CERTAIN CLASSES OF SOLUTIONS OF LAGERSTROM EQUATIONS

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1 SARAJEVO JOURNAL OF MATHEMATICS Vol.10 (22 (2014, DOI: /SJM CERTAIN CLASSES OF SOLUTIONS OF LAGERSTROM EQUATIONS ALMA OMERSPAHIĆ AND VAHIDIN HADŽIABDIĆ Absrac. This paper presens sufficien condiions for he exisence of soluions for cerain classes of Cauchy s soluions of he Lagersrom equaion as well as heir behavior. Behavior of inegral curves in he neighborhoods of an arbirary or inegral curve are considered. The obained resuls conain he answer o he quesion on approximaion of soluions whose exisence is esablished. The errors of he approximaion are defined by funcions ha can be sufficienly small. The heory of qualiaive analysis of differenial equaions and opological reracion mehod are used. 1. Inroducion Since inroduced in he 1950s by P. A. Lagersrom, he models of Lagersrom equaion were sudied by many auhors wih he help of variaional echniques (see 1] - 4]. Here we shall use he qualiaive analysis heory and he opological reracion mehod (5] - 11]. The Lagersrom equaion is used in asympoic reamen of viscous flow pas a solid a low Reynolds number. In general form i is given by he non-auonomous second-order differenial equaion: ( y + + y y = 0, n N, n 1. (1 The cases n = 2 and n = 3 represen he physically relevan seings of flow in wo and hree dimensions, respecively. We will consider he equaion (1 on inerval I = (a, b, where a < b < 0 or 0 < a < b +. Le Γ = {(y, D : y = ψ(, I}, 2010 Mahemaics Subjec Classificaion. 34C05. Key words and phrases. The Lagersrom equaion, behavior of soluions, approximaion of soluions.

2 68 ALMA OMERSPAHIĆ AND VAHIDIN HADŽIABDIĆ where D = I y I, I y R and ψ( C 2 (I, be an arbirary or inegral curve of equaion (1. We will esablish some sufficien condiions on he exisence and behavior of he classes of soluions of equaion (1 in a cerain region of he curve Γ, using he reracion mehod (mehod Wažewski. Le, r 2 C 1 (I, R + and ψ 0 = ψ( 0, ψ 0 = ψ ( 0, y 0 = y( 0, y 0 = y ( 0, 0 I. We shall consider he soluions y( of equaion (1 which saisfy on I, eiher one of he condiions y 0 ψ 0 < r 2 ( 0, y 0 ψ 0 < ( 0, (2 or Using subsiuion (y 0 ψ 0 2 r 2 2 ( 0 + (y 0 ψ 0 2 r 2 1 ( 0 < 1. (3 y = x, (4 where x = x( is a new unknown funcion, equaion (1 is ransformed ino a quasilinear sysem of equaions: x = ( n 1 + y x y = x (5 = 1 defined on Ω = I x D, I x R he open inerval, and curve Γ is ransformed ino a curve (ϕ(, ψ(,, I, where ϕ( = ψ (. We shall consider he behavior of he inegral curves (x(, y(, of he sysem (5 wih respec o he ses σ and ω: and σ = {(x, y, Ω : x ϕ( < (, y ψ( < r 2 (} { ω = (x, y, Ω : (x ϕ(2 r 2 1 ( The boundary surfaces of σ and ω are, respecively: X i = { (x, y, Clσ Ω : H 1 i (x, y, + } (y ψ(2 2( 1. := ( 1 i (x ϕ( ( = 0 }, i = 1, 2, Y i = { (x, y, Clσ Ω : H 2 i (x, y, W = := ( 1 i (y ψ( r 2 ( = 0 }, i = 1, 2, { } (x ϕ(2 (y ψ(2 (x, y, Clω Ω : H(x, y, := 2( + 2( 1= 0,

