperbolic remains as
hyperbolic, parabolic remains parabolic and vice versa. It can be
observed that

ab
bc
!
=

_x _y
_x _y
!
AB
BC
!
_x _x
_y _y
!
Taking the determinant on both sides gives
b2 − a c = (_x_y − _y_x)2  
B2 − AC
_
13

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