A Parameter-uniform Method for Two Parameters Singularly Perturbed Boundary Value Problems via Asymptotic Expansion

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1 Appl. Math. Inf. Sci. 7, No. 4, (2013) 1525 Applied Mathematics & Information Sciences An International Journal A Parameter-uniform Method for Two Parameters Singularly Perturbed Boundary Value Problems via Asymptotic Expansion D. Kumar 1,, A. S. Yadaw 2 and M. K. Kadalbajoo 3 1 Department of Mathematics, B.I.T.S. Pilani, , India 2 Department of Pharmacology & Systems Biology Mount Sinai School of Medicine, New York, NY Department of Mathematics & Statistics, I.I.T. Kanpur, , India Received: 11 Dec. 2012, Revised: 13 Jan. 2013, Accepted: 4 Feb Published online: 1 Jul Abstract: An approximate method for two parameters singularly perturbed boundary value problems having boundary layers at both end points is given. The method is motivated by the asymptotic behavior of the solution. In the outer region the solution of the problem is approximated by the zeroth order asymptotic expansion while in the inner region the solution of the problem is obtained by using B- spline collocation method. The method is iterated on the transition point of the boundary layer region. To demonstrate the applicability of the method two test examples are considered. Keywords: Singular perturbation, two parameters, boundary layer, asymptotic expansion, B-spline collocation method. 1. Introduction The boundary value problems for ordinary differential equations in which one or more small positive parameters multiplying the derivatives arise in the field of physics and applied mathematics. The problems with one small positive parameter multiplying to highest derivative have considered by many authors [1]. Motivated by asymptotic expansion [2, 3] a second order boundary value problem with two small parameters multiplying to the highest and second highest derivative is considered. This type of problems arise in chemical reactor theory, engineering, biology, lubrication theory etc. Consider the problem Ly(x) ǫy (x) µa(x)y (x)+b(x)y(x) = f(x), (1) x [0,1], with y(0) = α, y(1) = β, (2) where ǫ and µ are small positive parameters satisfying ǫ/µ 2 0 as µ 0. The functions a(x), b(x) and f(x) are sufficiently smooth satisfying a(x) a > 0, b(x) b > 0. In 1967, Malley gave the asymptotic solution of two parameters singularly perturbed boundary value problems and demonstrate the roll of ǫ and µ on the solution. After three decades, various mathematician [4, 5, 6] gave the numerical solution of two parameters problems. Malley [2,3,7,8,9] examined the nature of asymptotic solution of the continuous problem where the ratio of µ 2 to ǫ was identified as significant. In [10, 11], the standard upwind finite difference operator on two different choices of Shishkin mesh was shown to be parameter-uniform of first order. In [12] parameter-uniform methods on a uniform mesh were constructed. Vulanović [13], used the higher order finite difference scheme on a piecewise uniform mesh both of Shishkin and Bakhavalov type for solving quasi-linear boundary value problems with small parameters. 2. Solution of the problem 2.1. Asymptotic solution Consider the asymptotic expansion of (1) with (2) of the form y(x,ǫ,µ) = (y 0 +(ǫ/µ)y 1 +(ǫ/µ) 2 y 2 +O((ǫ/µ) 3 )) (3) Corresponding author dkumar2845@gmail.com

