Significance of Dirichlet Series Solution for a Boundary Value Problem

Transcription

1 IOSR Journal of Engneerng (IOSRJEN) ISSN (e): 5-3 ISSN (p): Vol. 6 Issue 6(June. 6) V PP Sgnfcance of Drchlet Seres Soluton for a Boundary Value Problem Achala L. Nargund* and Pramod S.* *P.G. Department of Mathematcs and Research Centre n Appled Mathematcs M.E.S College of Arts Commerce and Scence Bengaluru-563 Abstract: Drchlet seres s an exponental power seres ntroduced by P. G. L. Drchlet a German mathematcan. Usng ths power seres we study the boundary-layer equaton of flow over a nonlnearly stretchng sheet n the presence of magnetc feld. The soluton obtaned by ths method s n good agreement wth exstng solutons. Keywords: Boundary Layer Equatons Adoman decomposton method Homotopy Analyss Method Drchlet Seres Newton-Raphson Method. I. INTRODUCTION The boundary layer flow caused by a contnuously stretchng sheet s often encountered n many engneerng and ndustral processes. These flows are exstng n the polymer ndustry n the manufacture of sheetng materal through an extruson process and coolng of an nfnte metallc plate. Most boundary layer models can be reduced to systems of nonlnear ordnary dfferental equatons whch are usually solved ether by numercal methods or by analytc methods. Analytc methods have sgnfcant advantages over pure numercal methods n provdng more convergent solutons. In the modern developments of the theory of power seres a great part has been played by a varety of methods of summaton of oscllatng seres whch we assocate wth the names of Frobenus Holder Cesaro Borel Lndelof MttagLeffler and Le Roy. Of these defntons the smplest and the most natural s that whch defnes the sum of an oscllatng seres as the lmt of the arthmetc mean of ts frst n partal sums. Ths defnton was generalzed n two dfferent ways by Holder and by Cesaro who thus arrved at two systems of defntons the complete equvalence of whch has been establshed only recently by Knopp Schnee Ford and Schur. The range of applcaton of Cesaro s methods s lmted n a way whch forbds ther applcaton to the problem of the analytcal contnuaton of the functon represented by a Taylor s seres. A power seres outsde ts crcle of convergence dverges too crudely for the applcaton of such methods: more powerful though less delcate methods such as Borel s are requred. But Cesaro s methods have proved of hghest value n the study of power seres on the crcle of convergence and closely connected problems of the theory of Fourer seres. And t s natural to suppose that n the theory of Drchlet Seres where we are dealng wth seres whose convergence or dvergence s of a much more delcate character than s n general that of power seres they wll fnd a wder feld of applcaton [5] [6]. Drchlet seres soluton s an exponental seres soluton whch are most useful especally n obtanng derved quanttes than pure numercal schemes. The accuracy and the unqueness of the soluton so obtaned can be confrmed usng other equally powerful sem numercal schemes. Ths method explaned by P. L. Sachdev [9]. Sachdev. et. al () proposed the Drchlet seres soluton approach to unform flow past a sem-nfnte flat plate. Here they have found the soluton of free conventon flows n saturated porous meda and also soluton of Falkner-Skan equaton n the axsymmetrc flow due to stretchng flat surface []. Sachdev. et. al (5) proposed the Drchlet seres solutons to the boundary value problems for thrd order nonlnear ordnary dfferental equatons over an nfnte nterval. In ths paper they found the soluton of Falkner-Skan equaton occurs n the magneto hydrodynamcs []. Awang. et. al [] have studed the boundary layer equaton of flow over a nonlnearly stretchng sheet n the presence of chemcal reacton and a magnetc feld usng Adoman Decomposton Method (ADM). S. B. Sathyanarayana and L. N. Achala [] nvestgated the soluton by Homotopy Analyss Method (HAM) for the boundary layer flow of a vscous flow over a nonlnearly stretchng sheet n the absence of a chemcal reacton and n presence of magnetc feld. We are nterested n applyng the Drchlet Seres Soluton method to obtan an approxmate analytc soluton of the boundary layer vscous flow over a nonlnearly stretchng sheet n the Internatonal organzaton of Scentfc Research 8 P a g e

