Asymptotic Solutions of Fifth Order Critically Damped Nonlinear Systems with Pair Wise Equal Eigenvalues and another is Distinct

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Qus Journals Journal of Rsarch in Applid Mahmaics Volum ~ Issu (5 pp: -5 ISSN(Onlin : 94-74 ISSN (Prin:94-75 www.usjournals.

1 Qus Journals Journal of Rsarch in Applid Mahmaics Volum ~ Issu (5 pp: -5 ISSN(Onlin : ISSN (Prin: Rsarch Papr Asympoic Soluions of Fifh Ordr Criically Dampd Nonlinar Sysms wih Pair Wis Eual Eignvalus and anohr is Disinc Md. Mahafujur Rahaman M. Abul Kawsr Dparmn of Businss Adminisraion Z.H. Sikdr Univrsiy of Scinc & Tchnology Bangladsh Dparmnof Mahmaics Islamic Univrsiy Bangladsh Rcivd 8March 5; AccpdApril 5 Th auhor(s 4. Publishd wih opn accss a ABSTRACT : Th Krylov-Bogoliubov-Miropolskii (KBM mhod is on of h mos usd chnius o invsiga ransin bhavior of vibraing sysms. Iniially h mhod was dvlopd for obaining h priodic soluions of scond ordr nonlinar diffrnial sysms wih small nonlinariis. Afrward many rsarchrs hav sudid and modifid h mhod for achiving soluions of highr ordr nonlinar sysms. In his aricl w hav modifid h KBM mhod o xaminh asympoic soluions of fifh ordr criically dampd nonlinar sysms. Kywords: -KBM priodic soluion nonlinariy asympoic soluioncriically dampd sysm. I. INTRODUCTION In nonlinar diffrnial uaions on of mos usd chnius o xamin wakly nonlinar oscillaory and non-oscillaory diffrnial sysms is h Krylov-Bogoliubov-Miropolskii (KBM ([] [] mhod which was firs dvlopd by Krylov and Bogoliubov [] o find h priodic soluions of scond ordr nonlinar diffrnial sysms wih small nonlinariis. SubsunlyBogoliubov and Miropolskii [4] improvd and jusifid his mhod in mahmaical rms. I was furhr xndd by Popov [5] o dampd oscillaory nonlinar sysms. Lar on ovr-dampd nonlinar sysms wr xamind bymury and Dkshaulu [6]using Bogoliubov s mhod. An asympoic soluion of a scond ordr criically dampd nonlinar sysmwas poind ou by Saar [7]. Morovr a nw chniu was inroducd by Shamsul [8] o obain approxima soluions of scond ordr boh ovr-dampd and criically dampd nonlinar sysms. Furhr Osiniskii [9] invsigad h hird ordr nonlinar sysms using Bogoliuvob s mhod and imposd som rsricions upon h paramrs which mad h soluions ovr-simplifid and rvald incorrc rsuls. Howvr Mulholland [] rmovd hs rsricions o obain dsird rsuls. Consunly hsoluions of nonlinar sysms wr ransformd by Bojadziv [] o a hr dimnsional diffrnial sysm. Saar []also assssdh soluions of hird ordr ovr-dampd nonlinar sysms. Shamsul []analyzd h soluions of hird ordr ovr-dampd sysms whos ignvalus wr ingral mulipl. Shamsul and Saar [4] inroducd a unifid KBM mhod o obain approxima soluions of hird ordr dampd and ovr-dampd nonlinar sysms. Kawsr and Akbar [5] suggsd an asympoic soluion for h hird ordr criically dampd nonlinar sysm wih pair wis ual ignvalus. Kawsr and Saar [6] proposd an asympoic soluion of a fourh ordr criically dampd nonlinar sysm wih pair wis ual ignvalus. Th KBM mhod was furhr xndd by Akbr and Tanzr [7] o obain soluions for h fifh ordr ovr-dampd nonlinar sysms wih cubic nonlinariy. Rahaman and Rahman [8] xpoundd h analyical approxima soluions of fifh ordr mor criically dampd sysms in h cas of smallr riply rpad roos. Th aim of his aricl is o obain h asympoic soluions of fifh ordr criically dampd non-linar sysms by xnding h KBM mhod. Th rsuls obaind by h prurbaion mhod wr compard wih hos obaind by h fourh ordr Rung-Kua mhod whichdmonsrad prfc coincidn wih h numrical soluions. *CorrspondingAuhor:Md. MahafujurRahaman Dparmn of Businss Adminisraion Z.H. Sikdr Univrsiy of Scinc & Tchnology Bangladsh Pag

