1. Introduction We consider the following self-adjoint singularly perturbed boundary value problem: ɛu + p(x)u = q(x), p(x) > 0,

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1 TWMS J. Pure Appl. Math. V., N., 00, pp. 6-5 NON-POLYNOMIAL SPLINE APPROXIMATIONS FOR THE SOLUTION OF SINGULARLY-PERTURBED BOUNDARY VALUE PROBLEMS J. RASHIDINIA, R. MOHAMMADI Abstract. We consider the self-adjoint singularly perturbed two-point boundary value problems. We know that the numerical methods for solution of such problems based on nonpolynomial spline in grid points, can produce the fourth order method only. So that we look for an alternative to obtain higher order methods. We develop the non-polynomial spline in off-step points to rise the order of accuracy. Based on such spline, the purposed new methods are fourth, sixth and eighth-order accurate. These methods are applicable to problems both in singular and non-singular cases. The convergence analysis of the new eight-order method is proved. We applied the presented methods to test problems which have been solved by other existing methods in our references, for comparison of our methods with the existing methods. Numerical results are given to illustrate the efficiency of our methods. Keywords: Self-adjoint singularly-perturbed boundary value problem, Non-polynomial spline, Convergence analysis, Numerical conclusion. AMS Subject Classification: 4B5; F05; 65D0; 65L.. Introduction We consider the following self-adjoint singularly perturbed boundary value problem: ɛu + p(x)u = q(x), p(x) > 0, u(a) = η, u(b) = η, () where η, η are given constants and ɛ is a small positive parameter such that 0 < ɛ and p(x), q(x) are small bounded real functions. This problems occur naturally in various fields of science and engineering, for example, combustion, nuclear engineering, control theory, elasticity, fluid mechanics, fluid dynamics, quantum mechanics, optimal control, chemical-reactor theory, hydrodynamics, convection-diffusion process, geophysics, etc. A few notable examples are boundary-layer problems, WKB Theory, the modeling of steady and unsteady viscous flow problems with large Reynolds number and convective heat transport problems with large Peclet number. Secondly, the occurrence of sharp boundary-layers as ɛ, the coefficient of highest derivative, approaches zero creates difficulty for most standard numerical schemes. The application of splines for the numerical solution of singularly-perturbed boundary- value problems has been described in many papers []-[5], [7],[8],[]-[6]. Recently Aziz and khan [, ] and Khan et. al. [6] solve this problem by cubic spline in comparison, quintic spline and sextic spline with assuming p(x) = constant. A generalized scheme based on quartic non-polynomial spline functions was proposed by Tirmizi et. al. [6]. In this paper non-polynomial sextic spline relations have been derived using off-step points. In Section, we derive the formulation of our spline function approximation. We present the School of Mathematics, Iran University of Science and Technology, Tehran, Iran, rashidinia@iust.ac.ir Department of Mathematics, University of Neyshabour, Neyshabour, Iran, rez.mohammadi@gmail.com Manuscript received 8 November

2 J. RASHIDINIA, R. MOHAMMADI: NON-POLYNOMIAL SPLINE APPROXIMATIONS FOR... 7 spline relations to be used for discretization of the given problem (). In section, we present the formulation of our method. We drive boundary formulas in section 4. In section 5, convergence of the new eight-order method is stablished. Finally, in section 6, numerical evidence is included to demonstrate the efficiency of the method.. Non-polynomial spline functions We introduce the set of grid points in the interval [a, b] x 0 = a, x i = a + (i )h, h = b a N, i =,,..., N, x N = b. Definition. A non-polynomial spline function S l (x), interpolating to a function u(x) on [a, b] defined as: () In each subinterval [x l, x l+ ], S l (x) is a polynomial of degree at most six. () The first-fifth derivatives of S l (x) are continuous on [a, b]. () S l (x ) = u(x ), l = ()N. For each segment [x, x l+ ], l =,,..., N the non-polynomial spline S l (x), is define as S l (x) = a l + b l (x x l ) + c l (x x l ) + d l (x x l ) + e l (x x l ) 4 + +f l sin τ(x x l ) + g l cos τ(x x l ), l = 0,,,..., N, () where a l, b l, c l, d l, e l, f l and g l are constants and τ is free parameter. We further require that the values of the first-, second-, third-, fourth- and fifth-order derivatives be the same for the pair of segments that join at each point (x l, u l ). To derive expression for the coefficients of () in terms of u we first define: F and F l+ (i) S l (x ) = u,, (ii) S l (x l+ ) = u l+ (iii) S l (x ) = m, (iv) S l (x ) = M, (v) S l (x l+ ) = M l+, (vi) S (4) l (x ) = F, (vii) S (4) l (x l+ ) = F l+. From algebraic manipulation we get the following expression: a l =, u l+, m, M, M l+, 96τ 4 θ[ cos θ + θ( + θ ) sin θ] {[7θ(8 + θ ) + θ(9 + 4θ + 5θ 4 ) cos θ+ +( 5 48θ 7θ 4 + θ 6 ) sin θ]f [6θ(9 + 4θ + 5θ 4 ) + 44θ(8 + θ ) cos θ+ +( 04 96θ + 4θ 4 + θ 6 ) sin θ]f l+ + τ 4 {6h sin θ( θ + 5θ 4 )m + 4h [6θ( cos θ) + (9 + 6θ + θ 4 ) sin θ]m + h [44θ( cos θ) + (84 4θ θ 4 ) sin θ]m l+ + +[ 5( cos θ) + (04 + 8θ + 78θ 4 ) sin θ]u + [ 5( cos θ) + ( θ + +8θ 4 ) sin θ]u l+ }}, ()

