Analytical Solution for the Time-Dependent Emden-Fowler Type of Equations by Homotopy Analysis Method with Genetic Algorithm

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1 Applied Maheaics hp:// ISSN Online: ISSN Prin: Analyical Soluion for he Tie-Dependen Eden-Fowler Type of Equaions by Hooopy Analysis Mehod wih Geneic Algorih Waleed Al-Hayani Laheeb Alzubaidy Ahed Enesar Deparen of Maheaics College of Copuer Science and Maheaics Universiy of Mosul Mosul Iraq Deparen of Sofware Engineering College of Copuer Science and Maheaics Universiy of Mosul Mosul Iraq How o cie his paper: Al-Hayani W. Alzubaidy L. and Enesar A. (7) Analyical Soluion for he Tie-Dependen Eden-Fowler Type of Equaions by Hooopy Analysis Mehod wih Geneic Algorih. Applied Maheaics hps://doi.org/.436/a Received: Deceber 3 6 Acceped: May 7 Published: May 5 7 Copyrigh 7 by auhors and Scienific Research Publishing Inc. This work is licensed under he Creaive Coons Aribuion Inernaional License (CC BY 4.). hp://creaivecoons.org/licenses/by/4./ Open Access Absrac In his paper Hooopy Analysis ehod wih Geneic Algorih is presened and used o obain an analyical soluion for he ie-dependen Eden-Fowler ype of equaions and wave-ype equaion wih singular behavior a =. The advanage of his single global ehod eployed o presen a reliable fraework is uilized o overcoe he singulariy behavior a he poin = for boh odels. The ehod is deonsraed for a variey of probles in one and higher diensional spaces approiae-eac soluions are obained. The resuls obained in all cases show he reliabiliy and he efficiency of his ehod. Keywords Hooopy Analysis Mehod Geneic Algorih Eden-Fowler Equaion Wave-Type Equaion Adoian Polynoials Noise Ters Padé Approians Sipson Rule. Inroducion In any previous works of he diffusion of hea perpendicular o he surfaces of parallel planes are odeled by he hea equaion r r y + af g y + h = y < L < < T r> () or equivalenly r y + y + af( ) g( y) + h( ) = y < L < < T r> () f( ) g( y) + h( ) is he nonlinear hea source ( ) y is he e- DOI:.436/a May 7

2 peraure and is he diensionless ie variable. For he seady-sae case and for given by h = Equaion () is he Eden-Fowler equaion [] [] [3] r = f ( ) and n f ( ) = and g( y) y y + y + af g ( y) = y ( ) = y y = (3) g y are soe given funcions of and y respecively. For = Equaion (3) is he sandard Lane-Eden equaion ha was used o odel he heral behavior of a spherical cloud of gas acing under he uual aracion of is olecules [] and subjec o he classical laws of herodynaics. For oher special fors of g( y ) he well-known Lane- Eden equaion was used o odel several phenoena in aheaical physics and asrophysics such as he heory of sellar srucure he heral behavior of a spherical cloud of gas isoheral gas spheres and heory of herionic currens [] [] [3]. A subsanial aoun of work has been done on his ype of probles for various srucures of g( y ) in []-[7]. On he oher hand he wave ype of equaions wih singular behavior is given by r ( ) r y + af g y + h = y < L < < T r> (4) or equivalenly r y + y + af( ) g( y) + h( ) = y < L < < T r> (5) will be eained as well f( ) g( y) + h( ) is a nonlinear source is he diensionless ie variable and y( ) is he displaceen of he wave a he posiion and a ie. The singulariy behavior ha occurs a he poin = is he ain difficuly in he analysis of Equaions () and (5). Wazwaz [8] used he Adoian decoposiion ehod (ADM) o ge an analyical soluion for he ie-dependen Eden-Fowler ype of equaions and wave-ype equaion wih singular behavior. The ain objecive of his paper is o apply Hooopy Analysis Mehod (HAM) o obain approiae-eac soluions for differen odels for he iedependen Eden-Fowler ype of equaions and wave-ype equaion wih singular behavior a =. While he VIM [9] [] requires he deerinaion of Lagrange uliplier in is copuaional algorih HAM is independen of any such requireens HAM handles linear and nonlinear ers in a siple and sraighforward anner wihou any addiional requireens. Also in his paper we apply Geneic Algorih (GA) o obain an approiae soluion of he sae equaions. In wha follows we give a brief review of Hooopy analysis ehod and Geneic algorih.. Analysis of he Hooopy Analysis Mehod To describe he basic ideas of he HAM we consider he following differenial 694

