Research A rticle
An Asymptotic-Numerical Hybrid Method for Solving
Singularly Perturbed Linear Delay Differential Equations
Süleyman Cengizci
Institute of A pplied Mathematics, Middle Eas t Technical Uni versity, 06800 Ankara, Turkey
Correspondence should be addressed to S
¨
uleyman Cengizci; cengizci.sule yman@metu.edu.tr
Received April ; Accepted January ; Published February
AcademicEditor:PatriciaJ.Y.Wong
Copyright © S
¨
uleyman Cengizci. is is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
In this work, approximations to the solutions of singularly perturbed second-order linear delay dierential equations are studied. We
rstly use two-term Taylor series expansion for the delayed convection term and obtain a singularly p erturbed ordinary dierential
equation (ODE). Later, an ecient and simple asymptotic method so called Successive Complementary Expansion Method (SCEM)
is employed to obtain a uniformly valid approximation to this corresponding singularly perturbed ODE. As the nal step, we
employ a numerical procedure to solve the resulting equations that come from SCEM procedure. In order to show eciency of this
numerical-asymptotic hybrid method, we compare the results with exact solutions if possible; if not we compare with the results
that are obtained by other reported methods.
1. Introduction
Almost all physical phenomena in nature are modelled using
dierential equations, and singularly perturbed problems are
vital class of these kinds of problems. In general, a singular
perturbation problem is dened as a dierential equation
that is controlled by a p ositive small parameter 0<1
that exists as multiplier to the highest derivative term in the
dierential equation. As tends to zero, the solution of
problem exhibits interesting behaviors (rapid changes) since
the order of the equation reduces. e region where these
rapid changes occur is called inner region and the region
in which the solution changes mildly is called outer region.
As mentioned in [, ], these kinds of problems arise in
almost all applied natural sciences. Some of these can be given
as mechanical and electrical systems, celestial mechanics,
uid and solid mechanics, electromagnetics, particle and
quantum physics, chemical and biochemical reactions, and
economics and nancial mathematics. Various methods are
employed to solve singular perturbation problems analyt-
ically, numerically, or asymptotically such as the method
of matched asymptotic expansions (MMAE), the method
of multiple scales, the method of WKB approximation,
Poincar
´
e-Lindstedt method and periodic averaging method.
Rigorous analysis and applications of these methods can be
found in [–].
Modelling automatic systems oen involve the idea of
control because feedback is necessar y in order to maintain
a stable state. But much of this feedback require a nite
time to sense information and react to it. A general way for
describing this process is to formulate a delay dierential
equation (dierence-dierential equation). Delay dierential
equations (DDE) are widely used for modelling problems
in population dynamics, nonlinear optics, uid mechan-
ics, mechanical engineering , evolutionary biology, and even
modelling of HIV infection and human pupil-light reex.
One can refer to [–] for general theory and applications
of DDEs.
In this paper, we study an important class of delay dier-
ential equations: singularly perturbed linear delay dierential
equations. A singularly perturbed delay dierential equation
is a dierential equation in which the highest-order derivative
is multiplied by a positive small parameter and involving at
least one delay term. We restrict our attention to singularly
perturbed second-order ordinary delay dierential equations
that contains the delay in convection term. Various methods
have been used to solve singularly perturbed DDEs such as
nite dierence methods [, , ], nite element methods
Hindawi
International Journal of Differential Equations
Volume 2017, Article ID 7269450, 8 pages
https://doi.org/10.1155/2017/7269450
International Journal of Dierential Equations
[, ], homotopy perturbation met hod [, ], reproducing
kernel method [, ], spline collocation methods [, ],
and asymptotic approaches [, ]. We use an asymptotic-
numericalhybridmethodinordertonduniformlyvalid
approximations to singularly perturbed ODEs. At the rst
step, two-term Taylor series expansion is used to vanish
delayed term. Secondly, to obtain a uniformly valid approxi-
mation an ecient and easily applicable asymptotic method
so called Successive Complementary Expansion Method
(SCEM) that was introduced in [] is employed. Finally, a
numericalapproachisusedtosolveresultingequationsthat
come from SCEM process.
2. Description of the Method
In this section, we rst give a short overview of asymptotic
approximations and then explain Successive Complementary
Expansion Method by which we obtain highly accurate
approximations to solutions of singularly perturbed linear
DDEs.