2 Sgnfcance of Drchlet Seres Soluton for a Boundary Value Problem absence of a chemcal reacton and n presence of magnetc feld. Comparson of the present soluton wth the solutons obtaned by Awang. et. al [] n the absence of a chemcal reacton and S. B. Sathyanarayana and L. N. Achala [] by HAM were also made. II. GOVERNING EQUATIONS Here we consder the steady two-dmensonal ncompressble flow of an electrcally conductng vscous flud past a nonlnearly sem-nfnte stretchng sheet under the nfluence of a constant transverse appled magnetc feld. The magnetc Reynolds number s assumed small and neglgble n comparson to the appled magnetc feld. The governng boundary layer equatons are u x v y = () u u u B u v = v u m x y y () where x and y are dstances along and perpendcular to the sheet respectvely u and v are component of the velocty along x and y drectons respectvely v knematc vscosty flud densty electrcal m conductvty B strength of the magnetc feld. The correspondng boundary condtons for nonlnear stretchng sheet are as follow u ( x ) = a x cx v ( x ) = (3) u a s y (4) where d and c are constants. We ntroduce the smlarty transformatons [] a y u a x f c x g = = ' ( ) ' ( ) (5) m cx B (6) m v = a f ( ) g ( ) N = a / a m substtutng equatons (5) and (6) n () and () we obtan the followng ordnary dfferental equatons f ''' ff '' f ' N f ' = (7) g ''' fg '' 3 f ' g ' f '' g N g ' = (8) subject to boundary condtons (3) and (4) f ( ) = f ' ( ) = f ' ( ) = (9) g ( ) = g ' ( ) = g ' ( ) = () where f and g are functons related to the velocty fled N s a magnetc parameter and the prmes denote dfferentaton wth respect to. Internatonal organzaton of Scentfc Research 9 P a g e

3 Sgnfcance of Drchlet Seres Soluton for a Boundary Value Problem III. DIRICHLET SERIES APPROACH TO BOUNDARY VALUE PROBLEM OVER AN INFINITE INTERVAL We assume the Drchlet seres solutons to equatons (7) and (8) satsfyng f '( ) = and g '( ) = [3] [4] n the form f ( ) = b d e () = g ( ) = l p e () = where > d and p are parameters. Substtutng () and () n (7) and (8) we get 3 [ N ] b d e [ k k ] b b d e = (3) k k = = k = and 3 [ N N ] l p e b d e = = = k = [6 k 3 k ] b l d p e = k k k (4) For = equatons (3) and (4) reduces to [ N ] b d e = (5) bd =. (6) To avod trval soluton we set b = l =. Hence from equatons (5) and (6) we can wrte = N (7) = (8) where b and d. Substtutng equatons (7) and (8) n equatons (3) and (4) we obtan recursve relatons for the coeffcents b and l ( = ) = [ ] (9) k k ( )( N ) k = b k k b b and 3 d k = [ ] ( ). () k k ( )( N ) k = p l k k b l If the seres () and () converge absolutely when > for some then these seres converge absolutely and unformly n the half plane R e ( ) R e ( ) such that f ' ( ) = and g ' ( ) = [3] [4]. A general dscusson of the convergence etc. of the Drchlet seres () and () can be found n [5] [6]. The unknown parameters d p are determned from the remanng boundary condtons (9) and () at Internatonal organzaton of Scentfc Research P a g e

4 Sgnfcance of Drchlet Seres Soluton for a Boundary Value Problem = as follows N f ( ) = b d () = f ' ( ) = ( ) b d () = g ( ) = l p (3) = g ' ( ) = ( ) l p. (4) = The numercal values of d and p are calculated by applyng Newton - Raphson method to equatons () () and (4) (Table ). Substtutng equatons (7) and (8) n equatons () and () then the Drchlet Seres solutons for (7) and (8) wll be N f ( ) = b d e (5) = g ( ) = l p e (6) = where b and l can be calculated by usng equatons (9) and (). The shear stress at the surface s gven by 3 f '' ( ) = b d. (7) = The exact soluton of (7) subject to boundary condtons (9) s [6] f ( ) = [ exp ( N )] (8) N f '' ( ) = N. (9) Obtaned Drchlet seres soluton (5) and exact soluton (8) are compared graphcally for the dfferent values of N n Fgure. The exact shear stress gven n (9) s compared numercally wth our Drchlet seres shear stress (9) n Table. IV. NUMERICAL RESULTS AND CONCLUSIONS In the present paper we have gven approxmate analytc soluton of the boundary layer vscous flow over a nonlnearly stretchng sheet n the absence of a chemcal reacton and n presence of magnetc feld n the form of Drchlet seres (5) and (6). The calculated values of f ''( ) representng shear stress at the surface for the dfferent sets of values of N d and are gven n Table. Comparson of the values obtaned by the Drchlet seres method wth other methods gven n Table. The velocty f '( ) of exact soluton (8) and Drchlet seres soluton (5) are compared graphcally for the dfferent values of N n Fgure. Internatonal organzaton of Scentfc Research P a g e

