Singular Perturbations of a Boundary-Value Problem for a System of Nonlinear Differential Equations Citation Macki, Jack William (1964) Singular Perturbations of a Boundary-Value Problem for a System of Nonlinear Differential Equations. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/WXRP-3F53. https://resolver.caltech.edu/CaltechETD:etd-09302002-105202 Abstract NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. The nonlinear boundary-value problem [...] is examined, under the hypothesis that the degenerate problem [...], has a continuously differentiable solution. Under a series of assumptions concerned, for the most part, with the smoothness of the functions f and g, it is proved that, for [...] restricted to a small enough interval of the form [...], the above boundary-value problem has a solution of the form [...], where p and q are both 0(1) uniformly in t as [...] goes to zero, while [...] and [...] exhibit a boundary-layer type of behavior. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Mathematics and Physics) Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Minor Option: Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Erdélyi, Arthur Thesis Committee: Unknown, Unknown Defense Date: 1 April 1964 Record Number: CaltechETD:etd-09302002-105202 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-09302002-105202 DOI: 10.7907/WXRP-3F53 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 3827 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 30 Sep 2002 Last Modified: 19 Jan 2024 23:02 Thesis Files Preview PDF (Macki_jw_1964.pdf) - Final Version See Usage Policy. 1MB Repository Staff Only: item control page

Singular Perturbations of a Boundary-Value Problem for a System of Nonlinear Differential Equations Thesis by Jack William Macki In Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1964 (Submitted April 1, 1964) ii Acknowledgements I wish to express my appreciation to Frofessor Ae Erdélyi for his guidance and encouragement, and for the patience and care which he exercised in. reading the manuscript. I thank the National Science Foundation for contributing the major part of my financial support; and my wife and parents, for their encouragement and sacrifice. Finally, I would like to express my gratitude to Professors K. A. Bush, H. Sagan, and A. E. Labarre, for the inspiration and training I received from them as an undergraduate. ili Abstract The nonlinear boundary-value problem d at = f(x,y,t,e), dy eat = g(x,y,t,e), a,x(0,¢) + any (0,€) =a(e), b,x(1,¢) + b,y(1,¢) = Ble), is examined, under the hypothesis that the degenerate problem at = E(X452V52ts 0), b,x, (1) + boy, (1) = B(0), has a continuously differentiable solution. Under a series of assumptions concerned, for the most part, with the smoothness of the functions f and &, it is proved that, fore restricted toa small enough interval of the form 0<e < Eos the above boundary-value problem has a solution of the form x(t,e) It x(t) + ep(t,e) + eo(t,e), y(t,e) = yo(t) + eq(t,e) + T(t,e), where p and q are both O(1) uniformly in t as ¢« goes to zero, while p and tT exhibit a boundary-layer type of behavior. 0. INTRODUCTION | Perturbation methods have long been of in- portance in treating boundary-value problems which involve a small parameter, that is, boundary- value problems based on an equation of the form ox = F(x,t,¢), with x and F vectors, and e« a small parameter. If F depends regularly on ¢« near ¢ = 0, then one would expect that, under relatively mild conditions, the solution to a boundary-value problem based on the above equation would converge, as © goes to zero, to a solution of the degenerate (ec = 0) boundary- value problem. The requirement that F depend regularly on « near ce = 0, however, is not met for a large class of important equations. Of particular interest are those problems in which one or more of the components of F are of the form F(x.t,e)/e, with Fe a regular function of « neare = 0. The yeh equation in the system can then be written ext = F.(x,t,¢), so that the degenerate system is of lower order than the given system. This means that no solution to the degenerate system will in general satisfy all of the boundary conditions in the given problen; consequently, we cannot expect a solution to the given problem to converge uniformly to some solution of the degenerate system as ¢ goes to zero. In the type of problem with which we shall be concerned, the solution does converge uniformly to a solution of the degenerate system on every closed subset of the basic interval that does not contain a certain point, te This point of non-uniformity is deter- mined by the structure of the given system. Because of this non-uniform behavior, problems based on systems of the type described above must be treated by special means - the methods of the theory of singular perturbations. An extensive literature has developed concern- ing the existence of, and expansions for, solutions of singular perturbations of ordinary linear differ- ential cquations whose coefficients are analytic in é in some neighborhood of e = 0. Formal series solutions are constructed by substitution, setting the coefficients of each power of e to zero and ' recursively solving the resulting differential equations for the coefficients appearing in the formal series. True solutions are then construct- ed, and shown to be represented