Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 2012, Article ID 579431, 10 pages doi:10.1155/2012/579431 Research Article Numerical Treatment of Singularly Perturbed Two-Point Boundary Value Problems by Using Differential Transformation Method Nurettin Doğan,1 Vedat Suat Ertürk,2 and Ömer Akın3 1 Department of Computer Engineering, Faculty of Technology, Gazi University, Teknikokullar 06500 Ankara, Turkey 2 Department of Mathematics, Faculty of Arts and Sciences, Ondokuz Mayıs University, 55139 Samsun, Turkey 3 Department of Mathematics, Faculty of Arts and Sciences, TOBB University of Economics and Technology, Söğütözü, 06530 Ankara, Turkey Correspondence should be addressed to Nurettin Doğan, ndogan@ymail.com Received 13 March 2012; Accepted 24 March 2012 Academic Editor: Garyfalos Papaschinopoulos Copyright q 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, 2 Discrete Dynamics in Nature and Society 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 1–7. 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 8, non-uniform mesh compression spline numerical method 9, and the least squares methods based on the Bézier control points 10. The aim of our study is to introduce the differential transform method 11 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 11, 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 12–19. The basic principles of the differential transformation method can be described as follows. The differential transform of the kth derivative of a function fx is defined as follows.   1 dk fx , Fk  k! dxk xx 2.1 0 and the differential inverse transform of Fk is defined as follows: fx  ∞  k0 Fkx − x0 k . 2.2 In real applications, function fx is expressed by a finite series and 2.2 can be written as fx  N  k0 Fkx − x0 k . The following theorems that can be deduced from 2.1 and 2.2 are given 20. 2.3 Discrete Dynamics in Nature and Society 3 Theorem 2.1. If fx  gx ± hx, then Fk  Gk ± Hk. Theorem 2.2. If fx  agx, then Fk  aGk, where a is constant. Theorem 2.3. If fx  dm gx/dxm , then Fk  m Theorem 2.4. If fx  gxhx, then Fk  Theorem 2.5. If fx  xn , then Fk  k k1 0 k!/k!Gk m. Gk1 Hk − k1 . ⎧ ⎪ n ⎪ ⎪ x0 n−k , ⎪ ⎪ ⎪ ⎨ k ⎪ 1, ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0, k<n kn 2.4 k > n. Here n ∈ N, N is the set of natural numbers, and Wk is the differential transform function of wx. In the case of x0  0, one has the following result: Wk  δk − n  1, kn 0, k/  n. 2.5 Theorem 2.6. If fx  g1 xg2 x · · · gn−1 xgn x, then Fk  k k n−1   kn−1 0 kn−2 0 ··· k3  k2  k2 0 k1 0 G1 k1 G2 k2 − k1  · · · Gn−1 kn−1 − kn−2 Gn k − kn−1 . 2.6 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 21: εy′′ y  0; x ∈ 0, 1, 3.1 y1  1. 3.2 with the boundary conditions y0  0, 4 Discrete Dynamics in Nature and Society The exact solution for this problem is √  sin x/ ε yx  √ . sin 1/ ε 3.3 Taking the differential transform of both sides of 3.1, the following recurrence relation is obtained: 2  − Y k Y k ε k 1k 2 3.4 . The boundary conditions given in 3.2 can be transformed at x0  0 as follows: Y 0  0, N  Y k  1. 3.5 k0 Using 3.4 and 3.5 and by taking N  5, the following series solution is obtained: a 3 x 6ε yx  ax − a x5 120ε2   O x7 , 3.6 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 120ε2 . 1 − 20ε 120ε2 3.7 Then, by using the inverse transform rule in 2.2, we get the following series solution: yx  120ε2 20ε x− x3 1 − 20ε 120ε2 1 − 20ε 120ε2 1 x5 1 − 20ε 120ε2   O x7 . 3.8 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. Example 3.2. Secondly, we consider the following problem: εy′′ y  −x; x ∈ 0, 1, 3.9 with the boundary conditions y0  0, y1  0. 3.10 Discrete Dynamics in Nature and Society 5 N = 80 2 1.5 1 0.5 y 0 −0.5 −1 −1.5 0 0.2 0.4 0.6 0.8 1 x Figure 1: The approximate solution dotted curve versus the analytic solution solid curve for ε  2−9 . The exact solution for this boundary value problem is yx  −x √  sin x/ ε √ . sin 1/ ε 3.11 Taking the differential transform of 3.9, we have Y k 2  −δk − 1 − Y k . ε k 1k 2 3.12 Choosing x0  0, the boundary conditions given in 3.10 can be transformed to give Y 0  0, N  Y k  0. 3.13 k0 By using 3.12 and 3.13, and, by taking N  5, we get the following series solution:      a a 1 a 1 1 5 − − − x x7 x3 6ε 6ε 120ε2 120ε2 5040ε3 5040ε3     a 1 a 1 9 − x11 x − 39916800ε5 39916800ε5 362880ε4 362880ε4   1 a x13 6227020800ε6 6227020800ε6     a 1 − x15 O x7 , − 7 7 1307674368000ε 1307674368000ε yx  ax  − where, according to 2.1, a  y′ 0. 