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Special second order non symmetric fitted method for singular

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Special second order non symmetric fitted method for singular

  1. 1. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.11, 2012 Special Second Order Non Symmetric Fitted Method for Singular Perturbation Problems K. Phaneendra1* Y.N. Reddy2 Madhu Latha3 1. Department of Mathematics, Kakatiya Institute of Institute of Technology & Science, Warangal-506015, India 2. Department of Mathematics, National Institute of Technology, Warangal-506004, India 3. Department of Mathematics, National Institute of Technology, Warangal-506004, India * E-mail of the corresponding author: kollojuphaneendra@yahoo.co.inAbstractIn this paper, we present a special second order non symmetric fitted difference method for solving singularperturbed two point boundary value problems having boundary layer at one end. We introduce a fitting factor in thespecial second order non symmetric finite difference scheme which takes care of the rapid changes occur that in theboundary layer. The value of this fitting factor is obtained from the theory of singular perturbations. The discreteinvariant imbedding algorithm is used to solve the tridiagonal system obtained by the method. We discuss theexistence and uniqueness of the discrete problem along with stability estimates and the convergence of the method.We present the maximum absolute errors in numerical results to illustrate the proposed method.Keywords: Singularly perturbed two-point boundary value problem, Boundary layer, Fitting factor, Maximumabsolute error1. IntroductionDuring the last few years much progress has been made in the theory and in the computer implementation of thenumerical treatment of singular perturbation problems. Typically, these problems arise very frequently in fluidmechanics, fluid dynamics, chemical reactor theory, elasticity, aero dynamics and other domains of the great worldof fluid motion. The solution of this type of problem has a narrow region in which the solution changes rapidly andthe outside solution changes smoothly. However, the area of singular perturbations is a field of increasing interest toapplied mathematicians. Much progress has been made recently in developing finite element methods for solvingsingular perturbation problems. This type of problem was solved by Bellman (1964), Bender and Orszag (1978),Eckhaus (1973), Kevorkian and Cole (1981), Nayfeh (1973), O’Malley (1974), Van Dyke (1974), and numericallyby Ascher and Weis (1984), Kadalbajoo, Reddy (1989) and Kadalbajoo and Patidar (2003), Lin and Su (1989), Roos(1986), Vulanovic (1991). It is well known that standard discretization methods for solving singular perturbationproblems are unstable and fail to give accurate results when the perturbation parameter ε is small. Therefore, it isimportant to develop suitable numerical methods for these problems, whose accuracy does not depend on parameterε as presented in Doolan et al. (1980). The fitted technique is one such tool to reach these goals in an optimum way.There are two possibilities to obtain small truncation error inside the boundary layer(s). The first is to choose a finemesh there, whereas the second one is to choose a difference formula reflecting the behaviour of the solution(s)inside the boundary layer(s). Present work deals with the second approach. In this paper, we have presented a specialsecond order non symmetric fitted difference method for solving singularly perturbed problems. We introduce afitting factor in a special second order non symmetric finite difference scheme which takes care of the rapid changesoccur that in the boundary layer. This fitting factor is obtained from the theory of singular perturbations. The discreteinvariant imbedding algorithm is used to solve the tridiagonal system. The existence and uniqueness of the discreteproblem along with stability estimates are discussed. We have discussed the convergence of the method. Maximumabsolute errors in numerical results are presented to illustrate the proposed method.2. Description of the method2.1 Left-End Boundary Layer Problems 17