3 equaion: = N y k (6) N is a nonlinear operaor and denoe he independen variables y( ) is an unknown funcion and k( ) is a known analyic funcion. For sipliciy we ignore all boundary or iniial condiions which can be reaed in he siilar way. By eans of generalizing he radiional Hooopy ehod Liao [] consrucs he so called zero-order deforaion equaion ( q) L φ( q) y ( ) = qh N φ( q) k( ) ; ; q [ ] is an ebedding paraeer h is a nonzero auiliary paraeer L is an auiliary linear operaor y ( ) is an iniial guess of y( ) and φ ( q ; ) is an unknown funcion. I is iporan ha one has grea freedo o choose auiliary objecs such as h and L in HAM. Obviously when q = and q = i holds φ ; = y φ ; = y (8) respecively. Thus as q increases fro o he soluion ( q ; ) he iniial guess y ( ) o he soluion y( ). Epanding ( q ; ) Taylor series wih respec o q we have + = (7) φ varies fro φ in φ q ; = y + y q (9) y ( ) ( q ; ) =.! q q= () If he auiliary linear operaor he iniial guess he auiliary paraeer h and he auiliary funcion are so properly chosen he above series (9) converges a q = hen we have φ ( ) y ( ) y ( ) ; = + () which us be one of he soluions of he original nonlinear equaion as proved by Liao []. If h = Equaion (7) becoes + = ( q) L φ( q) y ( ) q N φ( q) k( ) ; + ; = which is used osly in he Hooopy perurbaion ehod []. According o Equaion () he governing equaion can be deduced fro he zero-order deforaion Equaion (7). We define he vecors { y( ) y( ) yi( ) } () y =. (3) i Differeniaing Equaion (7) ies wih respec o he ebedding paraeer q and hen seing q = and finally dividing he by! we have he socalled h-order deforaion equaion = L y χy hr y (4) 695

4 R ( ) { N φ ( q ; ) k( ) } =! q (5) q= and χ = (6) >. I should be ephasized ha y ( ) are governed by he linear Equaion (4) wih he linear boundary condiions ha coe fro he original proble which can be easily solved by sybolic copuaion sofwares such as Maple and Maheaica. 3. Geneic Algorihs Definiion Geneic Algorihs are search and opiizaion echniques based on Darwin s Principle of Naural Selecion. Definiion Geneic Algorih Operaors [3] [4] The siples for of geneic algorih involves hree ypes of operaors: selecion crossover and uaion. Selecion: This operaor selecs chroosoes in he populaion for reproducion. The fier he chroosoe he ore ies i is likely o be seleced o reproduce. Crossover: This operaor randoly chooses a locus and echanges he subsequences before and afer ha locus beween wo chroosoes o creae wo offspring. For eaple he srings and could be crossed over afer he hird locus in each o produce he wo offspring and. The crossover operaor roughly iics biological recobinaion beween wo single-chroosoe (haploid) organiss. Muaion: This operaor randoly flips soe of he bis in a chroosoe. For eaple he sring igh be uaed in is second posiion o yield. Muaion can occur a each bi posiion in a sring wih soe probabiliy usually very sall (e.g..). Algorih 3 A Siple Geneic Algorih [3] [4] Given a clearly defined proble o be solved and a bi sring represenaion for candidae soluions a siple GA works as follows:. Sar wih a randoly generaed populaion of n l-bi chroosoes (candidae soluions o a proble).. Calculae he finess f( ) of each chroosoe and in he populaion. 3. Repea he following seps unil n offspring have been creaed: (a) Selec a pair of paren chroosoes fro he curren populaion he probabiliy of selecion being an increasing funcion of finess. Selecion is done wih replaceen eaning ha he sae chroosoe can be seleced ore 696

