DDNSDiscrete Dynamics in Nature and Society1607-887X1026-0226Hindawi Publishing Corporation57943110.1155/2012/579431579431Research ArticleNumerical Treatment of Singularly Perturbed Two-Point Boundary Value Problems by Using Differential Transformation MethodDoğanNurettin1ErtürkVedat Suat2AkınÖmer3PapaschinopoulosGaryfalos1Department of Computer EngineeringFaculty of TechnologyGazi UniversityTeknikokullar, 06500 AnkaraTurkeygazi.edu.tr2Department of MathematicsFaculty of Arts and SciencesOndokuz Mayıs University55139 SamsunTurkeyomu.edu.tr3Department of MathematicsFaculty of Arts and SciencesTOBB University of Economics and TechnologySöğütözü, 06530 AnkaraTurkeyetu.edu.tr20122642012201213032012240320122012Copyright © 2012 Nurettin Doğan et al.This 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.

Differential transform method is adopted, for the first time, for solving linear singularly perturbed two-point boundary value problems. Four numerical examples are given to demonstrate the effectiveness of the present method. Results show that the numerical scheme is very effective and convenient for solving a large number of linear singularly perturbed two-point boundary value problems with high accuracy.

1. Introduction

Singularly perturbed second-order two-point boundary value problems, which received a significant amount of attention in past and recent years, arise very frequently in fluid mechanics, quantum mechanics, optimal control, chemical-reactor theory, aerodynamics, reaction-diffusion process, geophysics, and so forth. In these problems a small parameter multiplies to a highest derivative. A well-known fact is that the solution of such problems display sharp boundary or interior layers when the singular perturbation parameter ε is very small. Numerically, the presence of the perturbation parameter leads to difficulties when classical numerical techniques are used to solve such problems, and convergence will not be uniform. The solution varies rapidly in some parts and varies slowly in some other parts. There are thin transition boundary or interior layers where the solutions can change rapidly, while away from the layers the solution behaves regularly and varies slowly. There are a wide variety of techniques for solving singular perturbation problems (see ). Furthermore different numerical methods have been proposed by various authors for singularly perturbed two-point boundary value problems, such as non-uniform mesh tension spline methods , non-uniform mesh compression spline numerical method , and the least squares methods based on the Bézier control points .

The aim of our study is to introduce the differential transform method  as an alternative to existing methods in solving singularly perturbed two-point boundary value problems and the method is implemented to four numerical examples. The present method is the first time applied by the authors to singularly perturbed two-point boundary value problems.

The rest of the paper is organized as follows. In Section 2, we give a brief description of the method. In Section 3, we have solved four numerical examples to demonstrate the applicability of the present method. The discussion on our results is given in Section 4.

2. Fundamental of Differential Transform Method

In this section, the concept of the differential transformation method (DTM) is briefly introduced. The concept of differential transform was first introduced by Pukhov , who solved linear and nonlinear initial value problems in electric circuit analysis. This method constructs, for differential equations, an analytical solution in the form of a polynomial. It is a seminumerical and semianalytic technique that formulizes the Taylor series in a totally different manner. The Taylor series method is computationally taken long time for large orders. With this technique, the given differential equation and its related boundary conditions are transformed into a recurrence equation that finally leads to the solution of a system of algebraic equations as coefficients of a power series solution. This method is useful to obtain exact and approximate solutions of linear and nonlinear differential equations. No need to linearization or discretization, large computational work and round-off errors are avoided. It has been used to solve effectively, easily, and accurately a large class of linear and nonlinear problems with approximations. The method is well addressed in . The basic principles of the differential transformation method can be described as follows.

The differential transform of the kth derivative of a function f(x) is defined as follows.F(k)=1k![dkf(x)dxk]x=x0, and the differential inverse transform of F(k) is defined as follows:f(x)=k=0F(k)(x-x0)k. In real applications, function f(x) is expressed by a finite series and (2.2) can be written asf(x)=k=0NF(k)(x-x0)k. The following theorems that can be deduced from (2.1) and (2.2) are given .

Theorem 2.1.

If f(x)=g(x)±h(x), then F(k)=G(k)±H(k).

Theorem 2.2.

If f(x)=ag(x), then F(k)=aG(k), where a is constant.

Theorem 2.3.