3 CERTAIN CLASSES OF SOLUTIONS OF LAGERSTROM EQUATIONS 69 where ClS, (S = ω or S = σ is he se of all poins of closure of S. (x is a poin of closure of S, S a subse of a Euclidean space, if every open ball cenered a x conains a poin of S (his poin may be x iself. Le us denoe he angen vecor field o an inegral curve (x(, y(,, I, of he sysem (5 by T. The vecors Hi 1, H2 i and H are he exernal normals on surfaces X i, Y i and W, respecively. We have: ( ( T (x, y, = + y x, x, 1, ( Hi 1 ( = ( 1 i, 0, ( 1 i 1 ϕ, i = 1, 2, ( Hi 2 ( = 0, ( 1 i, ( 1 i 1 ψ r 2, i = 1, 2, 1 H(x, y, 2 and ( x ϕ = r 2 1, y ψ r 2 2, (x ϕ2 r 1 r 3 1 By means of scalar producs (y ψ2 r 2 3 P 1 i (x, y, = ( H 1 i, T on X i, P 2 i (x, y, = ( H 2 i, T on Y i, P (x, y, = ( 1 2 H, T on W, (x ϕ ϕ 2 (y ψ ψ 2. we shall esablish he exisence and behavior of inegral curves of he sysem (5 wih respec o he se σ and ω, respecively. Le us denoe wih S p (I, p {0, 1, 2}, class soluions (x (, y ( of he sysem (5 defined on I which depends of p parameers. We will say ha he class of soluions S p (I belongs o a se η (η = ω or η = σ if he graphs of funcions from S p (I are conained in η. In such a case we wrie S p (I η. For p = 0 we have noaion S 0 (I which means ha here is a leas one soluion (x (, y (, on I of sysem (5 whose graph lies in he se η. The resuls of his paper are based on he following lemmas (see 6], 9], which for he sysem (5 and ses σ and ω, have he form: Lemma 1. If i is, for he sysem (5, he scalar produc of P (x, y, < 0 on W (Pi k(x, y, < 0 on γσ = X 1 X 2 Y 1 Y 2, i = 1, 2, k = 1, 2, hen he sysem (5 has a class of soluions S 2 (I which belongs o a se ω, for every I, i.e. S 2 (I ω (S 2 (I σ.

4 70 ALMA OMERSPAHIĆ AND VAHIDIN HADŽIABDIĆ Lemma 2. If i is, for he sysem (5, he scalar produc of P (x, y, > 0 on W (Pi k(x, y, > 0 on γσ = X 1 X 2 Y 1 Y 2, i = 1, 2, k = 1, 2, hen he sysem (5 has a class of soluions S 0 (I which belongs o a se ω, for every I, i.e. S 0 (I ω (S 0 (I σ. Lemma 3. If i is, for he sysem (5, he scalar produc of Pi 1 (x, y, < 0 on X 1 X 2 and Pi 2(x, y, > 0 on Y 1 Y 2 (or vice versa, hen he sysem (5 has a class of soluions S 1 (I which belongs o a se σ for every I, i.e. S 1 (I σ. According o Lemma 1, he se W ( γσ = X 1 X 2 Y 1 Y 2 is a se of poins of sric enrance of inegral curves of he sysem (5 wih respec o he ses ω ( σ and Ω. Hence, all soluions of sysem (5 which saisfy condiion x 0 ϕ 0 ( 0, y 0 ψ 0 r 2 ( 0, (x 0 = x ( 0 also saisfy condiions x ( ϕ ( (, y ( ψ ( r 2 ( for every > 0, i.e. S 2 (I ω (S 2 (I σ. According o Lemma 2, he se W ( γσ = X 1 X 2 Y 1 Y 2 is a se of poins of sric exi of inegral curves of he sysem (5 wih respec o he ses ω ( σand Ω. Hence, according o T. Wazewski s reracion mehod 11], sysem (5 has a leas one soluions belonging o se ω ( σ for every I, i.e. S 0 (I ω (S 0 (I σ. According o Lemma 3, he se X 1 X 2 is a se of poins of sric enrance, and Y 1 Y 2 is a se of sric exi (or reversely of inegral curves of (5 wih respec o he ses σ and Ω. According o he reracion mehod sysem (5 has a one-parameer-class of soluions belonging o se σ for every I, i.e. S 1 (I σ. 2. Main resuls Theorem 1. Le Γ be an arbirary curve and, r 2 C 1 (I, R +. (a If ( ( + ψ ϕ + ϕ < + ψ + ( ϕ + r 2, (6 ( < r 2( (7 on γσ = X 1 X 2 Y 1 Y 2, hen all soluions y( of he problem (1, (2 saisfy he condiions (b If ( y( ψ( < r 2 (, y ( ψ ( < ( for > 0. (8 ( + ψ ϕ + ϕ > + ψ + + ( ϕ + r 2, (9 ( < r 2( (10

