Classical computational methods

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Classical computational

methods Daniele Boffi

Examples Plan of the Course PDE’s Classification

Classical computational methods

Daniele Boffi

Dipartimento di Matematica, Universit`a di Pavia http://www-dimat.unipv.it/boffi

Complexity and its Interdisciplinary Applications

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Some examples

A second order ODE

−u

⁰⁰

(x ) = f (x )

Solution can be explicitly determined (closed form solution)

u

⁰

(x ) = u

⁰

(x

₀

) + Z

x

x (0)

u

⁰⁰

(t) dt = u

⁰

(x

₀

) − Z

x

x (0)

f (t) dt

u(x ) = u(x

₀

) + Z

x

x (0)

u

⁰

(t) dt In general

u(x ) = α + βx − Z Z

f (t) dt

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Some examples

A second order ODE

−u

⁰⁰

(x ) = f (x )

Solution can be explicitly determined (closed form solution)

u

⁰

(x ) = u

⁰

(x

₀

) + Z

x

x (0)

u

⁰⁰

(t) dt = u

⁰

(x

₀

) − Z

x

x (0)

f (t) dt

u(x ) = u(x

₀

) + Z

x

x (0)

u

⁰

(t) dt

In general

u(x ) = α + βx − Z Z

f (t) dt

(4)

Some examples

A second order ODE

−u

⁰⁰

(x ) = f (x )

Solution can be explicitly determined (closed form solution)

u

⁰

(x ) = u

⁰

(x

₀

) + Z

x

x (0)

u

⁰⁰

(t) dt = u

⁰

(x

₀

) − Z

x

x (0)

f (t) dt

u(x ) = u(x

₀

) + Z

x

x (0)

u

⁰

(t) dt In general

u(x ) = α + βx − Z Z

f (t) dt

(5)

Some examples (cont’ed)

One dimensional convection equation (PDE)

∂u

∂t (x , t) − ∂u

∂x (x , t) = 0

Closed form solution u(x , t) = w (x + t)

(x,t) t

x

(x−T,t+T)

(6)

Some examples (cont’ed)

One dimensional convection equation (PDE)

∂u

∂t (x , t) − ∂u

∂x (x , t) = 0

Closed form solution u(x , t) = w (x + t)

(x,t) t

x

(x−T,t+T)

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Some examples (cont’ed)

One dimensional convection equation (PDE)

∂u

∂t (x , t) − ∂u

∂x (x , t) = 0

Closed form solution u(x , t) = w (x + t)

(x,t) t

x

(x−T,t+T)

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Some examples (cont’ed)

One dimensional wave equation

∂

²

u

∂t

²

(x , t) − c

²

∂

²

u

∂x

²

(x , t) = 0

Closed form solution

u(x , t) = w

1

(x + ct) + w

2

(x − ct)

Proof (x,t)

t

x

(x−cT,t+T) (x+cT,t+T)

(9)

Some examples (cont’ed)

One dimensional wave equation

∂

²

u

∂t

²

(x , t) − c

²

∂

²

u

∂x

²

(x , t) = 0

Closed form solution

u(x , t) = w

1

(x + ct) + w

2

(x − ct)

Proof (x,t)

t

x

(x−cT,t+T) (x+cT,t+T)

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Some examples (cont’ed)

One dimensional wave equation

∂

²

u

∂t

²

(x , t) − c

²

∂

²

u

∂x

²

(x , t) = 0

Closed form solution

u(x , t) = w

1

(x + ct) + w

2

(x − ct)

Proof (x,t)

t

x

(x−cT,t+T) (x+cT,t+T)

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Some examples (cont’ed)

One dimensional heat equation

∂u

∂t (x , t) − ∂

²

u

∂x

²

(x , t) = 0, x ∈ (0, 1), t > 0

Closed form solution

u(x , t) =

∞

X

j =1

u

0,j

e

^−(jπ)²^t

sin(j πx ),

where u

0

(x ) = u(x , 0) is the initial datum and

u

_0,j

= 2 Z

1

0

u

₀

(x ) sin(j πx ) dx

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Some examples (cont’ed)

One dimensional heat equation

∂u

∂t (x , t) − ∂

²

u

∂x

²

(x , t) = 0, x ∈ (0, 1), t > 0 Closed form solution

u(x , t) =

∞

X

j =1

u

0,j

e

^−(jπ)²^t

sin(j πx ),

where u

0

(x ) = u(x , 0) is the initial datum and

u

_0,j

= 2 Z

1

0

u

₀

(x ) sin(j πx ) dx

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Some examples (cont’ed)

Convection equation

∂u

∂t + div(~ βu) = 0 First order linear equation.

