A Computational Method for Solving Two Point Boundary Value Problems of Order Four

Transcription

1 Yoge Gupta et al, Int. J. Comp. Tec. Appl., Vol (5), - ISSN:9-09 A Computatonal Metod for Solvng Two Pont Boundary Value Problem of Order Four Yoge Gupta Department of Matematc Unted College of Engg and Management Allaabad-00(U.P.) Inda E-mal: yogegupta@unted.ac.n Pankaj Kumar Srvatava Department of Matematc Jaypee Inttute of Informaton Tecnology Noda-00(U.P.) Inda E-mal: pankaj.rvatava@jt.ac.n Abtract Preent paper portray a computatonal metod ung cubc B-plne to olve fourt order boundary value problem. Te propoed ceme frt appled for oluton of pecal cae fourt order boundary value problem. Te metod, ten, extended to oluton of non-lnear and ngular problem. Selected Example from te lterature are olved numercally ung computer program n MATLAB. Key word: Fourt order boundary value problem, Sngular boundary value problem, Cubc B-plne, Nodal pont, Maxmum abolute error.. Introducton Engneer are reearcng oluton to reolve many of today tecncal callenge. Numercal tecnque are ued to olve te matematcal model n engneerng problem. Many of te matematcal model of engneerng problem are expreed n term of Boundary Value Problem, wc are ordnary dfferental equaton wt boundary condton. Fourt-order Boundary Value Problem are n te matematcal modelng of two-dmenonal cannel wt porou wall, vcoelatc and nelatc flow, deformaton of beam, plate deflecton teory, beam element teory and a number of oter engneerng and appled matematc applcaton. Solvng uc type of boundary value problem analytcally poble only n very rare cae. Many reearcer worked for te numercal oluton of fourt order boundary value problem. Some numercal metod uc a fnte dfference metod, dfferental tranformaton metod, Adoman' decompoton metod, omotopy perturbaton metod, varatonal teraton metod, plne metod ave been developed for olvng uc boundary value problem. Two-pont and mult-pont boundary value problem for fourt order ordnary dfferental equaton ave attracted a lot of attenton recently. Many autor ave tuded te beam equaton under varou boundary condton and by dfferent approace. Conder moot approxmaton to te problem of bendng a rectangular clamped beam of lengt l retng on elatc foundaton. Te vertcal deflecton w of te beam atfe te ytem d w ( / ) ( ), k Dw D + = qx dx () w(0) = wl ( ) = w'(0) = w'( l) = 0. were D te flexural rgdty of te beam, and k te prng contant of te elatc foundaton, and te load qx ( ) act vertcally downward per unt lengt of te beam. Te detal of te mecancal nterpretaton are gven n []. Matematcally, te ytem () belong to a general cla of boundary problem of te form d y f( x) yx ( ) gx ( ), a x b dx + = < < () ya ( ) = A, yb ( ) = A, y'( a) = B, y'( b) = B () IJCTA SEPT-OCT 0 Avalable onlne@