5 CERTAIN CLASSES OF SOLUTIONS OF LAGERSTROM EQUATIONS 71 on γσ = X 1 X 2 Y 1 Y 2, hen a leas one soluion o he problem (1, (2 saisfies he condiions (8. (c If he condiions (6 and (10 or (7 and (9 are saisfied, hen he problem (1, (2 has a one-parameer class of soluions ha saisfy he condiions (8. Proof. We shall consider equaion (1 hrough he equivalen sysem (5. Consider a sysem of inegral curves of (5 respec o a se σ. For scalar producs Pi 1(x, y, on X i and Pi 2(x, y, on Y i we have, respecively: ( Pi 1 (x, y, = ( 1 i + y x + ( 1 i 1 ϕ ( = + ψ + (y ψ ( ] + ( 1 i + ψ ϕ + (y ψ ϕ ϕ r 1, Pi 2 (x, y, = ( 1 i x + ( 1 i 1 ψ r 2 = ( 1 i (x ϕ r 2. (a According o he condiions (6 and (7, he following esimaes are valid for Pi 1(x, y, on X i and Pi 2(x, y, on Y i, respecively: ( ( Pi 1 (x, y, + ψ + r ψ ϕ + ϕ + ϕ r 2 < 0, P 2 i (x, y, r 2 < 0. Accordingly, he se of γσ = X 1 X 2 Y 1 Y 2 is se of poins of sric enrance for he inegral curves of he sysem (5 respec o ses σ and Ω. Thus, all soluions of he sysem (5 ha saisfy he iniial condiions also saisfy he condiions y 0 ψ 0 r 2 ( 0, x 0 ϕ 0 ( 0, x( ϕ( < (, y( ψ( < r 2 ( for every > 0. As he y = x and ϕ = ψ, i is x 0 ϕ 0 = y 0 ψ 0, so, all he soluions of problems (1, (2 saisfies he condiions (8. (b Taking ino accoun he condiions (9 and (10, he following esimaes are valid for Pi 1(x, y, on X i and Pi 2(x, y, on Y i, respecively: ( ( Pi 1 (x, y, + ψ + r ψ ϕ + ϕ ϕ r 2 > 0, P 2 i (x, y, r 2 > 0.

6 72 ALMA OMERSPAHIĆ AND VAHIDIN HADŽIABDIĆ We conclude ha he se γσ is se of poins of sric exi inegral curves of he sysem (5 wih respec o he ses σ and Ω. Thus, according o he mehod of reracion T. Wažewski, sysem (5 has a leas one soluion in he se σ for all I. So he problem (1, (2 has a leas one soluion ha saisfies he condiions (8. (c In his case, he se (X 1 X 2 \ (Y 1 Y 2 is he se of poins of sric enrance and he se (Y 1 Y 2 \ (X 1 X 2 is he se of poins of sric exi (or vice versa inegral curves of he sysem (5 wih respec o he ses of σ and Ω. According o he reracion mehod sysem (5 has a one-parameer class of soluions in he se σ for every I. Thus, for he problem (1, (2, here is a one-parameer class of soluions ha saisfy he condiions (8. In he special case, when Γ is an inegral curve of equaion (1, from Theorem 1 i follows ha: Corollary 1. Le Γ be an inegral curve of (1 and, r 2 C 1 (I, R +. (a If ( 0 < + ψ + ( ϕ + r 2, ( < r 2( on γσ = X 1 X 2 Y 1 Y 2, hen all soluions y( of he problem (1, (2 saisfy he condiions y( ψ( < r 2 (, y ( ψ ( < ( for > 0. (b If ( 0 > + ψ + + ( ϕ + r 2, ( < r 2( on γσ = X 1 X 2 Y 1 Y 2, hen a leas one soluion o he problem (1, (2 saisfies he condiions (8. (c If he condiions (6 and (10 or (7 and (9 are saisfied, hen he problem (1, (2 has a one-parameer class of soluions ha saisfies he condiions (8. Le us now consider soluions y( of he equaion (1 ha saisfy condiion (3, where (ψ(,, I, is an arbirary inegral curve of he equaion (1. Theorem 2. Le funcions, r 2 C 1 (I, R + and ( ϕ + ψ r 2 2 ( + ( 2 ] ( ] 2 < 4 (r 2 ( + y ( + r 1( r 2(. (11 Then:

7 (i If (ii If CERTAIN CLASSES OF SOLUTIONS OF LAGERSTROM EQUATIONS 73 r 2( > 0, I, (12 hen all soluions y( of he problem (1, (3 saisfy he condiions (y ψ( 2 r 2 2 ( + (y ψ ( 2 r 2 1 ( < 1, for > 0. (13 r 2( < 0, I, (14 hen a leas one soluion o he problem (1, (3 saisfies he condiion (13, where y( = ψ( is he soluion he equaion (1. Proof. We shall consider equaion (1 hrough he equivalen sysem (5. Consider he inegral curves of he sysem (5 wih respec o a se of ω and Ω. For he scalar produc of P (x, y, = ( 1 2 H, T on W, we have: ( P (x, y, = + y x x ϕ 2 + x y ψ 2 (y ψ2 (x ϕ ϕ (y ψ ψ = 3 ( r 2 r y (x ϕ (x ϕ2 3 If we inroduce he noaion we have: 2 ( + y (y ψ2 3 r 2 (x ϕ ϕ 2 X = x ϕ, Y = y ψ, r 2 (x ϕ2 3 ] x ϕ (x ϕ (y ψ ϕ (y ψ ψ 2. ( ] P (x, y, = + y r 1 X 2 + XY r 2 Y 2 r 2 r 2 ( ] X + + y ϕ ( ] = + y r 1 X 2 + XY r 2 Y 2 r 2 r 2 ( ( ] + + ψ + y ϕ X.

8 74 ALMA OMERSPAHIĆ AND VAHIDIN HADŽIABDIĆ The following esimaes for P (x, y, on he surface W are valid: ( ] P (x, y, + y r 1 X 2 + r 2 X Y r 2 Y 2 r 2 + ϕ + ψ ] r 2 X Y r ( 1 ] = + y r 1 X 2 + ( ϕ + ψ r ] ] 2 + r 2 X Y + r 2 Y 2, r 2 ] P (x, y, p(y, r 1 X 2 q(y, r 2 + r 1 r 2 X Y + r 2 r 2 ] Y 2 L 1 ϕ + L 2 ψ + L 3 ] r 2 X Y r ] 1 = p(y, r 1 X 2 (L 1 ϕ + L 2 ψ + L 3 r 2 + q(y, r ] 2 r 2 X Y ] = r 2 Y 2. r 2 The righ-hand sides of he above inequaliies are he quadraic symmeric forms a 11 X 2 ± 2a 12 X Y + a 22 Y 2 where corresponding coefficiens a 11, a 12, a 22 are inroduced. (i Condiions (11 and (12 imply a 22 < 0, a 11 a 22 a 2 12 > 0, which, according o Sylveser s crierion, means ha P (x, y, < 0 on W. Consequenly, he se W is he se of poins of sric enrance o he inegral curves of he sysem (5 respec o he ses ω and Ω. Hence, all soluions of he sysem (5 ha saisfy he iniial condiion saisfy he inequaliy (x 0 ϕ 0 2 r 2 1 ( 0 (x( ϕ( 2 2( + + (y 0 ψ 0 2 r 2 2 ( 0 < 1, (15 (y( ψ(2 r 2 2 ( < 1, for > 0. (16 Since x 0 ϕ 0 = y 0 ψ 0, hen all he soluions of problems (1, (3 saisfy condiion (13.