N.B.: divergence operator div ~ v =

d

P

i =1

∂v_i

∂xi

This equation states the mass conservation of a body

occupying a region Ω ∈ R

^d

, with density u and velocity β ~

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Some examples (cont’ed)

Convection equation

∂u

∂t + div(~ βu) = 0 First order linear equation.

N.B.: divergence operator div ~ v =

d

P

i =1

∂v_i

∂xi

This equation states the mass conservation of a body

occupying a region Ω ∈ R

^d

, with density u and velocity β ~

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Some examples (cont’ed)

Laplace/Beltrami/Poisson equation

−∆u = f Second order linear equation.

N.B.: Laplace operator ∆v =

d

P

i =1

∂²v

∂x_i²

This equation states the diffusion of a homogeneous and

isotropic fluid occupying a region Ω ∈ R

^d

, as well as the

vertical displacement of an elastic membrane. Fundamental

equation for several models.

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Some examples (cont’ed)

Laplace/Beltrami/Poisson equation

−∆u = f Second order linear equation.

N.B.: Laplace operator ∆v =

d

P

i =1

∂²v

∂x_i²

This equation states the diffusion of a homogeneous and

isotropic fluid occupying a region Ω ∈ R

^d

, as well as the

vertical displacement of an elastic membrane. Fundamental

equation for several models.

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Some examples (cont’ed)

Heat equation

∂u

∂t − ∆u = f Second order linear equation

Wave equation

∂

²

u

∂t

²

− ∆u = 0

Second order linear equation

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Some examples (cont’ed)

Heat equation

∂u

∂t − ∆u = f Second order linear equation

Wave equation

∂

²

u

∂t

²

− ∆u = 0

Second order linear equation

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Some examples (cont’ed)

Burgers equation (d = 1)

∂u

∂t + u ∂u

∂x = 0 First order quasi-linear equation

Viscous Burgers equation (d = 1)

∂u

∂t + u ∂u

∂x = ε ∂

²

u

∂x

²

, ε > 0

Second order semi-linear equation

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Some examples (cont’ed)

Burgers equation (d = 1)

∂u

∂t + u ∂u

∂x = 0 First order quasi-linear equation

Viscous Burgers equation (d = 1)

∂u

∂t + u ∂u

∂x = ε ∂

²

u

∂x

²

, ε > 0

Second order semi-linear equation

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Plan of the Course

• Classification of Partial Differential Equations (PDE)

• Elliptic PDE’s

• Finite differences

• Finite elements

• Where the theory is elegant and complete. . .

•

From elliptic to hyperbolic PDE’s

• Convection-diffusion equation

• Finite differences, upwind

• Integrating along the characteristics

• Stabilization of finite elements

•

Parabolic equation

• Heat equation: space semidiscretization and evolution in time

• Stability of θ-method

•

Examples with Matlab

•

Conclusions and comments

•

Questions and answers

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Plan of the Course

•

Classification of Partial Differential Equations (PDE)

• Finite elements

•

From elliptic to hyperbolic PDE’s

•

Parabolic equation

•

Examples with Matlab

•

Conclusions and comments

•

Questions and answers

(23)

Plan of the Course

•

Classification of Partial Differential Equations (PDE)

• Finite elements

•

From elliptic to hyperbolic PDE’s

•

Parabolic equation

•

Examples with Matlab

•

Conclusions and comments

•

Questions and answers

(24)

Plan of the Course

•

Classification of Partial Differential Equations (PDE)

•

Elliptic PDE’s

• Finite elements

•

From elliptic to hyperbolic PDE’s

•

Parabolic equation

•

Examples with Matlab

•

Conclusions and comments

•

Questions and answers

(25)

Plan of the Course

•

Classification of Partial Differential Equations (PDE)

•

Elliptic PDE’s

• Finite elements

•

From elliptic to hyperbolic PDE’s

•

Parabolic equation

•

Examples with Matlab

•

Conclusions and comments

•

Questions and answers

(26)

Plan of the Course

•

Classification of Partial Differential Equations (PDE)

•

Elliptic PDE’s

• Finite elements

•

From elliptic to hyperbolic PDE’s

•

Parabolic equation

•

Examples with Matlab

•

Conclusions and comments

•

Questions and answers

(27)

