Research Article Numerical Treatment of Singularly Perturbed Two-Point Boundary Value Problems by Using Differential Transformation Method

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1 Discrete Dynmics in Nture nd Society Volume 202, Article ID 57943, 0 pges doi:0.55/202/57943 Reserch Article Numericl Tretment of Singulrly Perturbed Two-Point Boundry Vlue Problems by Using Differentil Trnsformtion Method Nurettin Doğn, Vedt Sut Ertürk, 2 nd Ömer Akın 3 Deprtment of Computer Engineering, Fculty of Technology, Gzi University, Teknikokullr Ankr, Turkey 2 Deprtment of Mthemtics, Fculty of Arts nd Sciences, Ondokuz Myıs University, 5539 Smsun, Turkey 3 Deprtment of Mthemtics, Fculty of Arts nd Sciences, TOBB University of Economics nd Technology, Söğütözü, Ankr, Turkey Correspondence should be ddressed to Nurettin Doğn, ndogn@ymil.com Received 3 Mrch 202; Accepted 24 Mrch 202 Acdemic Editor: Gryflos Ppschinopoulos Copyright q 202 Nurettin Doğn et l. This is n open ccess rticle distributed under the Cretive Commons Attribution License, which permits unrestricted use, distribution, nd reproduction in ny medium, provided the originl work is properly cited. Differentil trnsform method is dopted, for the first time, for solving liner singulrly perturbed two-point boundry vlue problems. Four numericl exmples re given to demonstrte the effectiveness of the present method. Results show tht the numericl scheme is very effective nd convenient for solving lrge number of liner singulrly perturbed two-point boundry vlue problems with high ccurcy.. Introduction Singulrly perturbed second-order two-point boundry vlue problems, which received significnt mount of ttention in pst nd recent yers, rise very frequently in fluid mechnics, quntum mechnics, optiml control, chemicl-rector theory, erodynmics, rection-diffusion process, geophysics, nd so forth. In these problems smll prmeter multiplies to highest derivtive. A well-known fct is tht the solution of such problems disply shrp boundry or interior lyers when the singulr perturbtion prmeter ε is very smll. Numericlly, the presence of the perturbtion prmeter leds to difficulties when clssicl numericl techniques re used to solve such problems, nd convergence will not be uniform. The solution vries rpidly in some prts nd vries slowly in some other prts. There re thin trnsition boundry or interior lyers where the solutions cn chnge rpidly,

2 2 Discrete Dynmics in Nture nd Society while wy from the lyers the solution behves regulrly nd vries slowly. There re wide vriety of techniques for solving singulr perturbtion problems see 7. Furthermore different numericl methods hve been proposed by vrious uthors for singulrly perturbed two-point boundry vlue problems, such s non-uniform mesh tension spline methods 8, non-uniform mesh compression spline numericl method 9, nd the lest squres methods bsed on the Bézier control points 0. The im of our study is to introduce the differentil trnsform method s n lterntive to existing methods in solving singulrly perturbed two-point boundry vlue problems nd the method is implemented to four numericl exmples. The present method is the first time pplied by the uthors to singulrly perturbed two-point boundry vlue problems. The rest of the pper is orgnized s follows. In Section 2, we give brief description of the method. In Section 3, we hve solved four numericl exmples to demonstrte the pplicbility of the present method. The discussion on our results is given in Section Fundmentl of Differentil Trnsform Method In this section, the concept of the differentil trnsformtion method DTM is briefly introduced. The concept of differentil trnsform ws first introduced by Pukhov, who solved liner nd nonliner initil vlue problems in electric circuit nlysis. This method constructs, for differentil equtions, n nlyticl solution in the form of polynomil. It is seminumericl nd seminlytic technique tht formulizes the Tylor series in totlly different mnner. The Tylor series method is computtionlly tken long time for lrge orders. With this technique, the given differentil eqution nd its relted boundry conditions re trnsformed into recurrence eqution tht finlly leds to the solution of system of lgebric equtions s coefficients of power series solution. This method is useful to obtin exct nd pproximte solutions of liner nd nonliner differentil equtions. No need to lineriztion or discretiztion, lrge computtionl work nd round-off errors re voided. It hs been used to solve effectively, esily, nd ccurtely lrge clss of liner nd nonliner problems with pproximtions. The method is well ddressed in 2 9.Thebsic principles of the differentil trnsformtion method cn be described s follows. The differentil trnsform of the kth derivtive of function f x is defined s follows. [ ] d k f x F k k! dx k x x 0, 2. nd the differentil inverse trnsform of F k is defined s follows: f x F k x x 0 k. 2.2 In rel pplictions, function f x is expressed by finite series nd 2.2 cn be written s N f x F k x x 0 k. 2.3 The following theorems tht cn be deduced from 2. nd 2.2 re given 20.