If f(x)=(dmg(x)/dxm), then F(k)=((m+k)!/k!)G(k+m).

Theorem 2.4.

If f(x)=g(x)h(x), then F(k)=k1=0kG(k1)H(k-k1).

Theorem 2.5.

If f(x)=xn, then F(k)={(nk)x0n-k,k<n1,k=n0,k>n. Here nN, N is the set of natural numbers, and W(k) is the differential transform function of w(x). In the case of x0=0, one has the following result: W(k)=δ(k-n)={1,  k=n0,  kn.

Theorem 2.6.

If f(x)=g1(x)g2(x)gn-1(x)gn(x), then F(k)=kn-1=0kkn-2=0kn-1k2=0k3k1=0k2G1(k1)G2(k2-k1)Gn-1(kn-1-kn-2)Gn(k-kn-1).

3. The Applications of Differential Transformation Method and Numerical Results

In order to evaluate the accuracy of DTM for solving singularly perturbed two-point boundary value problems, we will consider the following examples. These examples have been chosen because they have been widely discussed in the literature and also approximate solutions are available for a concrete comparison.

Example 3.1.

We first consider the following problem : εy′′+y=0;x[0,1], with the boundary conditions y(0)=0,  y(1)=1. The exact solution for this problem is y(x)=sin(x/ε)sin(1/ε). Taking the differential transform of both sides of (3.1), the following recurrence relation is obtained: Y(k+2)=-Y(k)ε  (k+1)(k+2). The boundary conditions given in (3.2) can be transformed at x0=0 as follows: Y(0)=0,  k=0NY(k)=1. Using (3.4) and (3.5) and by taking N=5, the following series solution is obtained: y(x)=ax-a6εx3+a120ε2x5+O(x7), where, according to (2.1), a=y(0). The constant a is evaluated from the second boundary condition given in (3.2) at x=1 as follows: a=120ε21-20ε+120ε2. Then, by using the inverse transform rule in (2.2), we get the following series solution: y(x)=120ε21-20ε+120ε2x-20ε1-20ε+120ε2x3+11-20ε+120ε2x5+O(x7). The evolution results for the exact solution (3.3) and the approximate solution (3.8) obtained by using the differential transform method, for ε=2-9, are shown in Figure 1.

The approximate solution (dotted curve) versus the analytic solution (solid curve) for ε=2-9.

Example 3.2.

Secondly, we consider the following problem: εy′′+y=-x;x[0,1], with the boundary conditions y(0)=0,  y(1)=0. The exact solution for this boundary value problem is y(x)=-x+sin(x/ε)sin(1/ε). Taking the differential transform of (3.9), we have Y(k+2)=-δ(k-1)-Y(k)ε  (k+1)(k+2). Choosing x0=0, the boundary conditions given in (3.10) can be transformed to give Y(0)=0,  k=0NY(k)=0. By using (3.12) and (3.13), and, by taking N=5, we get the following series solution: y(x)=ax+(-16ε-a6ε)x3+(1120ε2+a120ε2)x5+(-15040ε3-a5040ε3)x7+  (1362880ε4+a362880ε4)x9+(-139916800ε5-a39916800ε5)x11+(16227020800ε6+a6227020800ε6)x13+(-11307674368000ε7-a1307674368000ε7)x15+O(x7), where, according to (2.1), a=y(0).

The constant a is evaluated from the second boundary condition given in (3.10) at x=1 as follows: a=(1-210ε+32760ε2-3603600ε3+259459200ε4-10897286400ε5+217945728000ε6)/(-1+210ε-32760ε2+3603600ε-259459200ε4+10897286400ε5-217945728000ε6+1307674368000ε7).

Then, by using the inverse transform rule in (2.2), one can obtain the approximate solution. We do not give it because of long terms in the approximate solution.

In Figure 2, we plot the exact solution (3.11) and the approximate solution for ε=10-3.

The approximate solution (dotted curve) versus the analytic solution (solid curve) for ε=10-3.

Example 3.3.

