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BIJU PATNAIK UNIVERSITY OF TECHNOLOGY, ODISHA Lecture Notes On Prepared by, Dr. Subhendu Kumar Rath, BPUT, Odisha. DIFFERENCE EQUATION AND ITS APPLICATION
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Page 1: Lecture Notes On - BPUT Partial Differential Equations (in the example and are independent variable. s) PDE involves two or more independent variables ( , ) ( , ) Example: 2 2 x. t

BIJU PATNAIK UNIVERSITY OF TECHNOLOGY,

ODISHA

Lecture Notes

On

Prepared by,

Dr. Subhendu Kumar Rath,

BPUT, Odisha.

DIFFERENCE EQUATION AND

ITS APPLICATION

Page 2: Lecture Notes On - BPUT Partial Differential Equations (in the example and are independent variable. s) PDE involves two or more independent variables ( , ) ( , ) Example: 2 2 x. t

DIFFERENCE EQUATION AND ITS APPLICATION

Dr.Subhendu Kumar Rath

Page 3: Lecture Notes On - BPUT Partial Differential Equations (in the example and are independent variable. s) PDE involves two or more independent variables ( , ) ( , ) Example: 2 2 x. t

DEFINITIONAn expression which expresses a relation between an independent variable and successive values or Successive differences of dependent variable is called a difference equation.

Page 4: Lecture Notes On - BPUT Partial Differential Equations (in the example and are independent variable. s) PDE involves two or more independent variables ( , ) ( , ) Example: 2 2 x. t

EXAMPLES

Page 5: Lecture Notes On - BPUT Partial Differential Equations (in the example and are independent variable. s) PDE involves two or more independent variables ( , ) ( , ) Example: 2 2 x. t

ORDER AND DEGREE

Page 6: Lecture Notes On - BPUT Partial Differential Equations (in the example and are independent variable. s) PDE involves two or more independent variables ( , ) ( , ) Example: 2 2 x. t

SOLUTION A solution of a difference equation is any

function that satisfies it. The general solution of a difference

equation of order n is a solution that contains n arbitrary constants or n arbitrary function which are periodic with period equal to the interval of differencing.

Page 7: Lecture Notes On - BPUT Partial Differential Equations (in the example and are independent variable. s) PDE involves two or more independent variables ( , ) ( , ) Example: 2 2 x. t

The particular solution of a difference equation is a solution obtained by assigning particular values to the arbitrary constants or functions.

CONTD…..

Page 8: Lecture Notes On - BPUT Partial Differential Equations (in the example and are independent variable. s) PDE involves two or more independent variables ( , ) ( , ) Example: 2 2 x. t

LINEAR DIFFERENCE EQATION

Page 9: Lecture Notes On - BPUT Partial Differential Equations (in the example and are independent variable. s) PDE involves two or more independent variables ( , ) ( , ) Example: 2 2 x. t

CONTD….

Page 10: Lecture Notes On - BPUT Partial Differential Equations (in the example and are independent variable. s) PDE involves two or more independent variables ( , ) ( , ) Example: 2 2 x. t

APPLICATION Difference equation can be applicable in

the following areas. Numerical methods to solve partial

differential equation. Fourier series Algebra and Analysis

Page 11: Lecture Notes On - BPUT Partial Differential Equations (in the example and are independent variable. s) PDE involves two or more independent variables ( , ) ( , ) Example: 2 2 x. t

First Derivative Approximations Backward difference: (uj – uj-1) / Δx

Forward difference: (uj+1 – uj) / Δx

Centered difference: (uj+1 – uj-1) / 2Δx

Page 12: Lecture Notes On - BPUT Partial Differential Equations (in the example and are independent variable. s) PDE involves two or more independent variables ( , ) ( , ) Example: 2 2 x. t

Taylor Expansion u(x + Δx) = u(x) + u΄(x)Δx + 1/2 u˝(x)(Δx)

+ 1/6 u˝΄(x)(Δx) + O(Δx)

u(x – Δx) = u(x) – u΄(x)Δx + 1/2 u˝(x)(Δx) - 1/6 u˝΄(x)(Δx) + O(Δx)