3 Discrete Dynmics in Nture nd Society 3 Theorem 2.. If f x g x ± h x,thenf k G k ± H k. Theorem 2.2. If f x g x, thenf k G k,where is constnt. Theorem 2.3. If f x d m g x /dx m,thenf k m k!/k! G k m. Theorem 2.4. If f x g x h x,thenf k k k 0 G k H k k. Theorem 2.5. If f x x n,then ) n x n k 0, k < n k F k, k n 0, k > n. 2.4 Here n N, N is the set of nturl numbers, nd W k is the differentil trnsform function of w x. In the cse of x 0 0, one hs the following result: {, k n W k δ k n 0, k/ n. 2.5 Theorem 2.6. If f x g x g 2 x g n x g n x,then F k k k n k n 0 k n 2 0 k 3 k 2 k 2 0 k 0 G k G 2 k 2 k G n k n k n 2 G n k k n The Applictions of Differentil Trnsformtion Method nd Numericl Results In order to evlute the ccurcy of DTM for solving singulrly perturbed two-point boundry vlue problems, we will consider the following exmples. These exmples hve been chosen becuse they hve been widely discussed in the literture nd lso pproximte solutions re vilble for concrete comprison. Exmple 3.. We first consider the following problem 2 : εy y 0; x 0,, 3. with the boundry conditions y 0 0, y. 3.2

4 4 Discrete Dynmics in Nture nd Society The exct solution for this problem is y x sin x/ ε ) sin / ε ). 3.3 Tking the differentil trnsform of both sides of 3., the following recurrence reltion is obtined: Y k Y k 2 ε k k The boundry conditions given in 3.2 cn be trnsformed t x 0 0 s follows: Y 0 0, N Y k. 3.5 Using 3.4 nd 3.5 nd by tking N 5, the following series solution is obtined: y x x 6ε x3 20ε 2 x5 O x 7), 3.6 where, ccording to 2., y 0. The constnt is evluted from the second boundry condition given in 3.2 t x s follows: 20ε 2 20ε 20ε Then, by using the inverse trnsform rule in 2.2, we get the following series solution: y x 20ε 2 20ε 20ε x 20ε 2 20ε 20ε 2 x3 20ε 20ε 2 x5 O x 7). 3.8 The evolution results for the exct solution 3.3 nd the pproximte solution 3.8 obtined by using the differentil trnsform method, for ε 2 9,reshowninFigure. Exmple 3.2. Secondly, we consider the following problem: εy y x; x 0,, 3.9 with the boundry conditions y 0 0, y

5 Discrete Dynmics in Nture nd Society 5 2 N = y Figure : The pproximte solution dotted curve versus the nlytic solution solid curve for ε 2 9. x The exct solution for this boundry vlue problem is y x x sin x/ ε ) sin / ε ). 3. Tking the differentil trnsform of 3.9, we hve Y k 2 δ k Y k ε k k Choosing x 0 0, the boundry conditions given in 3.0 cn be trnsformed to give Y 0 0, N Y k By using 3.2 nd 3.3, nd, by tking N 5, we get the following series solution: y x x ) x 3 6ε 6ε ε ε ε ε 7 ) 20ε x ε ε ε ) ) 3 x ε x ε ε 6 ) x ε 7 ) x 5 O x 7), ) x where, ccording to 2., y 0.

6 6 Discrete Dynmics in Nture nd Society 4 N = 90 2 y x Figure 2: The pproximte solution dotted curve versus the nlytic solution solid curve for ε 0 3. The constnt is evluted from the second boundry condition given in 3.0 t x s follows: 20ε 32760ε ε ε ε ε 6) / 20ε 32760ε ε ε ε ε ε 7). 3.5 Then, by using the inverse trnsform rule in 2.2, one cn obtin the pproximte solution. We do not give it becuse of long terms in the pproximte solution. In Figure 2, we plot the exct solution 3. nd the pproximte solution for ε 0 3. Exmple 3.3. Thirdly, we consider the following problem 22 εy y 0; x 0, 3.6 subject to the boundry conditions y 0, y e /ε. 3.7 The exct solution for this problem is y x e x/ε. 3.8 Applying the opertions of the differentil trnsform to 3.6, we obtin the following recurrence reltion: k Y k Y k 2 ε k k

