J. Appl. Environ. Biol. Sci., 4(7S) , , TextRoad Publication

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1 J Appl Environ Biol Sci, 4(7S)379-39, 4 4, TexRoad Publicaion ISSN: Journal of Applied Environmenal and Biological Sciences wwwexroadcom Applicaion of Opimal Homoopy Asympoic Mehod o Convecive Radiaive Fin wih Temperaure Dependen Thermal Conduciviy H Ullah, S Islam, M Fiza, SA Khan Deparmen of Mahemaics, Abdul Wali Khan Universiy Mardan, Khyber Pakhunkhwa, Pakisan Deparmen of Compuer Science, Abdul Wali Khan Universiy Mardan, Khyber Pakhunkhwa, Pakisan Received: Sepember, 4 Acceped: November 3, 4 ABSTRACT Applicaion of Opimal Homoopy Asympoic Mehod (OHAM), a new analyic approximae echnique has been applied o convecive-radiaive fin wih emperaure dependen hermal conduciviy OHAM has he beauy o conrol he convergence of approximae soluions when i is compared wih oher mehods such as Homoopy Analysis Mehod (HAM), Adomian Decomposiion Mehod (ADM) and Homoopy Perurbaion Mehod (HPM) KEYWORDS: OHAM, Exac, ADM, HPM, HAM, Non-linear differenial equaions INTRODUCTION The engineering problems arising in convecive radiaive fins are nonlinear Some of analyic mehods are available in lieraure like Adomian Decomposiion Mehod (ADM) [], Variaional Ieraion Mehod (VIM)[], Differenial Transform Mehod (DTM) [3],Homoopy Perurbaion Mehod (HPM)[4,5] and Homoopy Analysis Mehod (HAM) [6] These mehods needs assume iniial soluion which is a problem The Perurbaion Mehods were sudied [7-9] for nonlinear boundary value problems (BVPs) These mehods conain a small parameer and are difficul o found Vasile Marinca e al inroduce Opimal Homoopy Asympoic Mehod (OHAM) [-4] for he soluion of nonlinear BVPs which is independen of he assumpion of small parameer and iniial guess soluion OHAM has been proved o be a powerful echnique for soluion of nonlinear BVPs [5-]The moivaion of his aricle is o use OHAM heory for convecive radiaive fins BVPs The srucure of aricle is as he basic idea of OHAM is discussed in secion and in secion 3 have been implemened o BVPs Basic Mahemaical Theory of OHAM Consider a general form of nonlinear differenial equaions L χ ς + s ς + N χ ς = () wih ( ( )) ( ) ( ) ( ), dχ ϒ χ, = dς () WhereL is linear, χ ( ς ) is unknown funcion, s( ς ) is known funcion, N ( χ ( ς )) The homoopy ϑ ( ς, m) : Ξ [,] Θ gives is nonlinear operaors * Corresponding Auhor: H Ullah, Deparmen of Mahemaics, Abdul Wali Khan Universiy Mardan, Khyber, Pakhunkhwa, Pakisan 379

2 Islam e al,4 ( ) ( ς ) ( ϑ ( ς, m) ) + s ( ς ) N ( ϑ ( ς, m) ) ( ) L ϑ ( ς, ) m m + s = ( ) K m (3) L +, (, m) ϑ ( ς, m), ϑ ς ϒ = ς,, (4) here m [ ] is embedding parameer, ϑ ( ς,m) is unknown funcion, ( ) funcion The soluion ϑ ( ς,m) varies from χ ( ς ) o ( ) L ( χ ( ς )) s ς χ (5) The auxiliary funcion is dχ + ( ) =, ϒ, = d ( ) 3 K m = ml + m l + m l + (6) 3 Where l, l, l 3, are opimal consans Using Taylor s series o ϑ ( ς,m) abou m ( ) = ( ) + ( ) i ϑ ς, m, l, l, l χ ς χ ς, l, l, l m, m i m i= K m is nonzero auxiliary χ ς, where u ( ) is zeroh order soluion for p = : i =,, (7) Using Eq (7) ino Eq ()-() and equaing he coefficien of like powers of m, we obain L χ ς = l N χ ς, ( ( )) ( ( )), (8) dχ ϒ χ, = d L χ ς L χ ς = l N χ ς + ( ( )) ( ( )) ( ( )) ( χ ( ς )) + ( χ ( ς ), χ ( ς )), l L N (9) dχ ϒ χ, =, d L χ ς L χ ς = l N χ ς + ( k ( )) ( k ( )) k ( ( )) k l ( χ ( ς )) + ( χ ( ς ) χ ( ς ) χ ( ς )) L N,,,, i k k i k i= () where N ( χ ( ς ), χ ( ς ),, χ ( ς )) k i dχk ϒ χk, =, k =,3,, d k i is he coefficien of k i m in he expansion series 38

