Application of Homotopy Analysis Method for Solving various types of Problems of Ordinary Differential Equations
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1 Iteratioal Joural o Recet ad Iovatio Treds i Coputig ad Couicatio IN: Volue: 5 Issue: 5 16 Applicatio of Hootopy Aalysis Meod for olvig various types of Probles of Ordiary Differetial Equatios V.P.Gohil Assistat Professor, Departet of Maeatics Goveret Egieerig College Bhavagar, Idia vpgohil99@gail.co Dr.G.A.Raabhatt Assistat Professor, Departet of Maeatics Goveret Egieerig College Bhavagar, Idia Abstract I is paper, various types of liear, o-liear, hoogeeous, o hoogeeous probles of ordiary differetial equatios discussed. Also show at hootopy aalysis eod applied successfully for solvig o hoogeeous ad o liear equatios. Keywords- hootopy aalysis eod, ordiaryy differetial equatio,liear, hoogeeous ***** I. INTRODUCTION It is well-kow at oliear ordiary differetial equatios (ODEs) ad partial differetial equatios (PDEs) for boudary-value probles are uch ore difficult to solve a liear ODEs ad PDEs, especially by eas of aalytic eods. I recet years, is eod (HAM) has bee successfully eployed to solve ay types of o liear, hoogeeous or o hoogeeous, equatios ad systes of equatios as well as probles i sciece ad egieerig ([4]). Very recetly, Ahad Bataieh ([]) preseted two odificatios of HAM to solve liear ad o liear ODEs. The HAM cotais a certai auiliary paraeter h which provides us wi a siple way to adjust ad cotrol e covergece regio ad rate of covergece of e series solutio. Moreover, by eas of e so-called h -curve, it is easy to deterie e valid regios of h to gai a coverget series solutio. Thus, rough HAM, eplicit aalytic solutios of o liear probles are possible. II. HOMOTOPY ANALYI METHOD We cosider e followig differetial equatios, N i i, t =, i = 1,,, N i are oliear operators at e represets e whole equatios, ad t are idepedet variables ad i, t are ukow fuctios respectively. By eas of geeralizig e traditioal hootopy eod, Liao costructed e so-called zero-order deforatio equatios 1 q L i, t, ; q i,, t = qh i N i [ i, t, ; q ] (1) q,1 is a ebeddig operators, h i are ozero auiliary fuctios, L is a auiliary liear operator, i,, t are iitial guesses of i, t ad i, t, ; q are ukow fuctios. It is iportat to ote at, oe has great freedo to choose auiliary objects such as h i ad L i HAM. Whe q = ad q = 1 we get by (1), i, t, ; = i,, t ad i, t, ; 1 = i, t Thus q icrease fro to 1, e solutios i, t, ; q varies fro iitial guesses i,, t to i,, t. Epadig i, t, ; q i Taylor series wi respect to, i, t, ; q = i,, t + =1 i,, t. q () i,, t = 1. i,t,;q, (3)! q q= If e auiliary liear operator, iitial guesses, e auiliary paraeter h i ad auiliary fuctios are properly chose a e series equtio () coverges at = 1. i, t, ; 1 = i,, t + =1 i,, t (4) This ust be oe of solutios of e origial oliear equatios. Accordig to (3), e goverig equatios ca be deduced fro e zero-order deforatio equatios (1). Defie e vectors i, = { i,, t, i,1, t, i,, t,., i,, t } Differetiatig (1) ties wi respect to e ebeddig paraeter q ad e settig q = ad fially dividig e by!. We have e so-called order deforatio equatios L i,, t χ i, 1, t = h i R i, ( i, 1) (5) R i, i, 1) = (5) 1. 1 N i [ i,t,;q ( 1)! q 1 q= IJRITCC May 17, ad (6) 16
2 Iteratioal Joural o Recet ad Iovatio Treds i Coputig ad Couicatio IN: Volue: 5 Issue: 5 16 III. χ =, 1 1, > 1 HOMOGENEOU LINEAR ORDINARY DIFFERENTIAL EQUATION Cosider hoogeeous liear differetial equatio u u u ubject to e iitial coditio u() 1, u'() 1 (8) To solve is syste (7) to (8) by HAM, first we choose iitial approiatio Ad e liear operator L ; q u ( ) 1 ; q Wi e property LC where C is itegral costat. We defie syste of o-liear operator as ; q ; q N ; q ; q (9) Usig e above defiitio, we costruct e zero-order deforatio equatios 1 q ; q qhn ; q Obviously, whe q ad q 1 we get ; u ad As q icrease to 1, varies fro Epadig ; q u( ) (7) (1) (11) u to u i Taylor series wi respect to q, ; q q 1 1 ; q! q q (1) (13) If e auiliary liear operator, iitial guesses, e auiliary paraeter h ad auiliary fuctios are properly chose a e series equatio (1) coverges at q 1. 1 u 1 i.e. This ust be oe of solutios of e origial o liear equatios as proved by Liao Defie e vectors,,,... IJRITCC May 17, 1 We have e so-called order deforatio equatios L h1 1 1! ; q q i.e. q h d c 1 1 h d c o 1 h N order approiatio ca be epressed by N 1 1 As N we get u assuptio of h IV. (14) (15) (16) (17) (18) (19) () wi soe appropriate NON HOMOGENEOU LINEAR ORDINARY Cosider o hoogeeous liear differetial equatio u u u ubject to e iitial coditio (1) 17
