Video 15: PDE Classification: Elliptic, Parabolic andHyperbolic Equations

David J. Willis

March 11, 2015

David J. Willis Video 15: PDE Classification: Elliptic, Parabolic and Hyperbolic EquationsMarch 11, 2015 1 / 20

Table of contents

1 PDE ClassificationEquation ClassificationPDE Classification

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References and Acknowledgements

The following materials were used in the preparation of this lecture:

1 Handbook of Grid Generation, Thompson, Soni and Weatherhill.

2 MIT Open Courseware 16.920 Course

The author of these slides wishes to thank these sources for making thecurrent lecture.

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Table of contents

1 PDE ClassificationEquation ClassificationPDE Classification

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PDE Classification Equation Classification

Equation Classification

Consider the following equation:

Ax2 + Bxy + Cy 2 = f

or

A

(x

y

)2

+ B

(x

y

)+ C =

f

y 2

Az2 + Bz + C = f ′

What does this equation represent?

Let’s plot it for several different A,B and C values.

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PDE Classification Equation Classification

Equation Classification

A = 2, B = 1, C = 2→ B2 − 4AC < 0

Paraboloid Surface – Forms and Ellipse

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PDE Classification Equation Classification

Equation Classification

A = 1, B = 2, C = 1→ B2 − 4AC = 0

Pure Parabolic Surface

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PDE Classification Equation Classification

Equation Classification

A = 1, B = 2, C = 1→ B2 − 4AC > 0

Hyperbolic Surface

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PDE Classification Equation Classification

Summary

The value of the discriminant: B2 − 4AC defines the equation

Contour shapes and the intersection of the shape with the x − yplane is:

B2 − 4AC < 0 : Circles and Ellipse (parabaloid)B2 − 4AC = 0 : Parallel lines (pure parabola)B2 − 4AC < 0 : X-lines (hyperbola)

a) Elliptic b) Parabolic c) Hyperbolic

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PDE Classification Equation Classification

Table of contents

1 PDE ClassificationEquation ClassificationPDE Classification

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PDE Classification PDE Classification

PDE Classification

Consider a general second order PDE:

A∂2u

∂x2+ B

∂2u

∂xy+ C

∂2u

∂y 2+ D

∂u

∂x+ E

∂u

∂y+ F = 0 (1)

This is identical in form to the equation for a conic section ( a generalform of the equation we considered earlier):

Ax2 + Bxy + Cy 2 + Dx + Ey + F = 0 (2)

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PDE Classification PDE Classification

PDE Classification

Aside: We can show that the characteristic equation:

A∂2u

∂x2+ B

∂2u

∂xy+ C

∂2u

∂y 2= 0 (3)

Analysed using a Fourier series expansion:

u =1

4π2

∑j

∑k

ujk expi(σx )j x expi(σy )k y (4)

Yields:

A

(σx

σy

)2

+ B

(σx

σy

)+ C = 0 (5)

Let’s have a look at how this equation applies to PDEs.

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PDE Classification PDE Classification

PDE Classification: Elliptic PDE

Start with Laplace’s or Poisson’s Equation

∂2φ

∂x2+∂2φ

∂y 2= 1 · ∂

2φ

∂x2+ 1 · ∂

2φ

∂y 2= 0

So, A = C = 1 and B = 0, so B2 − 4AC = −4 < 0

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PDE Classification PDE Classification

PDE Classification: Example

Let’s say you solve he membrane problem from last unit with a singlepoint force applied transverse to the membrane:

The point load is distributed throughout the membrane.

The point has base influence throughout the membrane.

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PDE Classification PDE Classification

PDE Classification: Example

Let’s add another point load:

The point load has influence throughout the membrane.

Deflection at any point is dependent on the forcing conditions at allpoints on the membrane.

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PDE Classification PDE Classification

PDE Classification: Parabolic PDE

Next, let’s consider the heat equation:

∂T

∂t− ∂2T

∂x2= f (6)

A = 1,C = 0,B = 0. Therefore, B2 − 4AC = 0

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PDE Classification PDE Classification

PDE Classification: Example

Consider the unsteady heat equation

The temperature of the bar is dependent on the temperature in thetime leading up to the current time.

The temperature in the future will depend on the temperature now.

Ie. If we turn up the heat now, it will only affect the temperature attimes after the current time.

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PDE Classification PDE Classification

PDE Classification: Hyperbolic PDE

Next, let’s consider the second order wave equation:

∂2u

∂t2− ∂2u

∂x2= f (7)

A = 1,B = 0,C = −1. Therefore, B2 − 4AC > 0

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PDE Classification PDE Classification

PDE Classification: Example

The wave equation propagates information to well defined zones ofinfluence.

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PDE Classification PDE Classification

What have we learned?

A second order (partial differential) equation can be classified into 3types:

Elliptic: Laplace and other similar equations. Elliptic equations havezones of influence and dependence that include the whole domain.Parabolic: Unsteady heat, parabolized Navier-Stokes, boundary layers,and other similar equations. Parabolic equations have zones ofinfluence that includes the downstream domain. The zone ofdependence is the upstream domain.Hyperbolic: Wave equation, supersonic Navier-Stokes, and othersimilar equations. Hyperbolic equations have zones of influence anddependence that converge to the point along characteristic lines(upstream & downstream).

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