Video 15: PDE Classification: Elliptic, Parabolic and Hyperbolic March Equations 11, / 20

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Sherman Edwards
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1 Video 15: : Elliptic, Parabolic and Hyperbolic Equations March 11, 2015 Video 15: : Elliptic, Parabolic and Hyperbolic March Equations 11, / 20

2 Table of contents 1 Video 15: : Elliptic, Parabolic and Hyperbolic March Equations 11, / 20

3 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 Course The author of these slides wishes to thank these sources for making the current lecture. Video 15: : Elliptic, Parabolic and Hyperbolic March Equations 11, / 20

4 Table of contents 1 Video 15: : Elliptic, Parabolic and Hyperbolic March Equations 11, / 20

5 Consider the following equation: A Ax 2 + Bxy + Cy 2 = f or ( ) x 2 ( ) x + B + C = f y y y 2 Az 2 + Bz + C = f What does this equation represent? Let s plot it for several different A, B and C values. Video 15: : Elliptic, Parabolic and Hyperbolic March Equations 11, / 20

6 A = 2, B = 1, C = 2 B 2 4AC < 0 Paraboloid Surface Forms and Ellipse Video 15: : Elliptic, Parabolic and Hyperbolic March Equations 11, / 20

7 A = 1, B = 2, C = 1 B 2 4AC = 0 Pure Parabolic Surface Video 15: : Elliptic, Parabolic and Hyperbolic March Equations 11, / 20

8 A = 1, B = 2, C = 1 B 2 4AC > 0 Hyperbolic Surface Video 15: : Elliptic, Parabolic and Hyperbolic March Equations 11, / 20

9 Summary The value of the discriminant: B 2 4AC defines the equation Contour shapes and the intersection of the shape with the x y plane is: B 2 4AC < 0 : Circles and Ellipse (parabaloid) B 2 4AC = 0 : Parallel lines (pure parabola) B 2 4AC < 0 : X-lines (hyperbola) a) Elliptic b) Parabolic c) Hyperbolic Video 15: : Elliptic, Parabolic and Hyperbolic March Equations 11, / 20

10 Table of contents 1 Video 15: : Elliptic, Parabolic and Hyperbolic March Equations 11, / 20

11 Consider a general second order PDE: A 2 u x 2 + B 2 u xy + C 2 u y 2 + D u x + E u y + F = 0 (1) This is identical in form to the equation for a conic section ( a general form of the equation we considered earlier): Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0 (2) Video 15: : Elliptic, Parabolic and Hyperbolic March Equations 11, / 20

12 Aside: We can show that the characteristic equation: A 2 u x 2 + B 2 u xy + C 2 u y 2 = 0 (3) Analysed using a Fourier series expansion: u = 1 4π 2 û jk exp i(σx ) j x exp i(σy ) ky j k (4) Yields: A ( σx σ y ) 2 + B ( σx σ y ) + C = 0 (5) Let s have a look at how this equation applies to PDEs. Video 15: : Elliptic, Parabolic and Hyperbolic March Equations 11, / 20

13 : Elliptic PDE Start with Laplace s or Poisson s Equation 2 φ x φ y 2 = 1 2 φ x φ y 2 = 0 So, A = C = 1 and B = 0, so B 2 4AC = 4 < 0 Video 15: : Elliptic, Parabolic and Hyperbolic March Equations 11, / 20

: Example Let s say you solve he membrane problem from last unit with a single point force applied transverse to the membrane: The point load is distributed

14 : Example Let s say you solve he membrane problem from last unit with a single point force applied transverse to the membrane: The point load is distributed throughout the membrane. The point has base influence throughout the membrane. Video 15: : Elliptic, Parabolic and Hyperbolic March Equations 11, / 20

15 : 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 all points on the membrane. Video 15: : Elliptic, Parabolic and Hyperbolic March Equations 11, / 20

16 : Parabolic PDE Next, let s consider the heat equation: A = 1, C = 0, B = 0. Therefore, B 2 4AC = 0 T t 2 T x 2 = f (6) Video 15: : Elliptic, Parabolic and Hyperbolic March Equations 11, / 20

: Example Consider the unsteady heat equation The temperature of

Ie. If we turn up the heat now, it will only affect the

17 : Example Consider the unsteady heat equation The temperature of the bar is dependent on the temperature in the time 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 at times after the current time. Video 15: : Elliptic, Parabolic and Hyperbolic March Equations 11, / 20

18 : Hyperbolic PDE Next, let s consider the second order wave equation: 2 u t 2 2 u x 2 = f (7) A = 1, B = 0, C = 1. Therefore, B 2 4AC > 0 Video 15: : Elliptic, Parabolic and Hyperbolic March Equations 11, / 20

19 : Example The wave equation propagates information to well defined zones of influence. Video 15: : Elliptic, Parabolic and Hyperbolic March Equations 11, / 20

20 What have we learned? A second order (partial differential) equation can be classified into 3 types: Elliptic: Laplace and other similar equations. Elliptic equations have zones 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 of influence that includes the downstream domain. The zone of dependence is the upstream domain. Hyperbolic: Wave equation, supersonic Navier-Stokes, and other similar equations. Hyperbolic equations have zones of influence and dependence that converge to the point along characteristic lines (upstream & downstream). Video 15: : Elliptic, Parabolic and Hyperbolic March Equations 11, / 20

8.6 Translate and Classify Conic Sections

8.6 Translate and Classify Conic Sections Where are the symmetric lines of conic sections? What is the general 2 nd degree equation for any conic? What information can the discriminant tell you about a