5 han once o becoe a paren. (b) Wih probabiliy pc (he crossover probabiliy or crossover rae ) cross over he pair a randoly chosen poin (chosen wih unifor probabiliy) o for wo offspring. If no crossover akes place for wo offspring ha are eac copies of heir respecive parens. (Noe ha here he crossover rae is defined o be he probabiliy ha wo parens will crossover in a single poin. There are also uli-poin crossover versions of he GA in which he crossover rae for a pair of parens is he nuber of poins a which a crossover akes place). (c) Muae he wo offspring a each locus wih probabiliy p (he uaion probabiliy or uaion rae ) and place he resuling chroosoes in he new populaion. If n is odd one new populaion eber can be discarded a rando. 4. Replace he curren populaion wih he new populaion. 5. Go o sep. Also see he flow char (Figure ). 4. Applicaions of he Mehod In his work we eaine si disinc odels wih singular behavior a = wo linear ie-dependen Lane-Eden ype of equaions wo linear odels of wave-ype equaion and wo nonlinear singular odels. To show he high accuracy of he soluion resuls copared wih he eac soluion we give he nuerical resuls applying he Geneic Algorih (GA) HAM ( n = 5) Padé approians (PA) of order [ pq ] and he nuerical soluion wih he Sipson rule (SIMP). Tweny poins have been used in he Sipson rule. The copuaions associaed wih he eaples were perfored using a Maple 3 package wih a precision of dgis. 4.. Tie-Dependen Lane-Eden Type Eaple. Firsly le us consider he following linear hoogeneous equaion y + y ( cos ) y = y (7) subjec o iniial condiions sin y = e y =. (8) To solve Equaions (7)-(8) by eans of he sandard HAM we choose he iniial approiaion sin y ( ) = e (9) and he linear operaor wih he propery L φ ( q) ( q ; ) φ( q ; ) ; = + L c c + = () () 697

6 Using he above definiion we consruc he zeroh-order deforaion equasar Generae a populaion of chroosoes of size N: N Calculae he finess of each chroosoe: f( ) f( ) f( N ) Is he erinaion Crlerion saisfied? Yes No Selec a pair of chroosoe for aing Wih he crossover probabiliy p c echange pars of he wo seleced chroosoes and creae wo offspring Wih he uaion probabiliy p randoly change he gene values in he wo offspring chroosoes Place he resuling chroosoes in he new populaion No Is he size of he new populaion equal o N? Yes Replace he curren chroosoe populaion wih he new populaion sop Figure. Flow Char of Geneic Algorih. c ( ) i i = are consans of inegraion. Furherore Equaion (7) suggess ha we define he nonlinear operaor as N φ ( q ; ) ( q ; ) φ( q ; ) = + φ ( q ; ) ( cos ) φ ( q ; ). () 698

7 ion as in (7) and (8) and he h-order deforaion equaion for is wih he iniial condiions = L y χy hr y (3) y = y = (4) R( y ) = ( y ) + ( y ) ( 6 4 cos ) y ( y ). + (5) Now he soluion of he h-order deforaion Equaion (3) for is ( ) = χ ( ) + ( y ). (6) y y h R We now successively obain y ( ) = h h 5 4 sin e 3 y ( ) = h + h h h ( h+ h) sin e 59 y3 ( ) = h h + h h e sin h + h ( hh h ) ( h+ h + h ) 5 5 sin e (7) and so on in his anner he res of he ieraions can be obained. Thus he approiae soluion in a series for is given by + y = y + y (8) Hence he series soluion when h = is sin y( ) = e.! 3! 4! (9) This series has he closed for as n = + sin y = e (3) which is he eac soluion of (7)-(8) copaible wih ADM [8]. Also his eaple is solved by using GA as follows: We ll choose si values of beween ( ) randoly and convering he fro he decial fora o he binary fora. =..3 =..5 = =