  2. 2. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.11, 2012Consider a linearly singularly perturbed two point boundary value problem of the form: εy ′′( x) + a( x) y ′( x) + b( x) y ( x) = f ( x) , x ∈ [0,1] (1)with the boundary conditions y (0 ) = α (2a) and y (1) = β (2b)where ε is a small positive parameter ( 0 < ε << 1 ) and α , β are given constants. We assume that a(x), b(x) and f(x)are sufficiently smooth functions and such that (1) with (2) has a unique solution in [0, 1]. Further more, we assumethat b(x) ≤ 0, a(x) ≥ M > 0 throughout the interval [0, 1], where M is some positive constant. This assumption impliesthat boundary layer exists in the neighborhood of x = 0.From the theory of singular perturbation s it is known that the solution of (1) - (2) is of the form x  a ( x ) b( x )  a(0) ∫   ε − dx a( x)   y ( x) = y 0 ( x) + (α − y 0 (0))e 0 + O(ε ) (3) a( x)where y0 ( x) is the solution of a ( x) y 0 ( x) + b( x) y 0 ( x) = f ( x) , y 0 (1) = β ′ (4)By taking Taylor’s series expansion for a(x) and b(x) about the point ‘0’ and restricting to their first terms, (3)becomes,  a (0) b (0)  −  ε − a (0)  x  y ( x) = y 0 ( x) + (α − y 0 (0))e   + O(ε ) (5)Now we divide the interval [0, 1] into N equal parts with constant mesh length h. Let 0= x1 , x 2 ,.......x N =1 be themesh points. Then we have x i = ih : i = 0, 1, 2, … , N.  a (0) b (0)  −  ε − a ( 0 )  ih From (5), we have y (ih) = y 0 (ih ) + (α − y 0 (0))e   + O(ε )therefore  a 2 ( 0 ) −εb ( 0 )  −  iρ  a (0) lim y (ih) = y 0 (0) + (α − y 0 (0))e   (6)h →0 hwhere ρ = . εFrom finite differences, we have  h 2 ( 4)  y i −1 − 2 y i + y i +1 = h 2  y i′′ +  y i  + O(h 6 )  (7)  12  4 h y i′′−1 − 2 y i′′ + y i′′+1 = h 2 y i( 4 ) + y i( 6 ) + ........................ 12Substituting h 2 yi( 4) from the above equation in (7), we have h2 y i −1 − 2 y i + y i +1 = ( y i′′−1 + 10 y i′′ + y i′′+1 ) + O(h 6 ) 12 (8)Now from the equation (1), we haveεy i′′+1 = −a i +1 y i′+! − bi +1 y i +1 + f i +1 *εy i′′ = −a i y i′ − bi y i + f iεy i′′−1 = −a i −1 y i′−! − bi −1 y i −1 + f i −1 * (9) where we approximate yi′* 1 and yi′* 1 using non symmetric finite differences + − 18
  3. 3. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.11, 2012 y i −1 − 4 y i + 3 y i +1y i′+1 = * − hy i′′ + O (h 2 ) 2h y i +1 − y i −1y i′ = + O(h 2 ) 2h − 3 y i −1 + 4 y i − y i +1y i′−1 = * + hy i′′ + O (h 2 ) 2h (10)Substituting (9) and (10) in (8) and simplifying we get ha ha  y − 2 y i + y i +1  a i −1 10a i ε + i −1 − i +1  i −1  2  24h (− 3 y i −1 + 4 y i − y i +1 ) + 24h ( y i +1 − y i −1 ) + 12 12  h  a i +1 b 10b b ( f + 10 f i + f i +1 )+ ( y i −1 − 4 y i + 3 y i +1 ) + i −1 y i −1 + i y i + i +1 y i +1 = i −1 24h 12 12 12 12Now introducing the fitting factor σ ( ρ ) in the above scheme ha ha  y i −1 − 2 y i + y i +1  a i −1 10a i σ ( ρ )ε + i −1 − i +1    24h (− 3 y i −1 + 4 y i − y i +1 ) + 24h ( y i +1 − y i −1 ) + 12 12  h2  (11) a i +1 bi −1 10bi bi +1 ( f i −1 + 10 f i + f i +1 )+ ( y i −1 − 4 y i + 3 y i +1 ) + y i −1 + yi + y i +1 = 24h 12 12 12 12The fitting factor σ ( ρ ) is to be determined in such a way that the solution of (11) converges uniformly to thesolution of (1)-(2). Multiplying (11) by h and taking the limit as h → 0 , and by using equation (6), we getσ =ρ a (0) coth ( )  