R. M O N A C O, R. R I G A N T I and M. G. Z A V A T T A R O (*)

Transcription

1 R i v. M a t. U n i v. P a r m a ( 6 ) 4 ( 2 1 ), R. M O N A C O, R. R I G A N T I and M. G. Z A V A T T A R O (*) Singular perturbation methods for the solution of some nonlinear boundary value problems (**) 1 - Introduction Several examples of nonlinear boundary value problems, exhibiting solutions of boundary layer type, arise in many areas, including chemical-reactor theory, optimal control theory, fluid dynamics and the physical theory of semiconducting devices. The mathematical model for such problems is given by a differential system of the form (1) A e dy dx 4g(x, y, e), xi4 [, 1] where y : IKR n is a n-dimensional unknown vector, A e is a n3n diagonal matrix, with non null elements a ii 4e q i, which are integer powers of a small parameter e : GeEe b 1, and where g4]g i (, i41, R, n is a function that is nonlinear in x, y and e. The solution y(x, e) is subjected to separated boundary conditions of type: (2) y r () 4a r (y s (), e) r41, R, m; s4m11, R, n y s (1) 4b s2m (y r (1), e) (*) Dipartimento di Matematica, Politecnico, Corso Duca degli Abruzzi 24, 1129 Torino, Italy. (**) Received May 8, 21. AMS classification 34 E 15, 76 P 5.

2 134 R. MONACO, R. RIGANTI and M. G. ZAVATTARO [2] where a r, b s2m are known invertible functions. In addition g i, a r, b s2m have the asymptotic expansions (3) C` D g i (x, y, e) a r (y s (), e) b s2m (y r (1), e) È F Q A! k4 C`D g ik (x, y) a rk (y s ()) b s2m, k (y r (1)) È F e k. If q i 4 for some value of index i, the corresponding solution components y i (x, e) of the boundary problem (1-2) (called slow variables) will converge uniformly in [, 1] as ek. On the contrary, the other fast variables, which have derivatives multiplied by a positive power q i of the small parameter e, have strong variation at the e q i wide boundary layers resulting at the right or left endpoint of I, as a consequence of the boundary conditions (2). Because of the nonuniform convergence of the fast variables as ek, the solution is constructed by means of composite asymptotic expansions. This classical method, originated from Prandtl s papers and developed by many Authors, see [1], [2], [3], [4] and related bibliography, allows to determine suitable boundary layer corrections, derived by rescaling the independent variable in the stretched variables t4 x, s4 12x e q i e q i and having asymptotic expansions as e K with terms tending to zero as t, skq. In this paper the conditions to be satisfied by functions g i and the structure of the composite solution to obtain a uniformly valid solution of problem (1-2) are studied. The e-order approximate solution is explicitly determined for some problems with one or two initial and/or terminal boundary layer. A formal solution for boundary layers problems of type (1-2), using the aforementioned composite expansions method, is derived by O Malley in [5], only when the function g is quasilinear, and precisely when it is linear in the fast variables. Here g is a nonlinear function of any component of the variable y: in this sense the following analysis may be considered as a generalization of the formal results presented in book [5]. Explicit results of the proposed analysis will be given in the last Section. In particular, in Section 2 the case q 1 4, q 2 41, in which the state vector y is composed by a slow variable and a fast one, is considered; in Section 3 the more general case q 1 4q 2 41 with e-order wide boundary layers at both endpoints is examined. In order to avoid inessential difficulties of treatment, in this paper the case n42 will be considered. In the last Section the above techniques are applied

3 [3] SINGULAR PERTURBATION METHODS FOR THE SOLUTION to determine approximate analytical solutions of two boundary value problems arising from Boltzmann like models. 2 - Problems with a single boundary layer Calling for simplicity y 1 4u, y 2 4v, g 1 4f, g 2 4g, consider the first-order system (1) rewritten here in the bidimensional form (4). / du 4f(x, u, v, e) dx e dv 4g(x, u, v, e) dx for the unknown functions u, v : I K R subjected to given boundary conditions (5) u()4a(v(), e), v(1)4b(u(1), e) Initial boundary layer Let us seek for a uniformly valid solution of problem (4-5) for x [, 1] and e K in the composite form: (6) u(x, e) 4U(x, e)1j(t, e) v(x, e) 4V(x, e)1h(t, e) where (U, V) is the outer solution, i.e. the solution outside the initial boundarylayer region, and the functions (j, h) of the streched variable t4x/e are the inner solution, i.e. the boundary layer correction terms which occur at the left end point, vanishing outside the boundary layer region and satisfying the matching conditions (7) (8) lim j(t, e) 4, tkq lim h(t, e) 4. tkq We assume that the functions (U, V), (j, h) have asymptotic expansions as ek

4 136 R. MONACO, R. RIGANTI and M. G. ZAVATTARO [4] of the type of Eq. (3): (9) CU(x, e) E CU k (x) E `V(x, e) ` Q `V ` ` A! k (x) ` e j(t, e) k4 `` j k (t) `` k. `D h(t, e) `F Dh k (t) F In the sequel we will omit, unless strictly necessary, the arguments of the functions U k, V k, j k and h k. Thus, the solution to problem (4-5) can be written in the form (1) Q u(x, e) 4! [U k 1j k ] e k Q, v(x, e) 4! [V k 1h k ] e k. k4 k4 The problem of convergence of such a procedure is widely treated in the literature quoted in the Introduction, specially in the book [4]. Here we will observe that it is assured if the perturbations U k, j k, V k, h k, kf1, are bounded (xi. In fact, in the applications shown in Section 4 this property is satisfied. We will show that solution (6) holds, and it can be approximated to the first order with respect to e, if the problem (4-5) satisfies the following conditions: l C1. A function W(u, x) exists, that is continuous and differentiable in a neighbourhood of U (x), such that (11) g(x, u, W(u, x), ) 4, GxG1 and such that a solution U (x) to the following nonlinear problem exists: (12) du dx 4f(x, U, W(U, ), ), U (1)4l with l root of the algebraic equation (13) U (1)4b 21 (W(U (1), 1) ). l C2. A constant c D exists, such that (14) g v f g v (x, U, V, ) G2c for GxG1, with V (x) 4W(U (x), x).

