# solution of partial differential equations (pdes)

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SOLUTION OF

Partial Differential Equations

(PDEs)

Mathematics is the Language of Science

PDEs are the expression of processes that occur across time & space: (x,t), (x,y), (x,y,z), or (x,y,z,t)

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Partial Differential Equations (PDE's)

A PDE is an equation which includes derivatives of an unknown function with respect to 2 or more

independent variables

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Partial Differential Equations (PDE's)PDE's describe the behavior of many engineering phenomena:

Wave propagation Fluid flow (air or liquid)

Air around wings, helicopter blade, atmosphereWater in pipes or porous mediaMaterial transport and diffusion in air or waterWeather: large system of coupled PDE's for momentum,

pressure, moisture, heat, Vibration Mechanics of solids:

stress-strain in material, machine part, structure Heat flow and distribution Electric fields and potentials Diffusion of chemicals in air or water Electromagnetism and quantum mechanics

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Partial Differential Equations (PDE's)

Weather Prediction heat transport & cooling advection & dispersion of moisture radiation & solar heating evaporation air (movement, friction, momentum, coriolis forces) heat transfer at the surface

To predict weather one need "only" solve a very large systems ofcoupled PDE equations for momentum, pressure, moisture, heat, etc.

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Conservacin de energa, masa, momento, vapor de agua,

ecuacin de estado de gases.

v = (u, v, w), T, p, = 1/ y q

360x180x32 x nvar

resolucin 1/tDuplicar la resolucin espacial supone incrementar el tiempo de cmputo en un factor 16

Modelizacin Numrica del Tiempo

D+0 D+1 D+2Condicin inicial H+0 H+24

D+3 D+4 D+5sbado 15/12/2007 00

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Partial Differential Equations (PDE's)

Learning Objectives

1) Be able to distinguish between the 3 classes of 2nd order, linear PDE's. Know the physical problems each class represents and the physical/mathematical characteristics of each.

2) Be able to describe the differences between finite-difference and finite-element methods for solving PDEs.

3) Be able to solve Elliptical (Laplace/Poisson) PDEs using finite differences.

4) Be able to solve Parabolic (Heat/Diffusion) PDEs using finite differences.

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Partial Differential Equations (PDE's)Engrd 241 Focus:Linear 2nd-Order PDE's of the general form

u(x,y), A(x,y), B(x,y), C(x,y), and D(x,y,u,,)

The PDE is nonlinear if A, B or C include u, u/x or u/y, or if D is nonlinear in u and/or its first derivatives.

ClassificationB2 4AC < 0 > Elliptic (e.g. Laplace Eq.)B2 4AC = 0 > Parabolic (e.g. Heat Eq.)B2 4AC > 0 > Hyperbolic (e.g. Wave Eq.)

Each category describes different phenomena. Mathematical properties correspond to those phenomena.

2 2 2

2 2u u uA B C D 0

x yx y

+ + + =

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Partial Differential Equations (PDE's)

Typical examples include

u u uu(x, y), (in terms of and )x y

Elliptic Equations (B2 4AC < 0) [steady-state in time] typically characterize steady-state systems (no time derivative)

temperature torsion pressure membrane displacement electrical potential

closed domain with boundary conditions expressed in terms of

A = 1, B = 0, C = 1 ==> B2 4AC = 4 < 0

2 22

2 2

0u uu u uD(x, y,u, , )x y x y

+ =

Laplace Eq. Poisson Eq.

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Elliptic PDEs Boundary Conditions for Elliptic PDE's:

Dirichlet: u provided along all of edge

Neumann: provided along all of the edge (derivative in normal direction)

Mixed: u provided for some of the edge and

for the remainder of the edge

Elliptic PDE's are analogous to Boundary Value ODE's

u

u

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Elliptic PDEs

Estimateu(x)from

LaplaceEquations

u(x=1,y)or

u/xgiven

onboundary

u(x,y =0) or u/y given on boundary

y

x

u(x,y =1) or u/y given on boundary

u(x=0,y)or

u/xgiven

onboundary

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Parabolic PDEs

Parabolic Equations (B2 4AC = 0) [first derivative in time ] variation in both space (x,y) and time, t typically provided are:

initial values: u(x,y,t = 0) boundary conditions: u(x = xo,y = yo, t) for all t

u(x = xf,y = yf, t) for all t

all changes are propagated forward in time, i.e., nothing goes backward in time; changes are propagated across space at decreasing amplitude.

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Parabolic PDEs Parabolic Equations (B2 4AC = 0) [first derivative in time ]

Typical example: Heat Conduction or Diffusion

(the Advection-Diffusion Equation)2

2u

x

u u1D : k D(x,u, )t x

= +

2 2

2 2u u

x y

u u u2D : k D(x, y,u, , )t x y

= + +

A = k, B = 0, C = 0 > B2 4AC = 0

2k u D= +

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Parabolic PDEs

Estimateu(x,t)from

the HeatEquation

u(x=1,t)given

onboundaryfor all t

u(x,t =0) given on boundaryas initial conditions for u/t

t

x

u(x=0,t)given

onboundaryfor all t

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Parabolic PDEs

x=L

An elongated reactor with a single entry and exit point and a uniform cross-section of area A.

