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A comparative study on a singular perturbation problem with two singular boundary points
APPLIED MAT~E~ATHC~ AND
CC~ ?UTAT[ON FA.SEVIER
Applied Mathematics and Computation 99 (1999) 179-193
A comparative study on a singular perturbation problem with two singular boundary points Abdul-Majid W a z w a z 1 Department of Mathematics and Computer Science, Saint Xavier University, 3700 West 103rd Street, Chicago, IL 60655, USA
Abstract
A singularly perturbed second-order differential equation with two singular boundary points will be investigated. The work will present a comparative study between the WKB-boundary layer analysis and the uniform approximations about the singular points. The exponential precision asymptotics are essential for this study. The matching concept will be applied in each type of analysis where dominant and recessive terms are matched independently. © 1999 Elsevier Science Inc. All rights reserved.
Keywords: Singular points," Exponential precision; WKB method," Uniform method," Matched asymptotics
I. Introduction
I n this p a p e r we are c o n c e r n e d with a c o m p a r a t i v e s t u d y o f the singularly perturbed problem
x(b - x ) { e f ( x ) u ~ z +g(x)u,] = su,
0 < x < b,
(1.1)
where u = u(x, E). T h e functions f ( x ) a n d g(x) a r e a s s u m e d real, positive, a n d several times c o n t i n u o u s l y differentiable in x. T h e p a r a m e t e r s is an eigenvalue p a r a m e t e r . T h e differential e q u a t i o n (1.1) is singular in t h a t the diffusion coefficient ex(b - x ) f ( x ) vanishes at the b o u n d a r y p o i n t s x = 0 a n d x = b.
I E-mail: [email protected].
0096-3003199/$ see front matter © 1999 Elsevier Science Inc. All rights reserved. PII: S0096-3003 (98)00006-X
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Equations of the form (1.1) usually arise in the first passage problems or extinction problems in population biology. Models like (1.1) have been the focus of an extensive study recently, and two distinct methods have been developed to approach similar models. In the first, a leading-order asymptotic solution of Eq. (1.1) is obtained by using the WKB method in the interior region between the boundary points and by using a boundary layer analysis near the singular boundary points x = 0 and x = b. These approximations are then related to each other by asymptotic matching. In the second approach, the uniform reduction method of [7] will be used. A leading-order asymptotic approximation about each singular boundary point will be constructed, such that each approximation is uniformly valid in one subdomain containing one singular point, and each approximation contains dominant and recessive terms as well. The uniform method rests upon using the comparison equation method and using the amplitude functions as well to justify the matching between the uniform asymptotic approximations. The two distinct approximations uniformly valid about the singular points were developed from two linear combinations of Whittaker's functions [1]. The dominant and recessive parts of these approximations will be matched independently. It is interesting to note that models like (1.1) are usually studied for a singular diffusion process such as extinction problems in population biology. Moreover, models like (1.1) with demographic stochasticity typically arise from discrete stochastic processes such as birth and death processes or branching processes, where u(x, ~) can be viewed as the moment generating function. Extensive background for this paper and for identical diffusion process models is found in the results of [~7]. In Eq. (1.1), the diffusion coefficient ~ ( b - x ) f ( x ) , with f ( x ) > 0, vanishes at regular singular points at the boundaries x = 0 and x = b. Since the small positive parameter e multiplies u..... we have a boundary layer type singular perturbation problem. This type of diffusion coefficient is typical of demographic type stochastic effects [3,4]. Moreover, the drift coefficient x(b - x)g(x), with g(x) > 0, does not give rise to any turning point in the domain of validity. Several identical perturbation models were extensively discussed in [2-7], with distinct drift coefficients that give rise to second-order interior turning points. It is the intention of this paper to conduct this comparative study between the classic WKB method and the uniform reduction method of [4] on a model that contains singular boundary points only. It is easily seen that simple or secondorder turning points do not appear in the domain of validity. In [2-8], several singular perturbation models were investigated where the diffusion coefficient in some of these models gives rise to an interior turning point and a singular point at the boundary. In several studies such as [2-7] in general, and in particular in [8], it was shown that exponential precision asymptotics, in the sense of Meyer [9], play a crucial role in the analysis of the matching process, due to the exponential
A.-M. Wazwaz / Appl. Math, Comput. 99 (1999) 17~193
181
behavior of the solution. Meyer [9] gave many illustrations of applications such as wave trapping by round islands, plasma physics, critical scattering in quantum mechanics, and geophysical fluid dynamics, where exponential precision is essential. Exponential precision asymptotics play an important role in optical tunneling. Meyer [9] emphasized that exponentially small terms be included in the analysis to obtain correct results. In [2], Hanson established the foundations of exponential asymptotic expansions by multiple scale techniques. The multiple scale approach of [2] justifies the complete asymptotic expansion and also removes the degeneracy of the usual, single Poincar6 algebraic asymptotic expansions that neglect the exponentially small terms. The Poincar6 algebraic asymptotic expansions hold only for the exponentially larger terms. The complete exponential asymptotic expansion contains both dominant and recessive (exponentially smaller) expansions [8]. The neglect of exponentially smaller terms means the neglect of one of the independent fundamental solutions for a second-order ODE like (1.1). In addition, exponential precision is needed to compute resonance conditions that we presented in many papers. Olver [10] points out that exponentially small terms may not be numerically small terms. However, Olver [10] points out the need to include exponentially small terms in the uniform approximation of certain Bessel functions [4], but he emphasized that this use should preferably be supported by error bounds. The Olver [10] approach as we indicated in [4] is fruitful for rigorously establishing error bounds for coming work, but this notion is beyond the scope of this paper. The intention of this paper is an application to many other papers by [2-8] where the complete asymptotic analysis was effectively employed in the sense of Meyer [9], Hanson [2] and by Hanson and Tier [3], and by many others, with promising results. 2. The WKB method and the boundary layer analysis
In this section we construct a leading-order asymptotic approximation to Eq. (1.1) by using the standard WKB method. In this method we first assume that the solution is of the form
u(x,e) = exp
(is) -~
y(t,e) dt
,
(2.1)
where y(x, e) has an asymptotic expansion of the form
y(x, e) = yo(x, e) + eyl (x, e) + O(e2).
(2.2)
Substituting Eqs. (2.1) and (2.2) into Eq. (1.1) yields 1 s -yo(Yoe+ ~ ) + y~(2yo + Fx) +Yo + O(e) - x(b - x ) f ( x ) '
(2.3)
A.-M. Wazwaz I Appl. Math. Comput. 99 (1999) 179-193
182
where F(x) is the squared natural metric of [3] defined
by
F(x) = f~g(t) dt, Ja f(t)
(2.4)
such that
F~(x) = f -g(x) ~>
0,
0
(2.5)
Equating the coefficients of powers of e to zero, we obtain to leading-order the solutions Y0 = 0
and
Y0 = -F~(x).