5 [5] SINGULAR PERTURBATION METHODS FOR THE SOLUTION l C3. For the same value of c also the following applies (15) g v (, U (), d, ) G2c for all values of d between V () and v()4a 21 (U (), ). In the above conditions, a 21 and b 21 are the inverse functions of those defined by Eq. (5), and computed for e 4. By using the asymptotic expansions (3), (9) the solution can be approximated to the first order with respect to e, i.e. by a truncation at N41 of Eq. (1), as it follows. Taking into account that the outer functions U, V satisfy the system (16). / du 4f(x, U, V, e) dx e dv 4g(x, U, V, e) dx the first two coefficients of their expansions satisfy respectively: (17) du dx 4f 4f(x, U, V, ) (18) 4g 4g(x, U, V, ) (19) du 1 dx 4f 14f u U 1 1f v V 1 1f e (2) dv dx 4g 14g u U 1 1g v V 1 1g e where f u, R, g e are partial derivatives calculated for e4: f u 4 f u (x, U, V, ), R, g e 4 g e (x, U, V, ). Now we require that the composite solution satisfies the system (4). In a right interval of x4, the boundary layer correction functions j(t, e), h(t, e) must sati-

6 138 R. MONACO, R. RIGANTI and M. G. ZAVATTARO [6] sfy the system (21) dj dt dh dt 4e] f(et, U(et, e)1j(t, e),v(et, e)1h(t, e), e)2f(et, U(et, e),v(et, e), e)( 4g(et, U(et, e)1j(t, e), V(et, e)1h(t, e), e)2g(et, U(et, e), V(et, e), e)fg(t). Inserting expansions (3), (9) into (21) and equating coefficients of like powers of e for finite values of t, give a hierarchy of equations for the boundary layer corrections j k, h k. For k4 we have the differential equations (22) dj dt 4 (23) dh dt 4g(, U (), V ()1h,)2g(, U (), V (), )4g(, U (), V ()1h, ). Taking into account condition (7), Eq. (22) gives j (t) 4const4j ()4 which shows that the solution u(x, e) for the slow variable, uniformly converges in I for e K. For k 4 1 we have the linear system (24) (25) dj 1 dt 4f(, U (), V ()1h, )2f(, U (), V (), ) dh 1 dt 4 dg(t) de N 4g v (, U (), V ()1h, )Qh 1 1A (t) e4 where the exponentially decaying term A (t) is given by (26) A (t) 4t[ g x (, U (), V ()1h (t), )2g x (, U (), V (), ) ] 1[tU 8()1U 1 ()][g u (, U (), V ()1h (t), )2g u (, U (), V (), )] 1[tV 8()1V 1 ()][g v (, U (), V ()1h (t), )2g v (, U (), V (), )] 1j 1 (t) g u (, U (), V ()1h (t), )1g e (, U (), V ()1h (t), ) 2g e (, U (), V (), ) where the primes denote partial derivatives with respect to et4x. By using the

7 [7] SINGULAR PERTURBATION METHODS FOR THE SOLUTION expansions (3) the boundary conditions (5) become (27) (28) U ()1eU 1 ()1R1ej 1 ()1R 4a (V (), h ())1ea 1 (V (), h (), V 1 (), h 1 ())1R V (1)1eV 1 (1)1R1h (1/e)1eh 1 (1/e)1R 4b (U (1))1eb 1 (U (1), U 1 (1))1R being a 4a(V (), h (), ), b 4b(U (1), ), a 1 4a v (V ()1h (), )[V 1 ()1h 1 () ]1a e (V ()1h (), ) b 1 4b u (U (1), ) U 1 (1)1b e (U (1), ) where a v, b u, a e, b e are partial derivatives at e4. The other terms of higher order a k, b k, k42, 3R, have analogous expressions, and are linear with respect to V k ()1h k () and U k (1). Condition (8) requires that, when ek, the terms vanish in the first member of Eq. (28), i.e. h k lim ek h k(1/e) 4, k4, 1, R Therefore, equating in Eqs. (27) and (28) the coefficients of like powers of e gives (29) (3) U ()4a(V ()1h (), ) V (1)4b(U (1), ) ) and for sufficiently small e: (31) U 1 ()1j 1 ()4a v (V ()1h (), )[V 1 ()1h 1 ()]1a e (V ()1h (), ) (32) V 1 (1)4b u (U (1), ) U 1 (1)1b e (U (1), ). Equations (17-2) and (22-25), when solved with the conditions provided by Eqs. (29) and (32), supply the unknown functions of the composite solution in the e-order approximation. Thus the solution can be determined as follows. Following condition C1, Eq. (18) can be written as a (unique) function V (x) 4W(U (x), x), and owing to condition (3) the terminal value of U (x) must sati-