2 1526 D. Kumar et al. : A Parameter-uniform Method for Two Parameters... +(z 0 +(ǫ/µ)z 1 +(ǫ/µ) 2 z 2 +O((ǫ/µ) 3 )) (4) = (y 0 +z 0 )+(ǫ/µ)(y 1 +z 1 ) (5) +(ǫ/µ) 2 (y 2 +z 2 )+O((ǫ/µ) 3 ) (6) = W 0 +(ǫ/µ)w 1 +(ǫ/µ) 2 W 2 +O((ǫ/µ) 3 ), (7) where the zeroth order asymptotic expansion W 0 is given by W 0 (x) = y 0 + z 0, where y 0 is the solution of the reduced problem µa(x)y 0(x)+b(x)y 0 (x) = f(x), y 0 (1) = β, (8) and y 1 (x) is the solution given by ǫy 1(x) µa(x)y 1(x)+b(x)y 1 (x) = µy 0(x), (9) y 1 (0) = 0, y 1 (1) = (ǫ/µ) 1 z 0 (1), (10) and z 0 is a layer correction given by d 2 z 0 /dτ 2 a(0)dz 0 /dτ = 0, z 0 (0) = α y 0 (0),z 0 ( ) = 0, (11) where τ = x/(ǫ/µ). The solution of Eq. (8) is given by ( 1 ) y 0 (x) = exp [ β x 1 x b(s) µa(s) ds f(s) µa(s) exp ( 1 x and the solution of Eq. (11) is given by z 0 (x) = (α y 0 (0))exp( a(0)x/(ǫ/µ)). ) b(s) µa(s) ds (12) ] ds (13), Thus the zeroth order asymptotic expansion W 0 (x) is obtained. It is easy to see that the zeroth order asymptotic expansion W 0 (x) of the problem (1) with (2) satisfies the following inequality: Theorem 1. The zeroth order asymptotic expansion W 0 (x) of the solutiony(x) of (1) with (2) satisfies y(x) W 0 (x) C(ǫ/µ), where ǫ/µ 2 0 asµ 0. Also the solution y(x) of (1) with (2) and its derivatives satisfy the following inequality: Theorem 2. Suppose a(x), b(x) and f(x) are sufficiently smooth and has derivatives at least of order k then we have y (i) (x) C[1+µ i exp( ν(1 x)/µ) (14) where b(x) a(x) ν. +ǫ i exp( αx/(ǫ/µ))], i = 0,1,...,k, (15) 2.2. Discrete solution We divide the interval [0, 1] into three non overlapping subintervals[0, k(ǫ/µ)], (k(ǫ/µ), 1 kµ) and[1 kµ, 1] where k be any positive integer such that kµ 1. The original problem is then divided into three equivalent problems (two inner regions and one outer region problem). To obtain the boundary conditions at the transition points k(ǫ/µ) and (1 kµ), we use the zeroth order asymptotic expansion W 0. Thus if W 0 (k(ǫ/µ)) = γ 1 and W 0 (1 kµ) = γ 2 then the three problems are P 1 :The left inner region problem (x [0,k(ǫ/µ)]) ǫy (x) µa(x)y (x)+b(x)y(x) = f(x), (16) y(0) = α, y(k(ǫ/µ)) = γ 1. (17) P 2 :The outer region problem (x (k(ǫ/µ),1 kµ)) ǫy (x) µa(x)y (x)+b(x)y(x) = f(x), (18) y(k(ǫ/µ)) = γ 1, y(1 kµ) = γ 2. (19) P 3 :The right inner region problem (x [1 kµ,1]) ǫy (x) µa(x)y (x)+b(x)y(x) = f(x), (20) y(1 kµ) = γ 2, y(1) = β. (21) After solving the inner and outer region problems, we combine their solutions to obtain an approximate solution of the problem (1) with (2) in the whole interval[0,1]. We change the value of k until the solution profiles do not differ much from iteration to iteration. For this we use the absolute error criteria y (m+1) (x) y (m) (x) σ, where σ is prescribed tolerance error bound. We use B-spline collocation method in the inner region and forp 1 we have x 0 = 0, x N = kǫ/µ, h 1 = kǫ/µn, x i = x 0 + ih 1, i = 1,2,...,N and for P 3 we have x 0 = 1 kµ, x N = 1, h 2 = kµ/n, x i = x 0 +ih 2, i = 1,2,...,N. For 0 < l 1 < l 2 < 1, we define L 2 [l 1,l 2 ] a vector space of all the square integrable function on [l 1,l 2 ] and let X be the linear subspace of L 2 [l 1,l 2 ]. Now define for i = 0,1,2,...,N (x x i 2 ) 3, x i 2 x x i 1, h 3 +3h 2 (x x i 1 )+3h(x x i 1 ) 2 B i (x) = 1 3(x x i 1 ) 3, x i 1 x x i, h 3 h 3 +3h 2 (x i+1 x)+3h(x i+1 x) 2 (22) 3(x i+1 x) 3, x i x x i+1, (x i+2 x) 3, x i+1 x x i+2, 0, otherwise. Let π = {x 0,x 1,...,x N } be the partition of [l 1,l 2 ]. We introduce four additional knots x 2 < x 1 < x 0 and x N+2 > x > x N. It is easy to check that each of the function B i (x) is twice continuously differentiable on the entire real line. Also 4, ifi = j, B i (x j ) = 1, ifi j = ±1, (23) 0, ifi j = ±2,