Plan of the Course

•

Classification of Partial Differential Equations (PDE)

•

Elliptic PDE’s

• Finite elements

•

From elliptic to hyperbolic PDE’s

•

Parabolic equation

•

Examples with Matlab

•

Conclusions and comments

•

Questions and answers

(28)

Plan of the Course

•

Classification of Partial Differential Equations (PDE)

•

Elliptic PDE’s

• Finite elements

• From elliptic to hyperbolic PDE’s

•

Parabolic equation

•

Examples with Matlab

•

Conclusions and comments

•

Questions and answers

(29)

Plan of the Course

•

Classification of Partial Differential Equations (PDE)

•

Elliptic PDE’s

• Finite elements

•

Parabolic equation

•

Examples with Matlab

•

Conclusions and comments

•

Questions and answers

(30)

Plan of the Course

•

Classification of Partial Differential Equations (PDE)

•

Elliptic PDE’s

• Finite elements

•

Parabolic equation

•

Examples with Matlab

•

Conclusions and comments

•

Questions and answers

(31)

Plan of the Course

•

Classification of Partial Differential Equations (PDE)

•

Elliptic PDE’s

• Finite elements

•

Parabolic equation

•

Examples with Matlab

•

Conclusions and comments

•

Questions and answers

(32)

Plan of the Course

•

Classification of Partial Differential Equations (PDE)

•

Elliptic PDE’s

• Finite elements

•

From elliptic to hyperbolic PDE’s

•

Parabolic equation

•

Examples with Matlab

•

Conclusions and comments

•

Questions and answers

(33)

Plan of the Course

•

Classification of Partial Differential Equations (PDE)

•

Elliptic PDE’s

• Finite elements

•

From elliptic to hyperbolic PDE’s

•

Parabolic equation

•

Examples with Matlab

•

Conclusions and comments

•

Questions and answers

(34)

Plan of the Course

•

Classification of Partial Differential Equations (PDE)

•

Elliptic PDE’s

• Finite elements

•

From elliptic to hyperbolic PDE’s

•

Parabolic equation

•

Examples with Matlab

•

Conclusions and comments

•

Questions and answers

(35)

Plan of the Course

•

Classification of Partial Differential Equations (PDE)

•

Elliptic PDE’s

• Finite elements

•

From elliptic to hyperbolic PDE’s

•

Parabolic equation

•

Examples with Matlab

•

Conclusions and comments

•

Questions and answers

(36)

Plan of the Course

•

Classification of Partial Differential Equations (PDE)

•

Elliptic PDE’s

• Finite elements

•

From elliptic to hyperbolic PDE’s

•

Parabolic equation

•

Examples with Matlab

•

Conclusions and comments

•

Questions and answers

(37)

Plan of the Course

•

Classification of Partial Differential Equations (PDE)

•

Elliptic PDE’s

• Finite elements

•

From elliptic to hyperbolic PDE’s

•

Parabolic equation

•

Examples with Matlab

•

Conclusions and comments

•

Questions and answers

(38)

Plan of the Course

•

Classification of Partial Differential Equations (PDE)

•

Elliptic PDE’s

• Finite elements

•

From elliptic to hyperbolic PDE’s

• Parabolic equation

•

Examples with Matlab

•

Conclusions and comments

•

Questions and answers

(39)

Plan of the Course

•

Classification of Partial Differential Equations (PDE)

•

Elliptic PDE’s

• Finite elements

•

From elliptic to hyperbolic PDE’s

•

Examples with Matlab

•

Conclusions and comments

•

Questions and answers

(40)

Plan of the Course

•

Classification of Partial Differential Equations (PDE)

•

Elliptic PDE’s

• Finite elements

•

From elliptic to hyperbolic PDE’s

•

Examples with Matlab

•

Conclusions and comments

•

Questions and answers

(41)

Plan of the Course

•

Classification of Partial Differential Equations (PDE)

•

Elliptic PDE’s

• Finite elements

•

From elliptic to hyperbolic PDE’s

• Stability ofθ-method

•

Examples with Matlab

•

Conclusions and comments

•

Questions and answers

(42)

Plan of the Course

•

Classification of Partial Differential Equations (PDE)

•

Elliptic PDE’s

• Finite elements

•

From elliptic to hyperbolic PDE’s

•

Parabolic equation

•

Examples with Matlab

•

Conclusions and comments

•

Questions and answers

(43)