8 14 R. MONACO, R. RIGANTI and M. G. ZAVATTARO [8] sfy Eq. (13). The solution of (17) is therefore (33) 1 U (x) 4l2f(s, U (s), W(U (s), s), ) ds. x The initial value U () obtained from (33), allows to determine from condition (29) the initial value h () of the boundary layer correction of the fast variable v(x, e). Integrating (23) with such initial data we find (34) h (t) 4a 21 (U () )2W(U (), ) t 1 [ g(, U (), W(U (), )1h (s), )2g(, U (), W(U (), ), ) ] ds that, if condition C3 is satisfied, is monotonically decreasing to zero for tkq (crf. [3], pp ) as prescribed by Eq. (8). Note that (34) is an integral equation for h which could presumably be solved by successive approximations, but not explicitely. The solution of the reduced problem is then determined, and it can be used to obtain the higher-order terms as it will be explained below. With regard to the outer solution of order e, we first observe that Replacing in (2) gives dv dx 4 W U du dx 1 W x 42 g u g v f 1W x. (35) V g v k2 g u g v f 1W x 2g u U 1 2g el. The terminal value V 1 (1) follows condition (32); so U 1 (1) must satisfy the algebraic equation (36) 1 k2 g u f 1W x 2g u U 1 2g 4b u (U (1), ) U 1 (1)1b e (U (1), ). g v g v elx41 Eq. (36) is a linearized version of (13), so it will be uniquely solvable if the solution of (13) is locally unique. If m is a solution of Eq. (36), it determines the terminal condition to assign to the differential equation (19). By substituting (35) we find

9 [9] SINGULAR PERTURBATION METHODS FOR THE SOLUTION the linear problem (37) with du 1 dx 4a(x, U ) U 1 1b(x, U ), U 1 (1)4m (38) a(x, U ) 4f u 2 f v g u g v, b(x, U ) 4 f v g v k2 g u g v f 1W x 2g el1f e that has the following solution: U 1 (x) 42expy a(s, U (s)) dsz{ expy2a(t, U (t)) dtzqb(s, U (s)) ds2m}. x x s Finally let us consider the e-order boundary layer correction terms. As j 1 must vanish when t KQ, we find from (24): (39) Q j 1 (t) 42[ f(, U (), V ()1h (s), )2f(, U (), V (), ) ] ds. t Once U 1 () and j 1 () are known, the boundary condition (31) gives the initial data of the differential equation (25) for the boundary layer correction term h 1, namely: h 1 ()4 U 1()1j 1 ()2a e (V ()1h (), ) a(v (), h (), ) 2V 1 () and, if joined to Eq. (25), defines an initial value linear problem with the solution t Q{ t h 1 (t) 4expy g v (, U (), V ()1h (s), ) dsz s A (s) expy2g v (, U (), V ()1h (t), ) dtz ds1h 1 ()} that, considering the hypotheses stated above, decreases exponentially to zero when tkq, as it is shown in ref. [3]. In conclusion, the e-order approximated

10 142 R. MONACO, R. RIGANTI and M. G. ZAVATTARO [1] solution of problem (4-5) is: (4) u 1 (x, e) v 1 (x, e) 4U (x)1eku 1 (x)1j 1g x e hl 4W(U (x), x)1h g x e h1ekv 1(x)1h 1g x e hl, and the higher-order terms can be found by proceeding as shown above. Generally the solution is not unique, and its multiplicity depends on the number of solutions of the algebraic equations (13) and (36) Terminal boundary layer In this subsection we derive conditions under which the fast variable v(x, e) has a terminal boundary layer in a left neighbourhood of x41. These conditions may be stated by introducing the simple transformation z412x of the space variable and by applying the calculations developed in 2.1 in order to obtain analogous results. However, since they will be used in the application which follows in Section 4.2, for sake of clarity we prefer to avoid this transformation and show the new conditions in terms of the original space variable x. Thus, let us modify conditions C1 R C3 as follows: l C1 -C2. A function W(u, x) exists, that satisfy (11) and such that a solution to the following problem exists du dx 4f(x, U, W(U, 1), ), U ()4l with initial condition defined by a solution l to the following algebraic equation U ()4a(W(U (), 1), ) and such that the derivative g v in (14) is strictly positive: g v (x, U, W(U, 1), ) Fc, with cr 1. C3. With the same value of c and with d between V (1) and v(1) 4b(U (1), ) we have also g v (1, U (1), d, ) Fc. Under these conditions the composite solution of the problem (4-5) has to be

11 [11] SINGULAR PERTURBATION METHODS FOR THE SOLUTION found in the following form (41) u(x, e) 4U(x, e)1x(s, e) v(x, e) 4V(x, e)1z(s, e) where x(s, e), z(s, e) are boundary layer corrections depending on the stretched variable s4 (12x)/e [, 1Q), endowed with asymptotic expansions for ek and such that vanish outside the terminal boundary layer. By proceeding in a similar way as above, with obvious modifications, problem (4-5) has in this case the following solution, approximated to e-order terms: where: u 1 (x, e) 4U (x)1eku 1 (x)1x 1g 12x v 1 (x, e) 4W(U (x), x)1z g 12x e e hl h1ekv 1 (x)1z 1g 12x e hl (42) x U (x) 4l1f(s, U (s), W(U (s), s), ) ds x x s U 1 (x)4expya(s, U (s)) dsz{ expy2 a(t, U (t)) dtzqb(s, U (s)) ds1m} with m4u 1 () root of and moreover: m4a e 1k a v g v g2 g u g v f 1W x 2g u m2g ehlx4 (43) Q x 1 (s) 4[ f(1, U (1), V (1)1z (s), )2f(1, U (1), V (1), ) ] ds s (44) s 2 z (s) 4b(U (1), )2W(U (1), 1) [ g(1, U (1), W(U (1), 1)1z (s), )2g(1, U (1), W(U (1), 1), ) ] ds