2 Yoge Gupta et al, Int. J. Comp. Tec. Appl., Vol (5), - ISSN:9-09 were f( x ) and gx ( ) are contnuou on [ ab, ] and A, B are fnte real arbtrary contant. Te analytcal oluton of () for arbtrary coce of f( x ) and gx ( ) cannot be determned. So, numercal metod are developed to overcome t lmtaton. Uman [] a formulated a mple condton tat guarantee te unquene of te oluton of te problem () and (). Among many numercal metod, a enumerated above, Splne metod ave been wdely appled for te approxmaton oluton of boundary value problem ncludng fourt order boundary value problem. (See [, ] and reference teren). Alo, Cubc B-plne a been ued to olve boundary value problem and ytem of boundary value problem [5,, 7], ngular boundary value problem [8] and alo, econd order perturbaton problem by ome autor. In te preent paper, we wll ue cubc B-plne to olve fourt order boundary value problem. T paper organzed a follow. In Secton, prelmnary reult and dervaton of cubc B- plne are preented; Secton contan cubc B-plne oluton of te pecal lnear fourt order boundary value problem baed on te reult n ecton ; general cae of te boundary value problem dcued n ecton ; ecton 5 deal wt non-lnear problem; wle ecton contan treatment of ngular problem. Secton 7 conclude te paper.. Dervaton for Cubc B-plne Te gven range of ndependent varable [ ab., ] For t range we cooe equdtant pont gven by π = { a = x0, x, x,... x n = b}.e. b a x = a + ( = 0,,... n) were =. Let u n defne S { ( π ) = pt () C [ ab, ] } uc tat pt () reduce to cubc polynomal on eac ub-nterval ( x, x + ).Te ba functon defned a ( x x ), f x [ x, x ] + ( x x ) + ( x x ) ( x x ), f x [ x, x] ( ) = ( + + ) + ( + ) ( x+ x), f x [ x, x+ ] x+ x f x x+ x+ B x x x x x ( ), [, ] 0, oterwe. Let u ntroduce four addtonal knot x < x < x0 and x > x > xn. From te above expreon, t obvou tat eac B ( x) C ( ). Alo, te value of B( x), B '( x) and B "( x ) at nodal pont are gven by Table I. Table I: Value of B ( x), B '( x) and B "( x ) at node B ( x ) B '( x ) B "( x ) x x / / / x / 0 / x + / / / x Snce eac B ( x) alo a pecewe cubc polynomal wt knot at π, eac B ( x) S( π ). Let Ω= { B, B0, B,..., B n + } and let B ( π ) = panω. Te functon n Ω are lnearly ndependent on[ ab,, ] tu B ( π ) ( n + ) - dmenonal. Alo B( π) = S( π) [9]. Let xbe ( ) te B-plne nterpolatng functon at te nodal pont and x ( ) B( π ).Ten xcan ( ) be wrtten a.terefore, for a gven functon yx ( ), = tere ext a unque cubc plne atfyng te nterpolatng condton: ( x) = cb ( x) = x ( ) = yx ( ) and '( a) = y'( a), '( b) = y'( b) () IJCTA SEPT-OCT 0 Avalable onlne@ 7

3 Yoge Gupta et al, Int. J. Comp. Tec. Appl., Vol (5), - ISSN:9-09 Let m = '( x ) and M = "( x ), we ave [0] ( v) m = '( x) y'( x) y ( x) (5) 80 ( v) ( v) M = "( x) y"( x) y ( x) + y ( x) 0 () M can be appled to contruct numercal dfference formulae for ( ) ( v) y ( x ), y ( x ) ( =,,..., n ) and ( v y ) ( x ) ( =,..., n ) a follow; ( ) ( ) M M ( x ) ( x ) ( ) ( v) = y ( x) + y ( x) (7) ( ) ( ) M M M ( x ) ( x ) = ( v) ( v) y ( x) y ( x) 70 M+ M+ + M M = M M M M M M y ( v) ( x ) (8) (9) Now, nce, ung Table I and above = equaton, we get approxmate value of ( yx ( ), y'( x ), y"( x ), ) ( v y ( x ) and y ) ( x ) a c + c + c+ yx ( ) = x ( ) (0) c+ c y'( x) = '( x) () c c + c+ y"( x) = "( x) () ( ) ( ) ( ) c ( ) + c + + c c y x x = () ( v) ( v) c+ c+ + c c + c y ( x) = ( x) (). Soluton of pecal cae fourt order boundary value problem Let yx ( ) = oluton of BVP = be te approxmate ( v y ) ( x) + f( x) yx ( ) = gx ( ) (5) Dcretzng BVP at te knot, we get ( v y ) ( x) + f( x) yx ( ) = gx ( ) ( =,... n ) () Puttng value n term of we get c ung equaton (0, ), c + c + + c c + c c c c f = g (7) Were f = f( x) an gd = gx ( ) are te value of f( x) and gxat ( ) te knot x. Smplfyng (7) become = ( c c c c c ) f( c c c ) g (8) T gve a ytem of ( n ) lnear equaton for ( =,... n ) n ( n + ) unknown vz. c ( =,0,... ). Remanng four equaton wll be obtaned ung te boundary condton a follow; ya ( ) = A c + c0 + c = A (9) yb ( ) = A cn + cn + c = A (0) y '( a) = B c + c = B () y '( b) = B c + c = B () n Te approxmate oluton yx ( ) = = obtaned by olvng te above ytem of ( n + ) lnear equaton n ( n + ) unknown ung equaton (8) and (9) to (). IJCTA SEPT-OCT 0 Avalable onlne@ 8