a 2 (0) − εb(0) ρ   2  2 a ( 0)    (12)which is a constant fitting factor.The tridiagonal system of the equation (11) is given by E i y i −1 − Fi y i + Gi y i +1 = H i , for i = 1,2,…,N-1 (13)where 1  ha ha  3a b 10a i a i +1 E j = 2  εσ + i −1 − i +1  − i −1 + i −1 −   + h  12 12  24h 12 24h 24h 2  ha ha  4a 10bi 4a i +1Fj =  εσ + i −1 − i +1  − i −1 −  + h2  12 12  24h  12 24h 1  ha ha  a b 10a i 3a i +1Gj =  εσ + i −1 − i +1  − i −1 + i +1 +  + h2  12 12  24h 12  24h 24hHj = 1 ( f i −1 + 10 f i + f i +1 ) 12where σ is given by (12). We solve this tridiagonal system by the discrete invariant imbedding algorithm.2.2 Right-end boundary layer problemFinally, we discuss our method for singularly perturbed two point boundary value problems with right-end boundarylayer of the underlying interval. To be specific, we consider a class of singular perturbation problem of the form:εy ′′( x) + a( x) y ′( x) + b( x) y ( x) = f ( x) , x ∈[0, 1] (14)with y(0)=α (15a)and y(1)= β (15b) 19
  4. 4. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.11, 2012where ε is a small positive parameter (0 < ε <<1) and α, β are known constants. We assume that a(x), b(x) and f(x)are sufficiently smooth functions in [0, 1]. We assume that a(x) ≤ M < 0 throughout the interval [0, 1], where M issome negative constant. This assumption merely implies that the boundary layer will be in the neighborhood of x =1.From the theory of singular perturbations the solution of (14)-(15) is of the form 1  a ( x) b( x)  a(1) ∫  ε a( x)    −   dxy ( x ) = y 0 ( x) + ( β − y 0 (1)) e x + O(ε ) a ( x) (16)where y0 ( x) is the solution of ′a ( x) y 0 ( x) + b( x) y 0 ( x) = f ( x) , y 0 (0) = α . (17)By taking the Taylor’s series expansion for a(x) and b(x) about the point ‘1’ and restricting to their first terms, (16)  a (1) b (1)    −  (1− x ) becomes y ( x) = y 0 ( x) + ( β − y 0 (1)) e  ε a (1)  + O (ε ) (18)Now we divide the interval [0, 1] into N equal parts with constant mesh length h.Let 0= x1, x2 ,.......xN =1 be the mesh points. Then we have xi = ih : i = 0, 1, 2, … , N.  a (1) b (1)   ε − a (1)  (1− ih )  From (18), we have y (ih) = y 0 (ih) + ( β − y 0 (1)) e   + O (ε ) .  a 2 (1) −εb (1)   1    − iρ    a (1)  ε Therefore lim y (ih) = y 0 (0) + ( β − y 0 (1)) e   h →0 (19) hwhere ρ = . εProceeding as in the left-end boundary layer problem, we get the fitting factor as σ = ρ a(0) coth ( )  a 2 (1) − εb(1) ρ   2  2a(1)   which is a constant fitting factor. The tridiagonal system of the equation (11) is given by (13) where Ei , Fi , Gi andH i are same as given in left-end boundary layer.3. Stability and convergence analysisTheorem 1. Under the assumptions ε > 0 , a ( x) ≥ M > 0 and b(x) < 0, ∀x ∈ [0, 1] , the solution to the system of thedifference equations (13), together with the given boundary conditions exists, is unique and satisfies y h , ∞ ≤ 2 M −1 f h , ∞ + ( α + β )where ⋅ h ,∞ is the discrete l ∞ − norm, given by x h ,∞ = max { xi }. 0≤i≤ NProof. Let Lh (.) denote the difference operator on left hand side of Eq. (13) and wi be any mesh function satisfyingL h ( wi ) = f iBy rearranging the difference scheme (13) and using non-negativity of the coefficientsE i , Fi and Gi , we obtain Fi wi ≤ H i + E i wi −1 + Gi wi +1Now using the assumption ε >0 and a i ≥ M , the definition of l∞ -norm and manipulating the above inequality, weget (w i +1 − 2 wi + wi −1 ) M  − wi +1 + 4 wi − 3 wi −1   bi −1 + 10bi bσε + w + wi + i +1 wi +1 h 2 12   2h  12 i −1  12 12 (20) 10M ( wi +1 − wi −1 ) M  3 wi +1 − 4 wi + wi −1    + Hi ≥ 0+ + 12 2h 12   2h   20