2

3

4

4

2

3

Page 13: Lecture Notes On - BPUT Partial Differential Equations (in the example and are independent variable. s) PDE involves two or more independent variables ( , ) ( , ) Example: 2 2 x. t

Taylor Expansionu΄(x) = u(x) – u(x – Δx) + O(Δx)

Δxu΄(x) = u(x + Δx) – u(x) + O(Δx)

Δxu΄(x) = u(x + Δx) – u(x – Δx) + O(Δx)

2Δx

2

Page 14: Lecture Notes On - BPUT Partial Differential Equations (in the example and are independent variable. s) PDE involves two or more independent variables ( , ) ( , ) Example: 2 2 x. t

Second Derivative Approximation Centered difference: (uj+1 – 2uj + uj-1) / (Δx)

Taylor Expansionu˝(x) = u(x + Δx) – 2u(x) + u(x – Δx) + O(Δx)

(Δx)

2

2

2

Page 15: Lecture Notes On - BPUT Partial Differential Equations (in the example and are independent variable. s) PDE involves two or more independent variables ( , ) ( , ) Example: 2 2 x. t

Function of Two Variables

u(jΔx, nΔt) ~ uj

Backward difference for t and x

(jΔx, nΔt) ~ (uj – uj ) / Δt

(jΔx, nΔt) ~ (uj – uj ) / Δx

n

n n-1

n-1

n

∂u∂t

∂u∂x

Page 16: Lecture Notes On - BPUT Partial Differential Equations (in the example and are independent variable. s) PDE involves two or more independent variables ( , ) ( , ) Example: 2 2 x. t

Function of Two Variables

Forward difference for t and x

(jΔx, nΔt) ~ (uj – uj ) / Δt

(jΔx, nΔt) ~ (uj – uj ) / Δx

n+1

n+1 n

n∂u∂t

∂u∂x

Page 17: Lecture Notes On - BPUT Partial Differential Equations (in the example and are independent variable. s) PDE involves two or more independent variables ( , ) ( , ) Example: 2 2 x. t

Function of Two Variables

Centered difference for t and x

(jΔx, nΔt) ~ (uj – uj ) / (2Δt)

(jΔx, nΔt) ~ (uj – uj ) / (2Δx)

n+1

n+1 n-1

n-1∂u∂t

∂u∂x

Page 18: Lecture Notes On - BPUT Partial Differential Equations (in the example and are independent variable. s) PDE involves two or more independent variables ( , ) ( , ) Example: 2 2 x. t

Partial Differential Equations

Partial Differential Equations (PDEs). What is a PDE? Examples of Important PDEs. Classification of PDEs.

Page 19: Lecture Notes On - BPUT Partial Differential Equations (in the example and are independent variable. s) PDE involves two or more independent variables ( , ) ( , ) Example: 2 2 x. t

Partial Differential Equations

s)t variableindependen are and example the(inst variableindependen moreor twoinvolves PDE

),(),(

:Example

2

2

tx

ttxu

xtxu

∂∂

=∂

A partial differential equation (PDE) is an equation that involves an unknown function and its partial derivatives.

Page 20: Lecture Notes On - BPUT Partial Differential Equations (in the example and are independent variable. s) PDE involves two or more independent variables ( , ) ( , ) Example: 2 2 x. t

Notation

.derivativeorder highest theoforder PDE theofOrder

),(

),(

2

2

2

=∂∂

∂=

∂∂

=

txtxuu

xtxuu

xt

xx

Page 21: Lecture Notes On - BPUT Partial Differential Equations (in the example and are independent variable. s) PDE involves two or more independent variables ( , ) ( , ) Example: 2 2 x. t

Examples of PDEsPDEs are used to model many systems in many different fields of science and engineering.

Important Examples: Laplace Equation Heat Equation Wave Equation

Page 22: Lecture Notes On - BPUT Partial Differential Equations (in the example and are independent variable. s) PDE involves two or more independent variables ( , ) ( , ) Example: 2 2 x. t

Laplace Equation

Used to describe the steady state distribution of heat in a body.

