Intrnational Journal of Enginring & Tchnology IJET-IJES Vol:3 o:05 5 Rational Approximation for th on-dimnsional Bratu Equation Moustafa Aly Soliman Chmical Enginring Dpartmnt, Th British Univrsity in Egypt El Shrouk City, Cairo, 837, Egypt -mail: moustafa.aly@bu.du.g Abstract-- Th solution of th on dimnsional planar Bratu quation is charactrizd by having two branchs. Th two branchs mt at a limit point that can b obtaind through th solution of an ign-rlation. In this papr a transformation is introducd to transform this highly non-linar problm to a linar on. Rational xprssions ar givn for th solution of th Bratu quation and th limit point. This is don using collocation and prturbation tchniqus. Indx Trm-- Bratu problm, collocation, rational function, prturbation. ITRODUCTIO Svral thortical studis and numrical mthods wr usd for th study of th Bratu problm [-6] which modls thrmal xplosion in a slab nclosur. Frank- Kamntskii [] formulatd th problm and obtaind analytical solutions for th cas of a slab and cylindr nclosur. Th numrical mthods usd for its solution includd th Chbychv psudospctral mthod [] B-splins [3], and variational mthods []. Mohsn [5] usd Grn s function and th rsulting intgral quation was discrtizd and th non-linar quations formd ar solvd. For mor rfrncs on th subjct, on can rfr to rfrncs [,5]. Th Bratu problm is givn by d u xp( u 0 ( u ( 0 ( du(0 0 Thr is a critical valu for th Frank-Kamntskii paramtr abov which thr is no solution to th Bratu problm. Blow this valu thr ar two solutions. Harly and Momoniat [6] hav shown how th Frank-Kamntskii paramtr can b obtaind in trms of th first intgrals. This problm has th analytical solution givn by u( x, log( z sc h ( z / x (3 ( Whr z is th solution of th ign-rlation z cosh( z / Boyd [] rcntly obtaind approximat rlations for this quation. H also showd that much information about th solution of th on-dimnsional Bratu quation can b obtaind by applying on point collocation. Th choic of th collocation points dpnd on what points w nd th solution to b accurat at. For th linar boundary valu problm (5 d u ( u 0 (6 u ( 0 (7 du(0 0 Th choic of th collocation points as th zros of Jacobi polynomials with a wight of ( x [7-9] will nsur that n th solution is corrct to th ordr of at th collocation n points and to th ordr of lswhr whr n is th polynomial ordr. W call this mthod standard collocation mthod [7-9].Th intgral (8 I u (9 0 will b corrct to th ordr of n In on point collocation, th collocation point at which th solution is valid up to th ordr of λ, is x / 5 0.7 (0 If w would lik th solution to b corrct to th ordr of at x=0, thn th collocation point should b 30705-99-IJET-IJES Octobr 03 IJES
Intrnational Journal of Enginring & Tchnology IJET-IJES Vol:3 o:05 55 x / 6 0.0883 ( This xplains why th choic of x 0.030705 ( mad by Boyd [] to obtain accurat prdiction of th limit point λ = 0.878576797890305. At this point z=.807 and u(0=.868. In this papr w first obtain an approximat solution to th ign- rlation (5 thn w transform th non-linar Bratu problm into a linar form and hnc apply a collocation mthod that nsurs that th polynomial solution is corrct to ( th ordr of n A (A to b dfind latr in quation (all through th domain of intrst.5 Finally w transform th polynomial solution into a rational form. Much can b oftn gaind by rcasting th solution into rational functions [0]. Th contribution of th prsnt work is twofold. First, w obtain approximat rlations to th valu of z in trms of λ and thus w can obtain th analytical solution xplicitly. For mor accurat solution, w can us this solution as initial guss for any numrical mthod. Th scond aim is to prsnt a rational function approximation to th solution. Th structur of this papr is as follows. Sction prsnts th approximat rlations for z. In Sction 3, w will apply th collocation mthod to a linar form of th boundary valu problm. Th collocation solution can b convrtd to a rational function form. Th fourth sction is dvotd to th rsults obtaind by th proposd mthod. Conclusions ar mad in Sction 5.. APPROXIMATE SOLUTIO TO THE EIGE-RELATIO Lt us writ quation (5 as A sc h ( A (3 Whr A z / ( W will study quation (3 for small A ( is small on th lowr branch and larg A ( is small on th uppr branch Expanding quation (3 for small A, w obtain 6 A /( A / A / A / 70 (5 Which can b put in a / Pad approximant form as A ( /5A /(( 3 /5A / 5A (6 This quation can b put into a quadratic form in A which can b solvd to giv A 3 / 30 ( 7 /5 / 900 ( /5 / 5 (7 Substituting for A in quation (7 from quation ( w obtain z ( 3 / 30 ( 7 /5 / 900 ( /5 / 5 (8 Th minus sign of th root is for th lowr branch of th solution. This solution is valid for th lowr branch to λ of 0.8. To xtnd this rlation to prdict accuratly th limit point and th valu of z at this point, w usd curv fitting with numrical solutions to obtain; On th lowr branch z whr ( 3 / 30 ( 7 /5 / 900 * ( 0.03 ( /5 / 5 (9 3 (- 0.007959 7 (0 and on th uppr branch z ( 3 / 30 ( 7 /5 / 900 * ( /5 / 5 ( 0.03 xp(.5(0.8785768 /( 3. ( For larg A, w can writ quation (3 30705-99-IJET-IJES Octobr 03 IJES
Intrnational Journal of Enginring & Tchnology IJET-IJES Vol:3 o:05 56 A / cosh( A A xp( A( xp( A ( log log( xp( A log A A (3 8 0.88 (7 Substituting quations (,5 into quation (3, w obtain ( A ( ( A C log ( A 8 8 (0.5 ( A (8 This is scond ordr quation in A from which w can obtain th valu of z ; Lt us approximat log A by ( A ( ( A log A (0.5 ( A and log( 8 ( xp( A ( (5 Whr and ar fitting paramtrs. Through numrical xprimntation, th following valus wr obtaind; 6 (6 z (( 3 6 C(3 6 (3 6 D (9 whr D C (3 6 (3 6 C(9 36 ( C(3 6 6 6 Th accuracy of th approximations drivd in this sction is shown in tabl I for th lowr branch and tabl (II for th uppr branch. Equation (9 is quit accurat for th lowr branch (tabl I. For th uppr branch on can us quation ( for λ 0.8 and quation (9 for λ< 0.8. λ Λ Tabl I Eign-paramtr z for th lowr branch z from Equation z Exact (9 0.0.005.005 0..065.06 0..0563.0563 0.5.788.797 0.8.5.575 0.87.6738.6776 0.8785768.807.8099 Tabl II Eign-paramtr z for th uppr branch z Exact z from Equation ( z from Equation (9 0.0 69.855 69.357 69.90 0. 5.9 5.785 5.78 0. 9.00 9.0 9. 0.5.5..83 0.8..39.37 0.87.97.966.97 0.87858.807.803.9 It would b also of intrst to find out an approximation for u(x/u(0 for small A and larg A For small A w hav from quations ( and (5; 30705-99-IJET-IJES Octobr 03 IJES
Intrnational Journal of Enginring & Tchnology IJET-IJES Vol:3 o:05 57 u( x log(cosh( Ax log( 0.5A x x u(0 log(cosh( A log( 0.5A (30 For larg A, w will hav; u( x log(cosh( Ax log(xp( Ax x u(0 log(cosh( A log(xp( A (3 otic that from quations (33 and ( v( 0 xp( u(0 / z (0 So that quation (37 bcoms d v A v 0 ( Th application of on point collocation at x / 5 givs th rlation umrical tsting in th nxt sction will show whn ths approximations can b usd..5( v A v 0 ( 3. METHOD DEVELOPMET Th Bratu problm is givn by quations (-3. Multiplying both sids by du/ and intgrating with rspct to x btwn 0 & x w obtain d u ( (xp( u xp( u(0 0 (3 W now us th transformation v xp( u / (33 To obtain du d v dv (3 d v v dv ( (35 u v Substituting quations (33-3 into quation (3, w obtain; dv ( ( (36 v v (0 v Substituting quations (36 into quation (35 and thn into quation (, w obtain d v v 0 v (0 (37 v ( (38 dv(0 0 (39 v is th valu of (x whr solution is thus A ( x v( x ( 0.A v at th collocation point and th (3 Th application of on point collocation at x / 6 givs th rlation.