3 J Appl Environ Biol Sci, 4(7S)379-39, 4 ς, ς, ς, k N ( ϑ ( ς, m, li )) = N ( ς ( ς )) + N k m, k, ς k i =,, 3, () I should be noed ha he convergence of Eq (7) depends uponl, l, If Eq (7) converges a q =, we have χ% ς, l, l, l χ ( ς ) χ ς, l, l, l ( ) ( ) = + i k i k () Subsiuing Eq () ino Eq (), so he residual is L ( χ% ( ς, l, l, li )) + s( ς ) R ( ς, l, l, li ) = + ( χ ( ς, l, l, li )) N % (3) Differen mehods like Galerkin s Mehod, Riz Mehod, Leas Squares Mehod and Collocaion Mehod are used for finding auxiliary consans, l, i =,,, m, Here we apply he Mehod of Leas Squares as d (,, ) ( ς,,, ) Q l l l = R l l l dς, k c (4) where c and d are wo disinc values The auxiliary consans l, i,,, m i Q Q Q = = = = (5) l l l i k = may be calculaed as from Eq (4) k These opimal consans can be used o find k h order approximae soluion 3 Applicaion of OHAM o convecive radiaive fin wih emperaure dependen hermal conduciviy The efficiency and effeciveness of OHAM formulaion are demonsraed by wo models Model 3 Consider he non-dimensional form of he emperaure dependen hermal conduciviy hrough a fin aken from [3] d u 4 n u ε u = d,(3) wih boundary condiions du ( ) u ( ) =, d = (3) Zeroh Order Problem: ( ) ( ) = (33) u ( ) =, ( ) u n u Is soluion is ec osh ( ) u ( ) = (35) + e Firs Order Problem: u ( ) + ( + l ) u ( ) u ( ) =, 4 lu ( ) ( + l ) u ( ) (36) u = (34) 38

4 Islam e al,4 u ( ) =, ( ) u = (37) Whose soluion is 3 e + e 5 9e 7 9 e e + + Cosh( ) u ( ) = C 5 5 5( + e ) e ( + e ) 45 + Cosh ( ) + Cosh( 4 ) (38) Second Order Problem: u = u + + C u + C 3 + 4C u u ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ( ) ( ) ( )) 4 C q + q q, (39) u ( ) =, ( ) (3) u = We obain he following soluion ( + e )( 6 e )( + e ) (e + e e e + e + e u ( ) = 9 e + 9 e e e 9 e 9 e e 9 ( + e ) e e e + e + e + e + e ) (5 e + e + 5e + 7 e + 7 e -8 e e e e + 3 e + 3 e e -7 6e e + 5 e e e -76 e - 8 e -6 e -8 e -5 e e + 4 e + 4 e - 7 e -5 e e e e + 7 e e e e e e e e e e e - 5 e - 7 e + 4 e + 4 e - 7 e e -8 e - 6 e -76 e e + 5 e + 5 e e + 3 e e + 3 e + 5 e + e + 5e e ( )+ e ( )- e ( ) e ( )-8 e ( )+ 8 e ( ) 6e (+e ) (e +e e e +e +e e +9e e e + e ) (3) C e 9e 9e e +9e e e +e +e + C C 38

5 J Appl Environ Biol Sci, 4(7S)379-39, 4 Adding Eqs (35), (38) and (3) forε = = n, we have (+e )( e ( + e ) (e +e e e + e + e u ( ) = e +9e e e 9 e 9e e C 9 9 ( + e ) e e e +e +e +e +e (5e + e +5e +7e +7e -8e -636e +386e e +3e +3e +46e -76e - 636e +5e e -636e -76e - 8e -6e -8 e -5e e +4e +4e -7e -5e +845e e +845e +7e +684e +574e -468e e +574e +684e +845e +386e +386e C e -5e -7e +4e +4e -7e e -8e -6e -76e -636e +5e +5e e +3e +46e +3e +5e +e +5e e (-979+6)+e ( )-e (4643+6) e (979+6)-8e ( )+8 e ( ) 6e (+e ) (5e )(+e ) (+e )+( e e +e + e ) (e + e +e 9e +e +e +9e +9e +e e + e +e e e e e ) (3) Using Eq (3) in (3) and applying procedure in Eqs (3)-(5), we have C = and C = Subsiuing hese in Eq (3),we obain C u( ) = Cosh ( ) Cosh ( 3) Cosh ( 4 ) +773 Cosh ( 5 ) Cosh ( 7) Cosh ( ) -4 ) Sinh ( ) 84 Sinh ( 4) (33) Soluion by Double Opimal Linearizaion Mehod (DOLM) [3] coshv u ( ) _ DOLM =, coshv where ( 5sinh ( n) + 5sinh ( 3n) + 3sinh ( 5n) ) ( n + ( n) ) ( n) ε V = n + 6 sinh cosh 3 (34) 383