9 CERTAIN CLASSES OF SOLUTIONS OF LAGERSTROM EQUATIONS 75 (ii Condiions (11, (14 imply a 22 > 0, a 11 a 22 a 2 12 > 0, which, according o Sylveser s crierion, means ha P (x, y, > 0 on W. Consequenly, W is he se of poins of sric exi inegral curves of he sysem (5 respec o he ses ω and Ω. Hence, according o he reracion mehod, problem (5, (15 has a leas one soluion ha saisfies he condiion (16. Consequenly, he problem (1, (3 has a leas one soluion ha saisfies he condiion (13. Remark. We noe ha he obained resuls also conain an answer o he quesion on approximaion of soluions y ( whose exisence is esablished. For example, he errors of he approximaion for soluions y ( and derivaive y ( in Theorem 1 are defined by he funcions ( and r 2 ( which end o zero as and r i ( < 0, (i = 1, 2, I. For example, for he funcions r i ( we can use ( = αe s and r 2 ( = βe p, s > 0, p > 0 wih parameers α and β, ha can be arbirarily small. In ha case curve Γ presen a good approximaion of soluion y ( in σ. References 1] D. S. Cohen, A. Fokas and P. A. Lagersrom, Proof of some asympoic resuls for a model equaion for low Reynolds number flow, SIAM J. Appl. Mah., 35 (1978, ] S. P. Hasings and J. B. McLeod, An elemenary approach o a model problem of Lagersrom, Siam J. Mah Anal., 40 (6 2009, ] S. Rosenbla and J. Shepherd, On he asympoic soluion of he Lagersrom model equaion, SIAM J. Appl. Mah., 29 (1975, ] N. Popovic and P. Szmolyan, A geomeric analysis of he Lagersrom model problem, J. Differ. Equaions, 199 (2004, ] B. Vrdoljak, On soluions of he Lagersrom equaion, Arch. Mah. (Brno, 24 (3 (1988, ] A. Omerspahić, Reracion mehod in he qualiaive analysis of he soluions of he quasilinear second order differenial equaion, Applied Mahemaics and Compuing (edied by M. Rogina e al., Deparmen of Mahemaics, Universiy of Zagreb, Zagreb, 2001, ] B. Vrdoljak and A. Omerspahić, Qualiaive analysis of some soluions of quasilinear sysem of differenial equaions, Applied Mahemaics and Scieific Compuing (edied by Drmač e al., Kluwer Academic/Plenum Publishers, 2002, ] B.Vrdoljak and A. Omerspahić, Exisence and approximaion of soluions of a sysem of differenial equaions of Volerra ype, Mah. Commun., 9 (2 (2004, ] A. Omerspahić and B. Vrdoljak, On parameer classes of soluions for sysem of quasilinear differenial equaions, Proceeding of he Conference on Applied Mahemaics and Scienific Compuing, Springer, Dordreh, 2005,

10 76 ALMA OMERSPAHIĆ AND VAHIDIN HADŽIABDIĆ 10] A. Omerspahić, Exisence and behavior soluions of a sysem of quasilinear differenial equaions, Crea. Mah. Inform., 17 (3 (2008, ] T. Wažewski, Sur un principe opologique de l examen de l allure asympoique des inegrales des equaions differenielles ordinaires, Ann. Soc. Polon. Mah., 20 (1947, (Received: April 15, 2013 (Revised: July 3, 2013 Faculy of Mechanical Engineering Universiy of Sarajevo Vilsonovo šeališe Sarajevo Bosnia and Herzegovina alma.omerspahic@mef.unsa.ba hadziabdic@mef.unsa.ba