asymptotically by the formal series. For rigorous expositions of the theory, with extensive bibliographies, see Turrittin (1) and Wasow (2). In recent years, a great deal of attention has been directed towards nonlinear problems. Those investigations to which this paper is related have concerned themselves with either the problem Po? ex" = g(x,x',t,c¢), x(0) =a, x(1) = 863 or the slightly more general problem Q.? x' = f(x,y,t,c) ey* = e(xsyot,e), x(0) =a, x(1) = 86. Since P. can be interpreted as the special case of Q, with (x,y,t,e) = y, we shall use the notation of Q. for discussing both problems. It is assumed that the degenerate system has a solution, a 5 o satisfying one of the two boundary conditions, with Le By (xo (t)sy_(t),t,0) non-zero and of constant sign on [0,1]. All investigations to date have indicated that the function By (x (t)syo(t),t,0) determines a single point of non-uniformity, te - if i Sy (X(t )5¥,(t),t, 0) < 0, then t, = 0, while if By (xo (t)syo(t),t,0) > 0, then t, =1. Since we want the solution to P. or Q. to converge to the hypothe- sized degenerate solution at every point except to it is clear that if we assume By (x (t)syg(t) st, 0) < 0, then we must assume x, (1) = ®8; while if we assume &y(xo(t),y(+),+,0) > O, we must assume %, (0) = a. The question of the existence of solutions to the problems P. and Q. under even relatively simple conditions is not a trivial one. For example, Coddington and Levinson (3) have shown that the problem P. based on the equation ex" = =x! = (xt)3 will in general fail to have a solution for small e- This has led most investigators to assume that g(x,y,t,e) is linear in y. The methods of attacking the problems P. 5 and Q, can be roughly classified into two groups. In order to simplify the discussion, let us con- sider the problem Pos and suppose that Xo satisfies the boundary condition at t = 1 so that te = 0 is our point of non-uniformity. The first method is characterized by the assumption that g(x,y,t,¢) is analytic in x and ec, linear in y, and differentiable to a certain degree in t, all within a certain region of (x,y,t,¢)- space containing the curve {(x,(t),x,"(t),t,0) r0ctet f The first step in this method is to construct a solution, x*(t), to the equation in P.» which differs little from Xu and satisfies the boundary condition at t = 1. The remaining boundary condition on x* is taken so as to keep x* close to Xe (e.g., x**(o) = x5"(0)). To construct the function x*(t), one “linearizes" the equation about x_ by writing fe) x* = Ky + @, and then solves the resulting nonlinear ‘problem in %. In solving the problem for @, use is made of the results for linear equations with analytic coefficients as described above. The function @, called the outer correction, is invariably found to converge uniformly to zero as « goes to zero. This approach, because of the powerful assumptions made, does a great deal more than show that @ goes uniformly to zero - it gives an asymptotic expansion in powers of e« for @, uni- formly valid for 0< t < i. The next step is to again "linearize" the equation, this time about x*, by writing the solution to the problem P. in the form x= x*¥ +. The resulting nonlinear problem for 4 is solved by assuming that 4(t,e) has a formal expansion in powers of WM =a -~ x*(o), and applying the same type of techniques used in determining the outer correction. Then it is shown that, for u sufficiently small, the formal series actuaily represents a solution of the problem for 4. The function $(t,c) is entirely a boundary-layer type of function, being significant only near t = 0, and converging exponentially to zero, at every point except t - 0, as e€ goes to Zero. 7 Wasow (4) has given a complete constructional procedure for obtaining the solution to P, by the above method, and Harris (5) has done the same for Qo: It should be remarked that the technique of linearization leads to the necessity of applying the previously developed theory of singular per- turbations of linear equations with analytic coefficients. The second method of attack - of which this paper is an example - is typified by much more limited assumptions on g, usually consisting of differentiability to a certain order in t,x, and ¥Y, and continuity ine at ¢ = 0. These compara- tively weak assumptions, of course, give less complete results - in general only one or two terms of the expansion of the solution. For examples of this approach, see Nagumo (6), von Mises (7), Levinson (8), Coddington and Levinson (9), Bri (10), and Erdélyi (11, 12, 13). The specific assumptions and techniques used by different investi- gators are quite diverse, and the results are just as much so. The investigations of Erdélyi are most 8 closely related to this paper, especially (12), where he considers the problem Poe One remarkable result of his investigation is that the condition that ~ be linear in y may be replaced by the condi- tion Byy = O(c). This paper is concerned with the problem Qos under the slightly more general boundary conditions: a, x( 0) * any (9) = a(e),b,x(1) + b,y(1) = @(e). The technique of linearigation described ahove is used, which means that singular perturbations of linear systems must be examined, since our assumptions are not strong enough to allow us to use the theory of linear equations with analytic coefficients. This is done in sections 1, 2, and 