12 144 R. MONACO, R. RIGANTI and M. G. ZAVATTARO [12] (45) with: s Q{ s z 1 (s) 4expy2g v (1, U (1), V (1)1z (s), ) dsz s B (s) expy g v (1, U (1), V (1)1z (t), ) dtz ds1z 1 ()} z 1 ()4b u (U (1), )[U 1 (1)1x 1 () ]1b e (U (1), )2V 1 (1), B (s) 4s[ g x (1, U (1), V (1)1z (s), )2g x (1, U (1), V (1), ) ] 1[sU 8 (1)1U 1 (1) ][g u (1, U (1), V (1)1z (s), )2g u (1, U (1), V (1), ) ] 1[sV 8 (1)1V 1 (1) ][g v (1, U (1), V (1)1z (s), )2g v (1, U (1), V (1), ) ] 1x 1 (s) g u (1, U (1), V (1)1z (s), )1g e (1, U (1), V (1)1z (s), ) 2g e (1, U (1), V (1), ). 3 - Problems with boundary layers at both endpoints In this Section we consider the case: n42, q 1 4q 2 41 where Eq. (1) is written in the form (46) e dy 4g(x, y, e), xi4 [, 1] dx where the small parameter e multiplies both the components of the derivative of the unknown vector y4 (y 1, y 2 ) T. To satisfy the conditions (47) y 1 () 4a(y 2 (), e) y 2 (1) 4b(y 1 (1), e), both the components y 1 and y 2 may undergo strong changes in an initial or terminal boundary layer. A suitable approximation to the solution of problem (46-47), uniformly valid in I for ek, must be searched in a composite form consisting of an outer approximation (Y 1, Y 2 ) valid outside the boundary layer regions for ExE1, plus initial or terminal boundary layer correction terms (j, h) or (x, z) which are to be defined on the basis of properties of the nonlinear function g

13 [13] SINGULAR PERTURBATION METHODS FOR THE SOLUTION (g 1, g 2 ). Namely, by assuming that g i, Y i have asymptotic expansions: Q i41, 2 : Y i (x, e)a! k4 Y ik (x) e k Q, g i (x, Y, e) A! g ik (x, Y) e k, Y4(Y 1, Y 2 ) k4 and by assuming the existence of the outer solution (Y 1 (x), Y 2 (x) ) of the reduced problem that is defined by Eq. (46) with e4, we will show that the boundary layer functions have to be choosen on the basis of the following P r o p o s i t i o n 1. If the jacobian matrix g y is non-singular in I, and two positive constants c 1, c 2 exist such that for GxG1 it results (48) with g 1 y 1 (x, Y 1, Y 2, ) Fc 1 ; g 2 y 2 (x, Y 1, Y 2, ) G2c 2 g 1 y 1 (1, d 1, Y 2 (1), )Fc 1 for all values of d 1 between Y 1 (1) and b 21 (Y 2 (1), ) and g 2 y 2 (, Y 1 (), d 2, ) G2c 2 for all values of d 2 between Y 2 () and a 21 (Y 1 (), ), then the solution of problem (46-47) has the form Q (49) y 1 (x, e)4! ky 1k (x)1x k4 kg hl 12x e k Q, y 2 (x, e)4! ky 2k (x)1h e k4 kg x e hle k where y 1 (x, e) and y 2 (x, e) are, for ek, non uniformly convergent respectively in a terminal and initial boundary layer. This Proposition may be replaced by the following equivalent one. If the jacobian matrix g y is non-singular in I and two po- exist such that for GxG1 it results P r o p o s i t i o n 2. sitive costants c 1, c 2 (5) g 1 y 1 (x, Y 1, Y 2, ) G2c 1 ; g 2 y 2 (x, Y 1, Y 2, ) Fc 2

14 146 R. MONACO, R. RIGANTI and M. G. ZAVATTARO [14] with for all values of d 1 g 1 y 1 (, d 1, Y 2 (), )G2c 1 between Y 1 () and a(y 2 (), ), and in addition g 2 y 2 (1, Y 1 (1), d 2, ) Fc 2 for all values of d 2 between Y 2 (1) and b(y 1 (1), ), then y 1 (x, e) and y 2 (x, e) have respectively an initial and terminal boundary layer, and the solution must be sought in the form (51) y 1 (x, e)4! k4 Q ky 1k (x)1j kg hl x e e k Q, y 2 (x, e)4! k4 ky 2k (x)1z kg 12x e hl e k. Equations (48) and (5) represent the hyperbolicity conditions of the jacobian of g(x, Y, e), if calculated in terms of the outer solution of the reduced problem, and are analogous to the ones introduced in [3] to study singularly perturbed linear and quasi-linear systems. P r o o f s. equations (52) The terms of the outer solution must satisfy, for G x G 1, the 4g(x, Y, ) dy (53) dx 4g y Y 1 1g e where g y is the jacobian matrix of g calculated for e4 and g e is the vector of partial derivatives with respect to e, calculated for e4. The zero and e-order boundary layer corrections must satisfy: (54) (55) (56) (57) dx ds 42g 1(1, Y 1 (1)1x, Y 2 (1), )1g 1 (1, Y 1 (1), Y 2 (1), ) dh dt 4g 2(, Y 1 (), Y 2 ()1h, )2g 2 (, Y 1 (), Y 2 (), ) dx 1 ds 42 g 1 y 1 (1, Y 1 (1)1x, Y 2 (1), )Qx 1 1B (s) dh 1 dt 4 g 2 y 2 (, Y 1 (), Y 2 ()1h, )Qh 1 1A (t)