Also used to describe the steady state distribution of electrical charge in a body.

0),,(),,(),,(2

2

2

2

2

2=

∂∂

+∂

∂+

∂∂

zzyxu

yzyxu

xzyxu

Page 23: Lecture Notes On - BPUT Partial Differential Equations (in the example and are independent variable. s) PDE involves two or more independent variables ( , ) ( , ) Example: 2 2 x. t

Heat Equation

∂∂

+∂∂

+∂∂

=∂

∂2

2

2

2

2

2 ),,,(

zu

yu

xu

ttzyxu α

The function u(x,y,z,t) is used to represent the temperature at time t in a physical body at a point with coordinates (x,y,z)

α is the thermal diffusivity. It is sufficient to consider the case α = 1.

Page 24: Lecture Notes On - BPUT Partial Differential Equations (in the example and are independent variable. s) PDE involves two or more independent variables ( , ) ( , ) Example: 2 2 x. t

Simpler Heat Equation

2

2 ),(),(x

txTt

txT∂

∂=

∂∂

T(x,t) is used to represent the temperature at time t at the point x of the thin rod.

x

Page 25: Lecture Notes On - BPUT Partial Differential Equations (in the example and are independent variable. s) PDE involves two or more independent variables ( , ) ( , ) Example: 2 2 x. t

Wave Equation

∂∂

+∂∂

+∂∂

=∂

∂2

2

2

2

2

22

2

2 ),,,(

zu

yu

xuc

ttzyxu

The function u(x,y,z,t) is used to represent the displacement at time t of a particle whose position at rest is (x,y,z) .

The constant c represents the propagation speed of the wave.

Page 26: Lecture Notes On - BPUT Partial Differential Equations (in the example and are independent variable. s) PDE involves two or more independent variables ( , ) ( , ) Example: 2 2 x. t

Classification of PDEsLinear Second order PDEs are important sets of equations that are used to model many systems in many different fields of science and engineering.

Classification is important because: Each category relates to specific engineering

problems. Different approaches are used to solve these

categories.

Page 27: Lecture Notes On - BPUT Partial Differential Equations (in the example and are independent variable. s) PDE involves two or more independent variables ( , ) ( , ) Example: 2 2 x. t

Linear Second Order PDEsClassification

Hyperbolic04

Parabolic04

Elliptic04

:follows as 4 on based classified is

and ,, ,, of functiona is D and of functions are C and B,A,

,0s)t variableindependen-(2 Elinear PDorder second A

2

2

2

2

>−

=−

<−

=+++

ACB

ACB

ACB

AC)(B

uuuyxyx

DuCuBuA

yx

yyxyxx

Page 28: Lecture Notes On - BPUT Partial Differential Equations (in the example and are independent variable. s) PDE involves two or more independent variables ( , ) ( , ) Example: 2 2 x. t

Linear Second Order PDEExamples (Classification)

0sin ,cos

sin ,sin

sin),( :solution possible One

EquationLaplace041,0,1

0),(),(EquationLaplace

2

2

2

2

2

=+

−==

==

=

⇒<−⇒===

=∂

∂+

∂∂

yyxx

xyy

xy

xxx

xx

x

uuyeuyeu

yeuyeu

yeyxu

EllipticisACBCBA

yyxu

xyxu

Page 29: Lecture Notes On - BPUT Partial Differential Equations (in the example and are independent variable. s) PDE involves two or more independent variables ( , ) ( , ) Example: 2 2 x. t

Linear Second Order PDEExamples (Classification)

HyperbolicisACBCBcA

ttxu

xtxuc

ParabolicisACBCBA

ttxu

xtxu

Equation Wave041 ,0 ,0

0),(),(Equation Wave

______________________________________EquationHeat

040 ,0 ,

0),(),(EquationHeat

22

2

2

2

22

2

2

2

⇒>−⇒−==>=

=∂

∂−

∂∂

⇒=−⇒===

=∂

∂−

∂∂

α

α

Page 30: Lecture Notes On - BPUT Partial Differential Equations (in the example and are independent variable. s) PDE involves two or more independent variables ( , ) ( , ) Example: 2 2 x. t

Boundary Conditions for PDEs To uniquely specify a solution to the PDE,

a set of boundary conditions are needed. Both regular and irregular boundaries are

possible.