2 Asympoic Soluions of Fifh Ordr Criically Dampd Non-linar Sysms wih Pair Wis Eual Eignvalus II. THE METHOD W ar going o propos a prurbaion chniu o solv fifh ordr non-linar diffrnial sysms of h form ( x v kx iv k x k x k4x k5 x f ( x x x x x iv ( v iv whr x and x sand for h fifh and fourh drivaivs rspcivly and ovr dos ar usd for h firs scond and hird drivaivs of x wih rspc o. k k k k4 k ar consans is a sufficinly small 5 paramr and f is h givn nonlinar funcion. As h unprurbd uaion ( is of ordr fiv so i has fiv ral ngaiv ignvalus whr four ignvalus ar pair wis ual and ohr on is disinc. Suppos h ignvalus ar. Whn h uaion ( bcoms linar and h soluion of h corrsponding linar uaion is x( ( a b ( c d h ( whr a b c d h ar consans of ingraion Whn following Shamsul [9] an asympoic soluion of h uaion ( is sough in h form x( ( a b ( c d h u( a b c d h... ( whr a b c d h ar h funcions of and hy saisfy h diffrnial uaions a A ( a b c d h... b B ( a b c d h... c C ( a b c d h... (4 d D ( a b c d h... h H ( a b c d h... Now diffrniaing ( fiv ims wih rspc o subsiuing h valu of x and h drivaivs v iv x x x x x in h original uaion ( uilizing h rlaions prsnd in (4 and finally bringing ou h cofficins of w obain A B ( D ( D B ( D C D ( D D ( D ( D H (5 ( D ( D ( D u f ( a b c d h ( ( iv whr f ( a b c d h f ( x x x x x and x( ( a b ( c d h ( W hav xndd h funcion f in h Taylor s sris (Saar [] Shamsul and Saar [] abou h origin in powr of. Thrfor w obain ( ( i j k f Fl ( a b c d h i j k l (6 Thus using (6 h uaion (5 bcoms A B ( D ( D B ( D C D ( D D ( D ( D H ( i jk ( D ( D ( D u Fl ( a b c d h i j k l *CorrspondingAuhor:Md. MahafujurRahaman (7 Pag

3 Asympoic Soluions of Fifh Ordr Criically Dampd Non-linar Sysms wih Pair Wis Eual Eignvalus Following h KBM mhod Mury and Dkshaulu [] Saar [] Shamsul[4] Shamsul and Saar ([5] ( [6] imposd h condiion hau dos no conain h fundamnal rms of f. Thrfor uaion (7 can b sparad for unknown funcions A B C D H andu in h following way: A B ( D ( D B ( D C D ( D D ( D ( D H (8 ( i jk Fl ( a b c d h i j k l ( and ( ( ( D D D u F ( i j k l a b c d h i j k l Now uaing h cofficins of from uaion (8 w obain i j k l F l ( a b c d h *CorrspondingAuhor:Md. MahafujurRahaman (9 A ( D ( D B ( D C ( D D ( D ( D H ( ( i jk B D ( D ( D ( D ( D i j k l F l ( a b c d h ( i jk Hr from uaions ( and ( drmining h unknown funcions A B C D and H. Thus o obain h unknown funcions A B C D and H w nd o impos som condiions (Shamsul [7] [8] [9] bwn h ignvalus. In his aricl w hav xamind soluions for h cass. As a rsul w will b abl o spara h uaion ( for unknown funcions B and D ; and solving hm for B and D subsiuing h valus of B and D ino h uaion ( and applying h condiions w can spara h uaion ( for hr unknown funcions B D and H ; and solving hm forh. As a b c d h ar proporional o small paramr hy ar slowly varying funcions of im and for firs approxima soluions; w may considr hm as consans in h righ sid. Mury and Dkshaulu [] firs mad his assumpion. Thrfor h soluions of h uaion (4 bcom a a A ( a b c d h d b b B ( a b c d h d c c C ( a b c d h d d d D ( a b c d h d h h H ( a b c d h d ( ( Pag