4 Yoge Gupta et al, Int. J. Comp. Tec. Appl., Vol (5), - ISSN:9-09 Numercal Example In t ecton we llutrate te numercal tecnque dcued n te prevou ecton by te followng two boundary-value problem: Problem. ( v) y + xy = (8 + 7 x + x ) e wt y(0) = y() = 0, y'(0) =, y'() = x () Te analytcal oluton yx ( ) = x( xe ) x.table II compare te numercal reult for problem of preent metod and numercal metod n []. Problem. ( v) y + y = wt y( ) = y() = 0, n n y'( ) = y'() = (co + co ) () Gven fourt order boundary value problem a analytcal oluton a nnn xn x+ cococo xco x yx ( ) = 0.5 co + co Comparon of numercal reult by preent metod and metod of [] demontrated n Table III. Table II: Max abolute error e for problem Preent metod Metod n[] /8.7E 8.5E 5 / 5.75E 9.9 E /.7E 9.5 E 8 Table III: Max abolute error e for problem Preent metod Metod n[] / 7.5E 9.0E. General cae lnear t order boundary value problem Conder te boundary value problem ( v) ( ) y ( x) + pxy ( ) ( x) + qxy ( ) " + rxy ( ) '( x) + txyx ( ) ( ) = ux ( ) (5) Subject to boundary condton gven by (). Let yx ( ) = be te approxmate = oluton of BVP. Dcretzg at knot ( v) ( ) y ( x) + py ( x) + qy "( x) + ry '( x) + tyx ( ) = u () Were p = px ( ), q = qx ( ), r = rx ( ), t = tx ( ), u = ux ( ). Puttng te value of dervatve ung (0-), we get c+ c+ + c c + c c+ c+ + c c + p c c + c+ c+ c c + c + c+ + q + r + t = u (7) On mplfcaton, t become c (c + c c + c ) + ( c p c + c c ) q( c c c+ ) r( c+ c ) t = ( c c c ) u (8) Now, te approxmate oluton obtaned by olvng te ytem gven by (8) and (9-). 5. Non-lnear t order boundary value problem Conder non-lnear fourt order BVP of te form ( v) ( ) y ( x) = f( x, y( x), y'( x), y"( x), y ( x)) (9) Subject to boundary condton gven n (). Let yx ( ) = be te approxmate = oluton of BVP. It mut atfy te BVP at knot. So, we ave ( v) ( ) y ( x) = f( x, y( x), y'( x), y"( x), y ( x)) (0) Ung (0-),we get /8.9E 7.8E 5 /.08E 8.7 E IJCTA SEPT-OCT 0 Avalable onlne@ 9

