Numerical Method for Blasius Equatio o a ifiite Iterval Alexader I. Zadori Omsk departmet of Sobolev Mathematics Istitute of Siberia Brach of Russia Academy of Scieces, Russia zadori@iitam.omsk.et.ru 1 Itroductio Blasius problem o a half-ifiite iterval is cosidered. This problem has a place uder mathematical modellig of viscid flow before thi plate. Blasius problem is a boudary value problem for a oliear third order ordiary differetial equatio o a half-ifiite iterval. This problem was ivestigated i may articles. For example, G.I. Shishki [1] studied asymptotic behavior of differetial ad differece solutios to get differece scheme with a fiite umber o odes for eough log iterval. We apply the method of markig of set of solutios, that satisfy the limit boudary coditio at ifiity to trasform a problem uder cosideratio to a problem for a fiite iterval [2]. To use that method we have problems, coected with oliearity of differetial equatio ad with ubouded coefficiet before secod derivative. We offer to do trasformatio of idepedet variable to avoid a problem, coected with boudless of that coefficiet. We cosider a Blasius problem as model oliear problem for applicatio of developig techic. 2 Case of a liear problem Cosider a problem: εu (x) + [a(x) + x]u (x) = f(x), (1) u(0) = A, lim u(x) = 0. (2) Suppose,that a(x) + x 0, ε > 0, fuctios a(x), f(x) are smooth eough, lim a(x) = a 0, lim f(x) = 0. Note, that domai is ubouded ad solutio u(x) ca be ubouded for a bouded fuctio f(x). Itegratig equatio (1), we get, that if a(x) + x α > 0, the u(x) A + 1 α x 0 f(s) ds. Supported by the Russia Foudatio of Basic Research uder Grat 04-01-00578 1
To avoid uboudedess of coefficiet before first derivative, itroduce ew variable t = x2 2. The problem (1)-(2) become a form: εu (t) + b(t)u (t) = F (t), where u(0) = A, lim u(t) = 0, (3) b(t) = a( 2t) 2t + 1 + 1 2t, F (t) = f( 2t)/(2t). So, coefficiet b(t) is bouded for t, separated from zero. We ca do some other replacemet of x to do fuctio b(t) uiformly bouded, but it is ot importat for method, cosidered below. By the ext equatio we mark solutios of differetial equatio (3), that satisfy the limit boudary coditio at ifiity [2, 3, 4]: εu (t) + g(t)u(t) = β(t), (4) where g(t) is solutio of sigular Cauchy problem for Riccati equatio: εg (t) + b(t)g(t) g 2 (t) = 0, lim g(t) = 1, (5) β(t) is solutio of sigular problem: εβ (t) + [b(t) g(t)]β(t) = εf (t), lim β(t) = 0. (6) Thak to coditio lim g(t) = 1 every solutio of equatio (4) teds to zero at ifiity ad equatio (4) picks that solutios of differetial equatio (3), that satisfy the limit boudary coditio at ifiity. Lemma 1 Let 0 < b 1 b(t) b 2. The b 1 g(t) b 2. Proof. At first prove, that g(t) > 0, v(t) = exp Obviously, v(t) is solutio of a problem: It follows from (7), that 0 t <. Defie t 0 ε 1 g(s)ds. εv + b(t)v (t) = 0, v(0) = 1, lim v(t) = 0, (7) v(t) > 0, v (t) < 0, 0 < t <. Takig ito accout that g(t) = εv (t)/v(t), we get g(t) > 0, 0 t <. Prove, that g(t) b 2. Cosider a problem: εg 2 + b 2 g 2 g 2 2 = 0, lim g 2 (t) = b 2 2
with solutio g 2 (t) = b 2. Let z = g 2 g. The z(t) is solutio of a problem: εz (t) [g 2 b(t) + g(t)]z(t) = (b(t) g 2 )g 2 0, lim z(t) = b 2 b 0. (8) If we suppose, that for some s z(s) < 0, the we shall have cotradictio. g(t) b 2. I the same maer we ca prove, that g(t) b 1. Lemma is proved. Usig a problem (6), we ca prove, that It follows, that β g 0 F (s) ds. Usig equatio (4), trasform a problem (3) to a problem for a fiite iterval: εu (t) + b(t)u (t) = f(t), 0 < t < L, u(0) = A, εu (L) + g(l)u(l) = β(l). (8) Prove, that problems (3) ad (8) have a same solutio for 0 