8 .8 = =. 6 6 ha is i i Now i i f ( i i) p( i i) f ( i i) is he series soluion of HAM given by (9) and ( i i) f ( i i) is he probabiliy of each chroosoes wih 6 f ( ) i = i i 6 f i i = i =.8. p = We arrange he chroosoes an ascending order and hen choose less han four chroosoes o find he bes soluion =. =. 3 3 =. =. =. =. 4 4 =. =. In his sep of GA crossover operaion (wo poins) is done as follows: cu = cu = 5 =. =. 3 3 =. =. =. =. 4 4 =. =. In he las sep of GA a uaion operaion (bi inverse = 3 ) and hen convering he fro he binary fora o he decial fora: =. =.65 =. = =. =.49 =. = =. =.5938 =. = =. =.875 =. = The opial soluion is found afer 5 generaion o converge o he eac soluion = E 4 and =.8743E 5. Afer eecue he Equaion (9) any ies by using GA as in Table we found he opial soluion. 7

9 Table. Opial soluion of geneic algorih for eaple. y Eac ( ) GA HAM(n = 5) PA [6/6] SIMP 7.96E4.9E Eaple. We consider he following linear inhoogeneous equaion subjec o iniial condiions 4 y + y ( 5+ 4 ) y = y (3) y = e y =. (3) To solve Equaions (3)-(3) by eans of he sandard HAM we choose he iniial approiaion and he linear operaor wih he propery L φ ( q) y = e (33) ( q ; ) φ( q ; ) ; = + L c + c = i i = are consans of inegraion. Furherore Equaion (3) suggess ha we define he nonlinear operaor as c ( q ; ) φ( q ; ) N φ( q ; ) = q ; φ ( q ; ) 4 ( 65 4 ). φ (34) (35) (36) Using he above definiion we consruc he zeroh-order deforaion equaion as in (7) and (8) and he h-order deforaion equaion for is wih he iniial condiions = L y χy hr y (37) y = y = (38) R y y y ( y ) = + ( 5+ 4 ) 4 ( y ) ( χ)( ) (39) Now he soluion of he h-order deforaion Equaion (37) for is 7

10 ( ) = χ ( ) + ( y ). (4) y y h R We now successively obain y ( ) = + he + + h y ( ) = h + h + h h h h e h h + h + h + h + h ( h + h) (4) and so on in his anner he res of he ieraions can be obained. Thus he approiae soluion in a series for is given by + y = y + y (4) = Hence he series soluion when h = is given by y( ) = e + noise ers.! 3! 4! (43) This series has he closed for as n y + e = + (44) which is he eac soluion of he proble (3)-(3) copaible wih ADM [8]. Noice ha he noise ers ha appear beween various coponens vanish in he lii. Using GA by he sae procedure as in eaple we ge he opial soluion is found afer 5 generaion o converge o he eac soluion = E5 and = 7.678E 5. Afer eecue he Equaion (43) any ies by using GA as in Table we found he opial soluion. 4.. Singular Wave-Type Equaions Eaple 3. Now we consider he inhoogeneous singular wave-ype equaion subjec o iniial condiions 3 5 y + y ( ) y = y (45) Table. Opial soluion of geneic algorih for eaple. y Eac ( ) GA HAM(n = 5) PA [6/6] SIMP 4.949E E

11 y = e y =. (46) To solve Equaions (45)-(46) by eans of he sandard HAM we choose he iniial approiaion ( ) y = e (47) and he linear operaor L φ ( q) ( q ; ) φ( q ; ) ; = + (48) wih he propery L c c + = i i = are consans of inegraion. Furherore Equaion (45) suggess ha we define he nonlinear operaor as c ( q ; ) φ( q ; ) N φ( q ; ) = q ; ( q ; ) 3 ( 5 4 ). φ Using he above definiion we consruc he zeroh-order deforaion equaion as in (7) and (8) and he h-order deforaion equaion for is = (49) L y χy hr y (5) wih he iniial condiions y = y = (5) R ( y ) = ( y ) + ( y ) ( 5+ 4 ) y 3 4 ( y ) ( χ)( ) 5 4. (5) Now he soluion of he h-order deforaion Equaion (5) for is ( ) = χ ( ) + ( y ). (53) y y h R We now successively obain y ( ) = + he + + h y ( ) = h + h + h h h h e h h + h + h + h + h ( h + h) and so on in his anner he res of he ieraions can be obained. Thus he approiae soluion in a series for is given by (54) 73

