Printer Friendly

An Asymptotic-Numerical Hybrid Method for Solving Singularly Perturbed Linear Delay Differential Equations.

1. Introduction

Almost all physical phenomena in nature are modelled using differential equations, and singularly perturbed problems are vital class of these kinds of problems. In general, a singular perturbation problem is defined as a differential equation that is controlled by a positive small parameter 0 < [epsilon] [much less than] 1 that exists as multiplier to the highest derivative term in the differential equation. As [epsilon] tends to zero, the solution of problem exhibits interesting behaviors (rapid changes) since the order of the equation reduces. The 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 [1, 2], these kinds of problems arise in almost all applied natural sciences. Some of these can be given as mechanical and electrical systems, celestial mechanics, fluid and solid mechanics, electromagnetics, particle and quantum physics, chemical and biochemical reactions, and economics and financial mathematics. Various methods are employed to solve singular perturbation problems analytically, numerically, or asymptotically such as the method of matched asymptotic expansions (MMAE), the method of multiple scales, the method of WKB approximation, Poincare-Lindstedt method and periodic averaging method. Rigorous analysis and applications of these methods can be found in [3-8].

Modelling automatic systems often involve the idea of control because feedback is necessary in order to maintain a stable state. But much of this feedback require a finite time to sense information and react to it. A general way for describing this process is to formulate a delay differential equation (difference-differential equation). Delay differential equations (DDE) are widely used for modelling problems in population dynamics, nonlinear optics, fluid mechanics, mechanical engineering, evolutionary biology, and even modelling of HIV infection and human pupil-light reflex. One can refer to [10-14] for general theory and applications of DDEs.

In this paper, we study an important class of delay differential equations: singularly perturbed linear delay differential equations. A singularly perturbed delay differential equation is a differential equation in which the highest-order derivative is multiplied by a positive small e parameter and involving at least one delay term. We restrict our attention to singularly perturbed second-order ordinary delay differential equations that contains the delay in convection term. Various methods have been used to solve singularly perturbed DDEs such as finite difference methods [9, 15,16], finite element methods [17,18], homotopy perturbation method [19,20], reproducing kernel method [21, 22], spline collocation methods [23, 24], and asymptotic approaches [25, 26]. We use an asymptotic-numerical hybrid method in order to find uniformly valid approximations to singularly perturbed ODEs. At the first step, two-term Taylor series expansion is used to vanish delayed term. Secondly, to obtain a uniformly valid approximation an efficient and easily applicable asymptotic method so called Successive Complementary Expansion Method (SCEM) that was introduced in [27] is employed. Finally, a numerical approach is used to solve resulting equations that come from SCEM process.

2. Description of the Method

In this section, we first 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 g([epsilon]) and h([epsilon]) that depend on a positive small parameter [epsilon]; then g([epsilon]) = 0(h([epsilon])) for [epsilon] [right arrow] 0 if there exist positive constants K and [[epsilon].sub.0] such that K [member of] (0, [[epsilon].sub.0]] with [absolute value of (g([epsilon]))] [??] K[absolute value of (h([epsilon]))] for [epsilon] [right arrow] 0, and g([epsilon]) = o(h([epsilon])) for [epsilon] [right arrow] 0 if [lim.sub. [epsilon][right arrow]0] (g([epsilon])/h([epsilon])) = 0. Let E be a set of real functions that depend on [epsilon], strictly positive and continuous in (0, [e.sub.[epsilon]]], such that [lim.sub.[epsilon][right arrow]0][delta](e) exists and [[delta].sub.1]([epsilon])[[delta].sub.2]([epsilon]) [member of] E for each [[delta].sub.1]([epsilon]), [[delta].sub.2]([epsilon]) [member of] E. A function [[delta].sub.i]([epsilon]) that satisfies these conditions is called order function. Given two functions [phi](x, [epsilon]) and [[phi].sub.a](x, [epsilon]) defined in a domain [OMEGA] are asymptotically identical to order [DELTA]([epsilon]) if their difference is asymptotically smaller than [delta]([epsilon]), where [delta]([epsilon]) is an order function; that is,

[phi](x, [epsilon]) - [[phi].sub.a](x, [epsilon]) = o([delta]([epsilon])), (1)

where [epsilon] is a positive small parameter arising from the physical problem under consideration. The function [[phi].sub.a](x, [epsilon]) is named as asymptotic approximation of the function [phi](x, [epsilon]). Asymptotic approximations in general form are defined by