t L. Cosider iitial value problem: εu (t) + g(t)u(t) = β(l), u(0) = A. (9) Takig ito accout equatios (5),(6), we prove, that solutio of a problem (9) satisfies to problems (3) ad (8). Problems uder cosideratio have uique solutio, therefor problems (3) ad (8) have a same solutio for 0 t L. So, problem (3) is exactly reduced to a problem (8), formulated for a fiite iterval. We have oly to fid coefficiets g(l) ad β(l). Fuctios g(t), β(t) as solutios of problems (5),(6) we ca sick as asymptotic series: g(t) ε g (t), β(t) ε β (t), (10) or i a form: I case of represetatio (11) we suppose, that g(t) (2t) /2 g, β(t) (2t) /2 β. (11) a( 2t) = f( 2t) = ( (2t) /2 a + O (2t) (N+1)/2), ( (2t) /2 f + O (2t) (N+1)/2). I both cases (10) ad (11) we got recurret formulas o β, g. I particular, i case of represetatio (11) we have: g 0 = 1, g 1 = a 0, g 2 = 1 + a 1, g 3 = εa 0 + 4a 0 + a 2 + 4a 0 a 1, β 0 = 0, β 1 = f 1, β 2 = f 2 β 1 ε 1 (g 3 a 2 ). 3
Make iverse trasformatio of idepedet variable ad get a ext problem for a fiite iterval: εu (x) + [a(x) + x]u (x) = f(x), 0 < x < L 0, ε u(0) = A, u (L 0 ) + g(l)u(l 0 ) = β(l 0 ). (12) L 0 Usig maximum priciple, we ca prove ext lemma. Lemma 2 Let ũ(t) is solutio of problem (12) with perturbed coefficiets g(l 0 ), β(l 0 ), g(l 0 ) g(l 0 ), β(l 0 ) β(l 0 ), g(l 0 ) σ > 0. The u(x) ũ(x) σ 1 ( u(l 0 ) + 1). Cosider a problem for umerical experimets: [ ] 2x u (x) + x + 1 + x u (x) = 0, u(0) = 1, lim u(x) = 0. We compare differet approaches for formulatio of boudary coditio at a fiite poit istead the limit boudary coditio at ifiity. Let = max vh ṽ, h where ṽ h is solutio of the scheme of upwid differeces for eough log iterval [0, L 0 ], L 0 = 100; v h is solutio of the same differece scheme o a shot iterval depedig o boudary coditio. I Table 1 error is represeted for differet approaches ad itervals. 3 Blasius problem Cosider Blasius problem: Let Table 1: Absolute errors for a liear problem L w (L) = 0 g (L) g 0 g 0 + g 1 /L g 0 + g 1 /L + g 2 /L 2 2 0.12e 01 0.95e 02 0.11e 02 0.44e-04 3 0.35e 03 0.20e 03 0.19e 04 0.41e-05 5 0.36e 07 0.13e 07 0.95e 09 0.84e-10 7 0.32e 12 0.86e 13 0.49e 14 0.23e-15 u (x) + u(x)u (x) = 0, u(0) = 0, u (0) = 0, u(x) = v(x) + x, The problem (13) ca be writte i a form: lim u (x) = 1. (13) w(x) = v (x). v (x) = w(x), v(0) = 0, 4
w (x) + [v(x) + x]w (x) = 0, w(0) = 1, Cosider iterative method for a problem (14): Lemma 3 Let The iterative method (15) coverges. lim w(x) = 0. (14) v (x) = w (x), v (0) = 0, w (x) + [v 1 (x) + x]w (x) = 0, 0 < x <, w (0) = 1, lim w (x) = 0. (15) v 0 (x) v(x), v 0 (x) v 1 (x), x > 0. Proof. Prove, that for every value of x sequeces v (x), w (x) are mootoe decreasig ad have low bouds. Low bouds. Let z (x) = v (x) v(x), p (x) = w (x) w(x). Compose a problem: z (x) = p (x), z (0) = 0, p (x) + [v 1 (x) + x]p (x) = z 1 (x)w (x), 0 < x <, p (0) = 0, lim p (x) = 0. By iductio prove, that for all x > 0 z (x) 0, p (x) 0. Accordig to coditios of lemma z 0 (x) 0. Let z 1 (x) 0. Prove, that p (x) 0. Suppose, that for some s p (s) < 0. The there is poit of egative miimum of fuctio p (x), it leads us to cotradictio. So, p (x) 0. It follows, that z (x) 0. Accordig to method of mathematical iductio for all z (x) 0, p (x) 0. It implies, that v (x) v(x), w (x) w(x). Mootoy. Prove, that sequeces v (x), w (x) are mootoe decreasig. Let z (x) = v (x) v +1 (x), p (x) = w (x) w +1 (x). The z (x) p (x) are solutios of a problem: z (x) = p (x), z (0) = 0, p (x) + [v 1 (x) + x]p (x) = z 1 (x)w +1(x), 0 < x <, p (0) = 0, lim p (x) = 0. Usig method of iductio, prove, that z (x) 0, p (x) 0. Accordig to coditios