)sin()0,(0),1(0),0(

0),(),(:EquationHeat 2

2

xxututu

ttxu

xtxu

π

α

===

=∂

∂−

∂∂

region of interest

x1

t

Page 31: Lecture Notes On - BPUT Partial Differential Equations (in the example and are independent variable. s) PDE involves two or more independent variables ( , ) ( , ) Example: 2 2 x. t

The Solution Methods for PDEs Analytic solutions are possible for simple

and special (idealized) cases only.

To make use of the nature of the equations, different methods are used to solve different classes of PDEs.

The methods discussed here are based on the finite difference technique.

Page 32: Lecture Notes On - BPUT Partial Differential Equations (in the example and are independent variable. s) PDE involves two or more independent variables ( , ) ( , ) Example: 2 2 x. t

Parabolic Equations Parabolic Equations Heat Conduction Equation Explicit Method Implicit Method Cranks Nicolson Method

Page 33: Lecture Notes On - BPUT Partial Differential Equations (in the example and are independent variable. s) PDE involves two or more independent variables ( , ) ( , ) Example: 2 2 x. t

Parabolic Equations

04 if parabolic is

and ,, y,x, of functiona is D yandx of functions are C and B,A,

,0) , st variableindependen-(2 linear PDEorder second A

2 =−

=+++

ACB

uuu

DuCuBuAyx

yx

yyxyxx

Page 34: Lecture Notes On - BPUT Partial Differential Equations (in the example and are independent variable. s) PDE involves two or more independent variables ( , ) ( , ) Example: 2 2 x. t

Parabolic Problems

solution.a specify uniquely toneeded are conditionsBoundary *)04( problem lic Parabo*

)sin()0,(0),1(),0(

),(),(:EquationHeat

2

2

2

=−

===

∂∂

=∂

ACB

xxTtTtT

xtxT

ttxT

π

xice ice

Page 35: Lecture Notes On - BPUT Partial Differential Equations (in the example and are independent variable. s) PDE involves two or more independent variables ( , ) ( , ) Example: 2 2 x. t

Finite Difference Methods

t

x

Divide the interval x into sub-intervals, each of width h

Divide the interval t into sub-intervals, each of width k

A grid of points is used forthe finite difference solution

Ti,j represents T(xi, tj) Replace the derivates by

finite-difference formulas

Page 36: Lecture Notes On - BPUT Partial Differential Equations (in the example and are independent variable. s) PDE involves two or more independent variables ( , ) ( , ) Example: 2 2 x. t

Finite Difference Methods

kTT

tTT

ttxT

tT

hTTT

xTTT

xtxT

xT

jijijiji

jijijijijiji

,1,,1,

2,1,,1

2,1,,1

2

2

2

2

),(

:for Formula DifferenceForward

2

)(2),(

:for Formula DifferenceCentral

formulas difference finiteby sderivative theReplace

−=

∆−

≈∂

∂∂∂

+−=

+−≈

∂∂

∂∂

++

+−+−

Page 37: Lecture Notes On - BPUT Partial Differential Equations (in the example and are independent variable. s) PDE involves two or more independent variables ( , ) ( , ) Example: 2 2 x. t

Solution of the Heat Equation• Two solutions to the Parabolic Equation

(Heat Equation) will be presented:

1. Explicit Method:

Simple, Stability Problems.

2. Crank-Nicolson Method:

Involves the solution of a Tridiagonal system of equations, Stable.