12 + y = y + y (55) = Hence he series soluion when h = is given by y( ) = e + noise ers.! 3! 4! (56) This series has he closed for as n y 3 e = + (57) which is he eac soluion of he proble (45)-(46) copaible wih ADM [8]. Noice ha he noise ers ha appear beween various coponens vanish in he lii. Using GA by he sae procedure as in eaple we ge he opial soluion is found afer 5 generaion o converge o he eac soluion = and =.879. Afer eecue he Equaion (56) any ies by using GA as in Table 3 we found he opial soluion Eaple 4. We now eaine he inhoogeneous wave-ype equaion subjec o iniial condiions y + y ( 8+ 9 ) y = y ( 8+ 9 ) (58) y = + y =. (59) To solve Equaions (58)-(59) by eans of he sandard HAM we choose he iniial approiaion y = + (6) and he linear operaor L φ ( q) ( q ; ) 4 φ( q ; ) ; = + (6) wih he propery c ( ) i L c 3 c + = (6) i = are consans of inegraion. Furherore Equaion (58) suggess ha we define he nonlinear operaor as Table 3. Opial soluion of geneic algorih for eaple 3. y Eac ( ) GA HAM(n = 5) PA [5/5] SIMP

13 ( q ; ) 4 φ( q ; ) N φ( q ; ) = q ; ( q ; ) 4 + ( 8+ 9 ) +. 4 φ (63) Using he above definiion we consruc he zeroh-order deforaion equaion as in (7) and (8) and he h-order deforaion equaion for is = L y χy hr y (64) wih he iniial condiions y = y = (65) 4 R y y y 4 ( y ) = + ( ) 4 ( y ) ( χ) ( ) Now he soluion of he h-order deforaion Equaion (64) for is (66) 4 4 ( ) = χ ( ) + ( y ). (67) y y h R We now successively obain 6 3 y ( ) = h h y ( ) = h + h + ( h h ) ( h + h ) y3 ( ) = h h h h + h + h (68) ( 3 h + h h) ( h + h + h) 6 and so on in his anner he res of he ieraions can be obained. Thus he approiae soluion in a series for is given by + y = y + y (69) Thus he series soluion when h = is given by y( ) = ! 3! 4! (7) This series has he closed for as n y = 3 e = + (7) which is he eac soluion of he proble (58)-(59) copaible wih ADM [8]. Using GA by he sae procedure as in eaple we ge he opial soluion is found afer 5 generaion o converge o he eac soluion = 4.693E6 and =.498E 6. Afer eecue he Equaion (7) any ies by using GA as in Table 4 we found he opial soluion 75

14 Table 4. Opial soluion of geneic algorih for eaple 4. y Eac ( ) GA HAM(n = 5) PA [6/6] SIMP 4.693E6.5E Nonlinear Models In wha follows we close his analysis by eaining wo nonlinear ie-dependen equaions Eaple 5. We now consider he nonlinear ie-dependen equaion y 5 y 4 6 e e y + y + + = y (7) subjec o iniial condiions y = y =. (73) To solve Equaions (7)-(73) by eans of he sandard HAM we choose he iniial approiaion y ( ) = (74) and he linear operaor L φ ( q) ( q ; ) 5 φ( q ; ) ; = + (75) wih he propery c ( ) i L c 4 + c = (76) i = are consans of inegraion. Furherore Equaion (7) suggess ha we define he nonlinear operaor as ( q ; ) 5 φ( q ; ) N φ ( q ; ) = e ( q ; ) φ ( q ; ) φ e. φ( q ; ) (77) Using he above definiion we consruc he zeroh-order deforaion equaion as in (7) and (8) and he h-order deforaion equaion for is = L y χy hr y (78) wih he iniial condiions y = y = (79) 76