4 Asympoic Soluions of Fifh Ordr Criically Dampd Non-linar Sysms wih Pair Wis Eual Eignvalus Euaion (9 is a non-homognous linar ordinary diffrnial uaion; hrfor i can b solvd by h wllknown opraor mhod. Subsiuing h valus of a b c d h andu in h uaions ( w shall g h comprhnsiv soluion of (. Thus h drminaion of h firs simad soluion is far-raching. III. EXAMPLE In his aricl w hav akn ino accoun h Duffing yp uaion of fifh ordr nonlinar diffrnial sysm as an xampl of h abov mhod: v iv x k x k x k x k x k x x ( 4 5 Comparing ( and ( w obain Thrfor iv f ( x x x x x x f a a c a h ac 6ach ah ( ( ( ( ( ( c c h ch h a b abc a d ( ( ( ( abh acd bc ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ad bcd bdh cd d h adh bch bh c d cdh dh ab b c abd b h b b d bd d ( ( Now comparing uaions (4 and ( w obain (4 i j k l F ( a b c d h a a c a h ac l ( i jk ( ( ( 6ach ah c c h ch h ( ( ( ( i j k l l F ( a b c d h a b abc a d ( i jk ( ( abh acd bc adh bch ( ( ( ( ( bh c d cdh dh ( ( ( i j k l ( i jk ( F ( a b c d h ab b c l abd b h ad bdh ( ( ( ( cd d h bcd ( ( (5 i j k l F ( a b c d h b b d bd d l ( i jk ( ( For uaion ( h uaions (9 o ( rspcivly bcom *CorrspondingAuhor:Md. MahafujurRahaman 4 Pag

5 Asympoic Soluions of Fifh Ordr Criically Dampd Non-linar Sysms wih Pair Wis Eual Eignvalus ( D ( D ( D u ab b c abd ( ( b h ad bcd bdh cd ( ( ( ( d h b b d bd d ( ( ( A ( D ( D B ( D C ( D D ( D ( D H a a c a h ac 6ach ( ( ( ( ( ( ( ah c c h ch h (7 B D ( D ( D ( D ( D a b abc a d abh acd ( ( ( ( bc adh bch bh c d ( ( ( ( ( ( cdh dh Sinc h rlaions among h ignvalus hn h uaion (8 can b sparad for h unknown funcionsb andd in h following way: B ( ( D ( D a b abc ( ( ( ( a d abh acd bc (9 adh bch bh ( ( ( D ( D ( D c d 6cdh dh Solving uaions (9 and ( w g ( ( B l a b l ( abc a d l ( acd bc ( l abh l ( adh bch l bh ( ( ( ( ( D p c d p cdh p dh ( whr l l ( ( ( ( l l 4 ( ( ( ( 6 l5 l 6 ( ( ( ( ( p p ( ( ( ( p ( ( Using h valus ofb andd in uaion (7 w obain (8 (6 *CorrspondingAuhor:Md. MahafujurRahaman 5 Pag

6 Asympoic Soluions of Fifh Ordr Criically Dampd Non-linar Sysms wih Pair Wis Eual Eignvalus A C ( D ( D ( D ( D ( D ( D H a a c ( a h ac 6ach ah c ( ( ( ( ( ( c h ch h ( ( ( ( 4 5 ( ( 6 ( l a b 8 ( l abc 8 ( l a d 4 ( l abh ( ( l ( adh bch ( ( l bh ( ( p c d 4 ( p cdh ( ( p dh ( ( Again applying h condiions in uaion ( hn w obain h following uaions for unknown funcionsa C and H A ( ( D ( D a a c ( ( ( ac 6ach a h ( ( l a b 8 ( l ( abc bc ( 8( ( l ( acd bc ( 4 ( l abh 4 ( ( ( l ( adh bch ( ( l bh ( ( 5 6 C D D c p c d c h ch 4 ( ( ( ( ( ( ( p cdh ( ( p dh ( ( ( D ( D H h (6 (4 (5 ( Solving uaions (4 (5 and (6 w obain A ( n a n a b ( n ac n acd { n a c 7 9 n a h n ( abc a d} { n ach ( n ( adh bch} ( n ah n bh n abh ( ( 5 (7 C ( rc r c d ( r c h r cdh ( r ch r dh (8 ( H mh (9 whr n n ( ( 4 ( ( 6 n n 4 n ( ( ( ( ( ( ( n6 4 ( ( 5 *CorrspondingAuhor:Md. MahafujurRahaman 6 Pag

