a
Reduce the equation Uxx - x´uyy = 0 to a
canonical form
Answers
MA 201: Second Order Linear PDE
Canonical Transformation
Lecture - 6
MA201(2016):PDE
• A second order PDE with two independent variables x and y is
given by
F(x, y, u, ux , uy , uxy , uxx , uyy ) = 0. (1)
What is the linear form?
• The unknown function u(x, y) satisfies an equation:
Auxx + Buxy + Cuyy + Dux + Euy + Fu + G = 0. (2)
Facts:
• The expression Lu ≡ Auxx + Buxy + Cuyy is called the Principal part
of the equation.
• Classification of such PDEs is based on this principal part.
MA201(2016):PDE
• At a point (x, y), the above equation is said to be
Hyperbolic if B
2
(x, y) − 4A(x, y)C(x, y) > 0
Parabolic if B
2
(x, y) − 4A(x, y)C(x, y) = 0
Elliptic if B
2
(x, y) − 4A(x, y)C(x, y) < 0
• Each category relates to specific problems
1. Laplace’s Equation
uxx + uyy = 0.
2. Wave Equation
utt − uxx = 0.
3. Heat Equation
ut = uxx.
MA201(2016):PDE
Methods and Techniques for Solving PDEs
• Change of coordinates: A PDE can be changed to an ODE or to an
easier PDE by changing the coordinates of the problem.
• Separation of variables: A PDE in n independent variables is reduced
to n ODEs.
• Integral transforms: A PDE in n independent variables is reduced to
one in (n − 1) independent variables. Hence, a PDE in two variables
could be changed to an ODE.
• Numerical Methods
MA201(2016):PDE
Change of coordinates: Canonical Transformations
• This can be achieved by introducing new coordinates
ξ = ξ(x, y), η = η(x, y).
• Compute the partial derivatives
MA201(2016):PDE
• Substitute these values into the original equation to obtain a new
form
where the new coefficients are as follows
Whether the form of PDE remains invariant even after
coordinate transformation?
MA201(2016):PDE
• It can be observed that
2A˜ B˜
B˜ 2C˜
=
ξx ξy
ηx ηy
2A B
B 2C
ξx ξy
ηx ηy
t
(3)
• Taking the determinant on both sides gives
B˜ 2 − 4A˜C˜ = (ξxηy − ξy ηx )
2
(B
2 − 4AC) = J
2
(B
2 − 4AC) (4)
• J is the Jacobian of the transformation and we select the
transformation (ξ, η) such that J 6= 0.
• Transformation ξ = ξ(x, y) and η = η(x, y) are called canonical
transformation or characteristics and the reduced form of the PDE is
called canonical form.
MA201(2016):PDE
Canonical Transformations: Hyperbolic PDE
• Hyperbolic PDE’s: B
2 − 4AC > 0
Natural Choice: A˜ = 0 and C˜ = 0
Leads to Algebraic Equations:
Aλ
2 + Bλ + C = 0, λ1 =
ξx
ξy
, λ2 =
ηx
ηy
.
Observations:
ξ(x, y) = c For ξ solve dy
dx = −λ1(x, y)
⇒ ξx + ξy
dy
dx = 0,
• For η solve dy
dx = −λ2(x, y).
• Canonical form for hyperbolic equation may be read as
uξη = φ(ξ, η, u, uξ, uη).
MA201(2016):PDE
Hyperbolic Equation with Constant Coefficients
• Assume that either A 6= 0 or C 6= 0.
• What can you say when A and C vanish simultaneously?
• For A 6= 0, canonical transformation (ξ, η) takes the form
ξ = y + λ1x, η = y + λ2x, λ1,2 =
−B ±
√
B2 − 4AC
2A
.
For A 6= 0 and C = 0, we have
ξ = y, η = y −
B
A
x.