Consider two continuous functions of real numbers ()
and () that depend on a positive small parameter ;then
() = (()) for →0if there exist positive constants
and
0
such that ∈(0,
0
] with |()| |()| for →0,
and () = (()) for →0if lim
𝜀→0
(()/()) = 0.Let
be a set of real functions that depend on ,strictlypositive
and continuous in (0,
0
],suchthatlim
𝜀→0
() exists and
1
()
2
() ∈ for each
1
(),
2
() ∈ .Afunction
𝑖
()
that satises these conditions is called order function.Given
two f unctions (,) and
𝑎
(,) dened in a domain
are asymptotically identical to order () if their dierence
is asymptotically smaller than (),where() is an order
function; that is,
(
,
)
−
𝑎
(
,
)
=
(
(
))
,
()
where is a positive small parameter arising from the
physical problem under consideration. e function
𝑎
(,)
is named as asymptotic approximation of the function (,).
Asymptotic approximations in general form are dened by
𝑎
(
,
)
=
𝑛
𝑖=1
𝑖
(
)
𝑖
(
,
)
,
()
where
𝑖+1
() = (
𝑖
()),as→0. Under these conditions,
the approximation () is named as generalized asymptotic
expansion. If the expansion () is written in the form of
𝑎
(
,
)
=
0
=
𝑛
𝑖=1
(0)
𝑖
(
)
(0)
𝑖
(
)
,
()
then it is called regular asymptotic expansion where the special
operator
0
is outer expansion operator of a given order ().
us, −
0
= (()). For more detailed information about
the asymptotic approximations, we refer the interested reader
to [–, , ].
Interesting behaviors occur when the function (,) is
not regular in so () or () is valid only in a restricted
region
0
∈, called the outer region. We introduce an
innerdomainwhichcanbeformallydenotedas
1
=−
0
andcorrespondinginnerlayervariable,locatednearthepoint
=
0
,as
=(−
0
)/(),with() being the order
ofthicknessoftheboundarylayer(theregioninwhichthe
rapid changes-behaviors occ ur). If a regular expansion can be
constructed in
1
, one can write down the approximation as
𝑎
(
,
)
=
1
=
𝑛
𝑖=1
(1)
𝑖
(
)
(1)
𝑖
(
)
,
()
where the inner expansion operator
1
, dened in
1
,isof
thesameorder() as the outer expansion operator
0
;that
is, −
1
= (()).us,
𝑎
=
0
+
1
−
1
0
()
is clearly uniformly valid approximation (UVA) [–].
Now let us consider se cond-order singularly perturbed
DDE in its general form (delay in the convection term):
(
)
+
(
)
(
−
)
+
(
)
(
)
=
(
)
,
()
where 0<1small parameter and 0<<1.Boundary
and interval conditions are given as
(
)
=
(
)
, −≤≤0,
(
1
)
=,
()
where (),(),(),and()are smooth functions, ∈R,
and is delay term.
As tends to zero, the order of the dierential equation
reduces and so a layer occurs in the solution. e sign of ()
on the interval [0,1] determines the type of the layer. If the
sign changes on the interval, interior layer behavior occurs
in the solution. If the sign of () does not change, there are
two possibilities: if () < 0 on [0,1],thenaboundarylayer
occursattherightend(nearthepoint=1)andif() > 0
on [0,1], then a boundary layer occurs at the le end (near
the point =0).
Using Taylor series expansion we linearize the convection
term; that is,
(−)=
() −
() and substituting it
into () one can reach
−
(
)
(
)
+
(
)
(
)
+
(
)
(
)
=
(
)
.
()
Letting =,where∈R
1−
(
)
(
)
+
(
)
(
)
+
(
)
(
)
=
(
)
()
is found and it is clear that () is a singularly perturbed ordi-
nary dierential equation for →0with the same bound-
ary and interval conditions as given by (). SCEM procedure
is applicable at this stage.
e uniformly valid SCEM approximation is in the
regular form given by
scem
𝑛
(
,
,
)
=
𝑛
𝑖=1
𝑖
(
)
𝑖
(
)
+Ψ
𝑖
(
)
,
()
where {
𝑖
()} is an asymptotic sequence and functions
Ψ
𝑖
(
) are the complementary functions that depend on .
International Journal of Dierential Equations
If the functions
𝑖
() and Ψ
𝑖
(
) depend also on ,the
uniformly valid SCEM approximation is cal led generalized
SCEM approximation and given by
scem
𝑛𝑔
(
,
,
)
=
𝑛
𝑖=1
𝑖
(
)
𝑖
(
,
)
+Ψ
𝑖
(
,
)
.
()
If only one-term SCEM approximation is desired, then one
seeks a uniformly valid SCEM approximation in the form of
scem
1
(
,
,
)
=
1
(
,
)
+Ψ
1
(
,
)
.