3 8 TWMS J. PURE APPL. MATH., V., N., 00 b l = 4τ θ {(4 + θ )(F F l+ ) + τ 4 [h (M M l+ ) + 4(u l+ u )]}, c l = 4τ θ[ cos θ + θ( + θ ) sin θ] {[4θ + 6θ(4 + θ ) cos θ + ( 48 θ + θ 4 ) sin θ]f [6θ(4 + θ ) + 4θ cos θ (4 + θ ) sin θ]f l+ + τ {6θ sin θ (8 + θ )m + h[4( cos θ) + θ(4 + θ ) sin θ]m + h[4( cos θ) 4θ sin θ]m l+ + τ[(48 + 6θ ) sin θ](u d l = 6τh [F l+ F + τ (M l+ e l = M )], u l+ )}}, 6θ[ 4( cos θ) + θ( + θ ) sin θ] {[θ cos θ + ( + θ ) sin θ]f + [ 6θ+ and +(6 + θ ) sin θ]f l+ + τ 4 sin θ[6hm + h (M f l = τ 4 sin θ (F i+ F ), + M l+ ) + 6(u u l+ )]}, g l = τ 4 θ[ 4( cos θ) + θ( + θ ) sin θ] {[ 4θ cos θ + (48 4θ + θ 4 ) sin θ ](F F l+ ) 8τ 4 sin θ [6hm + h (M where θ = τh and l =,,..., N. + M l+ ) + 6(u u l+ )]}, (4) From continuity of the first derivative at the point x = x l, we can obtain the following spline relation: m = 4τ θ sin θ( 4θ + θ + 48 sin θ ){ [9 + 8θ + 8(4 + 4θ + θ 4 ) cos θ 8(4 θ + θ 4 ) cos θ ( θ ) cos θ θ(96 0θ + θ 4 ) sin θ 9θ sin θ ]F + [ θ θ 4 + (4 + 6θ + θ 4 ) cos θ + 4( 4 + θ ) cos θ (4 8θ θ 4 ) cos θ 88θ sin θ θ(576 7θ θ 4 ) sin θ 88θ sin θ ]F + [ 576 4θ 4θ ( + θ ) cos θ + 4(4 + θ ) cos θ + 576θ sin θ + θ(88 + 6θ + θ 4 ) sin θ]f l+ ) + 6τ θ sin θ [θ ( 0 + θ ) cos θ 8(sin θ 8 sin θ)]m + 4τ θ sin θ [θ(84 + θ ) cos θ sin θ sin θ]m + τ θ [ 4( cos θ) + θ( + θ ) sin θ]m l τ 4 sin θ [ θ cos θ +