of lemma z 0 (x) 0.. It is follows from equatio o p (x), that if z 1 (x) 0, the p (x) 0 for x > 0. Coditio p (x) 0 implies, that z (x) 0. So, usig method of mathematical iductio, we proved, that for every z (x) 0, p (x) 0. We proved, that for every x > 0 sequeces v (x), w (x) are mootoe decreasig ad have low bouds. It s kow, that i this case sequeces uder cosideratio have the property of covergece. Lemma is proved. Cosider a case v 0 (x) = 0. Prove, that coditios of a lemma 3 are fulfilled. First coditio has a place, because v(x) 0. Verify the coditio v 0 (x) v 1 (x), x > 0. It follows from (15), that w 1 (x) 0 for x > 0. It implies, that v 1 (x) 0 = v 0 (x). So, coditios of lemma 3 are valid. 5
Cosider a questio of reductio Blasius problem to a fiite iterval. For every fixed problem (15) is liear ad we ca use results, obtaied i liear case. Secod equatio i (15) correspods to (1) with a(x) = v 1 (x). Trasformed to a fiite iterval problem (15) has a form: w v (x) = w (x), v (0) = 0, (x) + [v 1 (x) + x]w (x) = 0, 0 < x < L, 1 w (0) = 1, L w (L) + g (L)w (L) = 0. (16) Coefficiet g (L) ca be calculated o base of asymptotic series (11) as it was discussed for liear case. Cosider results of umerical experimets. Write differece scheme for a problem (16): w k+1 v k = v k 1 2w k + w k 1 h 2 w 0 = 1, 1 L + wk 1 + w k h, v 0 = 0, 2 + [v 1 k + x k ] wk+1 w k h w K w K 1 h + g (L)w K = 0, = 0, k = 1, 2,..., K 1. (17) Compare errors, correspodig to differet approaches for formulatio of boudary coditio istead of limit boudary coditio at ifiity. Let = max vh ṽ, h where ṽ h is solutio of problem (17) for eough log iterval [0, L 0 ], L 0 L, whe error of limit boudary coditio trasfer to poit L 0 is iessetial. Let L 0 = 100, h = 0.1 We cotiue iteratios, if max v k v 1 k > δ, δ = 10 13. Iitial iteratio is defied as v 0 (x) = 0. I Table 2 k error is preseted for differet L ad g (L) i compare with coditio w (L) = 0. I this Table g 0 = 1, g 1 = v (L). We have similar results i case h = 0.01. Table 2: Absolute errors for Blasius problem L w (L) = 0 g (L) = g 0 g (L) = g 0 + g 1 /L 3 0.83e 1 0.24e 1 0.19e 1 5 0.76e 3 0.15e 3 0.47e 4 7 0.34e 6 0.53e 7 0.97e 8 9 0.12e 10 0.15e 11 0.18e 12 It follows from umerical experimets, that usig special boudary coditio, based o equatio (4), we get more accurate results i compare with classical approach. Author thaks N.B. Koyukhova for useful proposals, that were took ito accout. 6
Refereces [1] G.I. Shishki Grid Approximatio of the Solutio to the Blasius Equatio ad of its Derivatives, Computatioal Mathematics ad Mathematical Physics, 41, 1, 37-54 (2001). [2] A.A. Abramov ad N.B. Koyukhova Trasfer of Admissible Boudary coditios From a Sigular Poits of Liear Ordiary Differetial Equatios, Sov. J. Numer.Aal. Math. Modellig, 1, 4, 245-265 (1986). [3] A.I. Zadori The Trasfer of the Boudary Coditio From the Ifiity for the Numerical Solutio to the Secod Order Equatios with a Small Parameter Siberia Joural of Numerical Mathematics, 2, 1, 21-36 (1999). [4] A.I. Zadori, O.V. Haria Numerical method for a system of liear equatios of secod order with a small parameter o a semi-ifiite iterval Siberia Joural of Numerical Mathematics, 7, 2, 103-114 (2004). [5] J.D. Kadilarov, L.G. Vulkov ad A.I. Zadori A method of lies approach to the umerical solutio of sigularly perturbed elliptic problems Lecture Notes i Computer Sciece, 1988, 451-458 (2001). [6] A.I. Zadori Reductio from a Semi-Ifiite Iterval to a Fiite Iterval of a Noliear Boudary Value Problem for a System of Secod-Order Equatios with a Small Parameter, Siberia Mathematical Joural, 42, 5, 884-892 (2001). 7