()
To improve the accuracy of approximation, () can be
iterated using (). It means that successive complementary
terms will be added to the ap proximation. To this end, second
SCEM approximation will be in the form of
scem
2
(
,
,
)
=
1
(
,
)
+Ψ
1
(
,
)
+
2
(
,
)
+Ψ
2
(
,
)
.
()
In this work, we s eek an approximation in our calculations in
the form of ().
Now, let us assume that problem () has a le boundary
layer (near the point =0)andlet
out
() be asymptotic
approximation to the outer solution and let Ψ(
) be the
complementary solution, where
=/is boundary layer
(stretching) variable. If approximations
out
(
,
)
=
1
(
)
+
2
(
)
+
2
3
(
)
+⋅⋅⋅,
Ψ
(
,
)
=Ψ
1
(
,
)
+Ψ
2
(
,
)
+
2
Ψ
3
(
,
)
+⋅⋅⋅
()
are substituted into () and if each term is balanced with
respect to the powers of (we balance just the terms (1) and
()),
(
)
1
(
,
)
+
(
)
1
(
,
)
=
(
)
,
1
(
1,
)
=,
1
(
,
)
+
(
)
2
(
,
)
+
(
)
2
(
,
)
=0,
2
(
1,
)
=0
()
are found. If the same procedure is applied for equations that
involve complementary functions
Ψ
1
(
,
)
+
(
)
Ψ
1
(
,
)
=
(
)
()
with the boundary conditions
Ψ
1
(
0,
)
=
(
0
)
−,
Ψ
1
1
,=0,
Ψ
2
(
,
)
−
(
)
Ψ
1
(
,
)
+
(
)
Ψ
2
(
,
)
+
(
)
Ψ
1
(
,
)
=0
()
with the boundary conditions
Ψ
1
(
0,
)
=−
2
(
0,
)
,
Ψ
1
1
,=0
()
being obtained and so () gives uniformly valid second
SCEM approximation.
3. Illustrative Examples
3.1. Le Boundary Layer Problem. Consider singularly per-
turbed DDE that exhibits a boundary layer at the le end of
the interval:
(
)
+
(
−
)
−
(
)
=0, 0≤≤1,
()
with the boundary conditions (0) = 1 and (1) = 1.e
exact solutio n of this problem is given by () = ((1 −
𝑚
2
)
𝑚
1
𝑥
+(
𝑚
1
−1)
𝑚
2
𝑥
)/(
𝑚
1
−
𝑚
2
),where
1,2
=(−1±
1 + 4( − ))/2( − ). As the rst step, we use two-term
Taylor expansion for
(−) =
() −
().Ifwe
substitute it into (), the problem turns into
(
−
)
(
)
+
(
)
−
(
)
=0, 0≤≤1.
()
In order to obtain a uniformly valid approximation (UVA),
we rst seek an outer solution which is valid far from the
boundary layer (the boundary layer is near the point =0for
this problem) and then using SCEM we add complementary
solutiontoit.Later,usingthesameidea,wewillgetmore
accurate approximations.
Outer Region Solutions.Letustake=−, assuming that
depends on , and adopt a solution for the outer region in
the form of
out
() =
1
(,)+
2
(,).Equation()turns
into
1
(
,
)
+
2
(
,
)
+
1
(
,
)
+
2
(
,
)
−
(
,
)
+
2
(
,
)
=0
()
and balancing the terms of the order (1) and (),we
reach the equations
1
(
,
)
−
1
(
,
)
=0,
1
(
1,
)
=1,
1
(
,
)
+
2
(
,
)
−
2
(
,
)
=0,
2
(
1,
)
=0.
()
One can easily nd the exact solutions of these equations as
1
(
,
)
=
𝑥−1
,
2
(
,
)
=
𝑥−1
(
1−
)
.
()
Itmeansthattheoutersolutionisoftheform
out
(
,
)
=
𝑥−1
+
𝑥−1
(
−1
)
.
()
International Journal of Dierential Equations
Complementary Solutions. No w applying the stretching trans-
formation
=/and adopting the complementary solution
as Ψ(
,) = Ψ
1
(,) + Ψ
2
(,),onecanreach
Ψ
1
(
)
+Ψ
2
(
,
)
+Ψ
1
(
,
)
+Ψ
2
(
,
)
−Ψ
1
(
,
)
−
2
Ψ
2
(
,
)
=0.
()
Balancing the terms of the order (1) and (),weobtain
Ψ
1
(
,
)
+Ψ
1
(
,
)
=0,
Ψ
1
(
0,
)
=1,Ψ
1
1
,=0,
()
Ψ
2
(
,
)
+Ψ
2
(
,
)
−Ψ
1
(
,
)
=0,
Ψ
2
(
0,
)
=−
−1
,Ψ
2
1
,=0.