Thirdly, we consider the following problem  ε  y′′+y=0;x[0,1] subject to the boundary conditions y(0)=1,  y(1)=e-1/ε. The exact solution for this problem is y(x)=e-x/ε. Applying the operations of the differential transform to (3.16), we obtain the following recurrence relation: Y(k+2)=-(k+1)Y(k+1)ε  (k+1)(k+2). By using the basic definitions of the differential transform and (3.17), the following transformed boundary conditions at x0=0 can be obtained: Y(0)=1,  k=0NY(k)=e-1/ε. By utilizing the recurrence relation in (3.19) and the transformed boundary conditions in (3.20), the following series solution up to 15-term is obtained: y(x)=1+ax-a2εx2+a6ε2x3-a24ε3x4+a120ε4x5-a5040ε5x6+a5040ε6x7-a40320ε7x8+a362880ε8x9-a3628800ε9x10+a39916800ε10x11-a479001600ε11x12+a6227020800ε12x13-a87178291200ε13x14+a1307674368000ε14x15-  O(x16), where a=y(0).

By taking N=15, the following equation can be obtained from (3.20): 1+a+a1307674368000ε14-a87178291200ε13+a6227020800ε12-a479001600ε11+a39916800ε10-a3628800ε9+a362880ε8-a40320ε7+a5040ε6-a720ε5+a120ε4-a24ε3+a6ε2-a2ε=e-1/ε.

From (3.22), a is evaluated as a=-(130767436800e-1/ε(-1+e1/ε)ε14)/(1-15ε+210ε2-2730ε3+32760ε4-360360ε5+3603600ε6-32432400ε7+259459200ε8-1816214400ε9+10897286400ε10-54486432000ε11+217945728000ε12-653837184000ε13+1307674368000ε14). By using this value of the missing boundary condition, the approximate solution can be obtained easily.

Comparison of the approximate solution with the exact solution (3.18) for ε=2-5 is sketched in Figure 3.

The approximate solution (dotted curve) versus the analytic solution (solid curve) for ε=2-5.

Example 3.4.

Finally, we consider the following problem [23, 24] -εy′′+y=ex;x[0,1] subject to the boundary conditions y(0)=0,  y(1)=0. Its exact solution is given by y(x)=11-ε[ex-1-e1-(1/ε)+(e-1)ex-(1/ε)1-e-1/ε]. By applying the fundamental mathematical operations performed by differential transform, the differential transform of (3.24) is obtained as Y(k+2)=-1/k!+(k+1)Y(k+1)ε(k+1)(k+2). The boundary conditions in (3.25) can be transformed at x0=0 as Y(0)=0,  k=0NY(k)=0. By using the inverse transformation rule in (2.2), the approximate solution is evaluated up to N=20. The first few terms of the series solution are given by y(x)=ax+(-16ε2+a6ε2-16ε)x3+(-124ε3+a24ε3-124ε2-124ε)x4+, where a=y(0).The solution obtained from (2.3) has yet to satisfy the second boundary condition in (3.25), which has not been manipulated in obtaining this approximate solution. Applying this boundary condition and then solving the resulting equation for a will determine the unknown constant a and eventually the numerical solution.

Graphical result for ε=1/1000 with comparison to the exact solution (3.26) is shown in Figure 4.

The approximate solution (dotted curve) versus the analytic solution (solid curve) for ε=1/1000.

4. Conclusion

In this study, the differential transformation method (DTM) has been employed, for the first time, successfully for solving linear singularly perturbed two-point boundary value problems. Four examples with boundary layers have been treated. This new method accelerated the convergence to the solutions. As it can be seen, this method leads to tremendously accurate results. It provides the solutions in terms of convergent series with easily computable components in a direct way without using linearization, discretization, or restrictive assumptions. The Mathematica software system has been used for all the symbolic and numerical computations in this paper.