  5. 5. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.11, 2012To prove the uniqueness and existence, let {u i }, {v i } be two sets of solution of the difference equation (13) satisfyingboundary conditions. Then wi = u i − v i satisfies Lh ( wi ) = f i where f i = 0 and w0 = wN = 0.Summing (20) over i = 1, 2, ……, N-1, we obtain w1 w N −1 w1 3 w N −1 1 N −1− σε 2 − σε 2 + a h ,∞ + a h ,∞ + ∑ bi −1 wi −1 + h h 24h 24h 12 i =1 (21)10 N −1 1 N −1 10 M 10M 3M M N −1 ∑ bi wi + ∑ bi +1 wi +1 − w1 − wN −2 − w1 − w N −1 + ∑ H i ≥ 012 i =1 12 i =1 24h 24h 24h 24h i =1Since ε > 0, a h, ∞ ≥ 0, bi < 0 and wi ≥ 0 ∀i, i = 1,2,......, N − 1, therefore for inequality (21) to hold, we must havewi = 0 ∀i, i = 1,2,.....N − 1 .This implies the uniqueness of the solution of the tridiagonal system of difference equations (13). For linearequations, the existence is implied by uniqueness. Now to establish the estimate, let wi = y i − l i ,where y i satisfies difference equations (13), the boundary conditions and l i = (1 − ih )α + (ih )β ,then w0 = w N = 0, and wi , i = 1,2,...N − 1L h ( wi ) = f iNow let wn = w h ,∞ ≥ wi , i = 0,1,......., N .Then summing (20) from i = n to N-1 and using the assumption on a(x), which gives ( wn − wn−1 ) w N −1 M  3 w N −1 + wn − 3 wn −1  1 N −1  +− σε − σε + b w + h 2 h 2 12  2h  12 i∑ i −1 i −1 =n  10 N −1 1 N −1 10M  w N −1 − wn − wn −1    ∑ bi wi + ∑ bi +1 wi +1 + (22)12 i = n 12 i = n 12   2h   M  − w N −1 − 3 wn + wn −1   N −1  + ∑ Hi ≥ 0+  12  2h  i =n Inequality (22), together with the condition on b(x) implies that M N −1 N wn ≤ h ∑ f i ≤ h ∑ f i ≤ f h, ∞ , 2 i=n i=0i.e., we have w n ≤ 2 M −1 f h,∞ (23)Also, we have y i = wi + l i y h, ∞ = max { y i } ≤ w h ,∞ + l h, ∞ ≤ wn + l h ,∞ . 0 ≤i ≤ N (24)Now to complete the estimate, we have to find out the bound on li l h ,∞ = max { l i } ≤ max { (1 − ih ) α + ih β } ≤ max { (1 − ih ) α + (ih ) β }, i.e., we have 0 ≤i ≤ N 0 ≤i ≤ N 0 ≤i ≤ N l h, ∞ ≤ α + β. (25)From Eqs. (23) – (25), we obtain the estimate y h ,∞ ≤ 2 M −1 f h ,∞ + (α + β ) .This theorem implies that the solution to the system of the difference equations (18) are uniformly bounded,independent of mesh size h and the perturbation parameter ε . Thus the scheme is stable for all step sizes.Corollary 1. Under the conditions for theorem 1, the error ei = y ( x i ) − y i between the solution y(x) of the continuesproblem and the solution y i of the discretized problem, with boundary conditions, satisfies the estimate e h,∞ ≤ 2M −1 τ h ,∞ , where 21
  6. 6. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.11, 2012  σh 2ε 2 ( 4)   ah 2 (3)  10ah 2 ( 3)   ah 2 (3) τi ≤ max  y ( x )  + max  y ( x )  + max  y ( x)  + max  y ( x)  xi −1 ≤ x ≤ xi +1 12 xi −1 ≤ x ≤ xi +1 72 xi −1 ≤ x ≤ xi +1 72 xi −1 ≤ x ≤ xi +1 72        Proof. Truncation error τ i in the difference scheme is given by  y − y i + y i −1   a  − 3 y i −1 + 4 y i − y i +1  10a i  y i +1 − y i −1 τ i = σε  i +1 2   − y i′′ + i −1    + hy i′′ − y i′−1  +  12   − y i′    h   12  2h   2h  a i +1  y i −1 − 4 y i + 3 y i +1 +   − hy i′′ − y i′+1   12  2h   σh 2ε 2 ( 4)   ah 2 (3)  10ah 2 ( 3)   ah 2 (3) τi ≤ max  y ( x )  + max  y ( x )  + max  y ( x)  + max  y ( x)  (26) xi −1 ≤ x ≤ xi +1 xi −1 ≤ x ≤ xi +1 72 xi −1 ≤ x ≤ xi +1 xi −1 ≤ x ≤ xi +1  12     72   72 One can easily show that the error ei , satisfiesLh (e( x i ) ) = Lh ( y ( x i ) ) − Lh ( y i ) = τ i , i = 1,2,..., N − 1And e 0 = e N = 0 .Then Theorem 1 implies that e h , ∞ ≤ 2 M −1 τ h , ∞ . (27)The estimate (26) establishes the convergence of the difference scheme for the fixed values of the parameter ε .Theorem 2. Under the assumptions ε > 0 , a( x) ≤ M < 0 and b(x)< 0, ∀x ∈ [0, 1] , the solution to the system of thedifference equations (13), together with the given boundary conditions exists, is unique and satisfies y h ,∞ ≤ 2M −1 f h ,∞ + (α + β ) .The proof of estimate can be done on similar lines as we did in theorem 