7 Asympoic Soluions of Fifh Ordr Criically Dampd Non-linar Sysms wih Pair Wis Eual Eignvalus n7 n 8 ( ( ( ( 6 n9 n ( ( ( ( n n ( ( ( ( ( r r ( ( ( ( 6 r r 4 r 5 ( ( ( ( ( ( r6 ( ( m ( Thus h soluion of h uaion (6 for u is u ab ( ( b c abd ( ( b h ( ( ad bcd ( ( ( bdh ( cd ( ( d h ( ( 9 b ( 4 5 b d ( bd ( ( ( d ( whr 4 ( ( 4 ( ( 4 ( ( 4 ( ( ( ( ( ( 4 ( ( ( ( 6 ( ( ( 4 ( ( ( ( ( ( *CorrspondingAuhor:Md. MahafujurRahaman 7 Pag

8 Asympoic Soluions of Fifh Ordr Criically Dampd Non-linar Sysms wih Pair Wis Eual Eignvalus ( ( 7 8 ( ( 9 ( ( ( ( 4 ( ( 4 ( ( 4 ( ( ( ( ( ( ( 4 ( ( ( ( ( 6 ( ( ( ( ( ( 4 ( ( ( ( ( ( ( 4 5 ( ( ( ( ( 7 4 ( ( ( ( ( ( 4 ( ( 6 8 ( ( ( ( ( ( 9 ( ( ( ( ( ( 4 ( ( 4 ( ( *CorrspondingAuhor:Md. MahafujurRahaman 8 Pag

9 Asympoic Soluions of Fifh Ordr Criically Dampd Non-linar Sysms wih Pair Wis Eual Eignvalus 4 ( ( 4 ( ( ( ( 4 ( ( 4 ( ( ( ( ( ( ( ( 5 4 ( ( ( ( ( 4 ( ( ( ( ( 4 ( ( ( ( 9 ( ( ( 4 ( 7 8 ( ( ( ( 9 4 ( ( ( ( ( ( ( ( ( 9 ( ( 4 ( ( ( 4 ( ( 9 4 ( ( 9 ( ( ( 4 ( ( ( ( ( *CorrspondingAuhor:Md. MahafujurRahaman 9 Pag

10 Asympoic Soluions of Fifh Ordr Criically Dampd Non-linar Sysms wih Pair Wis Eual Eignvalus 9 4 ( ( ( ( ( ( ( ( ( ( ( 4 ( ( ( 4 ( ( 4 4 ( ( ( ( 4 ( ( 5 6 ( ( ( ( ( ( ( ( 4 ( ( ( ( ( 7 4 ( ( ( ( ( Subsiuing h valus of A B C D and H from uaions (7 ( (8 ( and (9 ino uaion (4 w obain a ( n a n a b ( n ac n acd n ( a c b c ( 7 9 n a h n ( acd a d n ach ( ( ( n ( adh bch ( n ah n bh n abh ( ( 5 b l a b l ( abc a d l ( acd bc ( l ( adh bch l abh l bh ( ( c rc r c d ( r c h r cdh 5 ( r ch r dh ( 4 6 d p c d p cdh p dh h mh ( ( *CorrspondingAuhor:Md. MahafujurRahaman Pag

11 Asympoic Soluions of Fifh Ordr Criically Dampd Non-linar Sysms wih Pair Wis Eual Eignvalus Hr all of h uaions ( hav no accura soluions bu as a b c d h ar proporional o h small paramr hy ar slowly varying funcions of im. Accordingly i is possibl o rplac a b c d h by hir rspciv valus obaind in linar cas (i.. h valus of a b c d h obaind whn. in h righ hand sid of uaions (. Mury and Dkshaulu ([] []firs inroducd his sor of rplacmn o rsolv similar sor of nonlinar uaions. Thus h soluions of uaions ( ar a a ( na n7a b (( na c b c n6a h n a b c a d n a h n b h ( 8( ( 5 ( ( ( n4ach n ( adh bch na bh ( ( b b r ab r ( abc a d r4 abh ( r ( acd bc r5 ( adh bc h r6b h c c rc r c d ( rc h r5 cdh ( ( ( r4 ch r6 dh ( d dpc d pcdh pdh h h mh Hnc w obain h firs approxima soluion of h uaion ( as x( ( a b ( c d h u ( whr a b c d h ar givn by h uaions ( and u is givn by (. IV. RESULTS AND DISCUSSION I is usual o compar h prurbaion soluion o h numrical soluion o s h accuracy of h approxima soluion obaind by a crain prurbaion mhod. W hav compud x ( using ( in which a b c d h ar obaind from ( andu is calculad from uaion (. Th rsul obaind from ( for various valus of and h corrsponding numrical soluion obaind by a fourh ordr Rung-Kua mhod is prsnd in h followingfig. Fig. Fig.and Fig.4 rspcivly. *CorrspondingAuhor:Md. MahafujurRahaman Pag