6 Islam e al, OHAM PM Numerical DOLM u Fig Comparison of OHAM and exac resul for ε =, n = 9 8 u n5 n n u ( ) Fig OHAM resul for differen values of n and ε = 9 8 u n5 n n u ( ) Fig 3 OHAM resul for differen values of n and ε = 4 384

7 J Appl Environ Biol Sci, 4(7S)379-39, 4 9 u n5 n n u ( ) Fig 4 for differen values of n and ε = 6 Model 3 [3] Assuming he hermal conduciviy of he form ( β ) k k T = +, (3) where he consan β is a measure of he hermal conduciviy Using k = + εu k, (3) ε = βt where b The differenial equaions now become d ( ) 4 ( ) du + εu n u ( ) εu ( ) = d d, (33) wih boundary condiions du u ( ) = ( ) =, d (34) According o Eq (), we define he operaors 4 L u = u n u N u = ε u u + u u ( ) ( ( ) ( ) ( ) ( )) ( ( )) ( ) ( ) g ( ) =,, ( ) u ( ) u ( ) ( ) where and represen he firs and second derivaives of Zeroh Order Problem: u ( ) n u ( ) u ( ) = ( ), u = (37) ecosh ( ) Is soluion is u ( ) = =, (36) + e (38) u wih respec o Firs Order Problem: u ( ) u ( ) u ( ) C u ( ) ( u 3 = ) ( ) u ( ) + C ( u ( ) + u ( ) ), (39) u ( ) = ( ), u = (3), (35) 385

8 Islam e al,4 ( ) ( ) 4 8 -e(9 + e + 9e )Cosh + 4 Whose soluion is u ( ) = e(+e )(+e ) Cosh (3) 5 5( + e ) - e -45+Cosh ( 4 ) C Second Order Problem: u ( ) = u ( ) + Cu ( ) u ( ) + + C ( + u ( ) ) u ( ) 3 u ( ) + C ( + 4u ( ) ) u ( ) 4 C u ( ) u ( ) ( u ) ( ) ( u ( ) u + + ( )), (3) u ( ) = ( ) u = (33) We obain he following soluion u ( ) e = 8 7 ( + e ) (e )(+e ) (e +e +9e +9e e e e e + e + 9e 9e 9e + e + 9e e e (5+ 6 ) (7+ 6 ) (3+ 8 ) (5+ 8 ) e e +e +e (+ ) (3+ ) (5+ ) (7+ ) (4+ 3 ) (8+ 3 ) (3+ 4 ) 9 (5+ 4 ) (4+ 5 ) (8+ 5 ) (+ 6 ) (3+ 6 ) (e + e + e - 963e -963e + 9e - 3e -35e e e - 4e -84e -35e -35e -84e e + 68 e - 3 e + 3 e + 3 e - 3 e e + 35 e e + 9 e - 45 e + 9 e e - 4 e +744 e -96 e +346 e e e +744e - 963e -5e +88e -88e e + 88 e - 5 e - 4 e e e +346 e +346 e -96 e +744 e -4 e e +3465e +9e - 45e +3465e +35 e e -3e +3e +3 e +68 e - 35 e e - 3 e - 84 e + e + e + e e ( )-6 e ( ) - 6 e ( ) e ( ) + 6 e ( ) + e ( ) e ( ) - e (8 + 6 ) - e ( ) e (-4+)-36e (4+6 ) + e ( ) e ( ) + e ( ) C C 386