15 [15] SINGULAR PERTURBATION METHODS FOR THE SOLUTION where A (t) and B (s) are the following known functions of the zero-order solution: A (t)4tm g 2 x (,Y 1(),Y 2 ()1h (t), )2 g 2 x (,Y 1(),Y 2 (),)n 1kt dy 1 11()lm dx ()1Y g 2 (,Y 1 (),Y 2 ()1h (t),)2 g 2 (,Y 1 (),Y 2 y 1 y 1 (),)n (58) 1kt dy 2 dx ()1Y 21()lm g 2 y 2 (,Y 1 (),Y 2 ()1h (t),)2 g 2 y 2 (,Y 1 (),Y 2 (),)n 1 g 2 e (,Y 1(),Y 2 ()1h (t),)2 g 2 e (,Y 1(),Y 2 (),) (59) B (s)4sm g 1 x (1,Y 1(1)1x (s),y 2 (1),)2 g 1 x (1,Y 1(1),Y 2 (1),)(n 1ks dy 1 11(1)lm dx (1)1Y g 1 (1,Y 1 (1)1x (s),y 2 (1),)2 g 1 (1,Y 1 (1),Y 2 y 1 y 1 (1),)n 1ks dy 2 21(1)lm dx (1)1Y g 1 (1,Y 1 (1)1x (s),y 2 (1),)2 g 1 (1,Y 1 (1),Y 2 y 2 y 2 (1),)n 1 g 1 e (1,Y 1(1)1x (s),y 2 (1)),)2 g 1 e (1,Y 1(1),Y 2 (1),). Taking into account that terms x k, h k vanish outside the corresponding boundary layers, the boundary conditions (47) give for e K : (6) Y 1 ()4a(Y 2 ()1h (), ) (61) Y 2 (1)4b(Y 1 (1)1x (), ) (62) Y 11 ()4 a y 2 (Y 2 ()1h (), )Q [Y 21 ()1h 1 () ]1a e (63) Y 21 (1)4 b y 1 (Y 1 (1)1x (), )Q [Y 11 (1)1x 1 () ]1b e. According to the hypotheses, since the jacobian is non-singular two differentiable functions W 1, W 2 exist that satisfy Eq. (53) and represent the outer solution of the

16 148 R. MONACO, R. RIGANTI and M. G. ZAVATTARO [16] problem obtained by (46-47) for e K : (64) Y 1 (x) 4W 1 (x) Y 2 (x) 4W 2 (x). Eqs. (64) do not necessarily satisfy the boundary conditions. Therefore we must correct the solution of the reduced problem with boundary layer terms described by the differential equations (54-55). These equations, subject to initial conditions x (), h () deduced from (6) and (61), yield: (65) (66) s 2 t 1 x (s) 4b 21 (W 2 (1), )2W 1 (1) ] g 1 (1, W 1 (1)1x (s), W 2 (1), )2g 1 (1, W 1 (1), W 2 (1), ) )( ds h (t) 4a 21 (W 1 (), )2W 2 () ] g 2 (, W 1 (), W 2 ()1h (s), ), )2g 2 (, W 1 (), W 2 (), )( ds which decay to zero at infinity owing to the hypotheses stated in Proposition 1, and determine the boundary layer correction terms to account the non-uniform convergence of Y 1 and Y 2 near the end points. As regard to the e-order terms, we observe that equation (53) supplies the outer solution: Y 1 f.` Y 11 (67) ˆ` 4g y 21g eh dw dx 2g Y 21 where dw/dx is the vector of the derivatives of the known functions W 1 (x), W 2 (x). Inserting the terminal values of Y 11, Y 21 in Eqs. (62) and (63), the above outer solution can be solved with respect to x 1 () and h 1 (): (68) x 1 ()4 h 1 ()4 Y 21 (1)2b e b y 1 (Y 1 (1)1x (), ) Y 11 ()2a e a y 2 (Y 2 ()1h (), ) 2Y 11 (1) 2Y 21 () in order to determine (if the partial derivatives appearing in the right sides do not

