Research Article Numerical Solution of Singularly ...downloads.hindawi.com/journals/aaa/2014/731057.pdf · Research Article Numerical Solution of Singularly Perturbed Delay Differential

Research ArticleNumerical Solution of Singularly Perturbed Delay DifferentialEquations with Layer Behavior

F Ghomanjani1 A KJlJccedilman2 and F Akhavan Ghassabzade1

1 Department of Applied Mathematics Faculty of Mathematical Sciences Ferdowsi University of Mashhad Mashhad Iran2Department of Mathematics and Institute for Mathematical Research University Putra Malaysia (UPM)43400 Serdang Selangor Malaysia

Correspondence should be addressed to A Kılıcman akilicupmedumy

Received 4 September 2013 Accepted 24 December 2013 Published 16 January 2014

Academic Editor Aref Jeribi

Copyright copy 2014 F Ghomanjani et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We present a numerical method to solve boundary value problems (BVPs) for singularly perturbed differential-difference equationswith negative shift In recent papers the term negative shift has been used for delay The Bezier curves method can solve boundaryvalue problems for singularly perturbed differential-difference equations The approximation process is done in two steps Firstwe divide the time interval into 119896 subintervals second we approximate the trajectory and control functions in each subintervalby Bezier curves We have chosen the Bezier curves as piecewise polynomials of degree 119899 and determined Bezier curves on anysubinterval by 119899+1 control pointsThe proposedmethod is simple and computationally advantageous Several numerical examplesare solved using the presentedmethod we compared the computed result with exact solution and plotted the graphs of the solutionof the problems

1 Introduction

In recent years there has been a growing interest in thesingularly perturbed delay differential equation (see [1ndash4])A singularly perturbed delay differential equation is anordinary differential equation in which the highest derivativeis multiplied by a small parameter and involving at leastone delay term Such types of differential equations arisefrequently in applications for example the first exit timeproblem in modeling of the activation of neuronal variability[5] in a variety of models for physiological processes ordiseases [6] to describe the human pupil-light reflex [7] andvariational problems in control theory and depolarization inSteinrsquos model [8] Investigation of boundary value problemsfor singularly perturbed linear second-order differential-difference equations was initiated by Lange and Miura [59 10] they proposed an asymptotic approach in studyof linear second-order differential-difference equations inwhich the highest order derivative is multiplied by smallparameters Kadalbajoo and Sharma [11ndash14] discussed thenumerical methods for solving such type of boundary valueproblems Amiraliyev and Erdogan [15] and Amiraliyeva

and Amiraliyev [16] developed robust numerical schemes fordealing with singularly perturbed delay differential equation

In the present work we suggest a technique similar to theonewhichwas used in [17 18] for solving singularly perturbeddifferential-difference equation with delay in the followingform (see [13])

12059811991010158401015840

(119905) + 119886 (119905) 1199101015840

(119905 minus 120575) + 119887 (119905) 119910 (119905) = 119891 (119905) 0 lt 119905 lt 1

119910 (119905) = 120601 (119905) minus120575 le 119905 le 0

119910 (1) = 120574

(1)

where 120598 is small parameter 0 lt 120598 ≪ 1 and 120575 is alsoa small shifting parameter 0 lt 120575 ≪ 1 119886(119905) 119887(119905) 119891(119905)and 120601(119905) are assumed to be smooth and 120574 is a constantFor 120575 = 0 the problem is a boundary value problem fora singularly perturbed differential equation and then as thesingular perturbation parameter tends to zero the order ofthe corresponding reduced problem is decreased by one sothere will be one layer It may be a boundary layer or an

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 731057 4 pageshttpdxdoiorg1011552014731057

2 Abstract and Applied Analysis

interior layer depending on the nature of the coefficient ofthe convection term

The current paper is organized as follows In Section 2function approximationwill be introducedNumerical exam-ples will be stated in Section 3 Finally Section 4 will give aconclusion briefly