BenderC. M.OrszagS. A.Advanced Mathematical Methods for Scientists and Engineers1978New York, NY, USAMcGraw-Hillxiv+593International Series in Pure and Applied Mathematics538168ZBL0417.34001KevorkianJ.ColeJ. D.Perturbation Methods in Applied Mathematics198134New York, NY, USASpringerx+558Applied Mathematical Sciences608029ZBL0517.65032O'MalleyR. E.Jr.Introduction to Singular Perturbations1974New York, NY, USAAcademic Pressviii+206Applied Mathematics and Mechanics, Vol. 140402217ZBL0314.49006LiuC.The Lie-group shooting method for solving nonlinear singularly perturbed boundary value problemsCommunications in Nonlinear Science and Numerical Simulation201217415061521WangY.SuL.CaoX.LiX.Using reproducing kernel for solving a class of singularly perturbed problemsComputers & Mathematics with Applications201161242143010.1016/j.camwa.2010.11.0192754151ZBL1211.65142KadalbajooM. K.AroraP.B-splines with artificial viscosity for solving singularly perturbed boundary value problemsMathematical and Computer Modelling2010525-665466610.1016/j.mcm.2010.04.0122661752ZBL1202.65097KadalbajooM. K.KumarD.Initial value technique for singularly perturbed two point boundary value problems using an exponentially fitted finite difference schemeComputers & Mathematics with Applications20095771147115610.1016/j.camwa.2009.01.0102508545ZBL1186.65103MohantyR. K.AroraU.A family of non-uniform mesh tension spline methods for singularly perturbed two-point singular boundary value problems with significant first derivativesApplied Mathematics and Computation2006172153154410.1016/j.amc.2005.02.0232197921MohantyR. K.JhaN.A class of variable mesh spline in compression methods for singularly perturbed two point singular boundary value problemsApplied Mathematics and Computation2005168170471610.1016/j.amc.2004.09.0492170860ZBL1082.65550EvrenosogluM.SomaliS.Least squares methods for solving singularly perturbed two-point boundary value problems using Bézier control pointsApplied Mathematics Letters200821101029103210.1016/j.aml.2007.10.0212450645ZBL1160.34311PukhovG. E.Differential transformations and mathematical modelling of physical processes1986Kiev, UkraineNaukova DumkaGökdoğanA.MerdanM.YildirimA.The modified algorithm for the differential transform method to solution of Genesio systemsCommunications in Nonlinear Science and Numerical Simulation2012171455110.1016/j.cnsns.2011.03.0392825985AlomariA. K.A new analytic solution for fractional chaotic dynamical systems using the differential transform methodComputers & Mathematics with Applications20116192528253410.1016/j.camwa.2011.02.0432795001ZBL1221.65191ThongmoonM.PusjusoS.The numerical solutions of differential transform method and the Laplace transform method for a system of differential equationsNonlinear Analysis: Hybrid Systems20104342543110.1016/j.nahs.2009.10.0062645858ZBL1200.65062ChangS.-H.ChangI.-L.A new algorithm for calculating one-dimensional differential transform of nonlinear functionsApplied Mathematics and Computation2008195279980510.1016/j.amc.2007.05.0262381259ZBL1132.65062LiuH.SongY.Differential transform method applied to high index differential-algebraic equationsApplied Mathematics and Computation2007184274875310.1016/j.amc.2006.05.1732294941ZBL1115.65089LiuH.SongY.Differential transform method applied to high index differential-algebraic equationsApplied Mathematics and Computation2007184274875310.1016/j.amc.2006.05.1732294941ZBL1115.65089DoǧanN.ErtürkV. S.MomaniS.AkinÖ.YildirimA.Differential transform method for solving singularly perturbed Volterra integral equationsJournal of King Saud University - Science2011232232282-s2.0-7795500993410.1016/j.jksus.2010.07.013Ravi KanthA. S. V.ArunaK.Solution of singular two-point boundary value problems using differential transformation methodPhysics Letters A2008372264671467310.1016/j.physleta.2008.05.0192426613ZBL1221.34060ErtürkV. S.MomaniS.Comparing numerical methods for solving fourth-order boundary value problemsApplied Mathematics and Computation200718821963196810.1016/j.amc.2006.11.0752335049ZBL1119.65066SariM.Differential quadrature method for singularly perturbed two-point boundary value problemsJournal of Applied Sciences200886109110962-s2.0-4194908325710.3923/jas.2008.1091.1096Mokarram ShahrakiM.Mohammad HosseiniS.Comparison of a higher order method and the simple upwind and non-monotone methods for singularly perturbed boundary value problemsApplied Mathematics and Computation2006182146047310.1016/j.amc.2006.04.0072292055ZBL1106.65066KadalbajooM. K.PatidarK. C.Numerical solution of singularly perturbed two-point boundary value problems by spline in tensionApplied Mathematics and Computation20021312-329932010.1016/S0096-3003(01)00146-11920226ZBL1030.65087LorenzJ.Combinations of initial and boundary value methods for a class of singular perturbation problemsNumerical Analysis of Singular Perturbation Problems1979London, UKAcademic Press295315556523ZBL0429.65086