17 [17] SINGULAR PERTURBATION METHODS FOR THE SOLUTION vanish) the initial data for the linear equations (56-57) whose solutions j 1 (s), h 1 (t), according to the hypotheses, are exponentially vanishing for s, tkq. In this way the e-order approximate solution of problem (46-47) is determined; its component y 1 (x, e) has a terminal boundary layer, while y 2 (x, e) has an initial boundary layer. If on the contrary the function g(x, Y, e) satisfies the hypotheses of Proposition 2 with regard to the signs of the partial derivatives g i / y j defined in (5), by an analogous treatment we can construct a solution in the form (51) where the regions of non-uniform convergence are inverted, i.e. y 1 is characterized by an initial boundary layer, while y 2 has a terminal boundary layer. To complete the composite solution (51), we can obtain the boundary layer corrections terms by solving the following problems: (69) dj dt 4g 1(, Y 1 ()1j, Y 2 (), )2g 1 (, Y 1 (), Y 2 (), ) (7) dz ds 42g 2(1, Y 1 (1), Y 2 (1)1z, )1g 2 (1, Y 1 (1), Y 2 (1), ) (71) dj 1 dt 4 g 1 y 1 (, Y 1 ()1j, Y 2 (), )Qj 1 1A (t) (72) dz 1 ds 42 g 2 y 2 (1, Y 1 (1), Y 2 (1)1z, )Qz 1 1B (s) where A (t) and B (s) are derived from (58-59) by inverting the two components of g, and the initial conditions are respectively (73) (74) (75) (76) j ()4a(W 2 (), )2W 1 () z ()4b(W 1 (1), )2W 2 (1) j 1 ()4 a y 2 (W 2 (), )QY 21 ()1a e 2Y 11 () z 1 ()4 b y 1 (W 1 (1), )QY 11 (1)1b e 2Y 21 (1). r Higher-order terms in the obtained composite solutions can be calculated, as usual, by using all the terms of lower order appearing in the asymptotic expan-

18 15 R. MONACO, R. RIGANTI and M. G. ZAVATTARO [18] sions. Obviously, for the convergence of the method, the perturbation terms must satisfy the properties already recalled in Section Applications Singularly perturbed problems are well known in the context of kinetic theory and fluid dynamic equations. In particular boundary layer solutions for models of the Boltzmann equation can be found in [7]. In this Section the techniques here proposed are applied to determine approximate analytical solutions of two boundary value problems arising from stationary Boltzmann like equations. Contrary to ref. [7], these two problems are entirely treated at the microscopic scale. In particular, the theory developed in Section 3 is used to study a modified version of the well-known Ruijgrok-Wu model, and the techniques of Section 2 are applied, as a second example, to study a model for a mixture of reacting gases with two chemical species of disparate molecular masses A model of the Ruijgrok-Wu type Two-velocity models of the Boltzmann equation of the Ruijgrok-Wu type [9] have been studied by assuming a diffusive scaling of the kinetic equations describing the time and space evolution of a finite mass gas in a medium [1], [8]. In particular the model equations, at a molecular scale, will take into account: l absorbtion and scattering effects of the gas particles by the host medium; l particle source terms, distributed on the interval [, 1] of the x-axis; l elastic collisions between the gas particles. Then in the stationary case the kinetic equations may be written as (77). / e dy 1 dx 2e dy 2 dx 4g 1 (x, y 1, y 2, e) 42ay 1 1by 2 1egF(y 1 ) G(y 2 )2d 1 y 1 1s 1 (x) 42g 2 (x, y 1, y 2, e) 4ay 1 2by 2 2egF(y 1 ) G(y 2 )2d 2 y 2 1s 2 (x) where y 1 (x), y 2 (x) R 1 are the number densities of the molecules moving at equal speeds 6c on the x-axis, respectively; e41/c is the small parameter; F and G are bounded functions taking into account particle scattering; s 1 (x) and s 2 (x) are C 1 [, 1] positive functions representing sources; and a, b, g, d 1, 2 are nonnegative real constants. In particular, a and b represent cross sections of the par-

19 [19] SINGULAR PERTURBATION METHODS FOR THE SOLUTION ticle scattering with the host medium; g is the cross section of elastic particle collisions, and d 1, 2 are the total absorption cross sections. Let us seek for a solution of Eq. (77) satisfying the following boundary conditions (78) (79) y 1 ()4k y 2 () y 2 (1)4k 1 y 1 (1) describing either absorbtion of the gas molecules by the walls, if the non-negative constants k, k 1 are lower than unity, or pure specular reflection if k 4k Since the partial derivatives of g 1, g 2 satisfy hypothesis (5), the composite solutions to the problem (77)-(78, 79), truncated at the terms of order e, will have the form y 1 (x, e) 4Y 1 (x)1j (t)1e[y 11 (x)1j 1 (t) ]1O(e 2 ) y 2 (x, e) 4Y 2 (x)1z (s)1e[y 21 (x)1z 1 (s) ]1O(e 2 ) where j(t, e), z(s, e) are, respectively, initial and terminal boundary layer corrections to the outer solution Y(x, e) 4 (Y 1, Y 2 ) for the two densities. The latter one has zero-order terms that can be found as the unique solution of g(x, Y 1, Y 2, ) 4. By setting C4ad 2 1bd 1 1d 1 d 2 D, it is given by (8) Y 1 (x) 4W 1 (x) 4 1 C [ (b1d 2) s 1 (x)1bs 2 (x) ] (81) Y 2 (x) 4W 2 (x) 4 1 C [as 1(x)1 (a1d 1 ) s 2 (x) ] and is non-negative for all x [, 1], under the assumptions made for the above coefficients. Therefore, using the boundary conditions (73), (74) and integrating Eqs. (69), (7), the correction terms in the solution of the reduced problem can be found as (82) j (t) 4 [k W 2 ()2W 1 () ] e 2(a1d 1) t (83) z (s) 4 [k 1 W 1 (1)2W 2 (1) ] e 2(b1d 2) s.