Plan of the Course

•

Classification of Partial Differential Equations (PDE)

•

Elliptic PDE’s

• Finite elements

•

From elliptic to hyperbolic PDE’s

•

Parabolic equation

•

Examples with Matlab

•

Conclusions and comments

•

Questions and answers

(44)

Plan of the Course

•

Classification of Partial Differential Equations (PDE)

•

Elliptic PDE’s

• Finite elements

•

From elliptic to hyperbolic PDE’s

•

Parabolic equation

• Examples with Matlab

•

Conclusions and comments

•

Questions and answers

(45)

Plan of the Course

•

Classification of Partial Differential Equations (PDE)

•

Elliptic PDE’s

• Finite elements

•

From elliptic to hyperbolic PDE’s

•

Parabolic equation

•

Conclusions and comments

•

Questions and answers

(46)

Plan of the Course

•

Classification of Partial Differential Equations (PDE)

•

Elliptic PDE’s

• Finite elements

•

From elliptic to hyperbolic PDE’s

•

Parabolic equation

• Conclusions and comments

•

Questions and answers

(47)

Plan of the Course

•

Classification of Partial Differential Equations (PDE)

•

Elliptic PDE’s

• Finite elements

•

From elliptic to hyperbolic PDE’s

•

Parabolic equation

•

Questions and answers

(48)

Plan of the Course

•

Classification of Partial Differential Equations (PDE)

•

Elliptic PDE’s

• Finite elements

•

From elliptic to hyperbolic PDE’s

•

Parabolic equation

•

Examples with Matlab

• Questions and answers

(49)

Plan of the Course

•

Classification of Partial Differential Equations (PDE)

•

Elliptic PDE’s

• Finite elements

•

From elliptic to hyperbolic PDE’s

•

Parabolic equation

•

Examples with Matlab

(50)

Plan of the Course

•

Classification of Partial Differential Equations (PDE)

•

Elliptic PDE’s

• Finite elements

•

From elliptic to hyperbolic PDE’s

•

Parabolic equation

•

Examples with Matlab

•

Conclusions and comments

(51)

Plan of the Course

•

Classification of Partial Differential Equations (PDE)

•

Elliptic PDE’s

• Finite elements

•

From elliptic to hyperbolic PDE’s

•

Parabolic equation

•

Examples with Matlab

•

Conclusions and comments

(52)

Classification of (linear) PDE’s

The case of two variables (can be generalized) Lu ≡

A ∂

²

u

∂x

₁²

+ B ∂

²

u

∂x

₁

∂x

₂

+ C ∂

²

u

∂x

₂²

+ L.O.T .

Matrix associated with quadratic form

QF =

A

¹₂

B

1

2

B C

Note: A, B, and C might be functions themselves.

(53)

Classification of (linear) PDE’s

The case of two variables (can be generalized) Lu ≡

A ∂

²

u

∂x

₁²

+ B ∂

²

u

∂x

₁

∂x

₂

+ C ∂

²

u

∂x

₂²

+ L.O.T .

Matrix associated with quadratic form

QF =

A

¹₂

B

1

2

B C

Note: A, B, and C might be functions themselves.

(54)

Classification of (linear) PDE’s

The case of two variables (can be generalized) Lu ≡

A ∂

²

u

∂x

₁²

+ B ∂

²

u

∂x

₁

∂x

₂

+ C ∂

²

u

∂x

₂²

+ L.O.T .

Matrix associated with quadratic form

QF =

A

¹₂

B

1

2

B C

Note: A, B, and C might be functions themselves.

(55)

Classification of PDE’s (cont’ed)

Compute eigenvalues λ

_i

of QF

•

Elliptic equation: λ

1

λ

2

> 0

•

Parabolic equation: λ

₁

λ

₂

= 0

•

Hyperbolic equation: λ

₁

λ

₂

< 0

With the notation of quadratic forms: definite form,

semidefinite form, indefinite form, respectively.

(56)

Classification of PDE’s (cont’ed)

Compute eigenvalues λ

_i

of QF

•

Elliptic equation: λ

1

λ

2

> 0

•

Parabolic equation: λ

₁

λ

₂

= 0

•

Hyperbolic equation: λ

₁

λ

₂

< 0

With the notation of quadratic forms: definite form,

semidefinite form, indefinite form, respectively.