1.4. Numerical ExamplesTo demonstrate the applicability of the method we have applied the method to three linear singular perturbationproblems with left-end boundary layer and two linear singular perturbation problems with right-end boundary layer.These examples have been chosen because they have been widely discussed in literature and because approximatesolutions are available for comparison. The numerical solutions are compared with the exact solutions and maximumabsolute errors with and without fitting factor are presented to support the given method.Example 1. Consider the following homogeneous singular perturbation problem from Bender and Orszag [4]εy′′( x) + y′( x) − y ( x) = 0 , x∈[0,1] with y(0) = 1 and y(1) = 1.Clearly this problem has a boundary layer at x = 0 i.e., at the left end of the underlying interval.The exact solution is given by [(e m2 − 1)e m1x + (1 − e m1 )e m2 x ] y ( x) = where m1 = (−1 + 1 + 4ε ) /( 2ε ) and m 2 = (−1 − 1 + 4ε ) /(2ε ) [e m2 − e m1 ]The maximum absolute errors are presented in table 1 for different values of ε with and without fitting factor.Example 2. Now consider the following non-homogeneous singular perturbation problem from fluid dynamics forfluid of small viscosity εy′′( x) + y′( x) = 1 + 2 x ; x∈[0,1] with y(0)=0 and y(1)=1. The exact solution is given by ( 2ε − 1)(1 − e − x / ε )y ( x) = x( x + 1 − 2ε ) + . The maximum absolute errors are presented in table 2 for different values (1 − e −1 / ε )of ε with and without fitting factor.Example 3. Finally we consider the following variable coefficient singular perturbation problem from Kevorkian  x 1and Cole [3] εy ′′ + 1 −  y ′ − y = 0 , x∈[0,1] with y(0)=0 and y(1)=1.  2 2We have chosen to use uniformly valid approximation (which is obtained by the method given by Nayfeh [2] as our 22
  7. 7. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.11, 2012  x− x2 / 4  −  1 1  ε ‘exact’ solution: y ( x) = − e   2−x 2The maximum absolute errors are presented in table 3 for different values of ε with and without fitting factor.Example 4. Consider the following singular perturbation problem εy′′( x) − y′( x) = 0 ; x ∈[0,1]with y(0) = 1 and y(1) = 0. Clearly, this problem has a boundary layer at x=1. i.e., at the right end of the underlyinginterval.The exact solution is given by y ( x) = ( ) e ( x −1) / ε − 1 ( ) e −1 / ε − 1 . The maximum absolute errors are presented in table 4 fordifferent values of ε with and without fitting factor.Example 5. Now we consider the following singular perturbation problem εy′′( x) − y′( x) − (1 + ε ) y ( x) = 0 , x∈[0,1] with y(0) = 1+exp(-(1+ε)/ε) and y(1) =1+1/e.The exact solution is given by y(x) = exp(1 + ε )( x − 1) / ε + exp(− x)The maximum absolute errors are presented in table 5 for different values of ε with and without fitting factor.5. Discussions and conclusionsWe have described a special second order fitted difference method for solving singularly perturbed two pointboundary value problems. We have introduced a fitting factor in a special second order finite difference schemewhich takes care of the rapid changes occur that in the boundary layer. This fitting factor is obtained from the theoryof singular perturbations. Thomas algorithm is used to solve the tridiagonal system of the fitted method. Theexistence and uniqueness of the discrete problem along with stability estimates are discussed. We have presentedmaximum absolute errors for the standard examples chosen from the literature and also presented maximum absoluteerrors for the some of the examples with and without fitting factor to show the efficiency of the method whenε << h . The computational rate of convergence is also obtained by using the double mesh principle [4] definedbelow.Let Z h = max y h − y