3 Appl. Math. Inf. Sci. 7, No. 4, (2013) / and that B i (x) = 0 for x x i+2 and x x i 2. We can also see that 0, if i = j, B i(x j ) = ± 3 h, if i j = ±1, (24) 0, if i j = ±2, and B i (x j ) = 12 h 2, if i = j, 6 h, if i j = ±1, 2 0, if i j = ±2. (25) Each B i (x) is also a piecewise cubic at π and B i (x) X. Let Π = {B 1,B 0,B 1,...,B } and let Φ 3 (π) is span ofπ. ThenΠ is linearly independent on[l 1,l 2 ], thus Φ 3 (π) is(n+3)-dimensional. In factφ 3 (π) is a subspace of X. Let L be a linear operator whose domain is X and whose range is also inx. Now we define S(x) = c i B i (x), (26) where c i are unknown real coefficients. Here we have introduced two extra cubic B-splines, B 1 and B to satisfy the boundary conditions. Thus LS(x i ) = f(x i ), 0 i N, (27) and S(x 0 ) = α, S(x N ) = β. (28) On solving the collocation equations (27) and putting the values of B-spline functions B i and of derivatives at mesh points, we obtain a system of (N +1) linear equations in (N +3) unknowns ( 6ǫ+3µa i h+b i h 2 )c i 1 +(12ǫ+4b i h 2 )c i +( 6ǫ 3µa i h+b i h 2 )c i+1 = f i h 2. (29) The given boundary conditions become c 1 +4c 0 +c 1 = α, c N 1 +4c N +c = β. (30) Thus the Eqs. (29) and (30) lead to an (N + 3) (N + 3) system with(n +3) unknownsc 1,c 0,...,c N,c. Now eliminating c 1 from first equation of (29) and from first equation of (30) (36ǫ 12µa 0 h)c 0 +( 6µa 0 h)c 1 = h 2 f 0 α( 6ǫ+3µa 0 h+b 0 h 2 ). (31) Similarly, eliminating c from the last equation of (29) and from second equation of (30) (6µa N h)c N 1 +(36ǫ+12µa N h)c N = h 2 f N β( 6ǫ 3µa N h+b N h 2 ). (32) Coupling equations (31) and (32) with the second through (N 1)st equations of (29). Thus by the elimination of c 1 and c, we lead to a system Tx N = d N of (N + 1) linear equations in (N + 1) unknowns, where x N = (c 0,c 1,...,c N ) t and d N = (h 2 f 0 α( 6ǫ + 3µa 0 h + b 0 h 2 ),h 2 f 1,...,h 2 f N 1,h 2 f N β( 6ǫ 3µa N h + b N h 2 )). The elements t i,j of the tridiagonal matrixt are given by 36ǫ 12µa 0 h, i = j = 0, 6µa 0 h, i = 0,j = 1, 6ǫ+3µa i h+b i h 2, i = j +1,j = 0(1)N 2, 12ǫ+4b i h 2, i = j = 1(1)N 1, t i,j = (33) 6ǫ 3µa i h+b i h 2, i = j 1,j = 2(1)N, 6µa N h, i = N,j = N 1, 36ǫ+12µa N h, i = j = N, 0, i j > 1. It is easy to see that the matrix T is strictly diagonally dominant and hence nonsingular. Since T is nonsingular, we can solve the systemtx N = d N forc 0,c 1,...,c N and substitute into the boundary conditions (30) to obtain c 1 and c. Lemma 1. The B-splinesB 1,B 0,...,B defined in equation (22), satisfy the inequality B i (x) 10, 0 x 1. Proof. We know that B i(x) B i(x). At any node x i, we have B i = B i 1 + B i + B i+1 = 6 < 10. Also we have B i (x) 4 and B i 1 (x) 4, x i 1 x x i. Similarly B i 2 (x) 1 and B i+1 1, x i 1 x x i. Now for any point x i 1 x x i we have B i(x) = B i 2 + B i 1 + B i + B i Theorem 3. Let S(x) be the collocation approximation from the space of cubic splines Φ 3 (π) to the solution y(x) of the boundary value problem (1) (2). If f C 2 [l 1,l 2 ], then the parameter-uniform error estimate is given by y(x) S(x) Ch 2, where C is a positive constant independent of ǫ and N. Proof. To estimate the error y(x) S(x), let Y n be the unique spline interpolate from Φ 3 (π) to the solution y(x) of our boundary value problem (1) (2). If f(x) C 2 [l 1,l 2 ] then y(x) C 4 [l 1,l 2 ] and it follows from De Boor-Hall error estimates [14] that D j (y(x) Y n ) γ j h 4 j, j = 0,1,2, (34) where h is uniform mesh spacing and γ j s are constants independent ofhand N. Let Y n (x) = p i B i (x). (35)