(2.6)
The terms of order O(e °) give S
Y' -
x(b
- ~)g(~)
'
when y0 = 0
(2.7)
and
Y' =
s x ( b - x)g(x)
F~x(x)
~(~)
when y0 = -Fx(x).
(2.8)
We can easily conclude that the resulting values ofyt at Eqs. (2.7) and (2.8) are not integrable at the singular points at the boundaries x = 0 and x -- b. This leads to the conclusion that the standard W K B approximation (2.1) fails in small neighborhoods near the singular boundary points. Consequently, the W K B approximation (2.1) is valid everywhere in the subdomain 0 < x < b away from the singular boundary points x = 0 and x = b. For purposes of matching, that will be used later, it is convenient to define the function G(x) by s k c sG'(x) x(b - x)g(x) + x- + --'b- x (2.9) where k and c are constants given by k -
S
(2.10)
bg(O)
and
c-
S
bg(b)"
(2.11)
Combining Eqs. (2.1), (2.2) and Eqs. (2.6)-(2.9), the leading-order approximation valid in O(e) < x < b - O(e) is thus given by
Uo
+ Bexp
where A and B are constants of integration.
e
sa(x)
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183
The boundary layer analysis is now useful to determine the approximations in the boundary layer regions at the singular points x = 0 and x = b. This can be achieved by using the fast varying stretched variables X
Y - t' [(o)~
(2.13)
and b-x
(2.14)
Z = [.f(b)-----~'
respectively. As a result Eq. (1.1) is converted to the boundary layer equations Yurr + Yur + ku = 0
(2.15)
Zuzz - Zuz + cu = 0
(2.16)
and
valid at x = 0 and x = b respectively, and the constants k and c are given above by Eqs. (2.10) and (2.11). To determine the solutions of Eqs. (2.15) and (2.16), we employ the Liouville [11] normal transformations u ( Y , e) = v ( Y ) e -r/2
(2.17)
u(Z, e) = ?J(Z)e z/2
(2.18)
and
into Eqs. (2.15) and (2.16), respectively. Consequently, Eqs. (2.15) and (2.16) are converted to the normal forms -y
v=0
(2.19)
-2
v = 0,
(2.20)
and
each is a reduced form of Whittaker's equation [1] with solutions well defined by Whittaker's functions. Combining Eqs. (2.17)-(2.20), the solutions in the boundary layer regions x = 0 and x = b are given by ul = e-r/2[A1 W1(k, Y) + Bl W2(k, Y)]
(2.21)
u2 = e z/2 [A2WI(c, Z) + B2 W2(c, Z)],
(2.22)
and respectively, with AI, A2, B1, and B2 are constants of integration. The modified versions of Whittaker's functions Wt( a , R ) and W2(a,R) were chosen by [3,4] and are related to Whittaker's function Wk,0.5and Mk,0.s by
A.-M. Wazwaz I Appl. Math. Comput. 99 (1999) 179-193
184
(a, R) --
W~,0.5(R),
F(1 - k) ~ ( a , R ) - r(1 -k)Mk,0,(R) ~ r(1 + k~ cos(~k)~,0,(R).
(2.23)
It is worth noting that W~ and W2 have exponentially small and exponentially large behavior, respectively, for large R. The complete standard expansions of W1 and W2 can be found in [2~1]. In closing this section, the matching principle will be applied to the asymptotic approximations obtained above. For u(x,s) of (1.1) to be correctly represented by Ul, U0
g ( x ) ~ F(O) + F'(O)x + . . .
and therefore we obtain F(x) ~ F(O) c
--
+ Y
(2.25)
upon using Eqs. (2.5) and (2.13). We also note that x Y e - F'(0)
(2.26)
derived by using Eqs. (2.5) and (2.13). As x + 0 and by substituting Eqs. (2.25) and (2.26) into Eq. (2.12) we obtain
+ ~
(g~0)/k
g--~
•
The leading-order expansions for the related Whittaker's function of Eq. (2.23) are given by W1(a,R) ~ R ~ exp(-½R) + O(e), W2(a,R) ,-~ R-" exP(½R) + O(e),
(2.28)
<'(a,R) ,-~ -½R ~ e x p ( - ½R) + O(e), W2'(a,R ) ,.~ 1R-~ exP(½R) + O(e)
as R --+ oc, (for example, see [4], p. 954). Substituting expansions (2.28) into (2.21), the asymptotic expansion of ul becomes ul ~ AI y % - r + B1Y k.
(2.29)
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A.-M. Wazwaz / Appl. Math. Comput. 99 (1999) 179-193
The nonexponential varying terms of ul and u0
A.
(2.30)
Next, we equate the exponential varying terms of Eqs. (2.27) and (2.29) to find
F(O)e sG(O) (F (0))
A1 = exp
~)
g~B.
(2.31)
To match u0
X(x) ~ X(b) - f --g(b)'" ( b ) ~O_ x ) + "",
(2.32)
so that
F(x) ~ F(b)
- Z,
(2.33)
obtained by using Eq. (2.14). We also note that
b-x Z e -- F'(b)
(2.34)
by using Eq. (2.14). Accordingly, as x ---, b and by using Eqs. (2.33) and (2.34) we find
((F'(b))-cZ c
U0
F(b)e s G ( b ) ) ( b ) kF'~'wf(b)ezZ-c'( (o,, ~
+Bexp(
(2.35)
As Z ~ e~, using the asymptotic expansions of u2 in Eq. (2.22) yields
(2.36)
U2 "" A2 Ze -k B2eZZ -c.
Proceeding as before, we equate the nonexponential terms in Eqs. (2.35) and (2.36) to obtain
A2 = e"a(b)
(F'(b) )-CA
(2.37)
and by equating the exponential terms we find B2 = e x p ( F ( b ) e
s G ( b ) ) ( ! ) k ( r,,,,,,,cf(b)B ~o)) ~ .
(2.38)
This completes the asymptotic matching of every neighboring approximations. The two sets of constants A1, B~ and A2, B2 are determined in terms of one set of constants, namely A and B. We point out that the constants A and B can be
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186
completely determined by using specific boundary conditions. Further, it is worth noting that the exponential precision asymptotic we used is important for determining the sets of matching constants.