[[phi].sub.a](x, [epsilon]) = [n.summation over (i=1)][[delta].sub.i]([epsilon])[[phi].sub.i](x, [epsilon]), (2)

where [d.sub.i+1]([epsilon]) = o([[delta].sub.i]([epsilon])), as [epsilon] [right arrow] 0. Under these conditions, the approximation (2) is named as generalized asymptotic expansion. If the expansion (2) is written in the form of

[[phi].sub.a](x, [epsilon]) = [E.sub.0[phi]] = [n.summation over (i=1)][[delta].sup.(0).sub.i]([epsilon])[[phi].sup.(0).sub.i](x), (3)

then it is called regular asymptotic expansion where the special operator [E.sub.0] is outer expansion operator of a given order [delta]([epsilon]). Thus, [phi] - [E.sub.0][phi] = o([delta]([epsilon])). For more detailed information about the asymptotic approximations, we refer the interested reader to [3-8, 28, 29].

Interesting behaviors occur when the function [phi](x, [epsilon]) is not regular in [OMEGA] so (2) or (3) is valid only in a restricted region [[OMEGA].sub.0] e [OMEGA], called the outer region. We introduce an inner domain which can be formally denoted as [[OMEGA].sub.1] = [OMEGA] - [[OMEGA].sub.0] and corresponding inner layer variable, located near the point x = [x.sub.0], as [bar.x] = (x - [x.sub.0])/[eta]([epsilon]), with [eta]([epsilon]) being the order of thickness of the boundary layer (the region in which the rapid changes-behaviors occur). If a regular expansion can be constructed in [[OMEGA].sub.1], one can write down the approximation as

[[phi].sub.a](x, [epsilon]) = [E.sub.1[phi]] = [n.summation over (i=1)][[delta].sup.(1).sub.i]([epsilon])[[phi].sup.(1).sub.i]([bar.x]), (4)

where the inner expansion operator [E.sub.1], defined in [[OMEGA].sub.1], is of the same order [delta]([epsilon]) as the outer expansion operator [E.sub.0]; that is, [phi] - [E.sub.1][phi] = o([delta]([epsilon])). Thus,

[[phi].sub.a] = [E.sub.0][phi] + [E.sub.1][phi] - [E.sub.1][phi] - [E.sub.1][E.sub.0][phi] (5)

is clearly uniformly valid approximation (UVA) [28-30].

Now let us consider second-order singularly perturbed DDE in its general form (delay in the convection term):

[epsilon]y"(x) + p(x) y'(x - [delta]) + q (x) y(x) = r (x), (6)

where 0 < [epsilon] [much less than] 1 small parameter and 0< x < 1. Boundary and interval conditions are given as

y(x) = [phi](x), -[delta] [less than or equal to] x [less than or equal to] 0, y(1) = [gamma], (7)

where p(x), q(x), r(x), and [phi](x) are smooth functions, [gamma] [member of] R, and [delta] is delay term.

As [epsilon] tends to zero, the order of the differential equation reduces and so a layer occurs in the solution. The sign of p(x) 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 p(x) does not change, there are two possibilities: if p(x) < 0 on [0,1], then a boundary layer occurs at the right end (near the point x =1) and if p(x) > 0 on [0,1], then a boundary layer occurs at the left end (near the point x = 0).

Using Taylor series expansion we linearize the convection term; that is, y'(x - [delta]) = y'(x) - [delta]y"(x) and substituting it into (6) one can reach

([epsilon] - [delta]p(x)) y"(x) + p(x) y'(x) + q(x) y(x) = r(x). (8)

Letting [delta] = [kappa][epsilon]e, where [kappa] [member of] R

[epsilon](1 - [kappa]p (x)) y"(x) + p(x) y'(x) + q(x) y(x) =r(x) (9)

is found and it is clear that (9) is a singularly perturbed ordinary differential equation for [epsilon] [right arrow] 0 with the same boundary and interval conditions as given by (7). SCEM procedure is applicable at this stage.

The uniformly valid SCEM approximation is in the regular form given by

[y.sup.scem.sub.n](x, [bar.x], [epsilon]) = [n.summation over (i=0)][[delta].sub.i]([epsilon]) [[y.sub.i](x) + [[PSI].sub.i]([bar.x])], (10)

where {[[delta].sub.i]([epsilon])} is an asymptotic sequence and functions [[PSI].sub.i]([bar.x]) are the complementary functions that depend on [bar.x].