5 Sgnfcance of Drchlet Seres Soluton for a Boundary Value Problem Fgures represents the component of velocty u ( x y ) (5) and Fgures represents the component of velocty v ( x y ) (6) for the dfferent values N. Both u ( x y ) and v ( x y ) are obtaned by substtutng Drchlet seres (5) and (6) n equatons(5) and (6). We can observe that the component of velocty v ( x y ) movng upwards as N ncreases. The soluton obtaned by usng the Drchlet seres method matches wth the soluton obtaned by usng Adoman Decomposton Method [] Homotopy Analyss Method [] and the exact soluton [6]. We conclude that the Drchlet seres method s a strong analytc method to solve nonlnear dfferental equatons and t produces good convergent soluton. Table : Comparson of the values of f '' ( ) obtaned by the Drchlet seres method and exact soluton for the dfferent values of N and d. N d ' f ( ) ' f ( ) Drchlet Exact Table : Comparson of the values of f '' ( ) obtaned by usng the Drchlet Seres Method and other methods. N Drchlet Seres Exact ADM HAM Fgure : Velocty plot of f ( ) for dfferent N Internatonal organzaton of Scentfc Research P a g e

6 Sgnfcance of Drchlet Seres Soluton for a Boundary Value Problem Fgure : Velocty plot of f '( ) for dfferent N Fgure 3: u ( x y ) for N = Fgure 4: v ( x y ) for N = Internatonal organzaton of Scentfc Research 3 P a g e

7 Sgnfcance of Drchlet Seres Soluton for a Boundary Value Problem Fgure 5: u ( x y ) for N =.5 Fgure 6: v ( x y ) for N =.5 Fgure 7: u ( x y ) for N =9 Internatonal organzaton of Scentfc Research 4 P a g e

8 Sgnfcance of Drchlet Seres Soluton for a Boundary Value Problem Fgure 8: v ( x y ) for N =9 Fgure 9: u ( x y ) for N = 5 Fgure : v ( x y ) for N = 5 V. ACKNOWLEDGEMENT Internatonal organzaton of Scentfc Research 5 P a g e

9 Sgnfcance of Drchlet Seres Soluton for a Boundary Value Problem The authors are thankful to Vson Group of Scence and Technology (VGST) for provdng fnancal assstance through GRD 5 CISE. REFERENCES [] S. Awang Kechl and I. Hashm Seres soluton of flow over nonlnearly stretchng Sheet wth chemcal reacton and magnetc feld Physcs Letters A [] S. B. Sathyanarana and L. N. Achala Homotopy analyss method for flow over nonlnearly stretchng sheet wth magnetc Feld Journal of Appled Mathematcs and Flud mechancs ISSN Volume 3 Number pp. 5-. [3] T.K.Kravchenko A.I.Yablonsk Soluton of an nfnte boundary value problem for thrd order equaton Dfferental nye Uranenya vol. pp [4] T.K. Kravchenko A.I.Yablonsk A boundary value problem on a sem-nfnte nterval Dfferental nye Uranenya vol. 8() pp [5] S.Resz Introducton to Drchlet seres Camb. Unv. Press 957. [6] G.H. Hardy M. Resz The general theory of Drchlet s seres Cambrdge Unv. Press 95. [7] J.H. Merkn A note on the soluton of a dfferental equaton arsng n boundary layer theory J. Engg. Mathematcs [8] P.L. Sachdev Nonlnear ordnary dfferental equatons and ther applcatons Marcel Dekker INC 99. [9] P.L. Sachdev N. M. Bujurke and N. P. Pa Drchlet seres soluton of equatons arsng n boundary layer theory Mathematcal and computer modellng [] V. M. Falkner and S. W. Skan Solutons of the boundary layer equatons Phl. Soc. Mag. : [] P. L. Sachdev N. M. Bujurke and V. B. Awat Boundary value problems for thrd order nonlnear ordnary dfferental equatons Studes n Appled Mathematcs Vol. 5 No. 3 pp [] N. M. Bujurke R. B. Kudenatt and V. B. Awat Drchlet seres soluton of mhd flow over a nonlnear stretchng sheet Internatonal journal of appled mathematcs and engneerng scences Vol. 5 No January-June. [3] N. M. Bujurke R. B. Kudenatt and V. B. Awat An exponental seres method for the soluton of free convecton boundary layer flow n a saturated porous medum Amercan journal of computatonal mathematcs 4-. [4] N. M. Bujurke and V. B. Awat Approxmate analytcal solutons of MHD flow of a vscous flud on a nonlnear porous shrnkng sheet Bulletn of the nternatonal mathematcal vrtual nsttute vol [5] R. B.Kudenatt V. B.Awat and N. M.Bujurke Exact analytcal solutons of class of boundary layer equatons for a stretchng surface Appl Math Comp vol. 8 pp [6] F. T. Akyldz H. Bellout K. Vajravelu J. Math. Anal. Appl Internatonal organzaton of Scentfc Research 6 P a g e