Page 38: Lecture Notes On - BPUT Partial Differential Equations (in the example and are independent variable. s) PDE involves two or more independent variables ( , ) ( , ) Example: 2 2 x. t

Explicit Method

( )

),(),()21(),(),(

),(),(2),(),(),(

),(),(2),(),(),(

),(),(

2

2

2

2

2

thxTtxTthxTktxThkDefine

thxTtxTthxThktxTktxT

hthxTtxTthxT

ktxTktxT

xtxT

ttxT

++−+−=+

=

++−−=−+

++−−=

−+∂

∂=

∂∂

λλλ

λ

Page 39: Lecture Notes On - BPUT Partial Differential Equations (in the example and are independent variable. s) PDE involves two or more independent variables ( , ) ( , ) Example: 2 2 x. t

Explicit MethodHow Do We Compute?

meansthxTtxTthxTktxT ),(),()21(),(),( ++−+−=+ λλλ

T(x-h,t) T(x,t) T(x+h,t)

T(x,t+k)

Page 40: Lecture Notes On - BPUT Partial Differential Equations (in the example and are independent variable. s) PDE involves two or more independent variables ( , ) ( , ) Example: 2 2 x. t

Convergence and Stability

( )

.slowit makes This ansmaller th much is that means This

2

210)21( stability, guarantee To

magnified are errors unstable be Can

),(),()21(),(),(:usingdirectly computed be can),(

2

hk

hk

thxTtxTthxTktxTktxT

≤⇒≤⇒≥−

++−+−=++

λλ

λλλ

Page 41: Lecture Notes On - BPUT Partial Differential Equations (in the example and are independent variable. s) PDE involves two or more independent variables ( , ) ( , ) Example: 2 2 x. t

Example 1

( ) ( )

),(4),(7),(4),(

0),(),(4),(),(2),(16

0),(),(),(),(2),(

0),(),(

2

2

2

thxutxuthxuktxu

txuktxuthxutxuthxuk

txuktxuh

thxutxuthxut

txux

txu

++−−=+

=−+−++−−

=−+

−++−−

=∂

∂−

∂∂

Page 42: Lecture Notes On - BPUT Partial Differential Equations (in the example and are independent variable. s) PDE involves two or more independent variables ( , ) ( , ) Example: 2 2 x. t

Example 1),(4),(7),(4),( thxutxuthxuktxu ++−−=+

t=0

t=0.2

t=0.5t=0.7t=1.0

x=0.25

x=0.5

x=0.0

x=0.75

x=1.0

000

00

000

00

Sin(0.25π)

Sin(0. 5π)

Sin(0.75π)

Page 43: Lecture Notes On - BPUT Partial Differential Equations (in the example and are independent variable. s) PDE involves two or more independent variables ( , ) ( , ) Example: 2 2 x. t

Example 19497.0)2/sin(4)4/sin(70)0,5.0(4)0,25.0(7)0,0(4)25.0,25.0(

−=+−=+−=ππ

uuuu

t=0

t=0.2

t=0.5t=0.7t=1.0

x=0.25

x=0.5

x=0.0

x=0.75

x=1.0

000

00

000

00

Sin(0.25π)

Sin(0. 5π)

Sin(0.75π)

Page 44: Lecture Notes On - BPUT Partial Differential Equations (in the example and are independent variable. s) PDE involves two or more independent variables ( , ) ( , ) Example: 2 2 x. t

Crank-Nicolson Method

Method). Explicit the to(compared ,larger use can Weerror). of ionmagnificat (No stable is method The

equations.linear of system lTridiagonaa solving involves method The

kh→

Page 45: Lecture Notes On - BPUT Partial Differential Equations (in the example and are independent variable. s) PDE involves two or more independent variables ( , ) ( , ) Example: 2 2 x. t

Crank-Nicolson Method

22

2 ),(),(2),(),(

),(),(),(:lyrespective formulas and

their withsderivative partial second andfirst the Replace3. widthof lssubinterva into interval the Divide2. widthof lssubinterva into interval the Divide1.

method difference finite theon Based

hthxutxuthxu

xtxu

kktxutxu

ttxu

fferencecentral dibackward

kthx

++−−=

∂∂

−−≈

∂∂

Page 46: Lecture Notes On - BPUT Partial Differential Equations (in the example and are independent variable. s) PDE involves two or more independent variables ( , ) ( , ) Example: 2 2 x. t