If the functions [y.sub.i](x) and [[PSI].sub.i]([bar.x]) depend also on [epsilon], the uniformly valid SCEM approximation is called generalized SCEM approximation and given by

[y.sup.scem.sub.ng](x, [bar.x], [epsilon]) = [n.summation over (i=0)][[delta].sub.i]([epsilon]) [[y.sub.i](x, [epsilon]) + [[PSI].sub.i]([bar.x], [epsilon])]. (11)

If only one-term SCEM approximation is desired, then one seeks a uniformly valid SCEM approximation in the form of

[y.sup.scem.sub.1](x, [bar.x], [epsilon]) = [y.sub.1](x, [epsilon]) + [[PSI].sub.1]([bar.x], [epsilon]). (12)

To improve the accuracy of approximation, (12) can be iterated using (11). It means that successive complementary terms will be added to the approximation. To this end, second SCEM approximation will be in the form of

[y.sup.scem.sub.2](x, [bar.x], [epsilon]) = [y.sub.1](x, [epsilon]) + [[PSI].sub.1]([bar.x], [epsilon]) + [epsilon] ([y.sub.2](x, [epsilon]) + [[PSI].sub.2](x, [epsilon])). (13)

In this work, we seek an approximation in our calculations in the form of (13).

Now, let us assume that problem (9) has a left boundary layer (near the point x = 0) and let [y.sub.out](x) be asymptotic approximation to the outer solution and let [PSI]([bar.x]) be the complementary solution, where [bar.x] = x/[epsilon] is boundary layer (stretching) variable. If approximations

[mathematical expression not reproducible] (14)

are substituted into (9) and if each term is balanced with respect to the powers of [epsilon] (we balance just the terms 0(1) and O([epsilon])),

[mathematical expression not reproducible] (15)

are found. If the same procedure is applied for equations that involve complementary functions

[[PSI]".sub.1]([bar.x], [epsilon]) + p(x) [[PSI]'.sub.1]([bar.x], [epsilon]) = r(x) (16)

with the boundary conditions

[mathematical expression not reproducible] (17)

with the boundary conditions

[[PSI].sub.1](0, [epsilon]) = -[y.sub.2](0, [epsilon]), [[PSI].sub.1]([1/[epsilon]], [epsilon]) = 0 (18)

being obtained and so (13) gives uniformly valid second SCEM approximation.

3. Illustrative Examples

3.1. Left Boundary Layer Problem. Consider singularly perturbed DDE that exhibits a boundary layer at the left end of the interval:

[epsilon]y"(x) + y'(x - [delta]) - y(x) = 0, 0 [less than or equal to] x [less than or equal to] 1, (19)

with the boundary conditions y(0) = 1 and y(1) = 1. The exact solution of this problem is given by [mathematical expression not reproducible] As the first step, we use two-term Taylor expansion for y'(x - [delta]) = y'(x) - [delta]y"(x). If we substitute it into (19), the problem turns into

([epsilon] - [delta]) y"(x) + y(x) - y(x) = 0, 0 [less than or equal to] x [less than or equal to] 1. (20)

In order to obtain a uniformly valid approximation (UVA), we first seek an outer solution which is valid far from the boundary layer (the boundary layer is near the point x = 0 for this problem) and then using SCEM we add complementary solution to it. Later, using the same idea, we will get more accurate approximations.

Outer Region Solutions. Let us take [theta] = [epsilon] - [delta], assuming that [delta] depends on [epsilon], and adopt a solution for the outer region in the form of [y.sub.out](x) = [y.sub.1](x, [theta]) + [theta][y.sub.2](x, [theta]). Equation (20) turns into

[theta]([y".sub.1](x, [theta]) + [theta][y".sub.2](x, [theta])) + ([y'.sub.1](x, [theta]) + [theta][y'.sub.2](x, [theta])) -(y(x, [theta]) + [theta][y.sub.2](x, [theta])) = 0 (21)

and balancing the terms of the order O(1) and O([theta]), we reach the equations

[y'.sub.1](x, [theta]) - [y.sub.1](x, [theta]) = 0, [y.sub.1](1, [theta]) = 1, [y".sub.1](x, [theta]) + [y'.sub.2](x, [theta]) - [y.sub.2](x, [theta]) = 0, [y.sub.2](1,0) = 0. (22)