3. The uniform reduction method
In this section we construct leading-order approximations fil and /~2 of solutions to Eq. (1.1) uniformly valid into the subdomains/1 and/2, where I, = [0, b - O(~)), 12 = (O(e),b],
(3.1)
such that each subdomain includes only one singular boundary point. The asymptotic validity of the approximations /~1 and fi2 fails to hold in neighborhoods of the singular points x = b and x = 0, respectively. To achieve our goal, we will implement the uniform reduction theorem of [4]. As will be seen later, the method rests upon applying the comparison equation method and using the proper amplitude functions as well. We will prove later that the uniform variables y(x) and z(x), that will be used for our analysis, are defined by
y(x) =
F'(r) dr = F(x) - F(0),
z(x) =
F'(r) dr = F(b) - F(x),
(3.2)
valid only for the subintervals I1 and h, respectively, where F(x) is the natural metric of [3] given by Eq. (2.4). We point out that the equations given by Eq. (3.2) are consistent with the Langer [11] minimal properties given by y(0)=0, z(b) = O,
y'(x)>0, f ( x ) < 0.
(3.3)
We now begin our analysis by using the order-one uniform variables g(0) y(x) = f - ~ x ,
z(x)
(3.4)
g(bl ( b - x),
valid only for the subintervals lj and 12, respectively. In view of the uniform variables (3.4), a pair of comparison equations is constructed from the model Eq. (1.1), given by
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187
eyw~ + yWy + kw = 0
(3.5)
eZ~zz - z~z + c# = 0,
(3.6)
and where the constants k and c are defined above in Eqs. (2.10) and (2.11), respectively, but will be derived later in this section. It is important to note that the comparison equation (3.5) preserves the behavior of the model (1.1) in the subinterval Ii. Further, the original equation (1.1) matches the comparison equation (3.5) in the sense that both have one distinct zero at x = 0 at the subinterval I1. Similar conclusions can be used to relate the comparison equation (3.6) with the original equation (1.1) in the subinterval 12. In [4,5], it was formally proved that the solution of the original equation is asymptotically equivalent to the solution of the comparison equation, at the corresponding subinterval, upon introducing weakly varying amplitude functions. Following the uniform reduction method of [4,5], the uniform asymptotic representations ~ and ~2 take the form fq = M(y, e)w(y) + cN(y, C)Wy(y), valid for Ii
(3.7)
u2 = ~t(z, e)#(z) + e.~(z, e)~z(z),
(3.8)
and valid for 12,
where w(y) and ~(z) are the solutions of the comparison equations (3.5) and (3.6), respectively, and M, N, M, and N are the amplitude functions that will be used to justify the matching concept of the two uniform approximations. We point out that the amplitude functions areholomorphic with power series expansions as E--o0, where M and M are normalized such that M(0, e) = ~t(0, e) ---- 1. To determine the uniform approximations (3.7) and (3.8), our analysis will run in two directions. On one hand, we determine the functions w(y) and #(z) of the canonical equations (3_5) and (3.@ In the second, we determine the leading term expansions M0, M0, No, and No of the amplitude functions M, M, N, and N as will be seen later. To determine w(y) and ~(z) of the comparison equations (3.5) and (3.6), we first introduce the fast varying stretched variables Y =Y-,
(3.9)
and Z
(3.10)
in Eqs. (3.5) and (3.6) to obtain Ywrr + Ywr + kw = 0
(3.11)
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A.-M. Wazwaz / Appl. Math. Comput. 99 (1999) 179-193
and Zfvzz - Z~vz + c~ = 0,
(3.12)
respectively. The normal transformations w(Y) = e-r/2v(y)
(3.13)
~v(Z) = e-Z/Z~(Z)
(3.14)
and convert Eqs. (3.11) and (3.12) to the normal forms -~
v=0
(3.15)
-~
v=0,
(3.16)
and
respectively. We can easily observe that each of Eqs. (3.15) and (3.16) is a reduced form of Whittaker's equations [1]. Combining the solutions of Eqs. (3.15) and (3.16) with Eqs. (3.13) and (3.14) yields the solutions of the comparison equations Eqs. (3.5) and (3.6) given by w(y) = e-Y/2[CW1 (k, Y) + D ~ ( k , Y)]
(3.17)
W(Z) = e z/2 [CI WI(c, Z) + Dl W2(c, Z)],
(3.18)
and
where ~ and W2 are Whittaker's functions defined above by Eq. (2.23). Consequently, substituting Eqs. (3.17) and (3.18) into Eqs. (3.7) and (3.8) yields the uniform solutions ill(x, e) = e-r/2 {M(x, e)[CWl(k, Y) + DW2(k, Y)] + N(x, e)[C(W( - ½W1) + D(W2s - ½W2)]},
valid for 11
(3.19)
and ft2 (x, e) = e z/2 {M(x, e)[C1 W1(c, Z) + D1 W2(c, Z)]
q- iV(x, ~) [Ct (Wllt --b 1WI) + DI(W2s + lg/2)]}, valid for/2.
(3.20)
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We focus our concern now on determining the leading term expansions of the amplitude functions, therefore we follow [10]. We substitute Eqs. (3.7) and (3.8) in Eq. (1.1), using Eqs. (3.5) and (3.6) to eliminate W~yand ~zz in favor of wy and w, and vi,~and rb, respectively. We collect the coefficients of E"Wy, e"w, e"~z, and e-'~, and then equate these coefficients separately to zero on their domain of validity. As a result, the terms of order O(e°Wy) and O(e°~) give F~ = { Yx -Zx
for 11, for 12.
(3.21)
This implies that y(x) =
fo xF~(r) dr = F(x)
z(x) =
fx bFr(r) dr
- F(O)
(3.22)
= F(b) - F(x).