9 J Appl Environ Biol Sci, 4(7S)379-39, (+ ) (3+ ) (5+ ) (e )(+e ) (e +e +9e +9e e -e -e - (7+ ) (4+ 3 ) (8+ 3 ) (3+ 4 ) (5+ 4 ) (4+ 5 ) + e + e + 9e - 9e - 9e + e + (8+ 5 ) (+ 6 ) (3+ 6 ) (5+ 6 ) (7+ 6 ) (3+ 8 ) (5+ 8 ) 9e - e - e - e - e +e + e C (34) From Eqs (338), (33) and (334) by adding, we obain: (3 e( +e ) Cosh ( )- e(9 + e + 9e )Cosh ( ) u ( ) = C ( + e ) + (e+e ) ( +e )Cosh ( ) -e (45-Cosh ( ) e 8 7 ( + e ) (e )(+e ) (e +e +9e +9e -e -e -e - e + e + 9e - 9e - 9e + e + 9e - e - e (5+ 6 ) (7+ 6 ) (3+ 8 ) (5+ 8 ) -e - e + e +e (+ ) (3+ ) (5+ ) (7+ ) (4+ 3 ) (8+ 3 ) (3+ 4 ) 9 (5+ 4 ) (4+ 5 ) (8+ 5 ) (+ 6 ) (3+ 6 ) (e + e + e 963e 963e + 9e 3e 35e -963e e 4e 84e 35e 35e 84e e +68 e 3 e + 3 e + 3 e 3 e e + 35 e e + 9 e 45 e + 9 e e 4 e +744 e 96 e +346 e e e +744e 963e 5e +88e 88e e + 88 e 5 e 4 e e e +346 e +346 e 96 e +744 e 4 e e +3465e +9e 45e +3465e +35 e e 3e +3e +3 e +68 e 35 e e 3 e 84 e + e + e + e e (6874 ) 6 e ( ) 6 e ( ) e ( ) + 6 e ( ) + e ( ) e ( ) e (8 + 6 ) e ( ) e ( 4+) 36e (4+6 ) + e ( ) e ( ) + e ( ) C C (+ ) (3+ ) (5+ ) (e )(+e ) (e +e +9e +9e e e e (8+ 5 ) (+ 6 ) (3+ 6 ) (5+ 6 ) (7+ 6 ) (3+ 8 ) (5+ ) 9e e e e e +e +e (7+ ) (4+ 3 ) (8+ 3 ) (3+ 4 ) (5+ 4 ) (4+ 5 ) + e + e + 9e 9e 9e + e + Using he echnique as discussed in Eqs (3)- (5), we obain C (35) 387

10 Islam e al,4 C = , C = Eq (35) becomes u = Cosh Cosh 3 ( ) ( ) ( ) ( ) ( ) 7 ( ) 9 ( ) ( ) +687 Cosh Cosh Cosh 7 + Cosh ( ) + (496 ) Sinh (36) Soluion by Double Opimal Linearizaion Mehod (DOLM) [] coshv u ( ) _ DOLM =, (37) coshv where ( 5sinh ( n) + 5sinh ( 3n) + 3sinh ( 5n) ) ( n + ( n) ) ( n) ε V = n + 6 sinh cosh V 3 ( 3sinhV + 5sinh 3V ) ( V + V ) V ε = + 3 sinh cosh, u OHAM PM DOLM Numerical Fig 5Comparison of OHAM o Exac for n =, ε = u u ( ) Fig 6 Plo of wih respec o for n = 5 388

11 J Appl Environ Biol Sci, 4(7S)379-39, u u ( ) Fig 7 Plo of wih respec o for n = u u ( ) Fig 8 Plo of wih respec o for n = 5 4 RESULTS AND DISCUSSIONS The OHAM heory gives well correc soluion of he problems given in 3 For compuaional purpose we used Mahemaics 7Figs and 5 give he comparisons of OHAM and Exac soluions for Model 3 and 3 respecively While Figs -4 gives he variaion of u ( ) agains for differen values n a ε =, 4, 6 respecively for model 3 The behavior of soluion of he problem in model 3 can be seen in Figs 5-8 for differen values of ε a n = 5,,5 respecively To verify he accuracy of his mehod, OHAM resuls are compared wih exac soluion in Figs and 5 proving is effeciveness and efficiency 5 Conclusion In his aricle, OHAM has been proved a powerful ool o solve he boundary value problem arising from convecive-radiaive fin wih emperaure dependen hermal conduciviy We have operaed he mahemaical heory of OHAM o model 3 and 3, found i simpler, easy o convergence and conain less compuaional work Hence OHAM shows is poenial for solving srongly nonlinear BVPs 389

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