One can easily find the exact solutions of these equations as

[y.sub.1](x, [theta]) = [e.sup.x-1], [y.sub.2](x, [theta]) = [e.sup.x-1](1 - x). (23)

It means that the outer solution is of the form

[y.sub.out](x, [theta]) = [e.sup.x-1] + [theta][e.sup.x-1](x - 1). (24)

Complementary Solutions. Now applying the stretching transformation [bar.x] = x/[theta] and adopting the complementary solution as [PSI]([bar.x], [theta]) = [[PSI].sub.1]([bar.x], [theta]) + [theta][[PSI].sub.2] ([bar.x], [theta]), one can reach

[mathematical expression not reproducible] (25)

Balancing the terms of the order O(1) and O([theta]), we obtain

[[PSI]".sub.1]([bar.x], [theta]) + [[PSI]".sub.1]([bar.x], [theta]) = 0, [[PSI].sub.1](0, [theta]) = 1, [[PSI].sub.1]([1/[theta]]], [theta]) = 0, (26)

[[PSI]".sub.2]([bar.x], [theta]) + [[PSI]'.sub.1]([bar.x], [theta]) - [[PSI].sub.1]([bar.x], [theta]) = 0, [[PSI].sub.2](0, [theta]) = -[e.sup.-1], [[PSI].sub.2]([1/[theta]], [theta]) = 0. (27)

Here we are able to solve [[PSI].sub.1]([bar.x], [theta]) and [[PSI].sub.2]([bar.x], [theta]) exactly, but in many cases to obtain analytical solution to [[PSI].sub.1]([bar.x], [theta]) and [[PSI].sub.2]([bar.x], [theta]) is really tedious process, even for many problems it is impossible. The solutions may be given as

[[PSI].sub.1]([bar.x], [theta]) = [e - 1/e(1 - [e.sup.-1/[theta]])]([e.sup.-[bar.x]] - [e.sup.-1//[theta]]) (28)

and [[PSI].sub.2]([bar.x], [theta]) is given as the solution of (27). Since we solve (27) numerically (MATLAB bvp4c routine) and the others using an asymptotic scheme, our present method is a hybrid method. As a result, we obtain first two SCEM approximations to problem (19) as follows:

[mathematical expression not reproducible] (29)

3.2. Right Boundary Layer Problem. Consider singularly perturbed DDE that exhibits a boundary layer at the right end of the interval

[epsilon]y" (x) - y' (x - [delta]) + y(x) = 0, (30)

with the boundary and interval conditions

y(x) = 1, - [delta] [less than or equal to] x [less than or equal to] 0, y(1) = -1. (31)

Using two-term Taylor expansion for the convection term, we reach y'(x - [delta]) = y'(x) - [delta]y"(x) and applying it to (30) one can obtain

([epsilon] + [delta]) y"(x) - y' x) + y(x) = 0, 0 [less than or equal to] x [less than or equal to] 1 (32)

with the boundary conditions y(0) = 1 and y(1) = -1.

In order to obtain a uniformly valid approximation, we first seek an outer solution which is valid for far from the boundary layer (the boundary layer is near the point x = 1 for this problem) and then using SCEM we add complementary solution to it. Later, using the same idea, we will get more accurate approximations.

Outer Region Solutions. Let us take [theta] = [epsilon] + [delta] assuming that [delta] depends on [epsilon] and adopt an approximation for the outer region in the form of [y.sub.out](x) = [y.sub.1](x, [theta]) + [theta][y.sub.2](x, [theta]). Equation (32) turns into

[theta]([y".sub.1](x, [theta]) + [theta][y".sub.2](x, [theta])) - ([y'.sub.1](x, [theta]) + [theta][y'.sub.2](x, [theta])) + (y (x, [theta]) + [theta][y.sub.2] (x, [theta])} = 0; (33)

balancing the terms of the order 0(1) and O([theta]), we reach the equations

[y'.sub.1](x, [theta]) - [y.sub.1](x, [theta]) = 0, [y.sub.1](0, [theta]) = 1, [y".sub.1](x, [theta]) - [y'.sub.2](x, [theta]) + [y.sub.2](x, [theta]) = 0, [y.sub.2](0,[theta]) = 0. (34)