3 Iteratioal Joural o Recet ad Iovatio Treds i Coputig ad Couicatio IN: Volue: 5 Issue: 5 16 u() 1, u'() 1 L h1 () (9) To solve is syste (1) to ()by HAM, first we choose iitial approiatio u ( ) ; q R 1 1 Ad e liear operator 1! q q ; q (3) L ; q i.e Wi e property LC where C is itegral costat. We defie syste of o-liear operator as ; q ; q N ; q ; q (3) Usig e above defiitio, we costruct e zero-order deforatio equatios 1 q ; q qhn ; q Obviously, whe q ad q 1 we get ; u ad As q icrease to 1, varies fro Epadig ; q u( ) (4) (5) u to u i Taylor series wi respect to q, ; q q 1 1 ; q! q q (6) (7) If e auiliary liear operator, iitial guesses, e auiliary paraeter h ad auiliary fuctios are properly chose a e series equatio (6) coverges at q 1. 1 u 1 i.e. This ust be oe of solutios of e origial o liear equatios as proved by Liao Defie e vectors,,,... 1 We have e so-called order deforatio equatios (8) 1 1 h d c h d c o 1 1 h N order approiatio ca be epressed by N 1 1 As N we get u assuptio of h (31) (3) (33) (34) wi soe appropriate V. NON HOMOGENEOU NON LINEAR ORDINARY Cosider o hoogeeous o liear differetial equatio u u u u ubject to e iitial coditio u(), u'() 1 (35) (36) To solve is syste (35) to (36) by HAM, first we choose iitial approiatio u ( ) Ad e liear operator ; q L ; q IJRITCC May 17, 18
4 Iteratioal Joural o Recet ad Iovatio Treds i Coputig ad Couicatio IN: Volue: 5 Issue: 5 16 Wi e property LC where C is itegral costat. 1 h 1 d c We defie syste of o-liear operator as ; q ; q N ; q ; q ; q Now we will calculate (37) Usig e above defiitio, we costruct e zero-order 1 1 h R1 d c deforatio equatios 1 q ; q qhn ; q Obviously, whe q ad q 1 we get ; u ad As q icrease to 1, varies fro Epadig ; q u( ) (38) (39) u to u i Taylor series wi respect to q, ; q q 1 1 ; q! q q (4) (41) If e auiliary liear operator, iitial guesses, e auiliary paraeter h ad auiliary fuctios are properly chose a e series equatio (4) coverges at q 1. 1 u 1 i.e. This ust be oe of solutios of e origial o liear equatios as proved by Liao Defie e vectors,,,... 1 We have e so-called order deforatio equatios L h1 1 1! ; q q i.e. q (4) (43) (44) (45) IJRITCC May 17, o 1 h N order approiatio ca be epressed by N 1 1 As N we get u assuptio of h VI. (46) (47) (48) wi soe appropriate HOMOGENEOU NON LINEAR ORDINARY Cosider hoogeeous o liear differetial equatio u u u u ubject to e iitial coditio u(), u'() 1 (49) (5) To solve is syste (49) to (5) by HAM, first we choose iitial approiatio u ( ) Ad e liear operator ; q L ; q Wi e property LC where C is itegral costat. We defie syste of o-liear operator as ; q ; q N ; q ; q ; q (51) Usig e above defiitio, we costruct e zero-order deforatio equatios 1 q ; q qhn ; q (5) 19
5 Iteratioal Joural o Recet ad Iovatio Treds i Coputig ad Couicatio IN: Volue: 5 Issue: 5 16 Obviously, whe q ad q 1 we get 3 ; u ad 1 u( ) o (53) As q icrease to 1, varies fro u to u 1 h 3 Epadig ; q i Taylor series wi respect to q, N order approiatio ca be epressed by ; q q 1 1 ; q! q q (54) (55) If e auiliary liear operator, iitial guesses, e auiliary paraeter h ad auiliary fuctios are properly chose a e series equatio (54) coverges at q 1. 1 u 1 i.e. This ust be oe of solutios of e origial o liear equatios as proved by Liao Defie e vectors,,,... 1 We have e so-called order deforatio equatios L h1 1 1! ; q q i.e. q h d c 1 1 h d c (56) (57) (58) (59) (6) (61) N 1 1 A N WE GET u AUMPTION OF h VII CONCLUION (6) WITH OME APPROPRIATE Various types of hoogeeous, o hoogeeous, liear, o liear ordiary differetial equatio ca be solved easily by usig hootopy aalysis eod. REFERENCE [1] Abbasbady., The applicatio of hootopy aalysis eod to o liear equatios arisig i heat trasfer, Phys. Lett. A 36 (6) [] Bataieh A.ai, Noorai M..M., Hashi I., Modified hootopy aalysis eod for solvig systes of secodorder BVPs, Cou. Noliear ci. Nuer. iul., i press (doi:1.116/j.css.7.9.1). [3].J Liao., O e hootopy aalysis eod for o liear probles, Applied Maeatics ad Coputatio,vol.147,o.,pp ,4. [4].J. Liao ad Y. Ta, A geeral approach to obtai series solutios of o liear diferetial equatios, tudies i Applied Maeatics,vol.119,o.4,pp.97354,7. [5].J. Liao.,Beyod Perturbatio: Itroductio to e Hootopy Aalysis Meod, Chapa [6].J. Liao, Copariso betwee e hootopy aalysis eod ad hootopy perturbatio eod, Appl. Ma. Coput. 169 (5) [7] Z. Wag,I. Zou,H. Zhag.Applyig hootopy aalysis eodfor solvig diferetial-diferece equatio.phys Lett A 7;371:7-8 IJRITCC May 17,
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