2 Function Approximation

Consider the problem (1) Divide the interval [1199050 119905119891] into a

set of grid points such that

119905119894= 1199050+ 119894ℎ 119894 = 0 1 119896 (2)

where ℎ = (119905119891minus1199050)119896 119905119891= 1 1199050= 0 and 119896 is a positive integer

Let 119878119895= [119905119895minus1 119905119895] for 119895 = 1 2 119896 Then for 119905 isin 119878

119895 the

problem (1) can be decomposed to the following suboptimalcontrol problems

12059811991010158401015840

119895(119905) + 119886 (119905) 119910

minus1198962+119895(119905 minus 120575) + 119887 (119905) 119910

119895(119905) = 119891 (119905) 119905 isin 119878

119895

119910119895(119905) = 120601 (119905) minus120575 le 119905 le 119905

0 119895 = 1 2 119896

119910119896(1) = 120574

(3)

where 119910119895(119905)

We mention that 119910minus1198962+119895(119905 minus 120575) is defined where (119905 minus 120575) isin

[119905minus1198962+119895minus1 119905minus1198962+119895] Also

1198962=

120575

ℎ

120575

ℎisin N

([120575

ℎ] + 1)

120575

ℎnotin N

(4)

where [120575ℎ] denotes the integer part of 120575ℎLet119910(119905) = sum119896

119895=11205941

119895(119905)119910119895(119905)where1205941

119895(119905) is the characteristic

function of 119910119895(119905) for 119905 isin [119905

119895minus1 119905119895] It is trivial that [119905

0 119905119891] =

⋃119896

119895=1119878119895

Our strategy is using Bezier curves to approximate thesolutions 119910

119895(119905) by V

119895(119905) where V

119895(119905) is given below Individual

Bezier curves that are defined over the subintervals arejoined together to form the Bezier spline curves For 119895 =1 2 119896 define the Bezier polynomials V

119895(119905) of degree 119899 that

approximate the action of 119910119895(119905) over the interval [119905

119895minus1 119905119895] as

follows

V119895(119905) =

119899

sum

119903=0

119886119895

119903119861119903119899(119905 minus 119905119895minus1

ℎ) (5)

where

119861119903119899(119905 minus 119905119895minus1

ℎ) = (

119899

119903)1

ℎ119899(119905119895minus 119905)119899minus119903

(119905 minus 119905119895minus1)119903

(6)

is the Bernstein polynomial of degree 119899 over the interval[119905119895minus1 119905119895] and 119886119895

119903is the control points (see [17]) By substituting

(5) in (3) one may define 1198771119895(119905) for 119905 isin [119905

119895minus1 119905119895] as

1198771119895(119905) = 120598V10158401015840

119895(119905) + 119886 (119905) Vminus1198962+119895

119895(119905 minus 120575) + 119887 (119905) V

119895(119905) minus 119891 (119905)

(7)

Let V(119905) = sum119896119895=11205941

119895(119905)V119895(119905) where 1205941


function of V119895(119905) for 119905 isin [119905

119895minus1 119905119895] Beside the boundary con-

ditions on V(119905) at each node we need to impose continuitycondition on each successive pair of V

119895(119905) to guarantee the

smoothness Since the differential equation is of first orderthe continuity of 119910 (or V) and its first derivative give

V(119904)119895(119905119895) = V(119904)119895+1(119905119895) 119904 = 0 1 119895 = 1 2 119896 minus 1 (8)

where V(119904)119895(119905119895) is the 119904th derivative V

119895(119905) with respect to 119905 at

119905 = 119905119895

Thus the vector of control points 119886119895119903(119903 = 0 1 119899 minus 1 119899)

must satisfy (see [17])

119886119895

119899(119905119895minus 119905119895minus1)119899

= 119886119895+1

0(119905119895+1minus 119905119895)119899

(119886119895

119899minus 119886119895

119899minus1) (119905119895minus 119905119895minus1)119899minus1

= (119886119895+1

1minus 119886119895+1

0) (119905119895+1minus 119905119895)119899minus1

(9)

Ghomanjani et al [17] proved the convergence of thismethod where ℎ rarr 0

Now the residual function can be defined in 119878119895as follow

119877119895= int

119905119895

119905119895minus1

100381710038171003817100381710038171198771119895(119905)10038171003817100381710038171003817