3. In section 4, the nonlinear boundary-value problem is stated, along with the first assumptions on f and g, and the outer correction is obtained. In section 5 we derive Lhe boundary-value problem for the inner correction, state the additional assumptions that we require to solve this problem, and obtain the inner correction. The boundary-layer behavior of the inner correction is clearly exhibited. In section 6 we summarize our results, discuss some of their more important features, and give explicit formulas for the leading terms in 9 the expansions of the inner and outer corrections. Notation The independent variable t will always range over [0,1]. If f(t) is a bounded function of t on [0,1], we set lp = sup lect) |. oir O<t<1 A(t) = la, 4(t)] is a matrix, we set lat) le) lay y(t) I, 1,3 IAl= >. lag This definition, of course, includes i,j column matrices, which we shall call vectors. A column matrix of k elements will be called a _kevector. The transpose of a matrix A(t) will be denoted by av(c). Any matrix consisting solely of zeros will be simply written 0. For any square matrix, A(t), we will denote the determinant by det A(t). We shall say that the k-vector, p(t), is integrable if each component of p(t) is integrable b on [0,1], ana | e(s)ds will denote that k-vector, a each of whose components is the integral from a to b of the corresponding component of o(t). The collection of integrable k-vectors will be denoted 10 1 by L, (k), and for pel, (k) we define loll, =[ lo(s) lds. fe) The parameter ¢ will always be restricted to an open interval of the form 0<c¢< Eo The statement m(t,e) = O(afe)), m(t,e) a matrix or scalar, will mean that there exists a constant, K, such that |m(t,c)| < Kale) for O< t< 1, O<e < Eoe For two scalars, f(t,e) and g(t,e), the statement e({t,e)=[[f(t,e)]] means that g(t,e)=f(t,e)[14+0(e)]. Unless explicitly stated otherwise, z(t) and x(t) - h(t,e) will denote the 2-vectors | | and f£(t,e) y(t): | | respectively, and subscripted or super- g(t,e) scripted versions of z(t) and h(t,e) will denote the same 2-vectors with the components sub- or super- scripted. The symbols a and b will denote the Z 2 A Q- while A(e) will stand for the matrix [ } Oe 1 Py constant 2-vectors | and | | respectively, a “b In cases where it leads to no confusion, variables of integration have not been written - for example, je in place of J Talore as. For the convenience of the reader, a table of the more important notations and formulas used in this work follows. Symbol A(e) L(s,t,e) P(s,t,e) Wle,t,c) 7 lt,e) z.(t,€) h(2,t,¢) H(X,t,€) H*(X,t,€) u(e) a(b,t,e) 11 Formula (or Asymptotic Form) [oe cf] (or,0,)] enafeMo-t) - J'r,0.] cml | (ome0)] Le (bye) Boorte) + O(e) 2 page + Oe Ys 1 + O(e) -1 -s |2,(,(0),¥,(0), 0, e)|” = Oe”) 0, if a, # 03 -1-s, if a, = 0. an ‘renee a (uy,t,€), 6 (x,y,t,€) n(z(t) + K,t,€) - Ale )z'(t) - h (2 (¢),t,€)x h(z*(+) + X,t,€) h(2*(t),t,€) - h, (2*(t),t,€)X a(e) - a’z*(o) en if b, = 0, ete y ete VE ap b, #0. 12 1. THE LINEAR SYSTEM We are concerned with the existence and asymptotic behavior of solutions to initial- value and boundary-value problems connected with the linear system La Ola" = A(e)z’ = t(t.e), plt.e) € ~ Le(t, e),=1+e w( t,e) ) |Z (1-1) Assumption A There exists é, > 0 such that, for 0<e < €,3 £(t,e), plt,e))] _ (a) Lotte). Pte) = 0(1), (bo) 424(t,e) and w(t,€) are measurable, (0) ° c(t, aa f° (t, aa ae ee Cy and Po are differentiable functions of t, |; where p(t,e) po {tse} (tse) and Py2CysP,"9c,' are O(1). Assumption A will be assumed to hold throughout sections 1, 2, and 3. In order to convert the equation 1.1 into a useful integral equation, we shall construct matrices F(t,e), Glt,e), and E(t,e), auch that “Fo |z}=| G(t,e)z| +B(t,e)z, (1.2) -1 o? with E(t,e) small in a sense which will become F(t,eMale)z? - [ apparent below. We introduce the notation 13 L{s,t,¢) = ox] { (41R¢,)} P(s,t,e) = cma[e™No-t-| exye,)} 8 If we take L(+t,0,€), p,(t,€)L(t,0,€) F(t,¢€) = ’ ~ec (t,€)P(+,0,¢), P(t,0,€) and note that we must have G(t,e) = F(t,¢)A(e), then 0 » [-p!+p,(4+p,¢,) JL(+,0,€) E(t,e) = € . [ost+e (+p g) ] P(t,0,€), 0 We take =] 0 <¢, <min {ey> SUP lip, (t.ede(te)I"}, and restrict the number ¢ to 0<e < E 5 Since detG(t,e) = eL(t,0,¢)P(t,0,€)[1 + ep (t,e)e (t,e)], it is clear that G(t,¢) is non-singular for O<e<¢€,,0<t<1. 5? If we now multiply equation 1.1 by F(t,e) and use 1.2 to integrate the resulting equation, we get ~ upon multiplication by G7 *(t,e) - the following integral equation for z: 1 ? pi(t,€) T,(z) eg(t,e), -1/e || T(2) z(t) = h(t,e) +e » (1.3) 14 where ) P (0,€) h(t,e) = [itep (t, e)e( t; (t,e)]7 . (0,t, {.., oft, €), ep, (O,¢€)e oft. ot <) (0 (0,€)p (t, e),-ep,(t,e€). ( Yeo), 0, + -c,(0,€) P t,€ sf 0) pe Po% . T, fz] = [t+ep (trele (te) 1" | ° z(s)L(s,t,¢)ds, ° 0] Pg tPy tea (wo 4-P, 29) 1 t[et-c +e (4+p,¢,) v Tolz] = [ten (tee (te) 'f |? 1 ° 2(s)P(s,t,€)ds. * ° ° Oo} €e,p,-w We shall presently show that our choice of F, G, and E has given an integral equation which, for given z(0), has a unique solution of the form: a(t) = h(t,e) + O(el/n(t,e)]}). The following bounds are easily for O< t< 1, O<e< E43 anv) Pe(zysa | lels)las, x IT, (2) | <B,lall, IT, (z)] < eB,lzll. seen to hold (1.4) 15 Then, if we rewrite equation 1.3 as (z). a(t) = h(t,e) + [nt (2) }> the following bounds are immediate: t In, (z) | < cal lz(s) las, [T,(z) | A t B| \z(s) lds, 0 IT, (z) | = eC liz ll, for Og tel, O< e < ey. In the lemma and corollary below, we make a brief departure from our usual notation, in that z(t) and h(t) will represent the k-vectors [7 saee EK {)- x, (+) ty, (ey respectively, and we shall use the notation |lz||, = f° |z(s)|ds. This lemma and o (1.5) (1.6) its corollary are consequences of well-known results in the theory of integral equations, and are in- cluded here to keep this work as self-contained as possible. 