4 J. RASHIDINIA, R. MOHAMMADI: NON-POLYNOMIAL SPLINE APPROXIMATIONS FOR (sin θ + sin θ)]u + 48τ 4 sin θ [θ(48 + θ ) cos θ 4( sin θ + sin θ)]u 4τ 4 [ 4( cos θ) + θ( + θ ) sin θ]u l+ }. (5) Also continuity of the third derivative at the point x = x l, implies that: m = 4τ θ sin θ( θ + sin θ ){ [4 + (4 + 4θ + θ 4 ) cos θ + 4( + θ ) cos θ 8( θ ) cos θ θ(6 7θ ) sin θ 4θ sin θ ]F + [48 + 4θ + (4+ +6θ + θ 4 ) cos θ 48 cos θ (4 8θ θ 4 ) cos θ 4θ sin θ 6θ(8 + θ ) sin θ 4θ sin θ ]F + [ 4 θ ( + θ ) cos θ + 4 cos θ + 4θ sin θ + θ ( + θ ) sin θ]f l+ + τ sin θ [θ( 7θ ) cos θ 4 sin θ + 8θ sin θ]m + +4τ sin θ [ θ(4 + θ ) cos θ + 4 sin θ + θ sin θ]m + τ [ 4( cos θ) + θ( + θ ) sin θ]m l+ + 4τ 4 ( θ + sin θ ) sin θ(u u )}. (6) Finally, from continuity of the fifth derivative at the point x = x l, we have the following spline relation: m = 96τ θ cos θ sin θ {{(4 + 4θ + θ 4 ) cos θ + 8[( + θ ) cos θ θ sin θ ]} F [(4 + 6θ + θ 4 ) cos θ + ( 4 + 8θ + θ 4 ) cos θ 96θ θ cos sin θ ]F + +θ[θ( + θ ) cos θ 4 sin θ ]F l+ 6τ 4 cos θ sin θ [h (M + M )+ where l = ()N. +6(u u )]}, (7) From Eqs. (5) and (6) we have u u + u l+ = +44θ + (96 + 4θ θ 4 ) cos θ]f Also from Eqs. (5) and (7) we have 48τ 4 ( cos θ) {[48 θ + θ 4 (4 + θ ) cos θ]f + [ 96+ +τ θ ( cos θ)(m M θ {[ + θ + cos θ]f [ + ( + θ ) cos θ]f From Eqs. (8) and (9) we obtain τ ( cos θ)(m M + [48 θ + θ 4 (4 + θ ) cos θ]f l+ + + M l+ )}. (8) + [ + θ + cos θ]f l+ + M l+ )} = 0. (9)

5 40 TWMS J. PURE APPL. MATH., V., N., 00 h 4 F = [ + 5θ + ( + θ ) cos θ] {h {[4 θ + θ 4 4 cos θ](m + M l+ )+ +[ 4 + 5θ 4 + θ 4 + ( + θ ) cos θ]m + θ [ θ cos θ](u u + u l+ )}. (0) The elimination of F l s from (9) and (0) gives where h [u i 5 + β (u i + u l+ ) + γ u i +M i+ ) + γm i α = 4( cos θ) + θ ( + θ ) θ ( + θ, + cos θ) β = 48 + θ ( + 5θ ) + (48 + θ + θ 4 ) cos θ 6θ ( + θ, + cos θ) γ = 7 + θ ( + θ ) (6 + θ + 5θ 4 ) cos θ 6θ ( + θ, + cos θ) β = [ 4 + θ + (4 + θ ) cos θ] + θ, + cos θ + u i+ ] = α(m i 5 + M i+ ) + β(m i +, i = ()N, () γ = [ 6 + θ + ( + θ ) cos θ] + θ. () + cos θ When τ 0, that θ 0, then (α, β, γ) 0 (, 56, 46), (β, γ ) (8, 8) and the relations defined by () and () reduce into sextic polynomial spline functions.. Description of methods Now we consider the equation (). At the grid points x i, the proposed differential equation () may be discretized by ɛu i + p i u i = q i, () where p i = p(x i ) and q i = q(x i ). By using moment of spline in () we obtain ɛm i + p i u i = q i. (4) We discretize the given system (4) at the grid points x i, i =,,..., N and using the spline relation (). We obtain (N 4) linear algebraic equation in the (N) unknowns u i, i =,,..., N as ( ɛ + αh p i 5 )u i 5 +( ɛβ + βh p i+ )u i+ + ( ɛβ + βh p i )u i + ( ɛ + αh p i+ )u i+ +βq i+ + αq i+ where p i = p(x i ) and q i = q(x i ), with local truncation error + ( ɛγ + γp i )h 4 u i + + βq i + γq i + = h (αq i 5 ), i = ()N, (5) T i (h) = ɛ( + β + γ )u i + ɛ ( + β + γ )hu i + ɛ 8 (6α + 6β + 8γ 4 0β γ )h u i ɛ 48 (48α + 48β + 4γ 98 6β γ )h u i + ɛ (6α + 480β γ 706 8β γ )h 4 u (4) i ɛ 840 (7840α + 080β + 80γ 88 4β