2

119889119905 (10)

where sdot is the Euclidean norm and119872 is a sufficiently largepenalty parameter Our aim is solving the following problemover 119878 = ⋃119896

119895=1119878119895

min119896

sum

119895=1

119877119895

st 119886119895

119899(119905119895minus 119905119895minus1)119899

= 119886119895+1

0(119905119895+1minus 119905119895)119899

(119886119895

119899minus 119886119895


= (119886119895+1

1minus 119886119895+1

0) (119905119895+1minus 119905119895)119899minus1

V119895(119905) = 120601 (119905) minus120575 le 119905 le 119905

0 119895 = 1 2 119896

V119896(119905119891) = 120574

(11)

The mathematical programming problem (11) can be solvedby many subroutine algorithms Here we use Maple 12 tosolve this optimization problem

3 Numerical Results and Discussion

Consider the following examples which can be solved byusing the presented method

Example 1 First we consider the problem (see [11])

12059811991010158401015840

(119905) + 1199101015840

(119905 minus 120575) minus 119910 (119905) = 0 0 lt 119905 lt 1 (12)

under the boundary conditions

119910 (119905) = 1 minus120575 le 119905 le 0

119910 (1) = 1

(13)

Abstract and Applied Analysis 3

Table 1 The maximum error for 120598 = 01 and for different 120575 forExample 1

120575 Max error in [11] Max error of presented method001 001182463 00045003 001515596 00090006 002584799 00070008 008313177 00300

10

09

08

07

06

0 02 04 06 08 1

t

ApproximateExact

Figure 1 Graphs of the exact and computed solution of the BVPwith 120598 = 01 and 120575 = 001 for Example 1

A boundary layer exists on left side of the interval For thisproblem the exact solution is

119910 (119905) =(1 minus 119890

1198982) 1198901198981119905+ (1198901198981 minus 1) 119890

1198982119905

(1198901198981 minus 1198901198982) (14)

where

1198981=minus1 minus radic1 + 4 (120598 minus 120575)

2 (120598 minus 120575)

1198982=minus1 + radic1 + 4 (120598 minus 120575)

2 (120598 minus 120575)

(15)

Also we have plotted the graphs of the exact and computedsolution of the problem in Figure 1 The maximum errors areshown in Table 1

Example 2 Next we consider the problem (see [11])

12059811991010158401015840

(119905) minus 1199101015840

(119905 minus 120575) minus 119910 (119905) = 0 0 lt 119909 lt 1 (16)


119910 (119905) = 1 minus120575 le 119905 le 0

119910 (1) = minus1

(17)


120575 Max error of presented method001 0007003 0022006 0023008 0025

t

0 02 04 06 08 1

1

05

minus05

minus1

ApproximateExact


A boundary layer exists on right side of the interval For thisproblem the exact solution is

119910 (119905) =(1 + 119890

1198982) 1198901198981119905minus (1 + 119890

1198981) 1198901198982119905

(1198901198982 minus 1198901198981) (18)

where

1198981=1 minus radic1 + 4 (120598 + 120575)

2 (120598 + 120575)

1198982=1 + radic1 + 4 (120598 + 120575)

2 (120598 + 120575)

(19)

Also we have plotted the graphs of the exact and computedsolution of the problem in Figure 2Themaximum errors areshown in Table 2

4 Conclusions

We have described a numerical algorithm for solving BVPsfor singularly perturbed differential-difference equation withsmall shifts Here we have discussed both the cases by usingBezier curves when boundary layer is on the left side andwhen boundary layer is on the right side of the underlying


interval Numerical examples show that the proposedmethodis efficient and very easy to use

Conflict of Interests

The authors declare that they have no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors would like to thank the anonymous reviewersfor their careful reading constructive comments and nicesuggestions which have improved the paper very much

References

[1] K C Patidar and K K Sharma ldquo120598-uniformly convergent non-standard finite difference methods for singularly perturbeddifferential difference equations with small delayrdquo AppliedMathematics and Computation vol 175 no 1 pp 864ndash8902006