16 LEMMA 1.1 Let T be a linear mapping of L, (x) into L, (k), such that t Im(z)| < a®f ‘Je(s)las. ° Then, given any finite-valued integrahle k-vector, h(t), the equation z(t) = h(t) + T(z) has a unique solution zeL, (Ix), and t jz(t) - n(t)| <a*| “|n(s) lexpla*(t-s) las. re) Proof: Set Z_44t) = 0, Zp (t) = h(t) + T(z). It is clear that z cL,(k) for alln 2-1. The assumed bound on T, coupled with the linearity of T, gives Iz, - z,,| < A*llnll, » and an easy in- duction then shows that ax(aet hl, Ce)? inl, =< Janay (t)-2,(t) | n! ni Thus the sequence {z,(+) } is Cauchy for fixed t, which implies that the pointwise limit, z(t)= lim z(t), exists for 0< t<1. Furthermore, | = ° k-+1 le(t) ~ e,(t)l sl)” Ceeag-z ls Nally S fa"), n n 17 so the convergence is uniform, and zeL, (k). From the inequality InT(z,) - T(z)| < At lle az, it is clear that lim T(z) exists and is in fact equal to T(lim Zz). Taking the limit of the relation Zang, = At T(z,)s we see that z satisfies the given equation. A straightforward induction shows that Eee] w\ntl pt (t) - z(t) | <a (t-s)"In(s)Hs, .@] which implies = t le(t)-n(t) I<)" lznyy-z, Isa*] In(s) lexpla*(t-s) Jas. ie 1°] Finally, if z and z* are two solutions to the equation, then t Iz*(t) - 2(t)| < a! Iz(s) = z#(s) las. fe) By induction one can now show that +1, t Jz*(t)a2(t)| <A SW (s-s)P lem (s)-2(s) las < , fe) At n+1 <a lle*-zil,» for all n= 0. Hence o* = ze 18 COROLLARY 1.1 If, in addition to the hypotheses of the lemma, we assume: (a) |n(t)| is bounded for 0< t <1, (db) there exists 0<Bt<1, such that [T(z)| < B*llzll. then the solution to z =h + T(z) satisfies the inequality a(t) — h(t) < Belnii(a - B¥)72, Proof: Since |T(z)| < A*lzll, » the solution is bounded, its sup norm exists, and I'zll< llh |bB#l(z ||, so that |izil < {hil (4-Be)7!. Then |z-h| < B¥ligl] < < Bellnil (1-B*)74. 2- BASIC SOLUTIONS AND THE FUNDAMENTAL MATRIX Using the bounds 1.6, and writing A* = B + cA, B* = e(A + C), we see that we can apply the above lemma and corollary to equation 1.5. In order to use the corollary, we must have e¢(A + C) < 1, so we pick €, < min {ens(A + cyt, and restrict ¢€ to 3 O<ec<e., 3 19 The Basic Regular Solution Setting z(0) = [ 1 | in 1.3 and applying c(0,€) the lemma and corollary, we get the basic regular solution for the system 1.1, Zz (ts), where Zz fe) has the form: z(t) = Le, (t, gy [ECO t,e) * O(e), 14 Zz (Ose) = Le (oye) J The Basic Boundary-lLayer Solution The transformation a(t) = jeuttce ye t/* 4 coupled with the change of variable r=i-t, transforms the system i.1t into the system aX _ [-w(t-r,e), -c(1-r,ce) A(e) ~ Lep(i-r, ec), -1-¢€ 4(1 (ion,e) [X(2)s where X = ee . This system is or the same form as 1.1, and its coefficients satisfy the assumptions made above, so it can be transformed into an integral equation of the form of 1.5 with bounds of the form 1.6. We denote the constants appearing 20 in these bounds by A', B', and Ct, respectively, choose é, < min {e533 (A’ + atv, and restrict ¢ to Of € «< Sys Then it is clear that this system has a basic regular solution of the form: [-p, (1-rse Jef] (2, (2-88 Jo (1-s,¢€ ) - - w(1-s,¢)ds | + Ole), X(r,e) X(0,¢) = L-p (1,e) } Expressing the above solution in terms of z(t) and t, we get the basic boundary-layer solution for the system 1.1, Z (tse), of the form: “ 2 -ep (t, 0 . 2 tte) = ep. (tse) Wi1,tyeje7t/®+ (e*) ent le T 1 O(e) $ 2 (1,6) _ on eni/e . 1 where we have introduced the convenient notation t W(s,t,¢) = exp J (epee) ‘ From Zo and z, we can form the fundamental matrix U(t,e) = [2 (tye), zi (tse)] = L(0,t,¢)+0(e),[-ep (tye nn ~ ~ (2.1 oo (tye )L(0,t,€)+0(€),[W(1,t,€)40(e)] oe ® 21 Noting that detU(t,e) = [L(0,t,e)W(1,t,e) + Ole Je7t/", we take e€* so that det U(t,e) #4 0 for O< t <1, O0< ¢€ < e#, then set es < min(e,,e*), and restrict € to 0< ce < €,. 5 Using 2.1, it is easy to show that, for O<s<i, O< t <1, O< ex ees the matrix U(t,e)U7+(s,e) has the form: ~1/ 3¢ (s,¢) U(t,e)U (s, e) Leo tt (t, e), ene (s, eo, (t, ey {i L(s,t,e]]+ + °Post (t, (0 (s, e)e- “ED offs) Vr rw(s,tie 1128-8), (2.2) where, given a bounded function, f(s,t,¢), we say that a function gl(s,t,e) is of the form [(fl(s,t,e]] if g(s,t,e) = f(s,t,e)(1+0(e)) for O< e€ < Cg, O<s<i1,0<t<t. We point out that, if g(t,e) and w(t,e) can be written a(t,e) = u(t,e) + O(c), w(t,e) = wo(t,e) + Ofe), then in the formulas 2.1 and 2.2 we may replace # and w by L, and Wo? Henceforth, in order to obtain a certain 22 amount of economy of expression, we shall simply write U(t) for the matrix U(t,e). 