6 J. RASHIDINIA, R. MOHAMMADI: NON-POLYNOMIAL SPLINE APPROXIMATIONS FOR... 4 γ )h 5 u (5) i + ɛ (8470α β + 0γ β γ )h 6 u (6) i + + ɛ 50 (48476α β + 68γ β γ )h 7 u (7) i (6696α + 650β + 4γ β γ )h 8 u (8) i (87044α β + 88γ β γ )h 9 u (9) i + ɛ 090 ɛ ɛ ( α + 60β + 60γ β γ )h 0 u (0) i + O(h ), where u (k) i = u (k) (ξ i ) x i < ξ i < x i+. (6) By choosing suitable values of parameters β, γ, α, β and γ we obtain the following methods: () If we choose + β + γ = 0 and α + β + γ 4 β = 0 in (5) we obtain second-order schemes for the solution of (). () If we choose (β, γ, α, β, γ) = (4, 0, with truncation error T i = ɛ 80 h6 u (6) (ξ i ). 8, 9 () If we choose (β, γ, α, β, γ) = (4, 0, 0, with truncation error T i = ɛ 50 h8 u (8) (ξ i )., 5) in (5) we obtain a fourth-order scheme 5, 49 0 ) in (5) we obtain a sixth-order scheme (4) If we choose (β, γ, α, β, γ) = ( 8, 8, 465, , ) in (5) we obtain a eight-order scheme with truncation error T i = 79ɛ h0 u (0) (ξ i ), which is the the highest order among the methods for solution of (). 4. Development of boundary formula The relation (5) gives N 4 linear algebraic equations in the n unknowns u i, for i =,,..., n. We need four more equations, two at each end of interval, for the direct computation of u i. By using Taylor series and method of undetermined coefficients the boundary formulas associated with boundary conditions can be determined as follows. We drive the following equations to be associated with fourth-order method as: 0u 0 + 4u u u 5 u 0 u + 0u 4u 5 u 7 = h 48 [M 0 54M 75M M 5 ] h6 u (6) (x 0 )+ +O(h 7 ), i =, (7) = h 48 [0M 0 75M 0M 76M 5 M 7 ] h6 u (6) (x 0 ) + O(h 7 ), i =, (8) and for right boundary we obtain: u N u N + 0u N 4u N 5 u N 7 76M N 5 = h 48 [0M N 75M N 0M N M N 7 ] h6 u (6) (x N ) + O(h 7 ), i = N, (9)

7 4 TWMS J. PURE APPL. MATH., V., N., 00 0u N + 4u N u N u N 5 = h 48 [M N 54M N 75M N M N 5 ] h6 u (6) (x N ) + O(h 7 ), i = N. (0) In same manner for sixth-order method we derive the following boundary formula: 0u 0 + 4u u u 5 u 0 u + 0u 4u 5 u 7 89M 7 = h 440 [ 608M M 4059M + 009M 5 869M M 9 ] h8 u (8) (x 0 ) + O(h 9 ), i =, () = h 0960 [0784M M 6759M 706M M 9 ] h8 u (8) (x 0 ) + O(h 9 ), i =, () u N u N + 0u N 4u N 5 u N M N 706M N 5 89M N [0784M N 57059M N + 406M N 9 ] h8 u (8) (x N )+ +O(h 9 ), i = N, () = h 0u N + 4u N u N u N M N 5 869M N 7 = h 440 [ 608M N 4049M N + 6M N 9 ] h8 u (8) (x N )+ 4059M N + +O(h 9 ), i = N. (4) Finally for eight order method we develop the following boundary equation: 0u 0 + 4u u u 5 = h [ M M M M M M M M ] h0 u (0) (x 0 ) + O(h ), i =, (5) u 0 u + 0u 4u 5 u 7 = h [ M M M M M M M M ] h0 u (0) (x 0 ) + O(h ), i =, (6) u N u N + 0u N 4u N 5 u N 7 = h [ M N M N M N M N M N M N M N M N ]+