A mass balance is developed for a finite segment xalong the tank's longitudinal axis in order to derive a differential equation for concentration (V = A x).

xx=0

c(x,t) = concentrationat time, t, and distance, x.

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Parabolic PDEs

x=Lxx=0c c(x) c(x)V Qc(x) Q c(x) x DAt x x

= +

Flow in Flow out Dispersion in

c(x) c(x)DA x k Vc(x)x x x

+ + Dispersion out Decay

reaction2

2c c c Q cD kct t A xx

==> =

As t and x ==> 0

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Hyperbolic PDEs

Hyperbolic Equations (B2 4AC > 0) [2nd derivative in time ]

variation in both space (x, y) and time, t requires:

initial values: u(x,y,t=0), u/t (x,y,t = 0) "initial velocity" boundary conditions: u(x = xo,y = yo, t) for all t

u(x = xf,y = yf, t) for all t

all changes are propagated forward in time, i.e., nothing goes backward in time.

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Hyperbolic PDEs

Hyperbolic Equations (B2 4AC > 0) [2nd derivative in time]

Typical example: Wave Equation

A = 1, B = 0, C = -1/c2 ==> B2 4AC = 4/c2 > 0

Models vibrating string water waves voltage change in a wire

2 2

2 2 2u u

x t

u 1 u1D : D(x, y,u, , ) 0x c t

+ =

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Hyperbolic PDEs

Estimateu(x,t)from

the WaveEquation

u(x=1,t)given

onboundaryfor all t

u(x,t =0) and u(x,t=0)/t given on boundary

as initial conditions for 2u/t2

Now PDE is second order in time

t

x

u(x=0,t)given

onboundaryfor all t

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Numerical Methods for Solving PDEs

Numerical methods for solving different types of PDE's reflect the different character of the problems.

Laplace - solve all at once for steady state conditions Parabolic (heat) and Hyperbolic (wave) equations.

Integrate initial conditions forward through time.

Methods Finite Difference (FD) Approaches (C&C Chs. 29 & 30)

Based on approximating solution at a finite # of points, usuallyarranged in a regular grid.

Finite Element (FE) Method (C&C Ch. 31)Based on approximating solution on an assemblage of simply shaped (triangular, quadrilateral) finite pieces or "elements" which together make up (perhaps complexly shaped) domain.

In this course, we concentrate on FD applied to elliptic and parabolic equations.

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Finite Difference for Solving Elliptic PDE's

Solving Elliptic PDE's: Solve all at once Liebmann Method:

Based on Boundary Conditions (BCs) and finite difference approximation to formulate system of equations

Use Gauss-Seidel to solve the system 2 2

2 2y

0u u u uD(x, y,u, , )x x y

+ =

Laplace Eq.Poisson Eq.

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Finite Difference Methods for Solving Elliptic PDE's1. Discretize domain into grid of evenly spaced points2. For nodes where u is unknown:

w/ x = y = h, substitute into main equation

3. Using Boundary Conditions, write, n*m equations for u(xi=1:m, yj=1:n) or n*m unknowns.

4. Solve this banded system with an efficient scheme. Using Gauss-Seidel iteratively yields the Liebmann Method.

i 1, j i, j i 1, j 22

2 2

u 2u u

xO( x )

u

x ( ) + +

= +

2 22

2 2 2i 1, j i 1, j i, j 1 i, j 1 i, ju u u u 4uu u O(h )

x y h

+ ++ + +

+ = +

i, j 1 i, j i, j 1 22

2 2

u 2u u

yO( y )

u

y ( ) + +

= +

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Elliptical PDEs

The Laplace Equation2 2

2 2y

u u 0x

+ =

The Laplace molecule

i i+1i-1j-1

j+1

j

i 1, j i 1, j i, j 1 i, j 1 i, jT T T T 4T 0+ + + + + =If x = y then

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Elliptical PDEs The Laplace molecule:

T = 0o C

T = 100o C

T = 100o C

T = 0o C

11 12 13 21 22 23 31 32 33T T T T T T T T T

111213212223313233

-100T-T

TTT =TTTT

-4 1 0 1 0 0 0 0 01 -4 1 0 1 0 0 0 00 1 -4 0 0 1 0 0 01 0 0 -4 1 0 1 0 00 1 0 1 -4 1 0 1 00 0 1 0 1 -4 0 0 10 0 0 1 0 0 -4 1 00 0 0 0 1 0 1 -4 10 0 0 0 0 1 0 1 -4

100-200

00

-10000

-100

1,1 1,2 1,3

2,1 2,2 2,3

3,1 3,2 3,3

The temperature distribution can be estimated by discretizingt

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