()
Here we are able to solve Ψ
1
(
,) and Ψ
2
(,) exactly, but
inmanycasestoobtainanalyticalsolutiontoΨ
1
(
,) and
Ψ
2
(
,) is really tedious process, even for many problems it
is impossible. e solutions may be given as
Ψ
1
(
,
)
=
−1
1−
−1/𝜃
−
𝑥
−
−1/𝜃
()
and Ψ
2
(
,) is given as the solution of (). Since we
solve () numerically (MATLAB bvp4c routine) and the
others using an asymptotic scheme, our present method is
ahybridmethod.Asaresult,weobtainrsttwoSCEM
approximations to problem () as follows:
scem
1
(
,
,
)
=
𝑥−1
+
−1
1−
−1/𝜃
−
𝑥
−
−1/𝜃
,
scem
2
(
,
,
)
=
scem
1
(
,
,
)
+
𝑥−1
(
1−
)
+Ψ
2
(
,
)
.
()
3.2. Right Boundary Layer Problem. Consider singularly per-
turbedDDEthatexhibitsaboundarylayerattherightendof
the interval
(
)
−
(
−
)
+
(
)
=0,
()
with the boundary and interval conditions
(
)
=1, −≤≤0,
(
1
)
=−1.
()
Using two-term Taylor expansion for the convection term, we
reach
( − ) =
() −
() andapplyingitto()one
can obtain
(
+
)
(
)
−
(
)
+
(
)
=0, 0≤≤1
()
with the boundary conditions (0) = 1 and (1) = −1.
In order to obtain a uniformly valid approximation, we
rst seek an outer solution which is valid for far from the
boundary layer (the boundary layer is near the point =1for
this problem) and then using SCEM we add complementary
solutiontoit.Later,usingthesameidea,wewillgetmore
accurate approximations.
Outer Region Solutio ns.Letustake=+assuming that
depends on and adopt an approximation for the outer region
in the form of
out
() =
1
(,) +
2
(,).Equation()
turns into
1
(
,
)
+
2
(
,
)
−
1
(
,
)
+
2
(
,
)
+
(
,
)
+
2
(
,
)
=0;
()
balancing the terms of the order (1)and (),wereachthe
equations
1
(
,
)
−
1
(
,
)
=0,
1
(
0,
)
=1,
1
(
,
)
−
2
(
,
)
+
2
(
,
)
=0,
2
(
0,
)
=0.
()
One can easily nd the exact solutions of these equations as
1
(
,
)
=
𝑥
,
2
(
,
)
=
𝑥
.
()
Itmeansthattheoutersolutionisoftheform
out
(
,
)
=
𝑥
+
𝑥
.
()
Complementary Solutions. No w applying the stretching trans-
formation
= ( − 1)/ and adopting the complementary
solution as Ψ(
,) = Ψ
1
(,) + Ψ
2
(,) one can reach
Ψ
1
(
)
+Ψ
2
(
,
)
−Ψ
1
(
,
)
−Ψ
2
(
,
)
+Ψ
1
(
,
)
−
2
Ψ
2
(
,
)
=0.
()
Balancing the terms of the order (1) and () we obtain
Ψ
1
(
,
)
−Ψ
1
(
,
)
=0,
Ψ
1
−
1
,=0, Ψ
1
(
0,
)
=−1−,
()
Ψ
2
(
,
)
−Ψ
2
(
,
)
+Ψ
1
(
,
)
=0,
Ψ
2
−
1
,=0, Ψ
2
(
0,
)
=−1−.
()
e solutions are given as
Ψ
1
(
,
)
=
+1
−1/𝜃
𝑥
−1−−1
()
and Ψ
2
(
,) is given as the solution of (). us, we reach
uniformly valid SCEM approximations as
scem
1
(
,
,
)
=
𝑥
+
+1
−1/𝜃
𝑥
−1−−1,
scem
2
(
,
,
)
=
scem
1
(
,
,
)
+
𝑥
+Ψ
2
(
,
)
.