8 J. RASHIDINIA, R. MOHAMMADI: NON-POLYNOMIAL SPLINE APPROXIMATIONS FOR h0 u (0) (x N ) + O(h ), i = N, (7) 0u N + 4u N u N u N 5 = h [ M N M N M N M N M N M N M N M N ] h0 u (0) (x N ) + O(h ), i = N. (8) 5. Convergence analysis In this section, we investigate the convergence analysis of our eight order method. The eight order quintic spline solution of () is based on the linear equations given by (5) and boundary formulas (5)-(8). Let Z = (z i ) be N dimensional column vector and A = (a ij ) be an (N N) matrix such that z = (0ɛ h p 0 )η + h [ q q q q q q q q ], z = (ɛ h p 0 )η + h [ q q q q q q q q z i = h 465 (q i q i + 58q i + 688q i+ ], + q i+ ), i = ()N, z N = (ɛ h p N )η + h [ q N q N q N q N q N q N q N q N ], z N = (0ɛ h p N )η + h [ q N q N q N and q N q N q N q N a i,i±k = coefficient of u i k, k = ±, ±, q N ], It is easily seen that the system of N linear algebraic equations given by (5) and boundary formulas (5)-(8) may be written in the form (i) AU = Z + T, (ii) AU = Z. (9)

9 44 TWMS J. PURE APPL. MATH., V., N., 00 The vector T is the local truncation error vector and define as ɛh0 u (0) (x 0 ) + O(h ), i =, t i = ɛh0 u (0) (x 0 ) + O(h ), i =, ɛh0 u (0) (x i ) + O(h ), x i 5 < x i < x i+ i =,..., N, ɛh0 u (0) (x N ) + O(h ), i = N, ɛh0 u (0) (x N ) + O(h ), i = N. (0) The error equation is obtained in the usal manner in the form AE = T, () where E = (e i ), with e i = (u i u i ) and T = (t i ) are the N dimentional vectors. Now the matrix A may be written as where A 0 = A = ɛ(a 0 + A ) + h BP, () , () A = , (4)

10 J. RASHIDINIA, R. MOHAMMADI: NON-POLYNOMIAL SPLINE APPROXIMATIONS FOR b b b b 4 b 5 b 6 b b b b b 4 b 5 b 6 b B = b N b N b N b N 4 b N 5 b N 6 b N b N b N b N b N4 b N5 b N6 b N7 with (b, b, b, b 4, b 5, b 6, b 6 ) = (b N7, b N6, b N5, b N4, b N, b N, b N ) = ( , , , , , , ), (b, b, b, b 4, b 5, b 6, b 7 ) = (b N 7, b N 6, b N 5, b N 4, b N, b N, b N ) = ( , , , , , , ) and P = diag(p i ). (5) Theorem. The symmetric matrix A 0 os irreducible and monotone and A 0 84h 4 [5(b a)4 + 0(b a) h + 9h 4 ]. (6) This theorem proved by Usmani [7]. Theorem. Let u be the solution of singularly perturbed boundary value problem (), and u i be the numerical solution obtained from the difference scheme (9)(ii). Then, we have ɛ E = O(h 8 ) provided p < ( ω) where ω = [5(b a) 4 + 0(b a) h + 9h 4 ] and p = 5ωh 84h 4 Max p(x i ), a p(x i ) b. Proof. From () we have E = A T A T [ɛ(a 0 + A ) + h BP ] T ɛ I + A 0 A + ɛ h A 0 BP ] A 0 T. From (), (4) and (5) we have A + ɛ h A 0 BP ( + ɛ h p ). (7) Also from (0) we have T ɛh0 L 0, where L 0 = Max u (0) (x i ), a x i b.