20 152 R. MONACO, R. RIGANTI and M. G. ZAVATTARO [2] The above zero-order solution is now used to determine the further terms in the composite expansion of the solution. The outer terms of order e are given by Eq. (67), that for the present problem yields (84) Y 11 (x) 4 1 C [ (b1d 2) W8 1 (x)2bw8 2 (x)2gd 2 F(W 1 (x) ) G(W 2 (x) ) ] (85) Y 21 (x) 4 1 C [aw8 1(x)2 (a1d 1 )W8 2 (x)1gd 1 F(W 1 (x) ) G(W 2 (x) ) ] where the primes denote derivatives with respect to x of the previous solution terms. The values at the boundaries of these outer terms satisfy: Y 11 ()1j 1 ()4k Y 21 () Y 21 (1)1z 1 ()4k 1 Y 11 (1) and determine the initial conditions that must be assigned to the linear differential equations (71) and (72) for the unknown correction terms j 1 (t), z 1 (s). In our problem these equations are rewritten as (86) (87) dj 1 dt 42(a1d 1) j 1 (t)1a (t) dz 1 ds 42(b1d 2) z 1 (s)1b (s), where the non-homogeneous terms are known functions of the zero-order solution: A (t) 4g[F(W 1 ()1j (t) )2F(W 1 () ) ] G(W 2 () ) B (s) 4gF(W 1 (1) )[G(W 2 (1)1z (s) )2G(W 2 (1) ) ]. Therefore, integrating Eqs. (86) and (87) one obtains t j 1 (t) 4e 2(a1d 1) ty s z 1 (s) 4e 2(b1d 2) sy A (t) e (a1d 1) t dt1k Y 21 ()2Y 11 ()z B (t) e (b1d 2) t dt1k 1 Y 11 (1)2Y 21 (1)z which, under wide conditions on the nonlinear functions F(y 1 ) and G(y 2 ), are asymptotically vanishing as t, sk1q, as prescribed by the matching method to all terms of the boundary layer corrections.

21 [21] SINGULAR PERTURBATION METHODS FOR THE SOLUTION Mixture of reacting gases A two-velocity model for a finite mass mixture of gas particles has been proposed under suitable assumptions in ref. [11]. The gas mixture is formed by two chemical species of disparate molecular masses m A, m B with m A /m B 4e b 1, undergoing an irreversible chemical reaction (recombination) of type A1BKC. If in addition source terms distributed on the interval x [, 1] are present, then the kinetic equations derived in [11] are modified, in the stationary case, into the following two subsystems: (88) (89). /. / du 1 dx 2e dv 2 dx 2 du 2 dx e dv 1 dx 4f 1 (u 1, v 2 ) 4 1 c [2ku 1 v 2 1s 1 (x)] 42g 1 (u 1, v 2 ) 4 1 c [2ku 1 v 2 1s 4 (x)] 42f 2 (u 2, v 1 ) 4 1 c [2ku 2 v 1 1s 2 (x)] 4g 2 (u 2, v 1 ) 4 1 c [2ku 2 v 1 1s 3 (x)] where u 1 (x), u 2 (x): C [, 1]KR 1 are the densities of the particles of species A moving, respectively, along the positive and the negative directions of the x- axis; v 1 (x), v 2 (x): C [, 1]KR 1 are the densities of species B, defined analogously; s i (x) are, as in the previous application, C 1 [, 1] positive functions; k is the collisional frequency to be considered as a known positive constant, and c is a reference velocity. Let us assume that the effects of the molecular interactions at the walls x4 and x 4 1 can be described as (9) (91) u 1 ()4a v 2 (), v 2 (1)4b u 1 (1) u 2 ()4a 1 v 1 (), v 1 (1)4b 1 u 2 (1) with a, R, b 1 G1 are prescribed positive constants. If their value is chosen lower than unity, it denotes partial absorption by the walls, whereas a 4R4b 1 41 mean specular reflection of the two chemical species. Then Eq. (9) yields the boundary conditions for the subsystem (88), and Eq. (91) prescribes the boundary conditions for the subsystem (89).

22 154 R. MONACO, R. RIGANTI and M. G. ZAVATTARO [22] Since in the kinetic equations we have g 1 v 2 4 k c u 1 D, g 2 v 1 42 k c u 2 E, it follows that as ek the fast variables v 2 (x) and v 1 (x) have non uniform convergence, respectively, in a terminal and in an initial boundary layer. Therefore the N-order approximate solution to problem (88-9) must be searched in the form u 1 N (x, e)c! [Uk 1 (x)1x k (s)] e k, v 2 N (x, e)c! [Vk 2 (x)1z k (s)] e k, s4 12x k4 k4 e and the one related to problem (89-91) in the form u 2 N (x, e)c! [Uk 2 (x)1j k (t)] e k, v 1 N (x, e)c! [Vk 1 (x)1h k (t)] e k, t4 x e. k4 The solution to the reduced problem for (88-9) is determined as follows. Owing to (9) the zero-order terms must satisfy the boundary conditions k4 (92) U 1 ()4a V 2 (), V 2 (1)1z ()4b U 1 (1). The equation g 1 (U 1, V 2 ) 4 is satisfied by (93) where U 1 V 2 4W(U 1 (x), x) 4 s 4(x) ku 1 (x) () is determined by Eq. (92) as U 1 ()4o a s 4 () k. This initial condition is used to calculate U 1 (x) from Eq. (42), with the result (94) U 1 (x) 4o a s 4 () k 1 1 x [s 1 (t)2s 4 (t)] dt. c Since it is required to be strictly positive, the assumed source terms must satisfy for each x [, 1] the additional condition (95) x x s 1 (t)dtd s 4 (t) dt2co a s 4 () k The zero-order corrections in the terminal boundary layer are x (s) f and.