()
International Journal of Dierential Equations
T : Results of le layer problem for =10
−3
and = 0.5.
exact
scem
1
exac𝑡
−
scem
1
scem
2
exact
−
scem
2
Method []
0.0000 1.0000000 1.0000000 0.0000000 0.0000000 0.0000000 1.0000000
0.0010 0.4538692 0.45379572 7.3488 − 05 0.4538692 9.2048 − 09 0.3171426
0.0020 0.3803509 0.38019363 1.5732 − 04 0.3803510 9.4728 − 08 0.3603879
0.0030 0.3707303 0.37055161 1.7865 − 04 0.3707303 1.2683 − 07 0.3666117
0.0040 0.3697488 0.36956596 1.8289 − 04 0.3697489 1.3552 − 07 0.3678324
0.0050 0.3699358 0.36975214 1.8364 − 04 0.3699359 1.3753 − 07 0.3683816
0.0100 0.3717605 0.37157669 1.8379 − 04 0.3717606 1.3821 − 07 0.3708603
0.0150 0.3736230 0.37343923 1.8378 − 04 0.3736231 1.3848 − 07 0.3733556
0.0200 0.3754949 0.37531110 1.8376 − 04 0.3754950 1.3869 − 07 0.3758674
0.1000 0.4067525 0.40656966 1.8281 − 04 0.4067526 1.4163 − 07 0.4071563
0.2000 0.4495085 0.44932896 1.7958 − 04 0.4495086 1.4362 − 07 0.4499552
0.4000 0.5489761 0.54881164 1.6450 − 04 0.5489762 1.3978 − 07 0.5495225
0.6000 0.6704540 0.67032005 1.3394 − 04 0.6704541 1.2051 − 07 0.6711221
0.8000 0.8188125 0.81873075 8.1795 − 05 0.8188126 7.7685 − 08 0.8196300
0.9000 0.9048826 0.90483742 4.5197 − 05 0.9048826 4.4056 − 08 0.9057869
1.0000 1.0000000 1.0000000 0.0000000 0.0000000 0.0000000 1.0000000
T : Results of le layer problem for =10
−4
and = 0.5.
Exact
scem
1
exact
−
scem
1
scem
2
exact
−
scem
2
Method []
0.0000 1.0000000 1.0000000 0.0000000 1.0000000 0.0000000 1.0000000
0.0001 0.4534718 0.4534644 7.3497 − 06 0.4534717 1.7565 − 10 0.3181581
0.0002 0.3795465 0.3795307 1.5740 − 05 0.3795464 1.1979 − 09 0.3612116
0.0003 0.3695746 0.3695566 1.7877 − 05 0.3695745 1.2808 − 09 0.3670360
0.0004 0.3682570 0.3682386 1.8301 − 05 0.3682569 1.3520 − 09 0.3678237
0.0005 0.3681105 0.3680921 1.8377 − 05 0.3681105 1.3718 − 09 0.3679338
0.0010 0.3682659 0.3682475 1.8392 − 05 0.3682658 1.3794 − 09 0.3682020
0.0015 0.3684501 0.3684316 1.8392 − 05 0.3684500 1.3806 − 09 0.3684702
0.0020 0.3686343 0.3686159 1.8392 − 05 0.3686343 1.3794 − 09 0.3687384
0.1000 0.4065880 0.4065696 1.8294 − 05 0.4065879 1.4170 − 09 0.4066914
0.2000 0.4493469 0.4493289 1.7971 − 05 0.4493469 1.4373 − 09 0.4494485
0.4000 0.5488281 0.5488116 1.6462 − 05 0.5488281 1.3990 − 09 0.5489215
0.6000 0.6703334 0.6703200 1.3405 − 05 0.6703334 1.2061 − 09 0.6704094
0.8000 0.8187389 0.8187307 8.1865 − 06 0.8187389 7.7755 − 10 0.8187855
0.9000 0.9048420 0.9048374 4.5237 − 06 0.9048419 4.4095 − 10 0.9048674
1.0000 1.0000000 1.0000000 0.0000000 1.0000000 0.0000000 1.0000000
4. Conclusion
In this paper, singularly perturbed second-order linear delay
dierential equations that have a delay in the convection
term are considered. Firstly, the delayed terms are linearized
using two-term Taylor series expansion. Later, an ecient
asymptotic method so cal led Successive Complementary
Expansion Method (SCEM) is employed so as to obtain a
uniformly valid approximation scheme. At the last stage,
the equations that come from the SCEM process are solved
by a numerical procedure and so the present method is an
asymptotic-numerical hybrid method. e method is easily
applicable since it does not require any matching principle
in contrast to the well-known method matched asymptotic
expansions (MMAE). Highly accurate approximations are
obtained in only few iterations and moreover boundary con-
ditions are not satised asymptotically, but exactly. In Tables
and , exact solution, present method approximations, and
approximations that are obtained by the method given in []
are compared and to show the eciency of present method,
results are supported by Figures , , and . In Figures
and , the delay eects are compared and since the right