7/24/2019 Finite Element Legendre Wavelet Galerkin Solution of Non

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Finite Element Legendre Wavelet Galerkin Solution of Non-Fourier Heat

Transfer moel in Fin with Internal Heat Generation and Periodi !oundar"

#ondition

$%strat&

'e"ward&

() Introdution*) +athematial formulation of the ,ro%lem

The parabolic and hyperbolic heat transfer equations for the n are given as

cT

t=k

2T

x2

hP

A c(TT )+q

, (1)

c [ Tt+2T

t2 ]=k

2T

t2

hP

Ac(TT )

hP

A c

T

t+q+

q

t , (2)

subjected to the initial conditions

T(x ,0 )=T , (3)

T(x ,0)

t =0

, (4)

Boundary conditions

T(0, t)=Tb ,m+A (Tb ,mT)cos (t) ()

T(L ,t) x

=0, (!)

"here q

=q(1+ (TT ))

#ntroducing the di$ensionless variable and si$ilarity criteria

X=x

L,Fo=

k t

c L2,=

TTTb , mT

, N=h P L

2

k Ac,G=

qAc

h P(Tb ,mT), =

k

c L2

7/24/2019 Finite Element Legendre Wavelet Galerkin Solution of Non

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G= (Tb ,mT ) ,=c L

2

k % (&)

The syste$ of equations (1)'(!) governing the process of parabolic and hyperbolic

heat transfer in n can be "ritten in di$ensionless for$ as follo"s

Fo=

2

X2N2 (1G G ) +N

2G , ()

2

Fo2+ {1+ N2 (1G G )}

Fo=

2

X2N2 (1G G)+N

2G ,

(*)

(x ,0 )=0 , (10)

(X ,0) Fo

=0, (11)

(0,Fo )=1+Acos ( Fo) (12)

(1,Fo)

X =0, (1+)

pplying nite di-erence in space coordinate, that can be "ritten in follo"ing for$

d i

Fo=

i+12i+i1h2

N2 (1G G) i+N2G ,1

7/24/2019 Finite Element Legendre Wavelet Galerkin Solution of Non

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%

%

d9

Fo

=102

9+

8

h

2 N2 (1G G)9+N

2G , for the values of the "e are using the

sy$$etric condition (1,Fo)

X =2110+139+17897=0 (/ive point

for$ula)% Thi0s gives us10=

1

21(139+17897 )

d

Fo=[d 1 Fo ,

d2

Fo,

d3

Fo, ,

d9

Fo] and = [1, 2 ,3 ,,9]

d

d Fo=A1+

(14)

This i$plies that

1'23h2'5 13h2 6 6 6 6 6 6 67

13h2 '23h2'5 13h2 6 6 6 6 6 67

6 13h2 '23h2'5 13h2 6 6 6 6 67

6 6 13h2 '23h2'5 13h2 6 6 6 67

6 6 6 13h2 '23h2'5 13h2 6 6 67

6 6 6 6 13h2 '23h2'5 13h2 6 67

6 6 6 6 6 13h2 '23h2'5 13h2 67

6 6 6 6 6 6 13h2 '23h2'

5 13h27 6 6 6 6 6 6 '*3(218h2)

+3(218h2) '2*3(218h2)'597

B:28;

7/24/2019 Finite Element Legendre Wavelet Galerkin Solution of Non

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(0 )=0 ,

(0) Fo

=0,

1'23h2'5 13h2 6 6 6 6 6 6 67

13h2 '23h2'5 13h2 6 6 6 6 6 67

6 13h2 '23h2'5 13h2 6 6 6 6 67

6 6 13h2 '23h2'5 13h2 6 6 6 67

6 6 6 13h2 '23h2'5 13h2 6 6 67

6 6 6 6 13h2 '23h2'5 13h2 6 67

6 6 6 6 6 13h2 '23h2'5 13h2 67

6 6 6 6 6 6 13h2 '23h2'

5 13h27 6 6 6 6 6 6 '*3(218h2)

+3(218h2) '2*3(218h2)'597B:28;

7/24/2019 Finite Element Legendre Wavelet Galerkin Solution of Non

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!T=d*/)0 (A1,- ,.) is unknown onstant) The solution of Para%oli heat

euation is

(x , Fo )=!TP"(X) 1

*) For h",er%oli &5et us assu$e that

d2

d Fo2=!T"(X)

#ntegrating "ith respect to ? and using initial conditiond

d Fo=!TP"(X)

gain using #ntegrating "ith respect to ?

=!T

P2

"(X)

#$b#tit$ti%& th'()*$' o+ d

2

d Fo2,

d

d Fo

in a%ove euation .(301 we get

!T

"(X)+ 1+ N2 (1G G)!TP"(X)=A1!TP2"(X)+ dT"(X) 1

)bo(' 'q$)tio%2'd$c'd )#+o**o3i%& +o2m

!T+ {1+ N2 (1G G )}!TP=A1!TP2+ dT 1

!T

[ 4+ {1+ N2

(1G G )}P ]=A1!T

P2

+ dT

1 where I is identit" matri2 of

order of 4)

!T=A1!TP2 [ 4+ {1+ N2 (1G G )}P ]

1+ dT[ 4+ {1+ N2 (1G G )}P ]

1

A1!TP

2 [ 4+ {1+ N2 (1G G )}P ]1!T+ dT[ 4+ {1+ N2 (1G G )}P ]

1=0

-T=P2[ 4+{1+ N2 (1G G )}P ]

1

and .= dT[ 4+{1+N2 (1G G)}P ]

1

1 so

that the avo%e euation redues in s"lvestor matri2 form i)e)

A1!T

-T!T+.=0

Th'()*$' o+ $%k%o3% co%#t)%t i#

!T=d*/)0(A1,- , .)

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3h'2' d*/)0i# th'i%b$i*t +$%ctio%5ATLA1 The solution of Para%oli heat

euation is

(x , Fo )=!TP"(X) 1