3- INITIAL-VALUE AND BOUNDARY-VALUE PROBLEMS The Initial-Value Problem We consider the problem of obtaining solutions tO the system 1.1 with the initial conditions aXz(0) =a({e), b’z(0) = 68(e), where a and b are given constant vectors, and a and 8 are given functions of ¢. We assume that t a det] ft] # 0. Then it is clear that, for 0< ¢€ < Eps tj-41 _ “1;n\f a ale) z(t) = U(t)U “(0) ae | [ste solves the problem. Formula 2.2 can be used to get asymptotic estimates for gz. The Homogeneous Boundary-Value Problem We are now interested in solving the system 1.1 under the boundary conditions: a’2(0) =ale), b’2(1) = Ble), with a,b,a(e) and B(e) as before, except that we t. do not require det | pt #0. Writing z(t) = U(t)c, 23 with the @vector c to be determined, and apply- ing the boundary conditions, we get the follow- ing equation: a ELOY vp u(1) Je = (58) This equation has a unique solution for c if and only if tu(o) det] Sty (1) | = = [Lea,p, (0, )-anlI[Lb,+b50,(1,¢)I]L(0,1,¢)W(1,0,¢) is non-zero. Assumption B There exists ¢€' > 0 such that, for 0<e< e€'?3 (a) b, + b,c, (1,6) # 0, “(b) €a,p,(0,€) - a, # 0. We remark that, since p(t,e) = 0(1), if ay # 0 part (b) of the above assumption can always be satisfied by suitably restricting e. If Bo = 0, then part (b) is equivalent to assuming that po (0,¢) # 0 for € suitably restricted. We set €¢ = min(e’,eé_). Then it is clear 5 that, for 0<eé< E gs the matrix 24 a tu(o) vu (1) is non-singular, so that t -1 u(0) a(e) a(t) = UC (ey fe tytt) | Lele). solves the problen. The Non~ Homozeneous Boundary-Value Problem We consider the problen ' £(t,e) (t,¢e) aleyat = [2(fe) Pibsel oy jetaltye), a%z(0) =a(e), b’g(1) = Ble), (3.1) under the hypothesis that assumption B holds. We assume that he L, (2) for 0<e< Cg Using the method of variation of parameters, we get the following expression for 2: t -1 z(t) = U(t)e + | U(bIUT (s)a(t/e)n(s,e)ds, (3-2) fe) where the constant vector c is to be determined. Here we have used the fact that A~*(e) = A(1/e). From equation 3.2 we see that the vector X(t) = 2(t) ~ fuceyunt (sya (1/e )n(s,e)ds fe) is a solution of the homogeneous system, satisfying 25 the boundary conditions: t a’xX(0) =a(e), pb’ x(1) 1 (ec) - | v°u(1)u7+(s)a(1/e)n(s,e das. ° On the basis of assumption B, we can apply the theory developed above for the homogencous problem. The result is the following solution, for 0<e< ees to the non-homogeneous problem: z(t) =U (t)[ Se e607" ve) + t + | u(tyu7*(s)a(1/e)n(s,e as, (3.3) ° _ (e) y(e) = le (e)-Flu(r)um*sha (1/e n(sye Jas ° It is easy to verify that 3.3 can also be written in the form: -1 1 - | u(tu7t(s)a (1/e )nls,e as, (3.4) t where 26 1 t -1 (e 4] a’U(0)U7*(s)A(1/e )n(s,e)as yi(e) = ° . 8 (e) It will be useful later to have for reference the formula: atu(o)|7* [v%e_(1,€),-atz (0,e) = T T x L(1,0,¢ )W(0,1,¢) ; * TTeayp,(0,e}-a,TI1b, 4p (ENTT (3-5) Note that if by #byc. ( for € restricted to a small enough interval, then 1,€) is bounded away from zero the last factor is O(1) for Ao # 0, while it is O(l/ep, (0,€ )) for ao = 0. 4, THE NONLINEAR BOUNDARY-VALUE PROBLEM We are interested in the existence of, and asymptotic estimates for, solutions to the problem: A(e)z' = h(z,t,e), t t (4.4) a‘z(0) =a(e), b’z(1) = Ble) when we are given a continuously differentiable solution, z(t), to the degenerate problem: A(0)z? = h(z,t,0), b’2(1) = 8(0). (4.2) For simplicity, we shall assume a and b inde-~ 27 pendent of ¢. We write _Tf(x,y,t,e © h(zstse) = leley tie) ’ and introduce the notation for the matrix of first partial derivatives: h_(z,t,¢) = fi(x,y,t,e), f (xs¥stse}| ; Zz gr (x5ystse), Sy (xy stse) Assumption i The degenerate problem 4.2 has a continuously differentiable solution, z(t), and h, (z(t), t,¢) = = O(1). We assume that €, > 0 can be found so that 1 the following three assumptions hold. Assumption 2 There exists a region 9 = { (z,t,e)s0Ste1, O<e< e,/z -2z,(t)| < Y}s with y > 0, in which: (a) f and g, along with all of their first and second partial derivatives with respect to x and y, exist and are continuous functions of x, y, and +3 (b) all of the second partials of f and g with respect to x and y are O(1). 28 Assumption 3 - . By (xo (t) yo (t)stse) = -1 + Ofe), forO0<t<ti, O< € < €,- Assumption 4 fy (xo (t),y,(t),tr¢) and 6, (%q (t)syo(t),t,¢) can be written in the form: (x, (t)s,y(t),t,€) ul ty Pyg (tse) + O(e), g, (x(t) yg(t), tre) tN Bolte) + Oe), where Po and Bro are differentiable functions of t for 0<t<i, 0<e< €,>» while fio and Exo are O(1). On the basis of the results of sections 1 and 2, the above assumptions imply that there exists E, > 0 such that the linear system Ale) z* = h (z(t), t,€ )z (4.3) has a non-singular fundamental matrix, U(t), for O<€e< Eos O<t<i1. The form of this matrix is given by equation 2.1, with the obvious identifi- cation. Assumption 5 There exist E. > O and s > O such that, for 29 O<e< En, we have: |b, + D5By5(1,¢)| 7 = O(1), Ip, (Ose)|7* = o(e7§) . Assumption 6 There exists ey > 0, such that, for 0<€e< Ens O< t < 1, we have: (a) 68(e) =~ B(0) + Ofe). (b) h(zi(t),t,¢) = h(z(t),t,0) + O(c), Assumption 5 should be compared with Assumption B. The outer correction, @(t,¢), is defined by the solution z*(t,e) = Z(t) + @(t,e¢) to the problem: A(e)g*' = h(z*,t,e), : f t (4.4) c’z*(0) = c'z (0), b’z¥(1) = Ble), where e=[)| ita, = 0, cz a if 5 # 0. In order to be concise, we shall not continue to explicitly denote the e-dependence of %, but shall write S(t) for @(t,e). The above problem can be written in terms of % as follows: 30 Ate )g o°¥( 0) i h_(z_(t),t,e)@ + H(O(tht,e), z . (4.5) 0, o’@(1) = Ble) - B(0), where H(@(t),t,e) = ee + B(t),t,e) - ~ hy (aj(t),tre)B(t) - aledzt(t). We shall transform the problem 4.5 into an integral equation and apply the method of successive approxi- mations to show that the integral equation has a solution, On the basis of Assumptions 1 through 5, we can, for € restricted to a small enough interval - say 0< ce < fxs apply the results of section 3 to obtain an integrating factor for the problem 4.5. Using formula 3.3, we get the following integral equation for @: Ot) = 1H), (4.6) where oat TL] = [ (tut (s)a(1/e)H(B(s),s,¢)ds + ° . 