23 [23] SINGULAR PERTURBATION METHODS FOR THE SOLUTION z (s) whose initial condition, from Eq. (92), is z ()4b U 1 (1)2W(U 1 (1), 1). Therefore, integration of Eq. (44) yields (96) z (s) 4 [b U 1 (1)2W(U 1 (1), 1)] expg2 ku 1 (1) c sh. The reduced problem for (89-91) can be solved by satisfying the boundary conditions (97) U 2 ()4a 1 [V 1 ()1h () ], V 1 (1)4b 1 U 2 (1). By solving g 2 (U 2, V 1 ) 4 one has (98) V 1 (x) 4W(U 2 (x), x) 4 s 3(x) ku 2 (x) where U 2 (x) is determined by using Eq. (33) and the second of Eqs. (97): (99) U 2 (x) 4o s 3(1) kb 1 Again positivity of U 2 (x) requires that [s 3 (t)2s 2 (t)] dt. c x (1) 1 x 1 s 2 (t) dtd x s 3 (t) dt2co s 3(1) kb 1. The zero-order corrections in the initial boundary layer are j(t) f and h(t) that must satisfy the initial condition prescribed in the first of Eq. (97). By also applying Eq. (34) one obtains h (t) 4y U 2 () 2W(U 2 (), a 1 )z expg2 ku 2 () c It follows that the zero-order solutions for the fast variables, which satisfy the prescribed boundary conditions and are uniformly valid in the interval GxG1 under the action of source terms satisfying Eqs. (95), (1), are v 2 (x, e) 4 v 1 (x, e) 4 s 4(x) ku 1 (x) 1yb U 1 (1)2 s 4(1) ku 1 (1) s 3(x) 1y U (x) ku 2 2 () 2 s 3() a 1 ku 2 () th. 1 ku (1) z expy2 (12x)z ec 2 ku () x z expg2 ec h

24 156 R. MONACO, R. RIGANTI and M. G. ZAVATTARO [24] where U 1 (x), U 2 (x) are given by Eqs. (94) and (99). Once the (unique) solutions to the reduced problems have been determined, the e2order terms and the further ones in the composite expansions can now be calculated by a straightforward application of the procedure given in Section Conclusions This paper supplies a methodology which can be applied to determine an analytical, approximate solution to singularly perturbed boundary-value problems arising from the study of a wide class of nonlinear systems of ODE. In particular, in the present paper applications to Boltzmann s like models have been developed. Quantitative results can be obtained by solving numerically, if necessary, a pertinent set of nonlinear ordinary differential equations describing the boundary layer functions, i.e. Eqs. (34-44), (65-66) or (69-7) respectively. The above analysis also completes the treatment of singular perturbation techniques, that began with a previous paper [12] by considering initial value problems for a similar class of systems of ODE, with application to extended kinetic theory. Acknowledgments. This piece of research has been partially supported by the CNR «Progetto Coordinato» No and by PRIN 2, «Problemi matematici non lineari di propagazione e stabilità nei modelli del continuo» (Prof. T. Ruggeri). References [1] A. B. VASIL EVA, Asymptotic behaviour of solutions to certain problems involving nonlinear differential equations with a small parameter multiplying the highest derivatives, Russian Math. Surveys 18 (1963), [2] W. ECKHAUS, Matched asymptotic expansions and singular perturbations, North-Holland, Amsterdam [3] R. E. O MALLEY Jr., Introduction to singular perturbations, Academic Press, New York [4] A. B. VASIL EVA, V. F. BUTUZOV and L. V. KALACHEV, The boundary function method for singular perturbation problems, SIAM Studies in Appl. Math. 14, Philadelphia [5] R. E. O MALLEY Jr., Singular perturbation methods for ordinary differential equations, Springer Verlag, New York 1991.

25 [25] SINGULAR PERTURBATION METHODS FOR THE SOLUTION [6] D. R. SMITH, Singular-perturbation theory. An introduction with applications, Cambridge Univ. Press, Cambridge [7] R. CAFLISCH, Asymptotics of the Boltzmann equation and fluid dynamics. Kinetic theory and gas dynamics, , CISM Courses and Lectures 293, Springer, Wien [8] P. L. LIONS and G. TOSCANI, Diffusive limits for finite velocities Boltzmann kinetic models, Rev. Mat. Iberoamericana 13 (1997), [9] T. W. RUIJGROK and T. T. WU, A completely solvable model of the nonlinear Boltzmann equation, Phys. A 113 (1982), [1] E. GABETTA and G. PERTHAME, Scaling limits for the Ruijgrok-Wu model of the Boltzmann equation, Pubbl. IAN-CRN Pavia 111 (1998), [11] A. ROSSANI and R. MONACO, Soliton-type solutions of a discrete model of the Boltzmann equation for a mixture of reacting gases, Ann. Univ. Ferrara, sec. VII 37 (1991), [12] R. RIGANTI and A. ROSSANI, Singular perturbation techniques in the study of a dynamical system arising from the kinetic theory of atoms and photons, Appl. Math. Computation 96 (1998), A b s t r a c t This paper deals with some singular perturbation problems, described by two classes of nonlinear differential equations with nonlinear boundary conditions. Uniformly valid expansions composed of inner and outer solutions are used to solve initial and/or terminal boundary layer problems. In two examples, approximate solutions are explicitly derived for boundary value problems arising from stationary Boltzmann like model equations. * * *