[2] M K Kadalbajoo and K K Sharma ldquoNumerical treatment forsingularly perturbed nonlinear differential difference equationswith negative shiftrdquo Nonlinear Analysis Theory Methods andApplications vol 63 no 5 pp e1909ndashe1924 2005

[3] M K Kadalbajoo and K K Sharma ldquoParameter-uniform fittedmesh method for singularly perturbed delay differential equa-tions with layer behaviorrdquo Electronic Transactions on NumericalAnalysis vol 23 pp 180ndash201 2006

[4] P Rai and K K Sharma ldquoNumerical analysis of singularlyperturbed delay differential turning point problemrdquo AppliedMathematics and Computation vol 218 no 7 pp 3483ndash34982011

[5] C G Lange and RMMiura ldquoSingular perturbation analysis ofboundary value problems for differential-difference equationsV small shifts with layer behaviorrdquo SIAM Journal on AppliedMathematics vol 54 no 1 pp 249ndash272 1994

[6] M C Mackey and L Glass ldquoOscillations and chaos in physio-logical control systemrdquo Science vol 197 pp 287ndash289 1997

[7] A Longtin and J G Milton ldquoComplex oscillations in thehuman pupil light reflex with ldquomixedrdquo and delayed feedbackrdquoMathematical Biosciences vol 90 no 1-2 pp 183ndash199 1988

[8] V Y Glizer ldquoAsymptotic analysis and solution of a finite-horizon 119867

infincontrol problem for singularly-perturbed linear

systems with small state delayrdquo Journal of Optimization Theoryand Applications vol 117 no 2 pp 295ndash325 2003

[9] C G Lange and RMMiura ldquoSingular perturbation analysis ofboundary value problems for differential-difference equationsrdquoSIAM Journal on Applied Mathematics vol 42 no 3 pp 502ndash531 1982

[10] C G Lange and RMMiura ldquoSingular perturbation analysis ofboundary value problems for differential-difference equationsVI small shiftswith rapid oscillationsrdquo SIAM Journal onAppliedMathematics vol 54 no 1 pp 273ndash283 1994

[11] M K Kadalbajoo and K K Sharma ldquoNumerical analysis ofsingularly perturbed delay differential equations with layerbehaviorrdquo Applied Mathematics and Computation vol 157 no1 pp 11ndash28 2004

[12] M K Kadalbajoo and K K Sharma ldquoNumerical treatmentof a mathematical model arising from a model of neuronal

variabilityrdquo Journal of Mathematical Analysis and Applicationsvol 307 no 2 pp 606ndash627 2005

[13] M K Kadalbajoo and K K Sharma ldquoA numerical methodbased on finite difference for boundary value problems for sin-gularly perturbed delay differential equationsrdquo Applied Mathe-matics and Computation vol 197 no 2 pp 692ndash707 2008

[14] P Rai and K K Sharma ldquoNumerical study of singularlyperturbed differential-difference equation arising in the mod-eling of neuronal variabilityrdquo Computers amp Mathematics withApplications vol 63 no 1 pp 118ndash132 2012

[15] G M Amiraliyev and F Erdogan ldquoUniform numerical methodfor singularly perturbed delay differential equationsrdquo Comput-ersampMathematicswithApplications vol 53 no 8 pp 1251ndash12592007

[16] I G Amiraliyeva and G M Amiraliyev ldquoUniform differencemethod for parameterized singularly perturbed delay differen-tial equationsrdquoNumerical Algorithms vol 52 no 4 pp 509ndash5212009

[17] F Ghomanjani M H Farahi and M Gachpazan ldquoBeziercontrol points method to solve constrained quadratic optimalcontrol of time varying linear systemsrdquo Computational ampApplied Mathematics vol 31 no 3 pp 433ndash456 2012

[18] F Ghomanjani A Kılıcman and S Effati ldquoNumerical solutionfor IVP in Volterra type linear integro-differential equationssystemrdquo Abstract and Applied Analysis vol 2013 Article ID490689 4 pages 2013