11 46 TWMS J. PURE APPL. MATH., V., N., 00 By using (I + W ) W E ɛ A 0 T A 0 A +ɛ h A 0 BP ] ɛ A 0 T A 0 A +ɛ h A 0 BP, provided that A 0 A + ɛ h A 0 BP <. Therefore we obtain E provided W <, we get ωh 0 L [ ω( + ɛ h p )] O(h8 ), (8) provided ɛ p < ( ω). (9) 5ωh So that the given method for solving the boundary value problem () for sufficiently small h is a eight-order convergent method. 6. Numerical illustrations In order to test the viability of the proposed methods and to demonstrate their convergence computationally we have solved the following singularly perturbed boundary value problems which their exact solutions are known to us. The maximum absolute errors at the nodal points, max u(x i ) u i are tabulated in Tables. Example.Consider the following singularly perturbed boundary value problem [],[5],[6], []-[7]: with the exact solution ɛu + u = cos πx ɛ cos πx, u(0) = u() = 0 u(x) = [exp((x )/ ɛ) + exp( x/ ɛ)]/[ + exp( / ɛ)] cos πx. This problem has been solved using our methods with different values of N = 6,,, 8, 56 and ɛ = 6,..., 8, computed solutions are compared with the exact solutions in nodal points and the maximum absolute errors in solutions are tabulated in Table, and compared with the methods in Ref. [],[5],[6], []-[7] in Tables -. Our results shows the efficiency and accuracy of our methods.

12 J. RASHIDINIA, R. MOHAMMADI: NON-POLYNOMIAL SPLINE APPROXIMATIONS FOR ɛ N = 6 N = N = N = 8 N = 56 Our eight order method 6 6.9( 9).04( ) 6.86( 4).5( 5).07( 6) 8.48( 9).57( 0).7( 4).44( 5).08( 6) 5.86( 8).6( 9).9( ).44( 5).7( 5) 8 8.5( 7).84( 9) 4.97( ).6( 4).06( 4) Our sixth order method 6.8( 7) 6.74( 0) 4.5( ) 6.5( 4).00( 4) 4.08( 8).8( 0).00( ) 4.( 4).04( 4) 5.84( 7) 4.9( 9).8( ).7( ).68( 4) ( 6) 6.9( 8) 4.4( 0).09( ).0( 4) Our fourth order method 6.8( 4) 4.44( 5) 7.( 6) 9.97( 7).( 7) 5.84( 4).0( 4).07( 5).89( 6).8( 7).50( ).47( 4) 5.69( 5) 8.09( 6).07( 6) 8.7( ) 8.67( 4).50( 4).( 5).00( 6) Sixth order method in [6] 6.9( 7) 6.74( 0) 4.6( ) 6.8( 4).09( 4) 4.08( 8).9( 0).00( ) 4.6( 4) 9.60( 5) 5.84( 7) 4.9( 9).8( ).7( ).6( 4) ( 6) 6.4( 8) 4.4( 0).0( ).84( 4) Fifth order method in [4] 6.( 6) 6.45( 9).40( ).0( ) 8.88( 5).00( 6).68( 8).6( 0).09( ).90( 4) 8.89( 6).6( 7).0( 9).08( ) 9.07( 4) ( 5) 9.98( 7).8( 8).4( 0) 9.9( ) Table. The maximum absolute errors for example. Example. Consider following boundary value problem []: ɛu + u = x, u(0) =, u() = + exp( / ɛ) with the exact solution, u(x) = x + exp( x/ ɛ). This problem has been solved using our method with different values of N = 8, 6,,, 8 and ɛ = 6,..., 5. The maximum absolute errors in solutions are tabulated in Table and compared with the methods in Ref. [].

13 48 TWMS J. PURE APPL. MATH., V., N., 00 ɛ N = 6 N = N = N = 8 N = 56 Fourth order method in [] 6.57( 5) 8.79( 7) 5.( 8).0( 9).05( 0) 8.7( 6) 4.4( 7).6( 8).6( 9).00( 0).84( 5) 8.67( 7) 6.65( 8) 4.9( 9).78( 0) 8.0( 4).6( 6).( 7).54( 8) 9.44( 0) Fourth order method in [] ( 5).5( 6).58( 7) 9.87( 9) 6.7( 0).00( 5).4( 6) 7.74( 8) 4.8( 9).0( 0) 5.45( 5).4( 6).4( 7).4( 8) 8.9( 0) 8.8( 4).( 5) 7.68( 7) 4.8( 8).0( 9) The method in [4] ( ).0( ) 5.08( 4).7( 4).7( 5) 7.( ).79( ) 4.48( 4).( 4).80( 5) 6.58( ).66( ) 4.5( 4).04( 4).60( 5) 8 6.6( ).6( ) 4.0( 4).0( 4).5( 5) The method in [5] 6 4.4( ).0( ).54( 4) 6.5( 5).58( 5).68( ) 9.0( 4) 5.6( 5).40( 5).50( 6).45( ) 8.40( 4).08( 4) 5.0( 5).0( 5) 8.4( ) 8.( 4).0( 4) 5.06( 5).6( 5) The method in [6] 6.0( 4) 7.47( 6) 4.67( 7).90( 8) 4.9( 9).8( 4) 8.00( 6) 5.00( 7).4( 8).99( 9).60( 4).00( 5) 6.6( 7).9( 8).( 9) 8.4( 4).47( 5) 9.( 7) 5.77( 8).7( 9) The method in [7] ( ).77( ) 4.45( 4).( 4).78( 5) 5.68( ).4( ).55( 4) 8.89( 5).( 5) 4.07( ).0( ).54( 4) 6.5( 5).58( 5) ( ).75( ) 4.( 4).08( 4).7( 5) Table. The maximum absolute errors for example. Example. Finally consider the following boundary value problem [7]: ɛu + [ + x( x)]u = + x( x) + [ ɛ x ( x) ( x)] exp[ ]+ ɛ +[ ɛ x( x) ] exp[ x/ ɛ], u(0) = u() = 0,