One can easily find the exact solutions of these equations as

[y.sub.1](x, [theta]) = [e.sup.x], [y.sub.2](x, [theta]) = [e.sup.x]x . (35)

It means that the outer solution is of the form

[y.sub.out](x, [theta]) = [e.sup.x] + [theta][e.sup.x]x. (36)

Complementary Solutions. Now applying the stretching transformation [bar.x] = (x - 1)/[theta] and adopting the complementary solution as [PSI](x, [theta]) = [[PSI].sub.1]([bar.x], [theta]) + [[PSI].sub.2]([bar.x], [theta]) one can reach

[mathematical expression not reproducible] (37)

Balancing the terms of the order 0(1) and 0([theta]) we obtain

[[PSI]".sub.1]([bar.x], [theta]) - [[PSI]'.sub.1]([bar.x], [theta]) = 0, [[PSI].sub.1]([-1/[theta]], [theta]) = 0, [[PSI].sub.1](0, [theta]) = -1 - e, (38)

[[PSI]".sub.2]([bar.x], [theta]) - [[PSI]'.sub.2]([bar.x], [theta]) + [[PSI].sub.1]([bar.x], [theta]) = 0, [[PSI].sub.2](-[1/[theta]], [theta]) = 0, [[PSI].sub.2](0, [theta]) = -1 - e. (39)

The solutions are given as

[[PSI].sub.1]([bar.x], [theta]) = [e + 1/[e.sup.-1/[theta]]]([e.sup.[bar.x]] - 1) - e - 1 (40)

and [[PSI].sub.2]([bar.x], [theta]) is given as the solution of (39). Thus, we reach uniformly valid SCEM approximations as

[mathematical expression not reproducible] (41)

4. Conclusion

In this paper, singularly perturbed second-order linear delay differential 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 efficient asymptotic method so called 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. The 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 conditions are not satisfied asymptotically, but exactly. In Tables 1 and 2, exact solution, present method approximations, and approximations that are obtained by the method given in [9] are compared and to show the efficiency of present method, results are supported by Figures 1, 3, and 4. In Figures 2 and 5, the delay effects are compared and since the right layer problem does not have an exact solution, the first two SCEM approximations are compared in Table 3. As a result, the present method is a simple and very efficient technique for solving singularly perturbed linear DDEs.

http://dx.doi.org/10.1155/2017/7269450

Competing Interests

The author declares that there is no conflict of interests.

Acknowledgments

The author would like to express his sincere thanks to his master thesis advisors Dr. A. Eryilmaz and Dr. M. T. Atay for their valuable comments and suggestions.

References

[1] M. Kumar, "Methods for solving singular perturbation problem sarising in science and engineering," Mathematical and Computer Modelling, vol. 54, no. 1, pp. 556-575, 2011.

[2] K. K. Sharma, P. Rai, and K. C. Patidar, "A review on singularly perturbed differential equations with turning points and interior layers," Applied Mathematics and Computation, vol. 219, no. 22, pp. 10575-10609, 2013.

[3] A. H. Nayfeh, Perturbation Methods, John Wiley & Sons, 2008.

[4] E. J. Hinch, Perturbation Methods, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, UK, 1991.

[5] J. A. Murdock, Perturbations: Theory and Methods, vol. 27 of Classics in Applied Mathematics, SIAM, Philadelphia, Pa, USA, 1999.

[6] E. M. de Jager and J. F. Furu, The Theory of Singular Perturbations, vol. 42, Elsevier, 1996.

[7] P A. Lagerstrom, Matched asymptotic expansions, vol. 76 of Applied Mathematical Sciences, Springer New York, 1988.

[8] J. K. Kevorkian and J. D. Cole, Multiple Scale and Singular Perturbation Methods, vol. 114, Springer Science & Business Media, Berlin, Germany, 2012.

[9] F. Gemechis and Y. N. Reddy, "Terminal boundary-value technique for solving singularly perturbed delay differential equations," Journal of Taibah University for Sciences, vol. 8, no. 3, pp. 289-300, 2014.

[10] T. Erneux, Applied Delay Differential Equations, vol. 3 of Surveys and Tutorials in the Applied Mathematical Sciences, Springer, New York, NY, USA, 2009.

[11] E. Pinney, Ordinary Difference-Differential Equations, University of California Press, 1958.