(3.23)
and
The results (3.22) and (3.23) are consistent with our assumptions of the uniform variables given by Eq. (3.2) and with the Langer [11] minimal properties of Eq. (3.3). Terms of order (e°w) and (e°~,) give M o - h(y)Mo = 0
(3.24)
~-I; + h(z)Mo = 0,
(3.25)
and
so that Mo = exp
h(r) dr
(3.26)
and / ~ 0 = exp
(/0 z ) -
tt(r) dr
x(b
- x)g(x)yx(x)
x(b
- x)g(X)Zx(X)
(3.27)
,
where k
h(y) = ; +
(3.28)
and
ft(z)
c = z
s
(3.29) '
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190
It is important to note that the regularity conditions of h(y) as x ~ 0 and h(z) as x ~ b requires the selection of constants S
k-
bg(O)
c-
bg(b)'
(3.30)
and S
(3.31)
which is consistent with our assumption of k and c of (2.10) and (2.11). Terms of order (eWy) and (e#~) give No and N0 through the identities
No = Mo
yx(O) Moyx(x)
(3.32)
No = - f l o + zx(b)
(3.33)
and Mo~x(X) '
where the constants of integration have been chosen so that No and No are regular as x ~ 0 and x ~ b, respectively. As a result, the uniform approximations to leading-order (3.19) and (3.20) are rewritten by
~, (x, ,) = e-Y/2{M0[C~ (k, r) + DW~(k, r)] +No[C(W~'-½W~)+D(Wz'-½W2)]},
valid for I1
(3.34)
and
fi2 (x, e) = e z/2 {~¢0[C, WI(c, Z) + D1 W2(c, Z)] 1
+ N0 [G (W~'+ ½W~)+ D~ (W2' + ~ W2)]}, valid for/2
(3.35)
For purposes of matching, we need to derive explicit formulae that relate the weakly varying amplitude functions. To achieve this goal, it is useful to rewrite Eq. (2.9) in terms of the order-one uniform variables y(x) and z(x). Using Eq. (3.4) into Eq. (2.9) yields s yx zx sG'(x) - x(b - x)g(x) ~ k T ( ~ - C z - ~
(3.36)
In view of Eqs. (3.28), (3.29) and (3.36) we obtain z~ h(y)y~ = sG'(x) + Cz(x)
(3.37)
and
f,(z)zx = - s r ' ( x ) + k Y~ y(x)
"
(3.38)
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191
Inserting Eqs. (3.37) and (3.38) into Eqs. (3.26) and (3.27) yields 114o = exp(s[G(x) - G(0)]) (Z--~)Z(x)~
(3.39)
and \ x~x) / respectively. Furthermore, the uniform variable y(x) and z(x) are related through the identity z(x) + y(x) = F(b) - F(O)
(3.41)
and hence, the fast varying stretched variables are related through identity Z(x) + Y(x) -- F(b) - F(O) (3.42) e To accomplish our asymptotic analysis, it is essential to match the leadingorder uniform approximations (3.34) and (3.35). For u(x, e) to be correctly represented by fia and fi2, it is essential that the two approximations share validity on the overlapping subinterval (e, b - e). As Y ---* c~, the approximation (3.34) has the form fil ~ C (
yx(O) "~y%-r + DMoY-*
(3.43)
\yx(x)Mo]
derived from Eqs. (2.28) and (3.39). In a parallel manner, we use Eqs. (2.28) and (3.40) to find u2 ~ Cl~loZ c + 01 ( z x ( b ) )Z_Ce z \ zx(x)Mo ]
(3.44)
as Z --~ oc. Matching the exponential and nonexponential terms of Eqs. (3.43) and (3.44), and using Eq. (3.42) yield c = _ :~z~~ <0~:~ ~x, ( . i ~
- ~(0~1~~ 0 ,
~ < . ~ : ex~ ( ~ ( ~ - ~ ~(0~
)~,
(3.45) and D = exp (s[G(0) - G(b)])(Z(O))C(Y(b))kCl.
(3.46)
It is clear that one set of the constants of integration has been eliminated in favor of the other set. Consequently the remaining constants are completely determined by using proper boundary conditions. This completes the determination of the uniform approximations to leading order.
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A.-M. Wazwaz I Appl. Math. Comput. 99 (1999) 179 193
4. Conclusions and discussions An analytic comparison of two distinct methods, usually used in singular perturbation problem has been discussed. In a manner parallel to [3], we applied the standard WKB technique for the interior region between the singular boundary points. Two layer type approximations around the singular points have been developed by using proper stretched variables. The exponential precision asymptotics have been used in these approximations to accomplish the matching process. Following [4], the uniform reduction theorem was employed to obtain a two-part uniform approximation. Two uniform variables were used each valid in a subinterval that includes one singular point and excludes the other singular point. The fast varying stretched variables were implemented and amplitude functions were used in the uniform approximation to justify the matching. The two-part uniform approximation implemented the exponentially small terms as in the first method. From the discussion above, we can easily conclude that the uniform reduction method simplifies the calculations of the approximations, where we obtained two uniform approximations instead of three as is the case of the standard WKB method. The successful use of the uniform method and the exponential precision asymptotics demonstrate the validity of using the uniform approximations for singular points type of differential equations. It is important to note here that the asymptotic analysis strongly requires the power of the linear superposition for linear ODE. This means that, in specific cases, exponentially small terms are essential to be considered, and if neglected, one fundamental solution will be ignored. In [10], it was formally proved that exponentially small terms may not be numerically small terms, and its inclusion in the uniform expansion is essential, provided it is supported by error bounds. The Olver [10] approach as we indicated in [4] is fruitful for rigorously establishing error bounds for coming work, but this concept is beyond the scope of this paper. Recently, in [2] the importance of implementing the exponential precision asymptotics by using the multiple scale analysis was confirmed. The study of [2] removed the Poincar6 degeneracy due to ignoring exponentially small terms. For more details see [2]. In conclusion, we believe that the uniform reduction method of [7] is more effective over traditional techniques. The method reduces the size of calculations required by the WKB-boundary layer analysis. Moreover, the method introduces an easily calculable technique to determine uniform approximations of higher orders. The notion of the amplitude functions worked effectively in matching the dual uniform approximations. Acknowledgements The author gratefully acknowledges prior concerns to this model by colleagues Floyd B. Hanson and Charles Tier.
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References [1] M. Abramowitz, I. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Washington, DC, 1964. [2] F.B. Hanson, Singular point and exponential asymptotics, in: R. Wong (Ed.), Asymptotic and Computational Analysis: Conference in Honor of Frank W.J. Olver's 65th Birthday, Marcel Dekker, New York, 1990, pp. 21-40. [3] F.B. Hanson, C. Tier, An asymptotic solution of the first passage problem for singular diffusion in population biology, SIAM J. Appl. Math. 40 (1981) 113-132. [4] A.M. Wazwaz, F.B. Hanson, Matched uniform asymptotic approximations for a singular boundary point and an interior turning point, SIAM J. Appl. Math. 46 (1986) 943-961. [5] A.M. Wazwaz, On the asymptotic solutions of a differential equation with multiple singular points, J. Comp. Appl. Math. 34 (1991) 65-74. [6] A.M. Wazwaz, Asymptotic solutions of an eigenvalue problem with several second-order turning points, IMA J. Appl. Math. 48 (1992) 39-51. [7] A.M. Wazwaz, Resonance and asymptotic solutions of a singular perturbation problem, IMA J. Appl. Math. 49 (1992) 231-244. [8] A.M. Wazwaz, F.B. Hanson, Moments of extinction for a singular perturbation problem in the diffusion process, SIAM J. Appl. Math. 51 (1991) 233-265. [9] R.E. Meyer, Exponential asymptotics, SIAM Rev. 3 (1980) 225 235. [10] F.W.J. Olver, Asymptotic and Special Functions, Academic Press, New York, 1974. [11] R. Langer, The asymptotic solutions of ordinary differential equations of the second order with special reference to a turning point, Trans. Amer. Math. Soc. 67 (1949) 461-490.