0 ~ * Ls (e)-8(0)-s*otu(t yu" (s)a (1/e )H(Wl8), 8,0 Jas! ° 31 We remark that, on the basis of Assumption 5 and formula 3.5, both of the matrices ty(0) ‘ (0) ° lStuc1) btutt) | b are non-singular for 0<e< Eg. Using formulas 2.1, 2.2, and 3.5 (with the fact that the second component of c is non-zero), we obtain, for 0<¢€éQ<e O< t < i, the following 5? bounds: t -1 |} u(t)u7*(s)a(1/e)r(s)as| < x, |Irll, (4.7) ofu(o)i7 -1 luce Bey eds | “ft U(1)U7*(s)alt/e)r(s)as| < Kollz|l, where r(t) is any measurable, bounded two-vector. Since Assumption 6(b) implies that H(0,t,e) = O(c), the inequalities 4.7 may be applied to show that, for 0<t<i, 0<€eé< Ex, |TLH(O,t,¢)]]| < Re . (4.8) We now set Eg = min{€ sv /2R,[ 3BR(K,+K,)]~*}, and restrict € to 0<¢é< Ee. 32 We apply the method of successive approximations to equation 4.6, setting B_,{t) = 0, Gn it) = Td]. From the inequality 4.8, we see that, for 0< t<1, 0< € < €¢, we have so that (z)(t)+(t),t,¢) is inQ for O<t<1, O<e<e€,. Henceforth, for simplicity, we shall write "f is in © in place of "(f(t),t,e) is in A for 0< t<i1, 0O<e< ai Before we can proceed, we must have an estimate for H(X(t),t,e) - H(Y(t),t,¢) when X and Y are bounded, measurable vectors, with ze +’ X and Zo + X inQ. Applying the mean-value theorem twice, we get: [H(X(t),t,e) - H(x(t),t,e)| < < BIX(t) - v(t){maxf{|x(t)|,]x(e)]1, (4.9) where B is the constant, which exists on the basis of Assumption 2(b), bounding the second partials of f and & with respect to x and y inf. Then |H(Zo(t),t,e) - H(H_,(t),t,e)] < BR<e ~, and so, using equations 4.6 and 4.7, we have: | 0, (t) - Do(t)| < BeHe, |G, (t)| < Re (1+B*), 33 where we have set | i” B B( 4K, )Re. Since BH < B(K, +K,, ) Re ¢ < 1/3, we have co 1 + >. Hae)Rt < 2, ° so that |B, (+) <y for 0<e<e,. This implies that z, + g, is in Q, so we may continue. A straigntforward induction now shows that k k+1 Begg (t) - OB (t)] < 2°(Bey Re, |H.| < eRe <y, forall k>0, O0<e< Eg: In exactly the same way as with the lemma of section 1, we can argue the uniform convergence of the sequence {6 (t)} to a’ limit function, @(t), satisfying 4.6, with | B(t)-B,.(t)| < (1-2B*)“t2*(BH)**tRe < (2/3fRe, |2| < 2Re. Since all of the functions in the matrix U(t) are continuous functions of t (see formula 2.1) on the basis of Assumption 2, it is evident from equation 4.6 34 that f(t) is continuous for k > -1, hence @ is continuous. In fact, it is not difficult to show, using equations 2.1 and 4.6, along with Assumptions 2and 4, that @(t) is differentiable on [0,1] for O<ecd E gs 5. THE INNEH CORRECTION If X(t) is a two-vector, we define H#(X,t,¢) = h(z*+X,t,€ )-h(z*,t,€ )-h, (z*,t,€)X(t), where z* = Z + @ is the solution to the problem 4.4 (the existence of z* for 0<e«< eg was proved in the preceding section). The inner correction, W(t,e), is defined by writing the solution, 2(t,¢), to the nonlinear boundary-value problem 4.1 in the form z = z* + Ut. This results in the following problem for }: Afe)y' = h(z¥(t,e), tye )¥ + H¥(u,t,e), ‘ (5.1) a”y(0,¢) = ale) - atz#(0,e), d’v(1,€) = 0. In order to be concise, we shall - as we did with the outer correction - cease referring explicitly to the e-dependence of 4, and shall write (+t) for y(t,e). We choose to write y(t) in the form: _ | ep(t)| ve) = LEE] and introduce the notation u(e) =a(e) - atz#(o,e). 35 Applying the mean-value theorem, we see that h(2*(t),t,¢) = h (z(t), tse) + h*(t,€), where, by Assumption 2 and the fact that @(t) is Ofe) ami continuous, h*(t,e) is O(e) and continuous. Taking cognizance of the remark concerning the functions £(t,¢) and w(t,e) that follows formula 2.2, we see that, on the basis of Assumptions 1 through 6 and the above expansion for h, (z*(t),t,€), the system Afe}g! = h (2¥(t),t,€)z (5.2) nas a non-singular fundamental matrix for eé suitably restricted, say 0<¢e<e There will ° be no confusion if we call this matrix U(t). We point out that U(t) is given by formula 2.1 with the identification: £(t,€),p,(t,€) _ f(x (t),y(t),t,€), Eo bs€) o,(t,€) ,w,(t¢) Bol ts€), e' L148, (t(t) ,y#(t), t,€)] , where Poo and Exo are defined in Assumption 4. Just as we did for the outer correction, we can use the results of section 3 to restate the problem 5.1 as an integral equation. Using formula 3.4, we get the following equation for 36 the inner correction: b(t) = z,(t,e) + TTH*(¥,s,¢)], (5,3) where t -1 ator) = ote) hey. (5.4) bU(1) 0 a’u(o)) 7? [fatucojut (sa (t1/e )2(s)as Tl z] = U(t) m 0 . bU(1) 0 (5-5) 1 ~ | u(tyut(s)a(t/e)z(s)as. t We again point out that the matrix a’u(o)|7} p’u(1) exists for 0<e< Ens as a consequence of Assumption 5 and formula 3.5. We shall attempt to solve equation 5.3 by the method of successive approximations, setting #4 (t) = 0, ¥(t) = 2, (tye) + TTH*(0,s,e)] = zy (tse), and in general, Vey (t) = 2, (tye) + TTH*(¥,,8,€)]. We define the number m as follows: if ay = 0, m= -1 - 83 if An #0, m= 03 where the number s is defined in Assumption 5(b). 