Crank-Nicolson Method

( )

),(),(),( )21(),(

),(),(),(),(2),(

),(),(),(),(2),(

becomes ),(),( :ionHeat Equat

222

2

2

2

2

ktxuthxuhktxu

hkthxu

hk

ktxutxuthxutxuthxuhk

kktxutxu

hthxutxuthxu

ttxu

xtxu

−=+−++−−

−−=++−−

−−=

++−−

∂∂

=∂

Page 47: Lecture Notes On - BPUT Partial Differential Equations (in the example and are independent variable. s) PDE involves two or more independent variables ( , ) ( , ) Example: 2 2 x. t

Crank-Nicolson Method

),(),(),()21(),(

:becomes equation Heat then Define 2

ktxuthxutxuthxuhk

−=+−++−−

=

λλλ

λ

u(x-h,t) u(x,t) u(x+h,t)

u(x,t - k)

Page 48: Lecture Notes On - BPUT Partial Differential Equations (in the example and are independent variable. s) PDE involves two or more independent variables ( , ) ( , ) Example: 2 2 x. t

Crank-Nicolson Method

0,41,51,41,3

0,31,41,31,2

0,21,31,21,1

0,11,21,11,0

1,,1,,1

)21( )21( )21( )21(

:)1(fix equations of systema as expanded be can and)21( :as rewritten be can

),(),(),()21(),( :equation The

uuuuuuuuuuuuuuuu

juuuu

ktxuthxutxuthxu

jijijiji

=−++−

=−++−

=−++−

=−++−=

=−++−

−=+−++−−

−+−

λλλ

λλλ

λλλ

λλλ

λλλ

λλλ

Page 49: Lecture Notes On - BPUT Partial Differential Equations (in the example and are independent variable. s) PDE involves two or more independent variables ( , ) ( , ) Example: 2 2 x. t

Crank-Nicolson Method

hxxxuuhxhxhxhxx

uuuu

uuuu

uu

uuuu

ktxuthxutxuthxu

5 and at aluesboundary v theare and 4 and ,3 ,2 ,at

valuesre temperatuinitial theare and , , , where

2121

2121

:equations of system lTridiagonaa as expressed be can),(),(),()21(),(

001,51,0

0000

0,40,30,20,1

1,50,4

0,3

0,2

1,00,1

1,4

1,3

1,2

1,1

+=++++=

+

+

=

+−−+−

−+−−+

−=+−++−−

λ

λ

λλλλλ

λλλλλ

λλλ

Page 50: Lecture Notes On - BPUT Partial Differential Equations (in the example and are independent variable. s) PDE involves two or more independent variables ( , ) ( , ) Example: 2 2 x. t

Crank-Nicolson Method

etc. ,3at valuesre temperatucompute tostep above Repeat the and , , , compute To

2121

2121

)2( equations of system al tridiagonseconda Solve2at valuesre temperatu thecompute Toat and , , , valuesre temperatuThe

:produces system al tridiagon theof solution The

0

2,42,32,22,1

2,51,4

1,3

1,2

2,01,1

2,4

2,3

2,2

2,1

0

01,41,31,21,1

ktuuuu

uuuu

uu

uuuu

jktt

kttuuuu

+

+

+

=

+−−+−

−+−−+

=+=

+=

λ

λ

λλλλλ

λλλλλ

Page 51: Lecture Notes On - BPUT Partial Differential Equations (in the example and are independent variable. s) PDE involves two or more independent variables ( , ) ( , ) Example: 2 2 x. t

Example 2

]1,0[],1,0[for ),( find to25.0,25.0method Nicolson-Crank using Solve

)sin()0,(0),1(),0(

0),(),(

: PDE theSolve

2

2

∈∈==

===

=∂

∂−

∂∂

txtxukhUse

xxututu

ttxu

xtxu

π

Page 52: Lecture Notes On - BPUT Partial Differential Equations (in the example and are independent variable. s) PDE involves two or more independent variables ( , ) ( , ) Example: 2 2 x. t

Example 2Crank-Nicolson Method

( ) ( )