h / 2 , j = 0, 1, 2, ….., N-1 where y h is the computed solution on the mesh x j j j j j { } N 0 at the nodalpoint x j where x j = x j −1 + h, j = 1,2,.....N and y h / 2 is the computed solution at the nodal point x j on the mesh j{x } 2N j 0 where x j = x j −1 + h / 2, j = 1(1)2 N . In the same way we can define Z h / 2 by replacing h by h/2 and N by2N i.e., Z h / 2 = max y h / 2 − y h / 4 , j= 0, 1, 2, ….., 2N-1. j j j log Z h − log Z h / 2The computed order of convergence is defined as Order = . We have taken h = 2 −3 log(2)for finding the computed order of convergence and results are shown in Table 6.ReferencesAscher, U. & Weis, R. (1984), “Collocation for singular perturbation problem III: Non-linear problems withoutturning points”, SIAM J. Sci. Stat. Comput. 5, 811-829.Bellman, R. (1964), “Perturbation Techniques in Mathematics, Physics and Engineering”, New York, Holt, Rinehart,Winston.Bender, C.M., & Orszag, S.A. (1978), “Advanced Mathematical Methods for Scientists and Engineers”, New York,Mc. Graw-Hill.Doolan, E.P., Miller, J.J.H. & Schilders, W.H.A. (1980), “Uniform numerical methods for problems with initial andboundary layers”, Dublin, Boole Press.Eckhaus, W. (1973). Matched Asymptotic Expansions and Singular Perturbations, Amsterdam, Holland: NorthHolland Publishing company.Kadalbajoo, M. K and Reddy, Y. N. (1989), “Asymptotic and Numerical Analysis of singular perturbation problems:A Survey”, Applied Mathematics and computation 30, 223-259.Kadalbajoo, M. K. and Patidar, K.C. (2003), “Exponentially Fitted Spline in compression for the Numerical Solutionof Singular Perturbation Problems”, Computers and Mathematics with Application 46, 751-767. 23
  8. 8. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.11, 2012Kevorkian, J., & Cole, J. D. (1981), “Perturbation Methods in Applied Mathematics”, New York, Springer-Verlag.Lin, P. and Su, Y. (1989), “Numerical solution of quasilinear singularly perturbed ordinary differential equationwithout turning points”, App. Math. Mech. 10, 1005-1010.Nayfeh, A. H. (1973), “Perturbation Methods”, New York, Wiley.O’ Malley, R.E. (1974), “Introduction to Singular Perturbations”, New York, Academic Press.Roos, H.G. (1986), “A second order monotone upwind scheme”, Computing 36, 57-67.Van Dyke, M. (1974), “Perturbation Methods in Fluid Mechanics”, Stanford, California, Parabolic Press.Vulanovic, R. (1991), “A second order numerical method for nonlinear singular perturbation problems withoutturning points”, Zh. Vychisl. Mat. Mat. Fiz. 31, 522-532.Dr. K. Phaneendra He born at Nalgonda village in 28-08-1975. He completed his post graduation in M.Sc. AppliedMathematics and Ph.D. in Numerical Analysis from National Institute of Technology, Warangal, India. He published18 research articles in various International Journals. Presently, he is working as a senior assistant professor inmathematics at K.I.T.S., Warangal.Pro. Y.N.Reddy He completed his post graduation in M.Sc. Applied Mathematics from Regional EngineeringCollege, Warangal and Ph.D. in Numerical Analysis from IIT, Kanpur, India. He published 58 research articles invarious International Journals. He awarded 6 Ph.D.’s and one post doctorate under his supervision. Presently, he isworking as a professor in mathematics at N.I.T.., WarangalK. Madhu Latha She born at Warangal. She completed her post graduation in M.Sc. Applied Mathematics fromNational Institute of Technology, Warangal, India. She is perusing Ph.D. in Numerical Analysis under the supervisionof Prof. Y.N. Reddy, at N.I.T. Warangal. 24
  9. 9. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.11, 2012 25
  10. 10. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.11, 2012 26
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