Canonical Forms Or Normal Forms Math Worksheet
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2
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MODULE 3: SECOND-ORDER PARTIAL DIFFERENTIAL EQUATIONS
Lecture 2
Canonical Forms or Normal Forms
By a suitable change of the independent variables we shall show that any equation of the
form
Au
+ Bu
+ Cu
+ Du
+ Eu
+ F u + G = 0,
(1)
xx
xy
yy
x
y
where A, B, C, D, E, F and G are functions of the variables x and y, can be reduced to a
canonical form or normal form. The transformed equation assumes a simple form so that
the subsequent analysis of solving the equation will be become easy.
Consider the transformation of the indpendent variables from (x, y) to (ξ, η) given by
ξ = ξ(x, y), η = η(x, y).
(2)
Here, the functions ξ and η are continuously differentiable and the Jacobian
ξ
ξ
∂(ξ, η)
x
y
) ̸ = 0
J =
=
= (ξ
η
ξ
η
(3)
x
y
y
x
∂(x, y)
η
η
x
y
in the domain where (1) holds.
Using chain rule, we notice that
u
= u
ξ
+ u
η
x
ξ
x
η
x
u
= u
ξ
+ u
η
y
ξ
y
η
y
2
2
u
= u
ξ
+ 2u
ξ
η
+ u
η
+ u
ξ
+ u
η
xx
ξξ
ξη
x
x
ηη
ξ
xx
η
xx
x
x
u
= u
ξ
ξ
+ u
(ξ
η
+ ξ
η
) + u
η
η
+ u
ξ
+ u
η
xy
ξξ
x
y
ξη
x
y
y
x
ηη
x
y
ξ
xy
η
xy
2
2
u
= u
ξ
+ 2u
ξ
η
+ u
η
+ u
ξ
+ u
η
yy
y
y
ηη
yy
η
yy
ξξ
ξη
ξ
y
y
Substituting these expression into (1), we obtain
¯ A(ξ
+ ¯ B(ξ
+ ¯ C(η
, ξ
)u
, ξ
; η
, η
)u
, η
)u
= F (ξ, η, u(ξ, η), u
(ξ, η), u
(ξ, η)), (4)
x
y
ξξ
x
y
x
y
ξη
x
y
ηη
ξ
η
where
¯ A(ξ
2
2
, ξ
) = Aξ
+ Bξ
ξ
+ Cξ
x
y
x
y
x
y
¯ B(ξ
, ξ
; η
, η
) = 2Aξ
η
+ B(ξ
η
+ ξ
η
) + 2Cξ
η
x
y
x
y
x
x
x
y
y
x
y
y
¯ C(η
2
2
, η
) = Aη
+ Bη
η
+ Cη
.
x
y
x
y
x
y
An easy calculation shows that
¯ B
4 ¯ A ¯ C = (ξ
2
2
2
η
ξ
η
)
(B
4AC).
(5)
x
y
y
x
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