Submit your manuscripts athttpwwwhindawicom

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


interior layer depending on the nature of the coefficient ofthe convection term

The current paper is organized as follows In Section 2function approximationwill be introducedNumerical exam-ples will be stated in Section 3 Finally Section 4 will give aconclusion briefly

2 Function Approximation

Consider the problem (1) Divide the interval [1199050 119905119891] into a

set of grid points such that

119905119894= 1199050+ 119894ℎ 119894 = 0 1 119896 (2)

where ℎ = (119905119891minus1199050)119896 119905119891= 1 1199050= 0 and 119896 is a positive integer

Let 119878119895= [119905119895minus1 119905119895] for 119895 = 1 2 119896 Then for 119905 isin 119878

119895 the

problem (1) can be decomposed to the following suboptimalcontrol problems

12059811991010158401015840

119895(119905) + 119886 (119905) 119910

minus1198962+119895(119905 minus 120575) + 119887 (119905) 119910

119895(119905) = 119891 (119905) 119905 isin 119878

119895

119910119895(119905) = 120601 (119905) minus120575 le 119905 le 119905

0 119895 = 1 2 119896

119910119896(1) = 120574

(3)

where 119910119895(119905)

We mention that 119910minus1198962+119895(119905 minus 120575) is defined where (119905 minus 120575) isin

[119905minus1198962+119895minus1 119905minus1198962+119895] Also

1198962=

120575

ℎ

120575

ℎisin N

([120575

ℎ] + 1)

120575

ℎnotin N

(4)

where [120575ℎ] denotes the integer part of 120575ℎLet119910(119905) = sum119896

119895=11205941

119895(119905)119910119895(119905)where1205941


function of 119910119895(119905) for 119905 isin [119905

119895minus1 119905119895] It is trivial that [119905

0 119905119891] =

⋃119896

119895=1119878119895

Our strategy is using Bezier curves to approximate thesolutions 119910

119895(119905) by V

119895(119905) where V

119895(119905) is given below Individual

Bezier curves that are defined over the subintervals arejoined together to form the Bezier spline curves For 119895 =1 2 119896 define the Bezier polynomials V

119895(119905) of degree 119899 that

approximate the action of 119910119895(119905) over the interval [119905

119895minus1 119905119895] as

follows

V119895(119905) =

119899

sum

119903=0

119886119895

119903119861119903119899(119905 minus 119905119895minus1

ℎ) (5)

where

119861119903119899(119905 minus 119905119895minus1

ℎ) = (

119899

119903)1

ℎ119899(119905119895minus 119905)119899minus119903

(119905 minus 119905119895minus1)119903

(6)

is the Bernstein polynomial of degree 119899 over the interval[119905119895minus1 119905119895] and 119886119895

119903is the control points (see [17]) By substituting

(5) in (3) one may define 1198771119895(119905) for 119905 isin [119905

119895minus1 119905119895] as

1198771119895(119905) = 120598V10158401015840

119895(119905) + 119886 (119905) Vminus1198962+119895

119895(119905 minus 120575) + 119887 (119905) V

119895(119905) minus 119891 (119905)

(7)

Let V(119905) = sum119896119895=11205941

119895(119905)V119895(119905) where 1205941


function of V119895(119905) for 119905 isin [119905

119895minus1 119905119895] Beside the boundary con-

ditions on V(119905) at each node we need to impose continuitycondition on each successive pair of V

119895(119905) to guarantee the

smoothness Since the differential equation is of first orderthe continuity of 119910 (or V) and its first derivative give

V(119904)119895(119905119895) = V(119904)119895+1(119905119895) 119904 = 0 1 119895 = 1 2 119896 minus 1 (8)

where V(119904)119895(119905119895) is the 119904th derivative V

119895(119905) with respect to 119905 at

119905 = 119905119895

Thus the vector of control points 119886119895119903(119903 = 0 1 119899 minus 1 119899)

must satisfy (see [17])