37 Then, using formulas 2.1 and 3.5, we can show that loo (t)| = e7*] x, (tye) | < e"u(e)Maq(b,t,¢), (5.6) Irg(t)l = lyy(tye)] < eMule yuen*/®, where Mis a constant, and we have introduced the function: q(b,t,e) = e7t/e if b, = 0, a(b,t,e) = ee 4 emten te af b, # 0. The difference in the bound on p,(t) for the cases b, = 0 and b, # 0 is due to the term bz, (1,¢) occurring in the matrix on the right-hand side of formula 3.5. Assumption 7 There exists a region = { (zytye): O<e<e', |x-x,(t)|<2Re+Pe'*™q(b,t,€), ly-y,(t)|cenetpe™et/E, o<t< 1}, with P > 0, in which: (a) f, g, and all of their first and second partials with respect to x and y exist, with f, g, and their first partials continuous in x and y. 38 2 2 (b) The functions e¢ fox? © Byy? Pfyys &8y y y’ -m Pyy? Byy are all O(e ~). We shall simply say zi + Vi is in Q, if (2¥(t) + ¥,(t),t,e) is inO® for Os tsi; 0 < e€< €', and shall write: Hy ,te) = Hey) = [PEWS On the basis of the above assumption, if z2* + Vie and gz + Vint are in Qos we can apply the mean-value theorem twice to obtain: [F*(b FRY 1, [G*CY, )-GH(HL_y] S AM(t), (5-7) where e™u(t)=max{ {a(t} lo, (t)-05(t)[ slo (t)[ Ir, (t)-75(t) I, T(E) lo, (t)=o, (tbl Tle) [r, (tl-ro(tyl} with [e(t)f = max (|r, (t)], [u(#)]), [0(%)] = max (lo, (+)],]o,(4)]). If we write the transformation Tl z] appearing in equation 5.3 in the form T [2] Tz) =| 7 ||, Tp[z] then a very long calculation, using formulas 2.1, 39 2-2, and 3.5, shows that, for any bounded, measurable two-vector, z(t), the following bounds hold for Os t<l, O< € < 3 v4 |t,fz(s)a*(b.s,€)]| < Kellzlla(b,t,e), ITof2(s)q7(bys,€)]| < Kllzle"*/*. (5.8) We remark that K depends on b being in general 2) considerably smaller for by = 0 than for b, #7 0. Assumption 8 There exists Eg > 0 such that, for 0<e< Egi (a) ule)M < P/2, (b) 2u(e)AKM < 1/3. We set Ee, = Min[Ens€_5€'] and restrict € to O<ec< Eo We define Qe to be the region obtained from QO, by restricting « to 0<e< Eo» and remark that, since a is contained in Qo) the bound 5.7 is wlid for z* + Vie zt + Vind in ate We also point out that assumption 9(b) implies that, for 0<« < ey? co >. 27Tau(e akmyett < 1, Le] Under assumption 8(a), zt + ¥, is in Qf, so we may apply the bounds 5.6, 5.7, and 5.8 to obtain: 40 lo, -o | <MteTu(e )Mq(b, t,€), |r -Tol < wre, (e Men b/ lo, | < eu (14M )u(e )Mq(b,t,e), Ir, | < en (t4M)u (e \Me7t/® F) for 0<e€e~< E gs O< t< 1, where we have written M* = 2u(e¢)AKM. Assumption 8(b) implies that zt + 4 is in Q6> So we may continue. An easy induction now shows that t je/E < 2h) TMi (ede, lp -e, |La(b,t,¢)]7', et n n+1 -1 t lo. {late,t,6)1", [tr ]e /< < 2e"h(e)M. It is now a straightforward matter to prove that w(t) = lim u(t) = u(t) + DCH gg (t)-0,, (41 ie} is a solution to equation 5.3, with lo(t)iLaloyt,e)I72, |r (te’/® < eMule Ms Jo(t)-p (t)[La(dste)]7 1, |r(t)-r, (tle < < (1-2m*)7*22(meyP*t Mice yy < (2/ZfeM (eu. (5-9) Ad 6. CONCLUSIONS We can summarize the results of sections 4 and 5 in the following theorem. THEOREM 6.1 Under Assumptions 1 through 8, there exists e, > 0 such that the nonlinear boundary-value prohi em sede = nizatse)s (6.1) a’z(0) =ale), b’z(1) = Ble), has, for 0<e< Eo: a solution of the form z(t) = z(t) + B(t,e) + w(t,e), with B(t,e) = Oe), and Writing § = [°° |) p(t,e) = O(e%(e))q(d,t,e), T(t,e) = OleM(e))e7*/, tl where m= 0 if a, #0,m=-1-sif a, = 0, ule) =ale) - atz (0) + B(0,e)1, a(b,t,e) = e t/ ar by = 0, a(byt,ce) = e7t/ 4 enten“l/e ap b, # 0, -1 - [fy (O,e)|~~ = Ofe-*). Le Using the fact that B(t,e) = O(t,e) + O(e*), B(t,e) = TO, where Ti2] is defined by equation 4.6, we can show that B(t,e) = 1(t) + Ble) = Bo) - vI(1) , v, + boB oh 1 rE) L(1,t,¢€) Bo tre)L(1,t,€)-[r+e, (0,€) IW(0,t,¢)L(1,0,€)e where ° 1 1 ; He) = | Metidle yer be yep) taleols)oss€)-aledeg(9)1 + ~ s-t 0 é 'i(s,t,e)e! Ma (s)ivglo)sere)-eye(a) ds . L(s,t,e) = exe J frye (u)ayg(u) we) + to(uede,g(use) fan] , 5s t ~ We,tye) = emfe”'] {48 (x,(u),y,(u) we) ef (wee. (use) bu | , Ss Y= a,/a, if a, #0,y=O0if a, = 0. Similarly, noting that rn | 9 ° o(e"u?(e) en P/° b(t,e) = wb (tye) + bo (tse) = Z4(t5€ ), where Z1(t,e) is defined by equation 5.4, we can show + 43 that: _ ulelue(o tele | Pfyq (toe) w(t,e) = wea ty (0-6 )=a, “1 + wle)L(1,t.¢)W*(0,1,e)e/"f bp -eb ft, (1,6) 1 [éa,F,5(0,¢)-aj]lb, tbe, (1 €)1 Byo(tse)|* [atone “+ 9 0(6M2(2 en" t/e where L(s,t,€) is defined above and ay pte | W*(s,t,¢) = vo | {ite (x*(u,e),y# (use) use) - Ss - efyoltre)Bol0e)} a z¥(t,e) = 2 (t) + Blt,e). Note that the behavior of t(t,e) is significantly different in each of the four cases described by the possibilities a = 0, a, #03 b, = 0, b, # 0. 2 2 Our assumptions, for the most part, concern the smoothness of f and g and the possibility of ex- pansion of h(zo(t),t,e) about ¢ = 0 to terms of order €. Briefly, we assume that f and g are twice con- tinuously differentiable with respect to x and y, that f and gs and their first and second partials Aub are 0(1) in a region 2 (see Assumption 2) of (z,t,€)-space about the curve { (z(t), ,0)s0ctet}, and that the second partials of f and ge satisfy certain inequalities in a region oe (see Assumption 7) of (z,t,¢)-space. The region Q, is designed to contain the inner correction, and consequently is unbounded. In addition, we require that fy (x, (thy p(t), tre) and g(x (t),y (t),t.,e) differ by at most O(c) from differentiable functions with uniformly bounded derivatives, and assume that By (xy (t)sy¥g(t)stse) can