1,,1,,1

2

2

2

2

4 9 4),(),( 4),( 9),( 4

4

0),(),(4),(),(2),(16

),(),(),(),(2),(

0),(),(

−+− =−+−−=+−+−−

==

=−−−++−−

−−=

++−−

=∂

∂−

∂∂

jijijiji uuuuktxuthxutxuthxu

hkDefine

ktxutxuthxutxuthxuk

ktxutxuh

thxutxuthxut

txux

txu

λ

Page 53: Lecture Notes On - BPUT Partial Differential Equations (in the example and are independent variable. s) PDE involves two or more independent variables ( , ) ( , ) Example: 2 2 x. t

Example 2

)4/3sin(94 494)2/sin(494494)4/sin( 49 494

1,31,20,31,41,31,2

1,31,21,10,21,31,21,1

1,21,10,11,21,11,0

π

π

π

=+−⇒=−+−

=−+−⇒=−+−

=−⇒=−+−

uuuuuuuuuuuuu

uuuuuu

Page 54: Lecture Notes On - BPUT Partial Differential Equations (in the example and are independent variable. s) PDE involves two or more independent variables ( , ) ( , ) Example: 2 2 x. t

Example 2Solution of Row 1 at t1=0.25 sec

=

=

−−−

=

21151.029912.021151.0

)75.0sin()5.0sin()25.0sin(

94494

49

:equations of system al tridiagonfollowing theofsolution theis sec 25.0at PDE theof Solution The

1,3

1,2

1,1

1,3

1,2

1,1

1

uuu

uuu

t

πππ

Page 55: Lecture Notes On - BPUT Partial Differential Equations (in the example and are independent variable. s) PDE involves two or more independent variables ( , ) ( , ) Example: 2 2 x. t

Example 2: Second Row at t2=0.5 sec

21151.094 49429912.049449421151.0 49 494

2,32,21,32,42,32,2

2,32,22,11,22,32,22,1

2,22,11,12,22,12,0

=+−⇒=−+−

=−+−⇒=−+−

=−⇒=−+−

uuuuuuuuuuuuu

uuuuuu

Page 56: Lecture Notes On - BPUT Partial Differential Equations (in the example and are independent variable. s) PDE involves two or more independent variables ( , ) ( , ) Example: 2 2 x. t

Example 2Solution of Row 2 at t2=0.5 sec

=

=

=

−−−

=

063267.0089473.0063267.0

21151.029912.021151.0

94494

49

:equations of system al tridiagonfollowing theofsolution theis sec 5.0at PDE theof Solution The

2,3

2,2

2,1

1,3

1,2

1,1

2,3

2,2

2,1

2

uuu

uuu

uuu

t

Page 57: Lecture Notes On - BPUT Partial Differential Equations (in the example and are independent variable. s) PDE involves two or more independent variables ( , ) ( , ) Example: 2 2 x. t

Example 2Solution of Row 3 at t3=0.75 sec

=

=

=

−−−

=

018924.0026763.0018924.0

063267.0089473.0063267.0

94494

49

:equations of system al tridiagonfollowing theofsolution theis sec 75.0at PDE theof Solution The

3,3

3,2

3,1

2,3

2,2

2,1

3,3

3,2

3,1

3

uuu

uuu

uuu

t

Page 58: Lecture Notes On - BPUT Partial Differential Equations (in the example and are independent variable. s) PDE involves two or more independent variables ( , ) ( , ) Example: 2 2 x. t

Example 2Solution of Row 4 at t4=1 sec

=

=

=

−−−

=

0056606.00080053.00056606.0

018924.0026763.0018924.0

94494

49

:equations of system al tridiagonfollowing theofsolution theis sec 1at PDE theof Solution The

4,3

4,2

4,1

3,3

3,2

3,1

4,3

4,2

4,1

4

uuu

uuu

uuu

t

Page 59: Lecture Notes On - BPUT Partial Differential Equations (in the example and are independent variable. s) PDE involves two or more independent variables ( , ) ( , ) Example: 2 2 x. t

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