4 1528 D. Kumar et al. : A Parameter-uniform Method for Two Parameters... It follows immediately from the estimates (34) that LS(x i ) LY n (x i ) = f(x i ) LY(x i ) +Ly(x i ) Ly(x i ) λh 2, (36) where λ = ǫγ 2 + µγ 1 a(x) h + γ 0 b(x) h 2. Also LS(x i ) = Ly(x i ) = f(x i ). Let LY n (x i ) = ˆf n (x i ) for i = 1,2,...,N and ˆf n = (ˆf n (x 0 ), ˆf n (x 1 ),..., ˆf n (x N )) t. Now from the system Tx N = d N and (34), it is clear that the ith coordinate [T(x N y N )] i of T(x N y N ), where y N = (p 0,p 1,...,p N ) t, satisfies the inequality [T(x N y N )] i λh 4. (37) Since (Tx N ) i = h 2 f(x i ) and (Ty N ) i = h 2 ˆf n (x i ) for i = 0,1,2,...,N. The ith coordinate of [T(x N y N )] is theith equation ( 6ǫ+3µa i h+b i h 2 )δ i 1 +(12ǫ+4b i h 2 )δ i +( 6ǫ 3µa i h+b i h 2 )δ i+1 = ξ i, 1 i N 1,(38) where δ i = p i q i, 1 i N + 1 and ξ i = h 2 (f(x i ) ˆf n (x i )), 1 i N 1. Clearly ξ i λh 4. Let ξ = max ξ i. Also consider 1 i N 1 δ = (δ 1,δ 0,...,δ ) t, then we define e i = δ i and ẽ = max e i. Now equation (38) becomes 1 i N (12ǫ+4b i h 2 )δ i ξ i ξ i +(6ǫ b i h 2 )(δ i 1 +δ i+1 ) +3µa i h(δ i+1 δ i 1 ), 1 i N 1. (39) Taking absolute values with sufficiently small h, we obtain (12ǫ+4b i h 2 )e i ξ +2ẽ(6ǫ+3µa i h b i h 2 ). (40) Since 0 < a a(x) and 0 < b b(x), we get (12ǫ+4b h 2 )e i ξ +2ẽ(6ǫ+3µa h b h 2 ) (41) ξ +2ẽ(6ǫ+3µa h+b h 2 ). (42) In particular (12ǫ+4b h 2 )ẽ ξ +2ẽ(6ǫ+3µa h+b h 2 ). (43) Solving forẽ, we obtain(2b h 2 6µa h)ẽ ξ λh 4 or ẽ λh 3 2b h 6µa. (44) Now to estimate e 1,e 0,e N and e, we observe that the first and last equation of the the systemt(x N y N ) = h 2 ( f n ˆf n ) where f n = (f 0,f 1,...,f N ), gives e 0 and e N 2λb h 5 (36ǫ 12a hµ)(2b h 6µa ), (45) 2λb h 5 (36ǫ+12a hµ)(2b h 6µa ). (46) Now e 1 and e can be evaluated using boundary conditions δ 1 = (4δ 0 δ 1 ) and δ = (4δ N δ N 1 ) λh 3 ( 2b h 2 +9ǫ 3a ) hµ e 1 (2b h 6µa ) 9ǫ+3a, (47) hµ and e λh 3 (2b h 6µa ) ( 2b h 2 +9ǫ 3a hµ 9ǫ+3a hµ ). (48) Using valueλ = ǫγ 2 +µγ 1 b(x) h+γ 0 b(x) h 2 and since there exists a constant C such that e = max 1 i {e i} Ch 2. (49) The above inequality enables us to estimate S(x) Y n (x), and therefore y(x) S(x). In particular S(x) Y n (x) = (q i p i )B i (x). (50) Thus S(x) Y n (x) max q i p i B i (x). (51) Combining equations (49), (51) and using Lemma 1 we obtain S Y n Ch 2. Since y Y N γ 0 h 4 and y S y Y N + Y N S, we obtain y S Ch 2. Combining the results we get the required estimate. 3. Test examples and numerical results To demonstrate the efficiency of the method, two numerical examples are considered. Since the exact solution of the considered problems are given so the maximum absolute errors are estimated by using E N,ǫ = max 0 i N yn i S N i, where y N i is the exact solution and S N i is the computed solution. If the exact solution is not known then the maximum absolute errors can be obtained by using the double mesh principle. Example 1. Consider the boundary value problem ǫy (x) µy (x)+y(x) = x, y(0) = 1, y(1) = 0. The exact solution of the problem is given by y(x) = (1+µ)+(1 µ)em2 e m2 e m1 e m1x (52) + (1+µ)+(1 µ)em1 e m1 e m2 e m2x +x+µ, (53)

5 Appl. Math. Inf. Sci. 7, No. 4, (2013) / Table 1: Maximum absolute error at nodal points for Example 1 for ǫ = 10 6 and µ = 10 2 k Nodes E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-11 O max E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E+00

6 1530 D. Kumar et al. : A Parameter-uniform Method for Two Parameters... Table 2: Maximum absolute error at nodal points for Example 2 for ǫ = 10 6 and µ = 10 2 k Nodes E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E+00 O max E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E+00