CC~ ?UTAT[ON FA.SEVIER
Applied Mathematics and Computation 99 (1999) 179-193
A comparative study on a singular perturbation problem with two singular boundary points Abdul-Majid W a z w a z 1 Department of Mathematics and Computer Science, Saint Xavier University, 3700 West 103rd Street, Chicago, IL 60655, USA
Abstract
A singularly perturbed second-order differential equation with two singular boundary points will be investigated. The work will present a comparative study between the WKB-boundary layer analysis and the uniform approximations about the singular points. The exponential precision asymptotics are essential for this study. The matching concept will be applied in each type of analysis where dominant and recessive terms are matched independently. © 1999 Elsevier Science Inc. All rights reserved.
Keywords: Singular points," Exponential precision; WKB method," Uniform method," Matched asymptotics
I. Introduction
I n this p a p e r we are c o n c e r n e d with a c o m p a r a t i v e s t u d y o f the singularly perturbed problem
x(b - x ) { e f ( x ) u ~ z +g(x)u,] = su,
0 < x < b,
(1.1)
where u = u(x, E). T h e functions f ( x ) a n d g(x) a r e a s s u m e d real, positive, a n d several times c o n t i n u o u s l y differentiable in x. T h e p a r a m e t e r s is an eigenvalue p a r a m e t e r . T h e differential e q u a t i o n (1.1) is singular in t h a t the diffusion coefficient ex(b - x ) f ( x ) vanishes at the b o u n d a r y p o i n t s x = 0 a n d x = b.
I E-mail: [email protected].
0096-3003199/$ see front matter © 1999 Elsevier Science Inc. All rights reserved. PII: S0096-3003 (98)00006-X
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Equations of the form (1.1) usually arise in the first passage problems or extinction problems in population biology. Models like (1.1) have been the focus of an extensive study recently, and two distinct methods have been developed to approach similar models. In the first, a leading-order asymptotic solution of Eq. (1.1) is obtained by using the WKB method in the interior region between the boundary points and by using a boundary layer analysis near the singular boundary points x = 0 and x = b. These approximations are then related to each other by asymptotic matching. In the second approach, the uniform reduction method of [7] will be used. A leading-order asymptotic approximation about each singular boundary point will be constructed, such that each approximation is uniformly valid in one subdomain containing one singular point, and each approximation contains dominant and recessive terms as well. The uniform method rests upon using the comparison equation method and using the amplitude functions as well to justify the matching between the uniform asymptotic approximations. The two distinct approximations uniformly valid about the singular points were developed from two linear combinations of Whittaker's functions [1]. The dominant and recessive parts of these approximations will be matched independently. It is interesting to note that models like (1.1) are usually studied for a singular diffusion process such as extinction problems in population biology. Moreover, models like (1.1) with demographic stochasticity typically arise from discrete stochastic processes such as birth and death processes or branching processes, where u(x, ~) can be viewed as the moment generating function. Extensive background for this paper and for identical diffusion process models is found in the results of [~7]. In Eq. (1.1), the diffusion coefficient ~ ( b - x ) f ( x ) , with f ( x ) > 0, vanishes at regular singular points at the boundaries x = 0 and x = b. Since the small positive parameter e multiplies u..... we have a boundary layer type singular perturbation problem. This type of diffusion coefficient is typical of demographic type stochastic effects [3,4]. Moreover, the drift coefficient x(b - x)g(x), with g(x) > 0, does not give rise to any turning point in the domain of validity. Several identical perturbation models were extensively discussed in [2-7], with distinct drift coefficients that give rise to second-order interior turning points. It is the intention of this paper to conduct this comparative study between the classic WKB method and the uniform reduction method of [4] on a model that contains singular boundary points only. It is easily seen that simple or secondorder turning points do not appear in the domain of validity. In [2-8], several singular perturbation models were investigated where the diffusion coefficient in some of these models gives rise to an interior turning point and a singular point at the boundary. In several studies such as [2-7] in general, and in particular in [8], it was shown that exponential precision asymptotics, in the sense of Meyer [9], play a crucial role in the analysis of the matching process, due to the exponential
A.-M. Wazwaz / Appl. Math, Comput. 99 (1999) 17~193
181
behavior of the solution. Meyer [9] gave many illustrations of applications such as wave trapping by round islands, plasma physics, critical scattering in quantum mechanics, and geophysical fluid dynamics, where exponential precision is essential. Exponential precision asymptotics play an important role in optical tunneling. Meyer [9] emphasized that exponentially small terms be included in the analysis to obtain correct results. In [2], Hanson established the foundations of exponential asymptotic expansions by multiple scale techniques. The multiple scale approach of [2] justifies the complete asymptotic expansion and also removes the degeneracy of the usual, single Poincar6 algebraic asymptotic expansions that neglect the exponentially small terms. The Poincar6 algebraic asymptotic expansions hold only for the exponentially larger terms. The complete exponential asymptotic expansion contains both dominant and recessive (exponentially smaller) expansions [8]. The neglect of exponentially smaller terms means the neglect of one of the independent fundamental solutions for a second-order ODE like (1.1). In addition, exponential precision is needed to compute resonance conditions that we presented in many papers. Olver [10] points out that exponentially small terms may not be numerically small terms. However, Olver [10] points out the need to include exponentially small terms in the uniform approximation of certain Bessel functions [4], but he emphasized that this use should preferably be supported by error bounds. The Olver [10] approach as we indicated in [4] is fruitful for rigorously establishing error bounds for coming work, but this notion is beyond the scope of this paper. The intention of this paper is an application to many other papers by [2-8] where the complete asymptotic analysis was effectively employed in the sense of Meyer [9], Hanson [2] and by Hanson and Tier [3], and by many others, with promising results. 2. The WKB method and the boundary layer analysis
In this section we construct a leading-order asymptotic approximation to Eq. (1.1) by using the standard WKB method. In this method we first assume that the solution is of the form
u(x,e) = exp
(is) -~
y(t,e) dt
,
(2.1)
where y(x, e) has an asymptotic expansion of the form
y(x, e) = yo(x, e) + eyl (x, e) + O(e2).
(2.2)
Substituting Eqs. (2.1) and (2.2) into Eq. (1.1) yields 1 s -yo(Yoe+ ~ ) + y~(2yo + Fx) +Yo + O(e) - x(b - x ) f ( x ) '
(2.3)
A.-M. Wazwaz I Appl. Math. Comput. 99 (1999) 179-193
182
where F(x) is the squared natural metric of [3] defined
by
F(x) = f~g(t) dt, Ja f(t)
(2.4)
such that
F~(x) = f -g(x) ~>
0,
0
(2.5)
Equating the coefficients of powers of e to zero, we obtain to leading-order the solutions Y0 = 0
and
Y0 = -F~(x).