[12] R. D. Driver, Ordinary and Delay Differential Equations, vol. 20 of Applied Mathematics, Springer, Berlin, Germany, 1977.

[13] R. E. Bellman and K. L. Cooke, Differential-Difference Equations, Academic Press, New York, NY, USA, 1963.

[14] J. K. Hale, Functional Differential Equations, Springer, New York, NY, USA, 1971.

[15] M. K. Kadalbajoo and K. K. Sharma, "A numerical method based on finite difference for boundary value problems for singularly perturbed delay differential equations," Applied Mathematics and Computation, vol. 197, no. 2, pp. 692-707, 2008.

[16] R. Nageshwar Rao and P Pramod Chakravarthy, "A finite difference method for singularly perturbed differential-difference equations with layer and oscillatory behavior," Applied Mathematical Modelling, vol. 37, no. 8, pp. 5743-5755, 2013.

[17] S. Nicaise and C. Xenophontos, "Robust approximation of singularly perturbed delay differential equations by the hp finite element method," Computational Methods in Applied Mathematics, vol. 13, no. 1, pp. 21-37, 2013.

[18] H. Zarin, "On discontinuous Galerkin finite element method for singularly perturbed delay differential equations," Applied Mathematics Letters. An International Journal of Rapid Publication, vol. 38, pp. 27-32, 2014.

[19] F. Shakeri and M. Dehghan, "Solution of delay differential equations via a homotopy perturbation method," Mathematical and Computer Modelling, vol. 48, no. 3-4, pp. 486-498, 2008.

[20] Q. Wang and F. Fu, "Solving delay differential equations with homotopy analysis method," Communications in Computer and Information Science, vol. 97, no. 1, pp. 144-153, 2010.

[21] A. Akgul and A. Kiliman, "Solving delay differential equations by an accurate method with interpolation," Abstract and Applied Analysis, vol. 2015, Article ID 676939, 7 pages, 2015.

[22] F. Z. Geng, S. P Qian, and M. G. Cui, "Improved reproducing kernel method for singularly perturbed differential-difference equations with boundary layer behavior," Applied Mathematics and Computation, vol. 252, pp. 58-63, 2015.

[23] M. K. Kadalbajoo and D. Kumar, "Fitted mesh B-spline collocation method for singularly perturbed differential-difference equations with small delay," Applied Mathematics and Computation, vol. 204, no. 1, pp. 90-98, 2008.

[24] H. M. El-Hawary and S. M. Mahmoud, "Spline collocation methods for solving delay-differential equations," Applied Mathematics and Computation, vol. 146, no. 2-3, pp. 359-372, 2003.

[25] C. G. Lange and R. M. Miura, "Singular perturbation analysis of boundary value problems for differential-difference equations. v. small shifts with layer behavior," SIAM Journal on Applied Mathematics, vol. 54, no. 54, pp. 249-272, 1994.

[26] C. G. Lange and R. M. Miura, "Singular perturbation analysis of boundary-value problems for differential-difference equations. VI. Small shifts with rapid oscillations," SIAM Journal on Applied Mathematics, vol. 54, no. 1, pp. 273-283, 1994.

[27] J. Mauss and J. Cousteix, "Uniformly valid approximation for singular perturbation problems and matching principle," Comptes Rendus-Mecanique, vol. 330, no. 10, pp. 697-702, 2002.

[28] W. Eckhaus, Asymptotic Analysis of Singular Perturbations, vol. 9 of Studies in Mathematics and Its Applications, North-Holland, Amsterdam, The Netherlands, 1979.

[29] J. Cousteix and J. Mauss, Asymptotic Analysis and Boundary Layers, Scientific Computation, Springer, Berlin, Germany, 2007

[30] S. Cengizci, M. T. Atay, and A. Eryilmaz, "A uniformly valid approximation algorithm for nonlinear ordinary singular perturbation problems with boundary layer solutions," Springer-Plus, vol. 5, no. 1, article no. 280, 2016.

Suleyman Cengizci

Institute of Applied Mathematics, Middle East Technical University, 06800 Ankara, Turkey

Correspondence should be addressed to Suleyman Cengizci; cengizci.suleyman@metu.edu.tr

Received 7 April 2016; Accepted 11 January 2017; Published 8 February 2017

Academic Editor: Patricia J. Y. Wong

Caption: Figure 1: Left layer problem for [epsilon] = 0.01 and [delta] = 0.1[epsilon].