(57)

Classification of PDE’s (cont’ed)

Compute eigenvalues λ

_i

of QF

•

Elliptic equation: λ

1

λ

2

> 0

•

Parabolic equation: λ

₁

λ

₂

= 0

•

Hyperbolic equation: λ

₁

λ

₂

< 0

With the notation of quadratic forms: definite form,

semidefinite form, indefinite form, respectively.

(58)

Classification of PDE’s (cont’ed)

Consider operator

Lu ≡ A ∂

²

u

∂x

₁²

+ B ∂

²

u

∂x

1

∂x

2

+ C ∂

²

u

∂x

₂²

= 0 and look for change of variables

ξ = αx

₂

+ βx

₁

, η = γx

₂

+ δx

₁

so that Lu is a multiple of

_∂ξ∂η^∂²^u

(see wave equation)

Lu = (Aβ

²

+ Bαβ + C α

²

) ∂

²

u

∂ξ

²

+ (Aδ

²

+ Bγδ + C γ

²

) ∂

²

u

∂η

²

+ (2Aβδ + B(αδ + βγ) + 2C αγ) ∂

²

u

∂ξ∂η

(59)

Classification of PDE’s (cont’ed)

Consider operator

Lu ≡ A ∂

²

u

∂x

₁²

+ B ∂

²

u

∂x

1

∂x

2

+ C ∂

²

u

∂x

₂²

= 0 and look for change of variables

ξ = αx

₂

+ βx

₁

, η = γx

₂

+ δx

₁

so that Lu is a multiple of

_∂ξ∂η^∂²^u

(see wave equation)

Lu = (Aβ

²

+ Bαβ + C α

²

) ∂

²

u

∂ξ

²

+ (Aδ

²

+ Bγδ + C γ

²

) ∂

²

u

∂η

²

+ (2Aβδ + B(αδ + βγ) + 2C αγ) ∂

²

u

∂ξ∂η

(60)

Classification of PDE’s (cont’ed)

Lu = (Aβ

²

+ Bαβ + C α

²

) ∂

²

u

∂ξ

²

+ (Aδ

²

+ Bγδ + C γ

²

) ∂

²

u

∂η

²

+ (2Aβδ + B(αδ + βγ) + 2C αγ) ∂

²

u

∂ξ∂η If A = C = 0, trivial.

Suppose A 6= 0; we want

Aβ

²

+ Bαβ + C α

²

= 0, Aδ

²

+ Bγδ + C γ

²

= 0 When αγ 6= 0, divide first equation by α

²

, second one by γ

²

and solve for β/α and δ/γ, resp.

β/α = (2A)

⁻¹

(−B ± √

∆), δ/γ = (2A)

⁻¹

(−B ± √

∆)

∆ = B

²

− 4AC

(61)

Classification of PDE’s (cont’ed)

Lu = (Aβ

²

+ Bαβ + C α

²

) ∂

²

u

∂ξ

²

+ (Aδ

²

+ Bγδ + C γ

²

) ∂

²

u

∂η

²

+ (2Aβδ + B(αδ + βγ) + 2C αγ) ∂

²

u

∂ξ∂η If A = C = 0, trivial. Suppose A 6= 0; we want

Aβ

²

+ Bαβ + C α

²

= 0, Aδ

²

+ Bγδ + C γ

²

= 0 When αγ 6= 0, divide first equation by α

²

, second one by γ

²

and solve for β/α and δ/γ, resp.

β/α = (2A)

⁻¹

(−B ± √

∆), δ/γ = (2A)

⁻¹

(−B ± √

∆)

∆ = B

²

− 4AC

(62)

Classification of PDE’s (cont’ed)

Hyperbolic case

ξ = αx

2

+ βx

1

, η = γx

2

+ δx

1

β/α = (2A)

⁻¹

(−B ±

√

∆), δ/γ = (2A)

⁻¹

(−B ±

√

∆) For nonsingular change of variables, ∆ must be positive

α = γ = 2A, β = −B +

√

∆, δ = −B −

√

∆

Lu = −4A(B

²

− 4AC ) ∂

²

u

∂ξ∂η

As before, solution has the form u = p(ξ) + q(η) and the lines

ξ = constant and η = constant are called characteristics.