be expanded about ¢ = 0 to order ¢, with leading term -1. Although the requirement that g,(x,(t),y,(t),t,€) have leading term -1 appears rather restrictive, we can reduce a system of the form of 6.1 in which Gy (x9 (s)s¥,(s),s,€) = k(s) + O(e) to one in which By (xo (t),yo(t),t,e) has leading term -1, if k(s) is different from zero and of constant sign on [0,1]. This reduction is accomplish- ed by the change of variable [x(wranf Ptayan ifk <0, e) (9) Poecudanf Pitman] if k> 0, s 0 t qt ct il 45 coupled with the replacement of e€ by e's e| | [k(u)|duy™ . fe) If, in addition, we assume that k(s) is continuously differentiable, then the integrability and differ- entiability of the first and second partials of f and g are unaffected by the transformation. Of the remaining assumptions, one is especially worth discussing. In Assumption 9, we required that the quantity ule) = ale) - az, (0) + B(0,)] be sufficiently small. This means, roughly speaking, that the boundary condition which we impose at t= 0 (a’z(0) = a(€)) must be sufficiently close to that satisfied by the degenerate solution. To be more precise, the restrictions on u(e) are of the form: (a) ule)M < P/2 (b) au(e)AKM < 1/3, (6.2) where (in Assumption 7) we had required that: 2 -m ote lee lapel sel fyylrtleyylaltyylaleyyl < Ae in the region Qo: It is clear that, though we can L6 do little about the restriction 6.2(a), we could reduce the severity of 6.2(b) by replacing the constant A by a function of ¢,A(e), and requiring that A(e) = of(1). We also remark that, since “M= -t-s for a, = 0, while m = 0 for Bo # 0, the 2 above bound on the second partials is much more stringent - by a factor of e Stl if Bo = 0. We can specialize the problem 6.1 to the problem considered by Erdélyi (12) by taking f(x,y,t,¢) = vy as = b, = 0, m= <1. 2 A point by point comparison of assumptions reveals that he has assumed all of the hypotheses of Assumptions 2 and 7 hold in a single region D, where D can be interpreted as the union of and Oo In D, he has assumed that while we have assumed that, in 9 (a hounded region) Bry? Bxy? Syy = 001) and in 2, (an unbounded region) _ -1 _ - Bex = O(e ~), Bxy = O(1), Byy = O(e), On the other hand, we have assumed (Assumption 4) that 6, (x (t),y7,(t),t,¢) differs by at most O(c} Lo from a differentiable function with uniformly bounded derivative, which has no counterpart in Erdélyi’s hypotheses. Since this assumption is a direct consequence of our technique for con- verting a linear system into an integral equation, it is apparent that onr methods are not as effective as those of Erdélyi for the particular case of a second order equation with boundary conditions of the form x(o) =a(e), x(1) = 8(e). With the exception of these two relatively minor differences, our assumptions and results are equivalent to those of Erdélyi. Harris (5) treats a problem of the form 6.1 with a» = vy = 0, assuming that f and g are analytic in x and ¢€, linear in y, and C“ int. As far as they can be compared, the only important difference in our hypotheses is our replacement of linearity of f and g in y by the requirement that f and ¢g be yy yy O(c). Harris obtains a convergent expansion for the solution, whereas, because of our less stringent assumptions, we have obtained only the leading terms. With the exception of the formula for the second component of the outer correction, our expressions 4B for the leading terms of the inner and outer corrections can be reduced to his. (1) (3) (4) (5) ie) REFERENCES H. L. Turrittin, Asymptotic expansions of solutions of systems of ordinary differential equations. Contributions to the Theo of Nonlinear Oscillations II, Annals of Math- ematics Studies, 29 (1952), 81-116. W. Wasow, On the asymptotic solution of boundary value problems containing a parameter, J. Math. Phys., 23 (1944), 175-183. E. A- Coddington and N. Levinson, A boundary value problem for a nonlinear differential equation with a small parameter. Proc. Amer. Math. Sor., 3 (1952), 73-81. W. Wasow, Singular perturbations of boundary value problems for nonlinear differential equations of the second order, Comm. Pure Appl. Math., 9 (1956), 93-113. W. A. Harris, Jr., Singular perturbations of a boundary value problem for a nonlinear system of differential equations, Duke Math. "t M. Nagumo, Uber das Verhalten der Integrale von Ay" + f(x,y,y’,A) = 0 fir A> 0, Proc. Physico-Math. Soc. Japan, 21 (1939), 529-534. R- von Mises, Die Grenzgschichte in der Theorie der gewShnlichen Differentialgleichungen, Acta Universitatis Szegediensis Sect. Sci. Math., 12 (1950), 29-34. N. Levinson, Perturbations of discontinuous solutions of nonlinear systems of differential equations, Acta Math., 82 (1950), 71-105. (9) (12) (13) 50 E. A.» Coddington and N. Levinson, A boundary value problem for a nonlinear differential equation with a small parameter, Proc. Amer. Math. Soc., 3 (1952), 73-81. N. I. Bris, On boundary problems for the equation ey" = f(x,y,y',e) for small ¢’s, Dokl. Akad. Nauk, 95 (1954), 429-432. A. Erdélyi, An expansion procedure for singu- lar perturbations, Atti Accad. Sei. Torino, a Sci. Fis. Mat. Nat., 95 (1960-61), 651- 726 A. Erdélyi, On a nonlinear boundary value problem involving a small parameter, J. Australian Math. Soc., vol II, part 4 (1962), A. Erdélyi, An example in singular perturba- tious, Monatschefte ftir Mat., 66 (1962), 123- 128.