15 5 R( y ) = ( y ) + ( y ) ( 4 6 ) A B ( y ). + + (8) For he nonlinear ers φ( ; ) e q = A and = = B he corresponding Adoian polynoials [5] [6] are: e φ( q ; ) = A A = = y y e y e A = y + y! y e 3 A3 = y3 + yy + y 3! y e 4 A4 = y4 + y + yy3 + y y + y!! 4! y e (8) and B = y e y B = ye B = y + y 4! y e 3 B3 = y3 + yy + y 4 8 3! y e 4 B4 = y4 + y + yy3 + y y + y 4! 4 8! 6 4! y e (8) Now he soluion of he h-order deforaion Equaion (78) for is 5 5 ( ) = χ ( ) + ( y ). (83) y y h R We now successively obain 4 4 y ( ) = h h + h 6 3 y ( ) = h h + h + h h h + h h h + ( h + h) (84) and so on in his anner he res of he ieraions can be obained. Thus he approiae soluion in a series for is given by + y = y + y (84) = Thus he series soluion when h = is given by 77

16 y( ) = noise ers. 3 4 (85) This series has he closed for as n ( ) y = ln + (86) which is he eac soluion of he proble (7)-(73) copaible wih ADM [8]. Noice ha he noise ers ha appear beween various coponens vanish in he lii. Using GA by he sae procedure as in eaple we ge he opial soluion is found afer 5 generaion o converge o he eac soluion =.47 and =.895. Afer eecue he Equaion (85) any ies by using GA as in Table 5 we found he opial soluion Eaple 6. Finally we eaine he nonlinear hoogeneous equaion 6 4 y + y + ( 4 + ) y + 4y ln y = y (87) subjec o iniial condiions y = y =. (88) To solve Equaions (87)-(88) by eans of he sandard HAM we choose he iniial approiaion and he linear operaor wih he propery L φ ( q) y ( ) = (89) ( q ; ) 6 φ( q ; ) ; = + L c 5 c + = i i = are consans of inegraion. Furherore Equaion (87) suggess ha we define he nonlinear operaor as c ( q ; ) 6 φ( q ; ) N φ( q ; ) = q ; ( q ; ) + 4 φ( q ; ) ln φ( q ; ). Table 5. Opial soluion of geneic algorih for eaple 5. 4 φ y Eac ( ) GA HAM(n = 5) PA [6/6] (9) (9) (9) 78

17 Using he above definiion we consruc he zeroh-order deforaion equaion as in (7) and (8) and he h-order deforaion equaion for is = L y χy hr y (93) wih he iniial condiions y = y = (94) 6 4 R( y ) = ( y ) + ( y ) ( 4 ) y 4 A ( y ) (95) For he nonlinear er φ( q ; ) ln φ( q ; ) A Adoian polynoials [5] [6] are: A = yln y A = + ln y y A = ( + ln y ) y + y y! A = ( + ln y ) y + yy y y y 3! = he corresponding = A = + ln y y + y + yy y y + y y! y! y 4! Now he soluion of he h-order deforaion Equaion (93) for is We now successively obain (96) 6 6 ( ) = χ ( ) + ( y ). (97) y y h R 6 y ( ) = h + h 66 7 y ( ) = h + h + ( h + h) + h ( h + h) y3 ( ) = h + h + ( h + h ) h h + h h + h + h ( h h ) ( h h h) and so on in his anner he res of he ieraions can be obained. Thus he approiae soluion in a series for is given by = (98) + y = y + y (99) Thus he series soluion when h = is given by y( ) = ()! 3! 4! 79