5 Yoge Gupta et al, Int. J. Comp. Tec. Appl., Vol (5), - ISSN:9-09 c+ c+ + c c + c = c + c + c+ c+ c x,,, () f c c + c+ c+ c+ + c c, T eqn () togeter wt eqn (9-) gve a nonlnear ytem of equaton, wc olved to get te requred oluton of BVP.. Sngular t order boundary value problem Conder ngular fourt order BVP of te form ( v) γ ( ) y ( x) + y ( x) = f( xyx, ( )); x 0 x () Under te boundary condton y(0) = A, y () = A, y () = B, y (0) = 0. () Snce x = 0 ngular pont of eqn (), we frt modfy t at x = 0 to get tranformed problem a, ( v) ( ) y ( x) + pxy ( ) ( x) = rxy (, ) () were 0 x = 0 px ( ) = γ x 0 x (5) And f(0, y) x = 0 rxy (, ) = γ + f( xy, ) x 0 () Now, a n prevou ecton, let yx ( ) = be te approxmate oluton = of BVP. Dcretzg at knot, we get ( v) ( ) y ( x) + p( x) y ( x) = r( x, y( x)) (7) Puttng te value of dervatve ung (0-), c+ c+ + c c + c c+ c+ + c c + p c + c + c+ = rx (, ) (8) And boundary condton provde, y(0) = A c + c0 + c = A (9) y () = A c + c = A (0) n n n y () = B c c + c = B () T eqn (8) togeter wt eqn (9-) gve a nonlnear ytem of equaton, wc olved to get te requred oluton of BVP (). Numercal example Problem. d y y = e ( + x), 0 < x<, dx y(0) = 0, y() = ln, y'(0) =, y'() = /. () Te maxmum abolute error by our metod and by fnte dfference metod (Twzell []) for problem are preented n followng Table IV. Problem. d y d y 5 + = 5 y ( x y ) ( 7 x y ), x (0,), dx x dx () y(0) =, y () =, y () =, y (0) = Comparon of numercal reult by preent metod and tat of [] demontrated n Table V. Table IV: Max abolute error e for problem Preent metod Metod n[] /8.E-5 0.E- / 7.97E-7 0.E-5 / 5.8E-8 0.7E- Table IV: Max abolute error e for problem Preent metod Metod n[] /8.7E-.0E-0 /.E-7.75E-05 /.9E-8.8E-0 y (0) = 0 c c + c c = 0 () IJCTA SEPT-OCT 0 Avalable onlne@ 0

6 Yoge Gupta et al, Int. J. Comp. Tec. Appl., Vol (5), - ISSN: Concluon A numercal algortm for oluton of fourt order boundary value problem a been envaged. Te propoed metod a been extended to olve nonlnear and ngular problem a well. Te numercal reult demontrate tat te preent metod approxmate oluton better tan prevouly appled metod wt ame number of nterval. [] R. K. Sarma, C.P. Gupta, Iteratve oluton to nonlnear fourt-order dfferental equaton troug mult ntegral metod, Internatonal Journal of Computer Matematc, 8(989) 9. Reference [] E.L. Re, A.J. Callegar, D.S. Aluwala, Ordnary Dfferental Equaton wt Applcaton, Holt, Rneart and Wnton, New Cork, 97. [] R. A. Uman, Dcrete metod for boundary-value problem wt Engneerng applcaton, Matematc of Computaton, (978) [] M. Kumar, P. K. Srvatava, Computatonal tecnque for olvng dfferental equaton by cubc, quntc and extc plne, Internatonal Journal for Computatonal Metod n Engneerng Scence & Mecanc, 0( ) (009) [] M. Kumar, P. K. Srvatava, Computatonal tecnque for olvng dfferental equaton by quadratc, quartc and octc Splne, Advance n Engneerng Software 9 (008) -5. [5] N. Caglar, H. Caglar, B-plne metod for olvng lnear ytem of econd order boundary value problem, Computer and Matematc wt Applcaton 57 (009) [] H. Caglar, N. Caglar, K. Elfatur, B-plne nterpolaton compared wt fnte dfference, fnte element and fnte volume metod wc appled to two pont boundary value problem, Appled Matematc and Computaton 75 (00) [7] M. Degan, M. Laketan, Numercal oluton of nonlnear ytem of econd-order boundary value problem ung cubc B-plne calng functon, Internatonal Journal of Computer Matematc, 85(9) [8] M. Kumar, Y. Gupta, Metod for olvng ngular boundary value problem ung plne: a revew, Journal of Appled Matematc and Computng (00) [9] P. M. Prenter, Splne and varaton metod, Jon Wley & on, New York, 989 [0] F. Lang, Xao-png Xu, A new cubc B-plne metod for lnear fft order boundary value problem, Journal of Appled Matematc and Computng (0) 0-. [] Sraj-ul-Ilam, Ikram A. Trmz, Saadat Araf, A cla of metod baed on non-polynomal plne functon for te oluton of a pecal fourt-order boundary value problem wt engneerng applcaton, Appled Matematc and Computaton 7 (00) 9-80 [] E. H. Twzell, A two-grd, fourt order metod for nonlnear fourt order boundary value problem, Brunel Unverty department of matematc and Stattc Tecncal report TR//85 (985). IJCTA SEPT-OCT 0 Avalable onlne@