12 x x x Asympoic Soluions of Fifh Ordr Criically Dampd Non-linar Sysms wih Pair Wis Eual Eignvalus Prurbaion Rsul Numrical Rsul Figur : Comparison bwn prurbaion and numrical rsuls for..5.5 and.wih h iniial condiions a.45 b.5 c. d.5 h Prurbaion Rsul Numrical Rsul Figur : Comparison bwn prurbaion and numrical rsuls for.4. and.wih h iniial condiions a.4 b. c. d. h Prurbaion Rsul Numrical Rsul Figur : Comparison bwn prurbaion and numrical rsuls for.8..7 and.wih h iniial condiions a.45 b. c. d.5 h. *CorrspondingAuhor:Md. MahafujurRahaman Pag

13 x Asympoic Soluions of Fifh Ordr Criically Dampd Non-linar Sysms wih Pair Wis Eual Eignvalus Prurbaion Rsul Numrical Rsul Figur 4: Comparison bwn prurbaion and numrical rsuls for and.wih h iniial condiions a.4 b.5 c. d. h.5 Th corrsponding prurbaion andnumrical rsuls for various valus of ar dmonsrad in h following Tabl Tabl Tabl and Tabl 4 rspcivly. Tabl : Comparison bwn prurbaion andtabl : Comparison bwn prurbaion and Numrical rsuls numrical rsuls Tabl : Comparison bwn prurbaion andtabl 4: Comparison bwn prurbaion and Numrical rsuls numrical rsuls *CorrspondingAuhor:Md. MahafujurRahaman Pag