18 Table 6. Opial soluion of geneic algorih for eaple 6. y Eac ( ) GA HAM(n = 5) PA [6/6] This series has he closed for as n y = e () which is he eac soluion of he proble (87)-(88) copaible wih ADM [8]. Using GA by he sae procedure as in eaple we ge he opial soluion is found afer 5 generaion o converge o he eac soluion =.7 and =.935. Afer eecue he Equaion () any ies by using GA as in Table 6 we found he opial soluion. 5. Conclusions To our bes knowledge his is he firs resul on he applicaion of he HAM wih GA o solve odels of he ie-dependen Eden-Fowler ype of equaions and wave-ype equaion wih singular behaviors. The HAM wih GA has been successfully applied o solve odels of he ie-dependen Eden-Fowler ype of equaions and wave-ype equaion wih singular behaviors. The HAM wih GA has worked effecively o handle hese odels giving i a wider applicabiliy. The proposed schee of HAM has been applied direcly wihou any need for ransforaion forulae or resricive assupions. The soluion process of HAM is copaible wih hose ehods in he lieraure providing analyical approiaion such as ADM. The approach of HAM wih GA has been esed by eploying he ehod o obain approiae-eac soluions of si eaples. The resuls obained in all cases deonsrae he reliabiliy and he efficiency of his ehod. Acknowledgeens We would like o hank he referees for heir careful review of our anuscrip. References [] Davis H.T. (96) Inroducion o Nonlinear Differenial and Inegral Equaions. Dover Publicaions New York. [] Chandrasekhar S. (967) Inroducion o he Sudy of Sellar Srucure. Dover Publicaions New York. [3] Richardson O.U. (9) The Eission of Elecriciy fro Ho Bodies. London. [4] Adoian G. Rach R. and Shawagfeh N.T. (995) On he Analyic Soluion of Lane-Eden Equaion Found. Physics Leers

19 hps://doi.org/.7/bf87585 [5] Shawagfeh N.T. (993) Nonperurbaive Approiae Soluion for Lane-Eden Equaion. Journal of Maheaical Physics hps://doi.org/.63/.535 [6] Wazwaz A.M. () A New Mehod for Solving Differenial Equaions of he Lane-Eden Type. Applied Maheaics and Copuaion [7] Wazwaz A.M. (5) Adoian Decoposiion Mehod for a Reliable Treaen of he Eden-Fowler Equaion. Applied Maheaics and Copuaion [8] Wazwaz A.M. (5) Analyical Soluion for he Tie-Dependen Eden-Fowler Type of Equaions by Adoian Decoposiion Mehod. Applied Maheaics and Copuaion [9] Yıldırı A. and Öziş T. (9) Soluions of Singular IVPs of Lane-Eden Type by he Variaional Ieraion Mehod. Nonlinear Analysis [] Dehghan M. and Shakeri F. (8) Approiae Soluion of a Differenial Equaion Arising in Asrophysics Using he Variaional Ieraion Mehod. New Asronoy [] Liao S.J. (3) Beyond Perurbaion: Inroducion o he Hooopy Analysis Mehod. Chapan and Hall CRC Press Boca Raon. hps://doi.org/./ [] He J.H. (3) Hooopy Perurbaion Mehod: A New Nonlinear Analyical Technique. Applied Maheaics and Copuaion [3] Gen M. and Cheng R. (997) Geneic Algorihs and Engineering Design. John Wiley & Sons Hoboken. [4] Melanie M. (998) An Inroducion o Geneic Algorihs. MIT Press Cabridge. [5] Adoian G. (989) Nonlinear Sochasic Syses Theory and Applicaions o Physics. Kluwer Acadeic Publishers Dordrech. hps://doi.org/.7/ [6] Wazwaz A.M. () A New Algorih for Calculaing Adoian Polynoials for Nonlinear Operaors. Applied Maheaics and Copuaion Subi or recoend ne anuscrip o SCIRP and we will provide bes service for you: Acceping pre-subission inquiries hrough Eail Facebook LinkedIn Twier ec. A wide selecion of journals (inclusive of 9 subjecs ore han journals) Providing 4-hour high-qualiy service User-friendly online subission syse Fair and swif peer-review syse Efficien ypeseing and proofreading procedure Display of he resul of downloads and visis as well as he nuber of cied aricles Maiu disseinaion of your research work Subi your anuscrip a: hp://papersubission.scirp.org/ Or conac a@scirp.org 7