( v A v whr solution is thus 0 v is th valu of (x A ( x v( x ( 5A / ( v at th collocation point and th (5 A v( 0 (6 ( 5A / Substituting for A from quation ( and (0 into quation (6, w obtain th following rlation for and (0 v (0( v(0 5v(0 v. (7 Th maximum valu of occurs at v ( 0 0.( 0.5586 (8 u ( 0 ln( v(0.6586 (9 30705-99-IJET-IJES Octobr 03 IJES
Giving a valu of Intrnational Journal of Enginring & Tchnology IJET-IJES Vol:3 o:05 58 =0.879 (50 W now apply a modifid collocation mthod in which w collocat at x=0 in addition to th zros of Jacobi polynomials. For th on collocation point at x =0.7 th addition of th xtra point at x=0 will lad to th following quations..5( v A v 0 (5 v ( x r whr P v( x cp cp ( x (56 ( x ( x ( x xi (57 j x i ar th collocation points and c is a constant to b stimatd such that.5( v.0( v (0 A v (0 0 (5 W notic that th collocation quations at th zros of Jacobi polynomials ar not affctd by th us of xtra point. For th prsnt cas, th solution will b corrct to th scond ordr of A for th whol domain of x. W call th solution of th modifid collocation v (x A ( x A ( x (0. x v ( x v( (( 0.A (53 and ( A / 60 v (0 (5 A 0.A A 0.A Substituting for A from quation ( and solving for, w obtain, v (0 3 0.5v (0.5v (0 0.5v (0 3.5v (0 90.5v This givs a valu of 0. 88055 at v ( 0 0. 5508 and u ( 0. 96 (55 ow w introduc what w call rational collocation. For multi-point collocation, on would form a rational function v r (x from th solution v (x, and v (0 such that (0 v r (0 Thus v(0 cp (0 v (0 cp (0 (58 v (0( v(0 c ( v(0 / P (0 v (0 (59 This choic maks v r (x to b qual to v (x at th collocation points and at x 0 ow w apply th rational mthod to th on point collocation to obtain, u u r 60 A (0 (60 [5 A ] A u( x c( x x ( x c( x x (6 Choos c such that u r Thus c c (0 u(0 u (0 cx cx (6 u (0( u(0 ( u(0 / x ( u (0 (63 5A (5 A (6 30705-99-IJET-IJES Octobr 03 IJES
u r ( x Intrnational Journal of Enginring & Tchnology IJET-IJES Vol:3 o:05 59 A [5 ( 5x ] A [(5 A (3 ( x (65 W notic that this rational function is xact for trms up to A In addition as A, (x ] u r 0 for x ε [0, To summariz, for λ< 0.8 on th lowr branch w calculat z from quation (9, v(x from on point collocation quation (5and u(x from quation (33.For λ>0.8 on th lowr branch, w us quations (9,53 or 65,33. For th uppr branch, w us quations ( or 9,53 or 65,33. To gt bttr accuracy, w may go to two points or four points collocation mthods. Th cofficints for th scond drivativs at th collocation points can b obtaind from th routins in rfrncs [7-9].. RESULTS AD DISCUSSIO First w obtaind th ratio u(x/u(0using th xact solution whil vrifying numrically th accuracy of th standard collocation, th modifid collocation and th rational collocation mthods proposd in sction 3. W found that up to 0. 