14 Asympoic Soluions of Fifh Ordr Criically Dampd Non-linar Sysms wih Pair Wis Eual Eignvalus V. CONCLUSION In his aricl w hav modifid h KBM mhod and applid i o fifh ordr criically dampd nonlinar sysms. On h basis of h modificaionransin rsponss of nonlinar diffrnial sysms hav bn xamind. For fifh ordr criically dampd nonlinar sysms h soluions ar sarchd for such circumsancs whr. Thus h rsuls obaind for diffrn s of iniial condiions as wll as diffrn ignvalus hav shown xclln agrmn wih h numrical rsuls obaind by h Rung-Kua mhod. ACKNOWLEDGEMENTS Boh h auhors ar graful o Md. MizanurRahman Associa Profssor Dparmn of Mahmaics Islamic Univrsiy Bangladsh for his invaluabl commns and advic. REFERENCES [] N. N.Krylov and N. N.Bogoliubov Inroducion o Nonlinar Mchanics(Nw Jrsy Princon Univrsiy Prss 947. [] N. N.Bogoliubov and Y.Miropolskii Asympoic Mhods in h Thory of Nonlinar Oscillaions (Nw York Gordan and Brach 96. [] N. N.Krylov and N. N.Bogoliubov Inroducion o Nonlinar Mchanics(Nw Jrsy Princon Univrsiy Prss 947. [4] N. N.Bogoliubov and Y.Miropolskii Asympoic Mhods in h Thory of Nonlinar Oscillaions (Nw York Gordan and Brach 96. [5] I. P.Popov A Gnralizaion of h Bogoliubov Asympoic Mhod in h Thory of Nonlinar Oscillaions (in Russian Dokl. Akad. USSR [6] I. S. N.Mury and B. L.Dkshaulu and G.Krishna On an Asympoic Mhod of Krylov-Bogoliubov for Ovr-dampd Nonlinar Sysms J. Frank. Ins [7] M. A.Saar An asympoic Mhod for Scond Ordr Criically Dampd Nonlinar Euaions J. Frank. Ins [8] M. S.Alam Asympoic Mhods for Scond Ordr Ovr-dampd and Criically Dampd Nonlinar Sysms Soochow Journal of Mah [9] Z.Osiniskii Vibraion of a On Dgr Frdom Sysm wih Nonlinar Inrnal Fricion and Rlaxaion Procdings of Inrnaional Symposium of Nonlinar Vibraions Kiv Izad Akad Nauk USSR [] R. J. Mulholland Nonlinar Oscillaions of Third Ordr Diffrnial Euaion In. J. Nonlinar Mchanics [] G. N.Bojadziv Dampd Nonlinar Oscillaions Modld by a -dimnsional Diffrnial Sysm AcaMchanica [] M. A. Saar An Asympoic Mhod for Thr-dimnsional Ovr-dampd Nonlinar Sysms Gani J. Bangladsh Mah. Soc [] M. S. Alam On Som Spcial Condiions of Third Ordr Ovr-dampd Nonlinar Sysms Indian J. pur appl. Mah [4] M. S.Alam and M. A.SaarA Unifid KBM Mhod for Solving Third Ordr Nonlinar Sysms Indian J. pur appl. Mah [5] M. A.Kawsr and M. A. Akbar An Asympoic Soluion for h Third Ordr Criically Dampd Nonlinar Sysm in h Cas for Small Eual Eignvalus J. Mah. Forum XXII [6] M.A.Kawsr and M. A.Saar An Asympoic Soluion of a Fourh Ordr Criically Dampd Nonlinar Sysm wih Pair Wis Eual Eignvalus Rs. J. Mah. Sa. ( -. [7] S. T. A. Siddiuand M. A. Akbar Asympoic Soluion of Fifh Ordr Ovr-Dampd Nonlinar Sysms wih Cubic Nonlinariy Sudis in Mahmaical Scinc ( 4-4. [8] M. M. Rahaman and M. M. Rahman Analyical Approxima Soluions of Fifh Ordr Mor Criically Dampd Sysms in h cas of Smallr Triply Rpad Roos IOSR Journal of Mahmaics [9] M. S.Alam and M. B.Hossain On Som Spcial Condiions of n-h Ordr Non-oscillaory Nonlinar Sysms Communicaion of Koran Mah. Soc [] M. A.Saar An asympoic Mhod for Scond Ordr Criically Dampd Nonlinar Euaions J. Frank. Ins [] M. S.Alam and M. A.Saar An Asympoic Mhod for Third Ordr Criically Dampd Nonlinar Euaions J. Mah. andphy. Sci [] I. S. N.Mury and B. L.Dkshaulu Mhod of Variaion of Paramrs for Ovr-Dampd Nonlinar Sysms J. Conrol [] M. A.Saar An asympoic Mhod for Scond Ordr Criically Dampd Nonlinar Euaions J. Frank. Ins [4] M. S.Alam KBM Mhod for Crain Non-oscillaory Non-linar Sysm Journal of Bangladsh Acadmy of Scinc 5( 9-. [5] M. S.Alam and M. A.Saar An Asympoic Mhod for Third Ordr Criically Dampd Nonlinar Euaions J. Mah. andphy. Sci [6] M. S.Alam and M. A.SaarA Unifid KBM Mhod for Solving Third Ordr Nonlinar Sysms Indian J. pur appl. Mah [7] M. S.Alam KBM Mhod for Crain Non-oscillaory Non-linar Sysm Journal of Bangladsh Acadmy of Scinc 5( 9-. [8] M. S.Alam A Unifid Krylov-Bogoliubov-Miropolskii Mhod for Solving n-h Ordr Nonlinar Sysms J. Frank. Ins [9] M. S.Alam Mhod of Soluion o h n-h Ordr Ovr-dampd Nonlinar Sysms Undr Som Spcial Condiions Bull. Cal. Mah. Soc [] I. S. N.Mury and B. L.Dkshaulu Mhod of Variaion of Paramrs for Ovr-Dampd Nonlinar Sysms J. Conrol *CorrspondingAuhor:Md. MahafujurRahaman 4 Pag

15 Asympoic Soluions of Fifh Ordr Criically Dampd Non-linar Sysms wih Pair Wis Eual Eignvalus [] I. S. N.Mury and B. L.Dkshaulu and G.Krishna On an Asympoic Mhod of Krylov-Bogoliubov for Ovr-dampd Nonlinar Sysms J. Frank. Ins [] I. S. N.Mury and B. L.Dkshaulu Mhod of Variaion of Paramrs for Ovr-Dampd Nonlinar Sysms J. Conrol *CorrspondingAuhor:Md. MahafujurRahaman 5 Pag

On the Derivatives of Bessel and Modified Bessel Functions with Respect to the Order and the Argument

On the Derivatives of Bessel and Modified Bessel Functions with Respect to the Order and the Argument Inrnaional Rsarch Journal of Applid Basic Scincs 03 Aailabl onlin a wwwirjabscom ISSN 5-838X / Vol 4 (): 47-433 Scinc Eplorr Publicaions On h Driais of Bssl Modifid Bssl Funcions wih Rspc o h Ordr h Argumn