14 J. RASHIDINIA, R. MOHAMMADI: NON-POLYNOMIAL SPLINE APPROXIMATIONS FOR ɛ N = 8 N = 6 N = N = N = 8 Our eight order method 8.97( 9) 5.0( ).59( 4) 5.45( 5).76( 5) ( 8).7( 0).47( ) 8.( 5).( 5) 8.5( 7).84( 9) 4.96( ) 7.7( 5) 5.99( 5) 9.84( 6) 5.9( 8).6( 0).60( ).8( 4) 8 8.4( 5) 8.( 7).84( 9) 4.96( ) 8.60( 5) ( 4) 9.8( 6) 5.9( 8).6( 0).6( ) 5.0( ) 8.4( 5) 8.( 7).84( 9) 4.96( ) Our sixth order method ( 8) 4.8( 0).( ).70( 4).( 4) 6 6.7( 7) 5.50( 9).08( ).6( ).9( 4) 6.55( 6) 6.45( 8) 4.6( 0).0( ).7( 4) 5.0( 5) 6.70( 7) 5.49( 9).08( ).6( ) 8.6( 4) 6.55( 6) 6.44( 8) 4.6( 0).0( ) 56.55( ) 5.0( 5) 6.70( 7) 5.49( 9).08( ) ( ).6( 4) 6.55( 6) 6.44( 8) 4.6( 0) Our fourth order method 8 7.8( ).7( ) 6.0( 4).6( 4) 4.5( 5) 6.5( ).95( ).( ).04( 4) 7.88( 5).95( ) 7.0( ).( ) 5.84( 4).5( 4).95( ).9( ).9( ).( ).00( 4) 8 4.7( ).94( ) 7.0( ).( ) 5.5( 4) ( ).9( ).0( ).9( ).( ) 5 7.6( ) 4.7( ).95( ) 7.05( ).( ) The method in [] 8.0( 5) 7.0( 7) 4.8( 8).74( 9).7( 0) ( 5).96( 6).85( 7).5( 8) 7.4( 0).78( 4).8( 5) 7.54( 7) 4.67( 8).96( 9) 7.4( 4) 4.74( 5).96( 6).86( 7).6( 8) 8.70( ).78( 4).8( 5) 7.46( 7) 4.67( 8) 56 8.( ) 7.4( 4) 4.74( 5).98( 6).86( 7) 5.05( ).70( ).78( 4).8( 5) 7.46( 7) Table. The maximum absolute errors for example. with the exact solution, u(x) = + (x ) exp[ x/ ɛ] x exp[ ( x)/ ɛ]. This problem has been solved using our method with different values of N = 8, 6,,, 8 and ɛ = 6,..., 56 the maximum absolute errors in solutions are tabulated in Table 4.