8 on th lowr branch on point collocation is good nough. W hav to go to two points collocation up to th limit point and thn up to 0.0 on th uppr branch whr w should go for four points collocation. Th plots in Fig. show that on th lowr branch th ratio u(x/u(0 dos not chang much for λ< 0.8 and approachs th parabolic profil givn by quation (30. For λ< 0.0 on th uppr branch th ratio approachs th straight lin givn by quation (3. xt in Fig. & 3 w show th prformanc of diffrnt collocation mthods for λ=0.87 on th lowr branch and λ= 0.0 on th uppr branch. In Fig., w hav plottd xact solution and th numrical solution using on point standard, modifid and rational collocation mthods. Th profils obtaind from th modifid and rational collocation almost fall on th xact solution. For bttr accuracy w can go to two points collocation. Similarly in Fig.3 th two points modifid and rational collocation almost fall on th xact solution. For bttr accuracy w can go to four points collocation. Fig.. u(x/u(0 profils for diffrnt valus of 30705-99-IJET-IJES Octobr 03 IJES
Intrnational Journal of Enginring & Tchnology IJET-IJES Vol:3 o:05 60 Fig.. Th lowr branch u profils for th cas of =0.87 using diffrnt collocation mthods Fig. 3. Th uppr branch u profils for th cas of =0.0 using diffrnt collocation mthods 30705-99-IJET-IJES Octobr 03 IJES
Intrnational Journal of Enginring & Tchnology IJET-IJES Vol:3 o:05 6 5. COCLUSIOS Basd on a nw kind of transformation, it is possibl to transform th strongly nonlinar problms of Bratu quation containing an xponnt trm xp(u into a linar form. W studid th Bratu problm using diffrnt collocation mthods applid to th linar form. Th obtaind rsults ar found to b in good agrmnt with th xact solutions. By making us of orthogonal polynomials proprtis, w wr abl to rcast straightforward polynomials into a rational form. This is to b contrastd with th gnral procdur prsntd in rfrnc [0] which could not produc a rational form as good as th on prsntd in this papr. Low ordr polynomials wr good nough to giv accurat solutions. This is in agrmnt with th findings of Boyd [] that th solution is so smooth that th problm is not a strong tst for numrical mthods. Howvr currnt rsarch indicats that th mthods introducd in this papr can prov to b vry usful in trating th mor challnging problm of combustion raction in a sphr. REFERECES [] D.A. Frank-Kamntskii, Diffusion and Hat Exchang in Chmical Kintics, Princton Univrsity Prss, Princton, w Jrsy, 955. [] J.P. Boyd, On-point psudospctral collocation for th on dimnsional Bratu quation, Appl. Math. Comp., 7: 5553-5565, 0. [3] H. Caglar,. Caglar, M. Ozr, A. Valarstos, A.. Anagnostopoulos B-splin mthod for solving Bratu s problm, Int.J. Comput. Math., 87: 885-89, 00. [] M.I. Syam, Th modifid Broydn-variational mthod for solving nonlinar lliptic diffrntial quations, Chaos Solitons and Fractals, 3: 39-0, 007. [5] A. Mohsn, On th intgral solution of th on-dimnsional Bratu problm, Journal of Computational and Applid Mathmatics, 5:6-66,03 [6] C. Harly, E. Momoniat, Altrnat drivation of th critical valu of th Frank-Kamntskii paramtr in cylindrical gomtry, J. onlinar Math. Phys.,5: 69 76,008 [7] J. Villadsn and M.L. Michlsn, Solution of Diffrntial Equation Modls by Polynomial Approximation, Englwood Cliffs, w Jrsy, Prntic-Hall, 978. [8] B.A. Finlayson, onlinar Analysis in Chmical Enginring, w York, McGraw-Hill, 980. [9] M.A. Soliman, Th Mthod of Orthogonal Collocation, King Saud Univrsity Prss, 00. [0] J.P. Brrut, R. Baltnsprgr and H.D. Mittlmann, Rcnt Dvlopmnt in Barycntric Rational Intrpolation, in Trnds and Applications in Constructiv Approximation, Intrnat. Sr. umr. Math., 5: 7-5, 005 30705-99-IJET-IJES Octobr 03 IJES