Caption: Figure 2: Delay effect on left layer problem for [epsilon] = 0.01, [delta] = 0.01[epsilon], and [delta] = 0.9[epsilon].

Caption: Figure 3: Left layer problem for [epsilon] = 0.001 and [delta] = 0.9[epsilon].

Caption: Figure 4: Absolute errors in SCEM approximations for left layer problem.

Caption: Figure 5: Delay effect on right layer problem for [epsilon] = 0.01, [delta] = 0.5[epsilon], and [delta] = 2[epsilon].
Table 1: Results of left layer problem for [epsilon] =[10.sup.-3]
and [delta] = 0.5[epsilon].

x        [y.sub.exact]    [y.sup.scem.sub.1]

0.0000     1.0000000          1.0000000
0.0010     0.4538692          0.45379572
0.0020     0.3803509          0.38019363
0.0030     0.3707303          0.37055161
0.0040     0.3697488          0.36956596
0.0050     0.3699358          0.36975214
0.0100     0.3717605          0.37157669
0.0150     0.3736230          0.37343923
0.0200     0.3754949          0.37531110
0.1000     0.4067525          0.40656966
0.2000     0.4495085          0.44932896
0.4000     0.5489761          0.54881164
0.6000     0.6704540          0.67032005
0.8000     0.8188125          0.81873075
0.9000     0.9048826          0.90483742
1.0000     1.0000000          1.0000000

x        [absolute value of ([y.sub.exact] -    [y.sup.scem.sub.2]
                  [y.sup.scem.sub.1])]

0.0000                0.0000000                     0.0000000
0.0010               7.3488e - 05                   0.4538692
0.0020               1.5732e - 04                   0.3803510
0.0030               1.7865e - 04                   0.3707303
0.0040               1.8289e - 04                   0.3697489
0.0050               1.8364e - 04                   0.3699359
0.0100               1.8379e - 04                   0.3717606
0.0150               1.8378e - 04                   0.3736231
0.0200               1.8376e - 04                   0.3754950
0.1000               1.8281e - 04                   0.4067526
0.2000               1.7958e - 04                   0.4495086
0.4000               1.6450e - 04                   0.5489762
0.6000               1.3394e - 04                   0.6704541
0.8000               8.1795e - 05                   0.8188126
0.9000               4.5197e - 05                   0.9048826
1.0000                0.0000000                     0.0000000

x        [absolute value of ([y.sub.exact] -    Method [9]
                  [y.sup.scem.sub.2])]

0.0000                0.0000000                 1.0000000
0.0010               9.2048e - 09               0.3171426
0.0020               9.4728e - 08               0.3603879
0.0030               1.2683e - 07               0.3666117
0.0040               1.3552e - 07               0.3678324
0.0050               1.3753e - 07               0.3683816
0.0100               1.3821e - 07               0.3708603
0.0150               1.3848e - 07               0.3733556
0.0200               1.3869e - 07               0.3758674
0.1000               1.4163e - 07               0.4071563
0.2000               1.4362e - 07               0.4499552
0.4000               1.3978e - 07               0.5495225
0.6000               1.2051e - 07               0.6711221
0.8000               7.7685e - 08               0.8196300
0.9000               4.4056e - 08               0.9057869
1.0000                0.0000000                 1.0000000

Table 2: Results of left layer problem for [epsilon] = [10.sup.-4]
and [delta] = 0.5[epsilon].

X          Exact     [y.sup.scem.sub.1]

0.0000   1.0000000       1.0000000
0.0001   0.4534718       0.4534644
0.0002   0.3795465       0.3795307
0.0003   0.3695746       0.3695566
0.0004   0.3682570       0.3682386
0.0005   0.3681105       0.3680921
0.0010   0.3682659       0.3682475
0.0015   0.3684501       0.3684316
0.0020   0.3686343       0.3686159
0.1000   0.4065880       0.4065696
0.2000   0.4493469       0.4493289
0.4000   0.5488281       0.5488116
0.6000   0.6703334       0.6703200
0.8000   0.8187389       0.8187307
0.9000   0.9048420       0.9048374
1.0000   1.0000000       1.0000000

X        [absolute value of ([y.sub.exact] -    [y.sup.scem.sub.2]
                 [y.sup.scem.sub.1])]