(2.6)
The terms of order O(e °) give S
Y' -
x(b
- ~)g(~)
'
when y0 = 0
(2.7)
and
Y' =
s x ( b - x)g(x)
F~x(x)
~(~)
when y0 = -Fx(x).
(2.8)
We can easily conclude that the resulting values ofyt at Eqs. (2.7) and (2.8) are not integrable at the singular points at the boundaries x = 0 and x -- b. This leads to the conclusion that the standard W K B approximation (2.1) fails in small neighborhoods near the singular boundary points. Consequently, the W K B approximation (2.1) is valid everywhere in the subdomain 0 < x < b away from the singular boundary points x = 0 and x = b. For purposes of matching, that will be used later, it is convenient to define the function G(x) by s k c sG'(x) x(b - x)g(x) + x- + --'b- x (2.9) where k and c are constants given by k -
S
(2.10)
bg(O)
and
c-
S
bg(b)"
(2.11)
Combining Eqs. (2.1), (2.2) and Eqs. (2.6)-(2.9), the leading-order approximation valid in O(e) < x < b - O(e) is thus given by
Uo
+ Bexp
where A and B are constants of integration.
e
sa(x)
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The boundary layer analysis is now useful to determine the approximations in the boundary layer regions at the singular points x = 0 and x = b. This can be achieved by using the fast varying stretched variables X
Y - t' [(o)~
(2.13)
and b-x
(2.14)
Z = [.f(b)-----~'
respectively. As a result Eq. (1.1) is converted to the boundary layer equations Yurr + Yur + ku = 0
(2.15)
Zuzz - Zuz + cu = 0
(2.16)
and
valid at x = 0 and x = b respectively, and the constants k and c are given above by Eqs. (2.10) and (2.11). To determine the solutions of Eqs. (2.15) and (2.16), we employ the Liouville [11] normal transformations u ( Y , e) = v ( Y ) e -r/2
(2.17)
u(Z, e) = ?J(Z)e z/2
(2.18)
and
into Eqs. (2.15) and (2.16), respectively. Consequently, Eqs. (2.15) and (2.16) are converted to the normal forms -y
v=0
(2.19)
-2
v = 0,
(2.20)
and
each is a reduced form of Whittaker's equation [1] with solutions well defined by Whittaker's functions. Combining Eqs. (2.17)-(2.20), the solutions in the boundary layer regions x = 0 and x = b are given by ul = e-r/2[A1 W1(k, Y) + Bl W2(k, Y)]
(2.21)
u2 = e z/2 [A2WI(c, Z) + B2 W2(c, Z)],
(2.22)
and respectively, with AI, A2, B1, and B2 are constants of integration. The modified versions of Whittaker's functions Wt( a , R ) and W2(a,R) were chosen by [3,4] and are related to Whittaker's function Wk,0.5and Mk,0.s by
A.-M. Wazwaz I Appl. Math. Comput. 99 (1999) 179-193
184
(a, R) --
W~,0.5(R),
F(1 - k) ~ ( a , R ) - r(1 -k)Mk,0,(R) ~ r(1 + k~ cos(~k)~,0,(R).
(2.23)
It is worth noting that W~ and W2 have exponentially small and exponentially large behavior, respectively, for large R. The complete standard expansions of W1 and W2 can be found in [2~1]. In closing this section, the matching principle will be applied to the asymptotic approximations obtained above. For u(x,s) of (1.1) to be correctly represented by Ul, U0
g ( x ) ~ F(O) + F'(O)x + . . .
and therefore we obtain F(x) ~ F(O) c
--
+ Y
(2.25)
upon using Eqs. (2.5) and (2.13). We also note that x Y e - F'(0)
(2.26)
derived by using Eqs. (2.5) and (2.13). As x + 0 and by substituting Eqs. (2.25) and (2.26) into Eq. (2.12) we obtain
+ ~
(g~0)/k
g--~
•
The leading-order expansions for the related Whittaker's function of Eq. (2.23) are given by W1(a,R) ~ R ~ exp(-½R) + O(e), W2(a,R) ,-~ R-" exP(½R) + O(e),
(2.28)
<'(a,R) ,-~ -½R ~ e x p ( - ½R) + O(e), W2'(a,R ) ,.~ 1R-~ exP(½R) + O(e)
as R --+ oc, (for example, see [4], p. 954). Substituting expansions (2.28) into (2.21), the asymptotic expansion of ul becomes ul ~ AI y % - r + B1Y k.
(2.29)
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A.-M. Wazwaz / Appl. Math. Comput. 99 (1999) 179-193
The nonexponential varying terms of ul and u0
A.
(2.30)
Next, we equate the exponential varying terms of Eqs. (2.27) and (2.29) to find
F(O)e sG(O) (F (0))
A1 = exp
~)
g~B.
(2.31)
To match u0
X(x) ~ X(b) - f --g(b)'" ( b ) ~O_ x ) + "",
(2.32)
so that
F(x) ~ F(b)
- Z,
(2.33)
obtained by using Eq. (2.14). We also note that
b-x Z e -- F'(b)
(2.34)
by using Eq. (2.14). Accordingly, as x ---, b and by using Eqs. (2.33) and (2.34) we find
((F'(b))-cZ c
U0
F(b)e s G ( b ) ) ( b ) kF'~'wf(b)ezZ-c'( (o,, ~
+Bexp(
(2.35)
As Z ~ e~, using the asymptotic expansions of u2 in Eq. (2.22) yields
(2.36)
U2 "" A2 Ze -k B2eZZ -c.
Proceeding as before, we equate the nonexponential terms in Eqs. (2.35) and (2.36) to obtain
A2 = e"a(b)
(F'(b) )-CA
(2.37)
and by equating the exponential terms we find B2 = e x p ( F ( b ) e
s G ( b ) ) ( ! ) k ( r,,,,,,,cf(b)B ~o)) ~ .
(2.38)
This completes the asymptotic matching of every neighboring approximations. The two sets of constants A1, B~ and A2, B2 are determined in terms of one set of constants, namely A and B. We point out that the constants A and B can be
A.-M. Wazwaz / Appl. Math. Comput. 99 (1999) 179-193
186
completely determined by using specific boundary conditions. Further, it is worth noting that the exponential precision asymptotic we used is important for determining the sets of matching constants.