7 Appl. Math. Inf. Sci. 7, No. 4, (2013) / where m 1 = µ µ 2 +4ǫ 2ǫ, m 2 = µ+ µ 2 +4ǫ 2ǫ. For this problem the zeroth order asymptotic expansion W 0 is given by W 0 (x) = x exp((x 1)/µ) (54) + (1 exp( 1/µ))exp( x/(ǫ/µ)). (55) Example 2. Consider the boundary value problem ǫy (x) 2µy (x)+4y(x) = 1, y(0) = 0, y(1) = 1. The exact solution of the problem is given by 3+e m2 y(x) = 4(e m1 e m2 ) em1x 3+em1 4(e m1 e m2 ) em2x where m 1 = µ µ 2 +4ǫ ǫ, m 2 = µ+ µ 2 +4ǫ ǫ. For this example the zeroth order asymptotic expansion W 0 is given by W 0 (x) = (1/4)[1 exp( 2x/(ǫ/µ))] (56) + (3/4)[exp(2x/µ) exp( 2x/(ǫ/µ))] exp( 2/µ).(57) 4. Conclusion In this paper, we present an approximate method based on asymptotic expansion for two parameters singularly perturbed boundary value problems. The B-spline method is used in the inner region and for rest of the region we use the zeroth order asymptotic expansion approximation. To demonstrate the applicability of the method two numerical examples have been considered. For inner regions the absolute errors at nodal points are presented in the tables while for the outer region the maximum absolute error denoted by O max is presented at the middle of the table. It can be seen from the tables that the numerical solutions are very closed to exact solutions. The solution profiles for the considered examples for a fix µ and different values of ǫ are given in Figures 1 and ε = 10 3 ε = 10 4 ε = ε = 10 2 ε = 10 3 ε = Figure 2: Solution profile for Example 2 for µ = 10 4 using N = 32. References [1] S.M. Roberts, J. Math. Anal. Appl. 113, 411 (1988). [2] R.E. O Malley Jr., J. Math. Anal. Appl. 19, 291 (1967). [3] R.E. O Malley Jr., J. Math. Mech. 16, 1143 (1967). [4] small H.G. Roos and Z. Uzelac, Comput. Methods Appl. Math. 3, 443 (2003). [5] J.L. Gracia, E. O Riordan and M.L. Pickett, Appl. Numer. Math. 56, 962 (2006). [6] T. Linßand H.G. Roos, J. Math. Anal. Appl. 289, 355 (2004). [7] R.E. O Malley Jr., J. Math. Mech. 18, 835 (1969). [8] R.E. O Malley Jr., Introduction to Singular Perturbations (Academic Press, New York, 1974). [9] R.E. O Malley Jr., Singular Perturbation Methods for Ordinary Differential Equations (Springer, New York, 1990). [10] E. O Riordan, M.L. Pickett and G. I. Shishkin, Comput. Methods Appl. Math. 3, 424 (2003). [11] E. O Riordan, M.L. Pickett and G.I. Shishkin, Math. Comp. 75, 1135 (2006). [12] G.I. Shishkin and V.A. Titov, Chisl. Metody Mekh. Sploshn. Sredy 2, 145 (1976). [13] R. Vulanović, Computing 67, 287 (2001). [14] C. De Boor, A Practical Guide to Splines (Springer-Verlag, New York, 1978) Figure 1: Solution profile for Example 1 for µ = 10 4 using N = 32.

8 1532 D. Kumar et al. : A Parameter-uniform Method for Two Parameters... D. Kumar is presently employed as an assistant professor in the department of Mathematics at BITS Pilani, India. He obtained his PhD from IIT Kanpur (INDIA). He is an active researcher coupled with the teaching experience in various Institutes in India. He has published more than 10 research articles in reputed international journals of mathematical and engineering sciences. A. S. Yadaw is presently a postdoc fellow in the department of Pharmacology & Systems Biology Mount Sinai School of Medicine, New York. He obtained his PhD from IIT Kanpur (INDIA). He is an active researcher and has published more than 6 research articles in reputed international journals of mathematical and engineering sciences. M. K. Kadalbajoo is presently employed as a professor in the department of Mathematics at IIT Kanpur, India. He obtained his PhD from IIT Bombay (INDIA). He is an active researcher coupled with the vast (40 years) teaching experience at IIT Bombay and IIT Kanpur in India. He has been an invited speaker of number of conferences and has published more than 200 (Two hundred) research articles in reputed international journals of mathematical and engineering sciences.