0.0000                0.0000000                     1.0000000
0.0001               7.3497e - 06                   0.4534717
0.0002               1.5740e - 05                   0.3795464
0.0003               1.7877e - 05                   0.3695745
0.0004               1.8301e - 05                   0.3682569
0.0005               1.8377e - 05                   0.3681105
0.0010               1.8392e - 05                   0.3682658
0.0015               1.8392e - 05                   0.3684500
0.0020               1.8392e - 05                   0.3686343
0.1000               1.8294e - 05                   0.4065879
0.2000               1.7971e - 05                   0.4493469
0.4000               1.6462e - 05                   0.5488281
0.6000               1.3405e - 05                   0.6703334
0.8000               8.1865e - 06                   0.8187389
0.9000               4.5237e - 06                   0.9048419
1.0000                0.0000000                     1.0000000

X        [absolute value of ([y.sub.exact] -    Method [9]
                 [y.sup.scem.sub.2])]

0.0000                0.0000000                 1.0000000
0.0001               1.7565e - 10               0.3181581
0.0002               1.1979e - 09               0.3612116
0.0003               1.2808e - 09               0.3670360
0.0004               1.3520e - 09               0.3678237
0.0005               1.3718e - 09               0.3679338
0.0010               1.3794e - 09               0.3682020
0.0015               1.3806e - 09               0.3684702
0.0020               1.3794e - 09               0.3687384
0.1000               1.4170e - 09               0.4066914
0.2000               1.4373e - 09               0.4494485
0.4000               1.3990e - 09               0.5489215
0.6000               1.2061e - 09               0.6704094
0.8000               7.7755e - 10               0.8187855
0.9000               4.4095e - 10               0.9048674
1.0000                0.0000000                 1.0000000

Table 3: Results of right layer problem for [epsilon] = [10.sup.-3]
and [delta] = 0.5[epsilon].

x        [y.sup.scem.sub.1]   [y.sup.scem.sub.2]

0.0000       1.0000000             1.0000000
0.1000       1.1051709             1.1052261
0.2000       1.2214027             1.2215248
0.4000       1.4918246             1.4921230
0.6000       1.8221188             1.8226654
0.8000       2.2255409             2.2264311
0.9000       2.4596031             2.4607099
0.9980       2.6447479             2.6460768
0.9985       2.5290851             2.5303725
0.9990       2.2123501             2.2135226
0.9995       1.3490435             1.3499013
0.9996       1.0464630             1.0472103
0.9997       0.6768301             0.6774425
0.9998       0.2252993             0.2257469
0.9999       -0.3262616           -0.3260155
1.0000       -1.0000000           -1.0000000

X        [absolute value of ([y.sup.scem.sub.2] -    Numerical method
                   [y.sup.scem.sub.1])]

0.0000                   0.0000000                      1.0000000
0.1000                  5.5258e - 05                    1.1052262
0.2000                  1.2214e - 04                    1.2215250
0.4000                  2.9836e - 04                    1.4921233
0.6000                  5.4663e - 04                    1.8226660
0.8000                  8.9021e - 04                    2.2264322
0.9000                   0.0011068                      2.4607112
0.9980                   0.0013288                      2.6458021
0.9985                   0.0012873                      2.5299480
0.9990                  0.00117247                      2.2128733
0.9995                  8.5777e - 04                    1.3488799
0.9996                  7.4736e - 04                    1.0456402
0.9997                  6.1241e - 04                    0.6780147
0.9998                  4.4754e - 04                    0.2279200
0.9999                  2.4612e - 04                   -0.3247246
1.0000                   0.0000000                     -1.0000000
COPYRIGHT 2017 Hindawi Limited
No portion of this article can be reproduced without the express written permission from the copyright holder.
Copyright 2017 Gale, Cengage Learning. All rights reserved.

Article Details
Printer friendly Cite/link Email Feedback
Title Annotation:Research Article
Author:Cengizci, Suleyman
Publication:International Journal of Differential Equations
Article Type:Report
Date:Jan 1, 2017
Words:5127
Previous Article:Global Dynamics of a Periodic SEIRS Model with General Incidence Rate.
Next Article:Existence and Uniqueness of Solution of Stochastic Dynamic Systems with Markov Switching and Concentration Points.
Topics:

Terms of use | Privacy policy | Copyright © 2020 Farlex, Inc. | Feedback | For webmasters