3. The uniform reduction method
In this section we construct leading-order approximations fil and /~2 of solutions to Eq. (1.1) uniformly valid into the subdomains/1 and/2, where I, = [0, b - O(~)), 12 = (O(e),b],
(3.1)
such that each subdomain includes only one singular boundary point. The asymptotic validity of the approximations /~1 and fi2 fails to hold in neighborhoods of the singular points x = b and x = 0, respectively. To achieve our goal, we will implement the uniform reduction theorem of [4]. As will be seen later, the method rests upon applying the comparison equation method and using the proper amplitude functions as well. We will prove later that the uniform variables y(x) and z(x), that will be used for our analysis, are defined by
y(x) =
F'(r) dr = F(x) - F(0),
z(x) =
F'(r) dr = F(b) - F(x),
(3.2)
valid only for the subintervals I1 and h, respectively, where F(x) is the natural metric of [3] given by Eq. (2.4). We point out that the equations given by Eq. (3.2) are consistent with the Langer [11] minimal properties given by y(0)=0, z(b) = O,
y'(x)>0, f ( x ) < 0.
(3.3)
We now begin our analysis by using the order-one uniform variables g(0) y(x) = f - ~ x ,
z(x)
(3.4)
g(bl ( b - x),
valid only for the subintervals lj and 12, respectively. In view of the uniform variables (3.4), a pair of comparison equations is constructed from the model Eq. (1.1), given by
A.-M. Wazwaz / Appl. Math. Comput. 99 (1999) 179-193
187
eyw~ + yWy + kw = 0
(3.5)
eZ~zz - z~z + c# = 0,
(3.6)
and where the constants k and c are defined above in Eqs. (2.10) and (2.11), respectively, but will be derived later in this section. It is important to note that the comparison equation (3.5) preserves the behavior of the model (1.1) in the subinterval Ii. Further, the original equation (1.1) matches the comparison equation (3.5) in the sense that both have one distinct zero at x = 0 at the subinterval I1. Similar conclusions can be used to relate the comparison equation (3.6) with the original equation (1.1) in the subinterval 12. In [4,5], it was formally proved that the solution of the original equation is asymptotically equivalent to the solution of the comparison equation, at the corresponding subinterval, upon introducing weakly varying amplitude functions. Following the uniform reduction method of [4,5], the uniform asymptotic representations ~ and ~2 take the form fq = M(y, e)w(y) + cN(y, C)Wy(y), valid for Ii
(3.7)
u2 = ~t(z, e)#(z) + e.~(z, e)~z(z),
(3.8)
and valid for 12,
where w(y) and ~(z) are the solutions of the comparison equations (3.5) and (3.6), respectively, and M, N, M, and N are the amplitude functions that will be used to justify the matching concept of the two uniform approximations. We point out that the amplitude functions areholomorphic with power series expansions as E--o0, where M and M are normalized such that M(0, e) = ~t(0, e) ---- 1. To determine the uniform approximations (3.7) and (3.8), our analysis will run in two directions. On one hand, we determine the functions w(y) and #(z) of the canonical equations (3_5) and (3.@ In the second, we determine the leading term expansions M0, M0, No, and No of the amplitude functions M, M, N, and N as will be seen later. To determine w(y) and ~(z) of the comparison equations (3.5) and (3.6), we first introduce the fast varying stretched variables Y =Y-,
(3.9)
and Z
(3.10)
in Eqs. (3.5) and (3.6) to obtain Ywrr + Ywr + kw = 0
(3.11)
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A.-M. Wazwaz / Appl. Math. Comput. 99 (1999) 179-193
and Zfvzz - Z~vz + c~ = 0,
(3.12)
respectively. The normal transformations w(Y) = e-r/2v(y)
(3.13)
~v(Z) = e-Z/Z~(Z)
(3.14)
and convert Eqs. (3.11) and (3.12) to the normal forms -~
v=0
(3.15)
-~
v=0,
(3.16)
and
respectively. We can easily observe that each of Eqs. (3.15) and (3.16) is a reduced form of Whittaker's equations [1]. Combining the solutions of Eqs. (3.15) and (3.16) with Eqs. (3.13) and (3.14) yields the solutions of the comparison equations Eqs. (3.5) and (3.6) given by w(y) = e-Y/2[CW1 (k, Y) + D ~ ( k , Y)]
(3.17)
W(Z) = e z/2 [CI WI(c, Z) + Dl W2(c, Z)],
(3.18)
and
where ~ and W2 are Whittaker's functions defined above by Eq. (2.23). Consequently, substituting Eqs. (3.17) and (3.18) into Eqs. (3.7) and (3.8) yields the uniform solutions ill(x, e) = e-r/2 {M(x, e)[CWl(k, Y) + DW2(k, Y)] + N(x, e)[C(W( - ½W1) + D(W2s - ½W2)]},
valid for 11
(3.19)
and ft2 (x, e) = e z/2 {M(x, e)[C1 W1(c, Z) + D1 W2(c, Z)]
q- iV(x, ~) [Ct (Wllt --b 1WI) + DI(W2s + lg/2)]}, valid for/2.
(3.20)
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We focus our concern now on determining the leading term expansions of the amplitude functions, therefore we follow [10]. We substitute Eqs. (3.7) and (3.8) in Eq. (1.1), using Eqs. (3.5) and (3.6) to eliminate W~yand ~zz in favor of wy and w, and vi,~and rb, respectively. We collect the coefficients of E"Wy, e"w, e"~z, and e-'~, and then equate these coefficients separately to zero on their domain of validity. As a result, the terms of order O(e°Wy) and O(e°~) give F~ = { Yx -Zx
for 11, for 12.
(3.21)
This implies that y(x) =
fo xF~(r) dr = F(x)
z(x) =
fx bFr(r) dr
- F(O)
(3.22)
= F(b) - F(x).