15 50 TWMS J. PURE APPL. MATH., V., N., 00 ɛ N = 6 N = N = N = 8 Our eight order method ( 0) 6.00( ) 4.77( 5).( 6) 7.9( 9).( ).6( 4).94( 5).0( 7).74( 0).97( ) 9.0( 5) 8.9( 6) 5.08( 9) 9.8( ).5( 4) Our sixth order method 6.6( 8) 9.( ) 4.4( ) 7.4( 5).47( 7).00( 9) 5.04( ).78( 4).5( 6).05( 8) 6.07( ).75( ) 8.0( 5).05( 7) 7.( 0).57( ) Our fourth order method 6 5.5( 4) 7.79( 5).05( 5).7( 6).0( ).9( 4).70( 5).59( 6).6( ) 4.( 4) 6.9( 5) 9.45( 6) 8 5.7( ).08( ).74( 4).48( 5) The method in [7] 6.6( 8) 9.( ) 4.4( ).4( 5).47( 7).00( 9) 5.04( ).78( 4).5( 6).06( 8) 6.07( ).75( ) 8.0( 5).05( 7) 7.4( 0).57( ) Table 4. The maximum absolute errors for example. References [] Aziz, T., Khan, A., (00), A spline method for second-order singularly perturbed boundary-value problems, J. Comput. Appl. Math., 47(), pp [] Aziz, T., Khan, A., (00), Quintic Spline Approach to the Solution of a Singularly-Perturbed Boundary- Value Problem, J. Optim. Theory Appl., (), pp [] Chin, R.C.Y. and Krasny, R., (98), A hybrid asymptotic-finite element method for stifi two point boundary value problems, SIAM J. Sci. Stat. Comput., 4, pp [4] Jain, M.K. and Aziz, T., (98), Numerical Solution of Stiff and Convection-Diffusion Equations Using Adaptive Spline Function Approximation, Applied Mathematical Modeling, Appl. Math. Model., 7, pp [5] Kadalbajoo, M.K. and Bawa, R.K., (996), Variable-mesh difference scheme for singularly-perturbed boundary-value problem using splines, J. Optim. Theory Applic., 9, pp [6] Khan, A., Khan, I.. and Aziz, T., (006), Sextic spline solution of a singularly perturbed boundary-value problems, Appl. Math. Comput., 8, pp [7] Major, M.R., (985), Prog. Sci. Comput., Birkhauser, Boston, Massachusettes, 5, pp [8] Mohanty, R.K., Navnit Jha, Evans, D.J., (004), Spline in compression method for the numerical solution of singularly perturbed two-point singular boundary-value problems, Intern. J. Computer Math., 8, pp [9] Niijima, K., (980), On a three-point difference scheme for a singular perturbation problem without a first derivative term II, Mem.Numer.Math., 7, pp. -7. [0] Niijima, K., (980), On a three-point difference scheme for a singular perturbation problem without a first derivative term I, Mem.Numer.Math., 7, pp. -0. [] Rashidinia, J., (990), Applications of spline to numerical solution of differential equates, M.phil dissertation A.M.U.India. [] Rashidinia, J., (00), Intern. J. Engin. Sci.,, pp. -.

16 J. RASHIDINIA, R. MOHAMMADI: NON-POLYNOMIAL SPLINE APPROXIMATIONS FOR... 5 [] Rashidinia, J., Ghasemi, M., Mahmoodi, Z., (007), Spline approach to the solution of a singularly-perturbed boundary- value problems, Appl. Math. Comput., 89, pp [4] Surla, K., Herceg, D., and Cvetkovic, L., (99), A finally of exponential spline difference scheme, Review of Research, Faculty of Science, Mathematics Series, University of Novi Sad, 9, pp. -. [5] Surla, K., Stojanovic, M., (988), Solving singularly- perturbed boundary-value problems by splines in tension, J. Comput. Appl. Math., 4, pp [6] Surla, K., Vukoslavcevic, V., (995), A spline difference scheme for boundary value problems with a small parameter, Review of research, Faculty of science, Mathematics series, University of Novi Sad, 5, pp [7] Tirmizi, I. A., Haq, F., Siraj-Ul-Islam, (007), Appl. Math. Comput., to Apear. [8] Usmani, R. A., (99), The use of quartic splines in the numerical solution of a fourth-order boundary-value problem, J. Comput. Appl. Math., 44, pp Jalil Rashidinia - Associate professor, Head Department of Applied Mathematics in School of Mathematics, Iran University of Science and Technology Narmak, Tehran 6844, Iran since 006.He has done Ph.D. In Applied Mathematics (Numerical analysis),in A.M.U India on 994. Research Interests: Applied mathematics and computational sciences Numerical solution of ODE, PDE and IE Spline approximations.