7 Discrete Dynmics in Nture nd Society 7 By using the bsic definitions of the differentil trnsform nd 3.7, the following trnsformed boundry conditions t x 0 0 cn be obtined: Y 0, N Y k e /ε By utilizing the recurrence reltion in 3.9 nd the trnsformed boundry conditions in 3.20, the following series solution up to 5-term is obtined: y x x 2ε x2 6ε 2 x3 24ε 3 x ε 7 x ε x2 20ε 4 x ε 9 x ε 2 x3 x 6), ε 8 x ε 4 x5 O 5040ε 5 x6 5040ε 6 x7 x ε0 x ε3 3.2 where y 0. By tking N 5, the following eqution cn be obtined from 3.20 : ε ε ε ε ε ε ε ε ε 6 720ε 5 20ε 4 24ε 3 6ε 2 2ε e /ε From 3.22, is evluted s e /ε e /ε) ε 4) / 5ε 20ε ε ε ε ε ε ε ε ε ε ε ε ε 4) By using this vlue of the missing boundry condition, the pproximte solution cn be obtined esily. Comprison of the pproximte solution with the exct solution 3.8 for ε 2 5 is sketched in Figure 3.

8 8 Discrete Dynmics in Nture nd Society N = y Figure 3: The pproximte solution dotted curve versus the nlytic solution solid curve for ε 2 5. x Exmple 3.4. Finlly, we consider the following problem 23, 24 εy y e x ; x 0, 3.24 subject to the boundry conditions y 0 0, y Its exct solution is given by [ ] y x e x e /ε e e x /ε ε e /ε By pplying the fundmentl mthemticl opertions performed by differentil trnsform, the differentil trnsform of 3.24 is obtined s Y k 2 The boundry conditions in 3.25 cn be trnsformed t x 0 0s /k! k Y k ε k k 2 Y 0 0, N Y k By using the inverse trnsformtion rule in 2.2, the pproximte solution is evluted up to N 20. The first few terms of the series solution re given by y x x 6ε 2 6ε ) x 3 2 6ε 24ε 3 24ε 3 24ε ) x 4, 2 24ε 3.29

9 Discrete Dynmics in Nture nd Society 9.75 N = y x Figure 4: The pproximte solution dotted curve versus the nlytic solution solid curve for ε /000. where y 0.The solution obtined from 2.3 hs yet to stisfy the second boundry condition in 3.25, which hs not been mnipulted in obtining this pproximte solution. Applying this boundry condition nd then solving the resulting eqution for will determine the unknown constnt nd eventully the numericl solution. Grphicl result for ε /000 with comprison to the exct solution 3.26 is shown in Figure Conclusion In this study, the differentil trnsformtion method DTM hs been employed, for the first time, successfully for solving liner singulrly perturbed two-point boundry vlue problems. Four exmples with boundry lyers hve been treted. This new method ccelerted the convergence to the solutions. As it cn be seen, this method leds to tremendously ccurte results. It provides the solutions in terms of convergent series with esily computble components in direct wy without using lineriztion, discretiztion, or restrictive ssumptions. The Mthemtic softwre system hs been used for ll the symbolic nd numericl computtions in this pper. References C. M. Bender nd S. A. Orszg, Advnced Mthemticl Methods for Scientists nd Engineers, Interntionl Series in Pure nd Applied Mthemtics, McGrw-Hill, New York, NY, USA, J. Kevorkin nd J. D. Cole, Perturbtion Methods in Applied Mthemtics, vol. 34 of Applied Mthemticl Sciences, Springer, New York, NY, USA, R. E. O Mlley Jr., Introduction to Singulr Perturbtions, Applied Mthemtics nd Mechnics, Vol. 4, Acdemic Press, New York, NY, USA, C. Liu, The Lie-group shooting method for solving nonliner singulrly perturbed boundry vlue problems, Communictions in Nonliner Science nd Numericl Simultion, vol. 7, no. 4, pp , Y. Wng, L. Su, X. Co, nd X. Li, Using reproducing kernel for solving clss of singulrly perturbed problems, Computers & Mthemtics with Applictions, vol. 6, no. 2, pp , M. K. Kdlbjoo nd P. Aror, B-splines with rtificil viscosity for solving singulrly perturbed boundry vlue problems, Mthemticl nd Computer Modelling, vol. 52, no. 5-6, pp , 200.

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