(3.23)
and
The results (3.22) and (3.23) are consistent with our assumptions of the uniform variables given by Eq. (3.2) and with the Langer [11] minimal properties of Eq. (3.3). Terms of order (e°w) and (e°~,) give M o - h(y)Mo = 0
(3.24)
~-I; + h(z)Mo = 0,
(3.25)
and
so that Mo = exp
h(r) dr
(3.26)
and / ~ 0 = exp
(/0 z ) -
tt(r) dr
x(b
- x)g(x)yx(x)
x(b
- x)g(X)Zx(X)
(3.27)
,
where k
h(y) = ; +
(3.28)
and
ft(z)
c = z
s
(3.29) '
A.-M. Wazwaz I Appl. Math. Comput. 99 (1999) 179.193
190
It is important to note that the regularity conditions of h(y) as x ~ 0 and h(z) as x ~ b requires the selection of constants S
k-
bg(O)
c-
bg(b)'
(3.30)
and S
(3.31)
which is consistent with our assumption of k and c of (2.10) and (2.11). Terms of order (eWy) and (e#~) give No and N0 through the identities
No = Mo
yx(O) Moyx(x)
(3.32)
No = - f l o + zx(b)
(3.33)
and Mo~x(X) '
where the constants of integration have been chosen so that No and No are regular as x ~ 0 and x ~ b, respectively. As a result, the uniform approximations to leading-order (3.19) and (3.20) are rewritten by
~, (x, ,) = e-Y/2{M0[C~ (k, r) + DW~(k, r)] +No[C(W~'-½W~)+D(Wz'-½W2)]},
valid for I1
(3.34)
and
fi2 (x, e) = e z/2 {~¢0[C, WI(c, Z) + D1 W2(c, Z)] 1
+ N0 [G (W~'+ ½W~)+ D~ (W2' + ~ W2)]}, valid for/2
(3.35)
For purposes of matching, we need to derive explicit formulae that relate the weakly varying amplitude functions. To achieve this goal, it is useful to rewrite Eq. (2.9) in terms of the order-one uniform variables y(x) and z(x). Using Eq. (3.4) into Eq. (2.9) yields s yx zx sG'(x) - x(b - x)g(x) ~ k T ( ~ - C z - ~
(3.36)
In view of Eqs. (3.28), (3.29) and (3.36) we obtain z~ h(y)y~ = sG'(x) + Cz(x)
(3.37)
and
f,(z)zx = - s r ' ( x ) + k Y~ y(x)
"
(3.38)
A.-M. Wazwaz I Appl. Math. Comput. 99 (1999) 179-193
191
Inserting Eqs. (3.37) and (3.38) into Eqs. (3.26) and (3.27) yields 114o = exp(s[G(x) - G(0)]) (Z--~)Z(x)~
(3.39)
and \ x~x) / respectively. Furthermore, the uniform variable y(x) and z(x) are related through the identity z(x) + y(x) = F(b) - F(O)
(3.41)
and hence, the fast varying stretched variables are related through identity Z(x) + Y(x) -- F(b) - F(O) (3.42) e To accomplish our asymptotic analysis, it is essential to match the leadingorder uniform approximations (3.34) and (3.35). For u(x, e) to be correctly represented by fia and fi2, it is essential that the two approximations share validity on the overlapping subinterval (e, b - e). As Y ---* c~, the approximation (3.34) has the form fil ~ C (
yx(O) "~y%-r + DMoY-*
(3.43)
\yx(x)Mo]
derived from Eqs. (2.28) and (3.39). In a parallel manner, we use Eqs. (2.28) and (3.40) to find u2 ~ Cl~loZ c + 01 ( z x ( b ) )Z_Ce z \ zx(x)Mo ]
(3.44)
as Z --~ oc. Matching the exponential and nonexponential terms of Eqs. (3.43) and (3.44), and using Eq. (3.42) yield c = _ :~z~~ <0~:~ ~x, ( . i ~
- ~(0~1~~ 0 ,
~ < . ~ : ex~ ( ~ ( ~ - ~ ~(0~
)~,
(3.45) and D = exp (s[G(0) - G(b)])(Z(O))C(Y(b))kCl.
(3.46)
It is clear that one set of the constants of integration has been eliminated in favor of the other set. Consequently the remaining constants are completely determined by using proper boundary conditions. This completes the determination of the uniform approximations to leading order.
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4. Conclusions and discussions An analytic comparison of two distinct methods, usually used in singular perturbation problem has been discussed. In a manner parallel to [3], we applied the standard WKB technique for the interior region between the singular boundary points. Two layer type approximations around the singular points have been developed by using proper stretched variables. The exponential precision asymptotics have been used in these approximations to accomplish the matching process. Following [4], the uniform reduction theorem was employed to obtain a two-part uniform approximation. Two uniform variables were used each valid in a subinterval that includes one singular point and excludes the other singular point. The fast varying stretched variables were implemented and amplitude functions were used in the uniform approximation to justify the matching. The two-part uniform approximation implemented the exponentially small terms as in the first method. From the discussion above, we can easily conclude that the uniform reduction method simplifies the calculations of the approximations, where we obtained two uniform approximations instead of three as is the case of the standard WKB method. The successful use of the uniform method and the exponential precision asymptotics demonstrate the validity of using the uniform approximations for singular points type of differential equations. It is important to note here that the asymptotic analysis strongly requires the power of the linear superposition for linear ODE. This means that, in specific cases, exponentially small terms are essential to be considered, and if neglected, one fundamental solution will be ignored. In [10], it was formally proved that exponentially small terms may not be numerically small terms, and its inclusion in the uniform expansion is essential, provided it is supported by error bounds. The Olver [10] approach as we indicated in [4] is fruitful for rigorously establishing error bounds for coming work, but this concept is beyond the scope of this paper. Recently, in [2] the importance of implementing the exponential precision asymptotics by using the multiple scale analysis was confirmed. The study of [2] removed the Poincar6 degeneracy due to ignoring exponentially small terms. For more details see [2]. In conclusion, we believe that the uniform reduction method of [7] is more effective over traditional techniques. The method reduces the size of calculations required by the WKB-boundary layer analysis. Moreover, the method introduces an easily calculable technique to determine uniform approximations of higher orders. The notion of the amplitude functions worked effectively in matching the dual uniform approximations. Acknowledgements The author gratefully acknowledges prior concerns to this model by colleagues Floyd B. Hanson and Charles Tier.
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References [1] M. Abramowitz, I. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Washington, DC, 1964. [2] F.B. Hanson, Singular point and exponential asymptotics, in: R. Wong (Ed.), Asymptotic and Computational Analysis: Conference in Honor of Frank W.J. Olver's 65th Birthday, Marcel Dekker, New York, 1990, pp. 21-40. [3] F.B. Hanson, C. Tier, An asymptotic solution of the first passage problem for singular diffusion in population biology, SIAM J. Appl. Math. 40 (1981) 113-132. [4] A.M. Wazwaz, F.B. Hanson, Matched uniform asymptotic approximations for a singular boundary point and an interior turning point, SIAM J. Appl. Math. 46 (1986) 943-961. [5] A.M. Wazwaz, On the asymptotic solutions of a differential equation with multiple singular points, J. Comp. Appl. Math. 34 (1991) 65-74. [6] A.M. Wazwaz, Asymptotic solutions of an eigenvalue problem with several second-order turning points, IMA J. Appl. Math. 48 (1992) 39-51. [7] A.M. Wazwaz, Resonance and asymptotic solutions of a singular perturbation problem, IMA J. Appl. Math. 49 (1992) 231-244. [8] A.M. Wazwaz, F.B. Hanson, Moments of extinction for a singular perturbation problem in the diffusion process, SIAM J. Appl. Math. 51 (1991) 233-265. [9] R.E. Meyer, Exponential asymptotics, SIAM Rev. 3 (1980) 225 235. [10] F.W.J. Olver, Asymptotic and Special Functions, Academic Press, New York, 1974. [11] R. Langer, The asymptotic solutions of ordinary differential equations of the second order with special reference to a turning point, Trans. Amer. Math. Soc. 67 (1949) 461-490.