3.14 6 Discrete Dynamics in Nature and Society N = 90 4 2 y 0 −2 −4 −6 0 0.2 0.4 0.6 0.8 1 x Figure 2: The approximate solution dotted curve versus the analytic solution solid curve for ε  10−3 . 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 . 3.15 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 . Example 3.3. Thirdly, we consider the following problem 22 εy′′ y′  0; x ∈ 0, 1 3.16 y1  e−1/ε . 3.17 subject to the boundary conditions y0  1, The exact solution for this problem is yx  e−x/ε . 3.18 Applying the operations of the differential transform to 3.16, we obtain the following recurrence relation: Y k 2  − k 1Y k ε k 1k 1 . 2 3.19 Discrete Dynamics in Nature and Society 7 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, N  Y k  e−1/ε . 3.20 k0 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: a 2 x 2ε a a 3 a 4 a a x5 − x − x x6 x7 6ε2 24ε3 5040ε5 5040ε6 120ε4 a a a a x11 x8 x9 − x10 − 7 8 9 40320ε 362880ε 3628800ε 39916800ε10 a a a − x12 x13 − x14 479001600ε11 6227020800ε12 87178291200ε13   a 15 x − O x16 , 1307674368000ε14 yx  1 ax − 3.21 where a  y′ 0. By taking N  15, the following equation can be obtained from 3.20: 1 a a a a a − − 14 13 12 1307674368000ε 87178291200ε 6227020800ε 479001600ε11 a a a a a a a − − − 39916800ε10 3628800ε9 362880ε8 40320ε7 5040ε6 720ε5 120ε4 a a a  e−1/ε . − − 3 2 2ε 24ε 6ε 3.22 From 3.22, a is evaluated as   a  − 130767436800e−1/ε −1    e1/ε ε14 / 1 − 15ε 210ε2 − 2730ε3 − 360360ε5 32760ε4 3603600ε6 − 32432400ε7 259459200ε8 − 1816214400ε9 10897286400ε10 − 54486432000ε11 3.23 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. 8 Discrete Dynamics in Nature and Society N = 90 1 0.8 0.6 y 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 x 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 ; 3.24 x ∈ 0, 1 subject to the boundary conditions y0  0, y1  0. 3.25 Its exact solution is given by   1 1 − e1−1/ε e − 1ex−1/ε x yx  e − . 1−ε 1 − e−1/ε 3.26 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 1Y k εk 1k 2 1 3.27 . The boundary conditions in 3.25 can be transformed at x0  0 as Y 0  0, N  Y k  0. 3.28 k0 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 yx  ax  1 − 2 6ε  a 1 x3 − 6ε2 6ε  1 − 24ε3  a 1 1 x4 − − 24ε3 24ε2 24ε ··· , 3.29 Discrete Dynamics in Nature and Society 9 N = 100 1.75 1.5 1.25 1 y 0.75 0.5 0.25 0 0 0.2 0.4 0.6 0.8 1 x Figure 4: The approximate solution dotted curve versus the analytic solution solid curve for ε  1/1000. 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. 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. 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Lorenz, “Combinations of initial and boundary value methods for a class of singular perturbation problems,” in Numerical Analysis of Singular Perturbation Problems, pp. 295–315, Academic Press, London, UK, 1979. Advances in Advances in Operations Research Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Decision Sciences Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Journal of Applied Mathematics Algebra Hindawi Publishing Corporation http://www.hindawi.com Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Journal of Probability and Statistics Volume 2014 The Scientific World Journal Hindawi Publishing Corporation http://www.hindawi.com Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 International Journal of Differential Equations Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Volume 2014 Submit your manuscripts at http://www.hindawi.com International Journal of Advances in Combinatorics Hindawi Publishing Corporation http://www.hindawi.com Mathematical Physics Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Journal of Journal of Complex Analysis Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 International Journal of Mathematics and Mathematical Sciences Mathematical Problems in Engineering Mathematics Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Discrete Mathematics Journal of Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Discrete Dynamics in Nature and Society Journal of Function Spaces Hindawi Publishing Corporation http://www.hindawi.com Abstract and Applied Analysis Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Journal of International Journal of Stochastic Analysis Optimization Hindawi Publishing Corporation http://www.hindawi.com Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Volume 2014