119886119895

119899(119905119895minus 119905119895minus1)119899

= 119886119895+1

0(119905119895+1minus 119905119895)119899

(119886119895

119899minus 119886119895


= (119886119895+1

1minus 119886119895+1

0) (119905119895+1minus 119905119895)119899minus1

(9)

Ghomanjani et al [17] proved the convergence of thismethod where ℎ rarr 0

Now the residual function can be defined in 119878119895as follow

119877119895= int

119905119895

119905119895minus1

100381710038171003817100381710038171198771119895(119905)10038171003817100381710038171003817

2

119889119905 (10)

where sdot is the Euclidean norm and119872 is a sufficiently largepenalty parameter Our aim is solving the following problemover 119878 = ⋃119896

119895=1119878119895

min119896

sum

119895=1

119877119895

st 119886119895

119899(119905119895minus 119905119895minus1)119899

= 119886119895+1

0(119905119895+1minus 119905119895)119899

(119886119895

119899minus 119886119895


= (119886119895+1

1minus 119886119895+1

0) (119905119895+1minus 119905119895)119899minus1

V119895(119905) = 120601 (119905) minus120575 le 119905 le 119905

0 119895 = 1 2 119896

V119896(119905119891) = 120574

(11)

The mathematical programming problem (11) can be solvedby many subroutine algorithms Here we use Maple 12 tosolve this optimization problem

3 Numerical Results and Discussion

Consider the following examples which can be solved byusing the presented method

Example 1 First we consider the problem (see [11])

12059811991010158401015840

(119905) + 1199101015840

(119905 minus 120575) minus 119910 (119905) = 0 0 lt 119905 lt 1 (12)


119910 (119905) = 1 minus120575 le 119905 le 0

119910 (1) = 1

(13)




10

09

08

07

06

0 02 04 06 08 1

t

ApproximateExact



119910 (119905) =(1 minus 119890

1198982) 1198901198981119905+ (1198901198981 minus 1) 119890

1198982119905

(1198901198981 minus 1198901198982) (14)

where


2 (120598 minus 120575)


2 (120598 minus 120575)

(15)



12059811991010158401015840

(119905) minus 1199101015840

(119905 minus 120575) minus 119910 (119905) = 0 0 lt 119909 lt 1 (16)


119910 (119905) = 1 minus120575 le 119905 le 0

119910 (1) = minus1

(17)



t

0 02 04 06 08 1

1

05

minus05

minus1

ApproximateExact



119910 (119905) =(1 + 119890

1198982) 1198901198981119905minus (1 + 119890

1198981) 1198901198982119905

(1198901198982 minus 1198901198981) (18)

where

1198981=1 minus radic1 + 4 (120598 + 120575)

2 (120598 + 120575)

1198982=1 + radic1 + 4 (120598 + 120575)

2 (120598 + 120575)

(19)


4 Conclusions






Acknowledgment


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












10

09

08

07

06

0 02 04 06 08 1

t

ApproximateExact



119910 (119905) =(1 minus 119890

1198982) 1198901198981119905+ (1198901198981 minus 1) 119890

1198982119905

(1198901198981 minus 1198901198982) (14)

where


2 (120598 minus 120575)


2 (120598 minus 120575)

(15)



12059811991010158401015840

(119905) minus 1199101015840

(119905 minus 120575) minus 119910 (119905) = 0 0 lt 119909 lt 1 (16)


119910 (119905) = 1 minus120575 le 119905 le 0

119910 (1) = minus1

(17)



t

0 02 04 06 08 1

1

05

minus05

minus1

ApproximateExact



119910 (119905) =(1 + 119890

1198982) 1198901198981119905minus (1 + 119890

1198981) 1198901198982119905

(1198901198982 minus 1198901198981) (18)

where

1198981=1 minus radic1 + 4 (120598 + 120575)

2 (120598 + 120575)

1198982=1 + radic1 + 4 (120598 + 120575)

2 (120598 + 120575)

(19)


4 Conclusions






Acknowledgment


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Acknowledgment


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Numerical Solution of Singularly ...downloads.hindawi.com/journals/aaa/2014/731057.pdf · Research Article Numerical Solution of Singularly Perturbed Delay Differential