Partial Differential Equations

Dr Muhammad Al-Salamah, Industrial Engineering, KFUPM

Topic 9: Partial Differential Equations

•An equation involving partial derivatives of an unknown function of two or more independent variables is called a partial differential equation – PDE.

•An example of a PDE is

•The general form of a linear second order PDE is written as:

0

12

2

22

2

2

2

2

2

2

Dy

uC

yx

uB

x

uA

uy

uxy

x

u

Dr Muhammad Al-Salamah, Industrial Engineering, KFUPM

•Here, A, B, and C are functions of the independent variables x and y only.

•The part D is a function of x, y, u, u/x, and u/y.

•Depending on the values of A, B, and C, the 2–PDE is classified into one of three categories.

B2 – 4AC Category

< 0 Elliptic

= 0 Parabolic

> 0 Hyperbolic

Dr Muhammad Al-Salamah, Industrial Engineering, KFUPM

Finite Difference: Elliptic Equations

•Elliptic equations in engineering are typically used to characterize steady-state, boundary value problems.

THE LAPLACE EQUATION

•Elliptic equations are typically used to characterize steady–state systems.

•We will illustrate the solution of elliptic PDEs in the context of an example.

•This example is called the headed plate.

Dr Muhammad Al-Salamah, Industrial Engineering, KFUPM

•The plate has a thickness of z.

•The plate is insulated everywhere except its edges.

•The edges are fixed at prescribed temperatures.

Dr Muhammad Al-Salamah, Industrial Engineering, KFUPM

•At the steady state, over time period t

heat flow–in = heat flow–out

q(x) y z t + q(y) x z t = q(x + x) y z t

+ q(y + y) x z t

where q is the heat flux.

q(y)

q(x) q(x + x)

q(y + y)

Dr Muhammad Al-Salamah, Industrial Engineering, KFUPM

•After simplification, we arrive to the PDE

•We need a PDE in terms of temperature.

•The relation between heat flux q and temperature T is given by

•This equation is called Fourier’s law of heat conduction.

i

TCq

y

q

x

q

i

0

Dr Muhammad Al-Salamah, Industrial Engineering, KFUPM

•Substituting the value of q in the PDE, we get

•It is called the Laplace equation.

•The Laplace equation is an elliptic equation because

B2 – 4 A C = –4 < 0

02

2

2

2

y

T

x

T

Dr Muhammad Al-Salamah, Industrial Engineering, KFUPM

SOLUTION TECHNIQUES

•The solution method works by substituting the partial derivatives by the finite difference formulas.

The Laplace Difference Equation

•The centered difference formulas for the second derivatives are

2

1,,1,

2

2

2

,1,,1

2

2

2

and

2

y

TTT

y

T

x

TTT

x

T

jijiji

jijiji

Dr Muhammad Al-Salamah, Industrial Engineering, KFUPM

•After the substitution, the differential equation becomes

•Assume x = y and simplify:

•This relationship is called the Laplace difference equation.

04

022

,1,1,,1,1

2

1,,1,

2

,1,,1

jijijijiji

jijijijijiji

TTTTT

y

TTT

x

TTT

Dr Muhammad Al-Salamah, Industrial Engineering, KFUPM

41,1,,1,1

,

jijijijiji

TTTTT

Dr Muhammad Al-Salamah, Industrial Engineering, KFUPM

•Consider the simplest case where the boundary temperatures along the edges are set to fixed values:

Dr Muhammad Al-Salamah, Industrial Engineering, KFUPM

•At note (1, 1), the equation is written as

•Note T0,1 = 75 and T1,0 = 0. The equation becomes

•Similarly, the relationship can be written for all interior points and the result is a system of linear equations.

754

04

1,22,11,1

1,10,12,11,01,2

TTT

TTTTT

Dr Muhammad Al-Salamah, Industrial Engineering, KFUPM

1504

1004

1754

504

04

754

504

04

754

332332

33231322

231312

33322231

2332221221

13221211

323121

22132111

122111

TTT

TTTT

TTT

TTTT

TTTTT

TTTT

TTT

TTTT

TTT

•The system has 9 equations and 9 unknowns:

Dr Muhammad Al-Salamah, Industrial Engineering, KFUPM

The Liebmann Method

•Most numerical solutions of Laplace equation involve systems that are very large.

•For larger size grids, a significant number of terms will be zero.

•For this reason, approximation methods provide a viable approach for obtaining solutions.

Dr Muhammad Al-Salamah, Industrial Engineering, KFUPM

•Steps of the Liebmann method:

1. Set Toldi,j = 0 for the interior points.

2. Compute Tnewi,j:

3. Apply the over-relaxation formula

for 1 ≤ ≤ 2.

4. Go to (2) if .ε%100

)λ1(λ

4

new,

old,

new,

old,

new,

new,

old1,

old1,

old,1

old,1new

,

sji

jiji

jijiji

jijijijiji

T

TT

TTT

TTTTT

Dr Muhammad Al-Salamah, Industrial Engineering, KFUPM

Example

See the example and the solution in the book.

Dr Muhammad Al-Salamah, Industrial Engineering, KFUPM

Finite Difference: Parabolic Equations

• Parabolic equations in engineering are typically used to characterize time variable problems.

THE HEAT CONDUCTION EQUATION

• Conservation of energy can be used to develop an transit-state energy balance for the differential element in a long, thin insulated rod.

Dr Muhammad Al-Salamah, Industrial Engineering, KFUPM

•The heat conduction equation for the insulated rod is

•The temperature depends on location and time.

t

T

x

Tk

2

2

Dr Muhammad Al-Salamah, Industrial Engineering, KFUPM

EXPLICIT METHODS

•We can substitute the partial derivatives by their finite difference approximations:

•Substituting in the heat conduction equation:

t

TT

x

TTTk

t

TT

t

T

x

TTT

x

T

li

li

li

li

li

li

li

li

li

li

1

211

1

211

2

2

2

and

2

Dr Muhammad Al-Salamah, Industrial Engineering, KFUPM

•Simplifying:

2

111

where

2

x

tk

TTTTT li

li

li

li

li

Dr Muhammad Al-Salamah, Industrial Engineering, KFUPM

Example

See the example and the solution in the book.

Dr Muhammad Al-Salamah, Industrial Engineering, KFUPM

Convergence and Stability

•Convergence means that as x and t approach zero, the results of the finite difference method approach the true solution.•Stability means that errors at any stage of the computation are not amplified but are attenuated as the computation progresses.•The explicit method is both convergent and stable if ≤ ½, or

k

xt

2

2

1

Dr Muhammad Al-Salamah, Industrial Engineering, KFUPM

Solution to the previous example for = 0.735.

Dr Muhammad Al-Salamah, Industrial Engineering, KFUPM

Dr Muhammad Al-Salamah, Industrial Engineering, KFUPM

A SIMPLE IMPLICIT METHOD

• The difference between the explicit and implicit approximations is illustrated in this figure.

Dr Muhammad Al-Salamah, Industrial Engineering, KFUPM

li

li

li

li

li

li

li

li

li

li

li

li

TTTT

t

TT

x

TTTk

x

TTT

x

T

11

111

1

2

11

111

2

11

111

2

2

λλ21λ

2

2

• In the implicit method, the derivative is approximated by:

• The heat conduction equation becomes

• Simplifying:

Dr Muhammad Al-Salamah, Industrial Engineering, KFUPM

• The solution will involve the solution of a system of linear equations.

Dr Muhammad Al-Salamah, Industrial Engineering, KFUPM

Example

See the example and the solution in the book.