(63)

Classification of PDE’s (cont’ed)

Hyperbolic case

ξ = αx

2

+ βx

1

, η = γx

2

+ δx

1

β/α = (2A)

⁻¹

(−B ±

√

∆), δ/γ = (2A)

⁻¹

(−B ±

√

∆) For nonsingular change of variables, ∆ must be positive

α = γ = 2A, β = −B +

√

∆, δ = −B −

√

∆

Lu = −4A(B

²

− 4AC ) ∂

²

u

∂ξ∂η

As before, solution has the form u = p(ξ) + q(η) and the lines

ξ = constant and η = constant are called characteristics.

(64)

Classification of PDE’s (cont’ed)

Hyperbolic case

ξ = αx

2

+ βx

1

, η = γx

2

+ δx

1

β/α = (2A)

⁻¹

(−B ±

√

∆), δ/γ = (2A)

⁻¹

(−B ±

√

∆) For nonsingular change of variables, ∆ must be positive

α = γ = 2A, β = −B +

√

∆, δ = −B −

√

∆

Lu = −4A(B

²

− 4AC ) ∂

²

u

∂ξ∂η

As before, solution has the form u = p(ξ) + q(η) and the lines

ξ = constant and η = constant are called characteristics.

(65)

Classification of PDE’s (cont’ed)

Actually, when x

1

= t and x

2

= x, the change of variables x

⁰

= x − B

2A t, t

⁰

= t

maps our hyperbolic operator (A 6= 0) to a multiple of wave equation

∂

²

u

∂t

²

− c

²

∂

²

u

∂x

²

Hence, L is a wave operator in a frameset moving at speed

−B/(2A).

(66)

Classification of PDE’s (cont’ed)

Actually, when x

1

= t and x

2

= x, the change of variables x

⁰

= x − B

2A t, t

⁰

= t

maps our hyperbolic operator (A 6= 0) to a multiple of wave equation

∂

²

u

∂t

²

− c

²

∂

²

u

∂x

²

Hence, L is a wave operator in a frameset moving at speed

−B/(2A).

(67)

Classification of PDE’s (cont’ed)

Parabolic case

Lu = (Aβ

²

+ Bαβ + C α

²

) ∂

²

u

∂ξ

²

+ (Aδ

²

+ Bγδ + C γ

²

) ∂

²

u

∂η

²

+ (2Aβδ + B(αδ + βγ) + 2C αγ) ∂

²

u

∂ξ∂η

For β/α = −B/(2A) coefficient of

^∂_∂ξ²^u2

vanishes

But B/(2A) = 2C /B, so coefficient of

_∂ξ∂η^∂²^u

is zero as well

Everything can be written as a multiple of

^∂_∂η²^u2

(68)

Classification of PDE’s (cont’ed)

Parabolic case

Lu = (Aβ

²

+ Bαβ + C α

²

) ∂

²

u

∂ξ

²

+ (Aδ

²

+ Bγδ + C γ

²

) ∂

²

u

∂η

²

+ (2Aβδ + B(αδ + βγ) + 2C αγ) ∂

²

u

∂ξ∂η For β/α = −B/(2A) coefficient of

^∂_∂ξ²^u2

vanishes

But B/(2A) = 2C /B, so coefficient of

_∂ξ∂η^∂²^u

is zero as well

Everything can be written as a multiple of

^∂_∂η²^u2

(69)

Classification of PDE’s (cont’ed)

Parabolic case

Lu = (Aβ

²

+ Bαβ + C α

²

) ∂

²

u

∂ξ

²

+ (Aδ

²

+ Bγδ + C γ

²

) ∂

²

u

∂η

²

+ (2Aβδ + B(αδ + βγ) + 2C αγ) ∂

²

u

∂ξ∂η For β/α = −B/(2A) coefficient of

^∂_∂ξ²^u2

vanishes

But B/(2A) = 2C /B, so coefficient of

_∂ξ∂η^∂²^u

is zero as well

Everything can be written as a multiple of

^∂_∂η²^u2

(70)

Classification of PDE’s (cont’ed)

Parabolic case

Lu = (Aβ

²

+ Bαβ + C α

²

) ∂

²

u

∂ξ

²

+ (Aδ

²

+ Bγδ + C γ

²

) ∂

²

u

∂η

²

+ (2Aβδ + B(αδ + βγ) + 2C αγ) ∂

²

u

∂ξ∂η For β/α = −B/(2A) coefficient of

^∂_∂ξ²^u2

vanishes

But B/(2A) = 2C /B, so coefficient of

_∂ξ∂η^∂²^u

is zero as well

Everything can be written as a multiple of

^∂_∂η²^u2