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computers &

matWics

~gplloltronr

PERGAMON Computers and Mathematics with Applications 46 (2003) 463-478

www.elsevier.com/locate/camwa

Asymptotic

Numerical

Methods

for Singularly

Perturbed

Fourth-Order

Ordinary

Differential

Equations

of Reaction-Diffusion

Type

V.

SHANTHIAND

N.

RAMANUJAM

Department of Mathematics, Bharathidasan University Tiruchirappalli, 620 024, Tamil Nadu, India (Received April 2001; revised and accepted December 2001)

Abstract-Singularly perturbed boundary value problems (BVPs) for fourth-order ordinary dif- ferential equations (OD%) with a small positive parameter multiplying the highest derivative of the form

--El/iv(zj

+ b(z)y’yz)

- c(~)9(z)

= --ft2),

z E D := (0, l), y(0) = p, y(1) = q, y”(0) = --T, y”(1) = -s, O<E<l,

are considered. The given fourth-order BVP is transformed into a system of weakly coupled systems of two second-order ODES, one without the parameter and the other with the parameter E multiplying the highest derivative, and suitable boundary conditions. In this paper, computational methods for solving this system are presented. In these methods, we first find the zero-order asymptotic approx- imation expansion of the solution of the weakly coupled system. Then the system is decoupled by replacing the first component of the solution by its zero-order asymptotic approximation expansion of the solution in the second equation. Then the second equation is solved by the fitted operator method, fitted mesh method, and boundary value technique. Error estimates are derived and examples are provided to illustrate the methods. @ 2003 Elsevier Science Ltd. All rights reserved.

Keywords-Fourth-order ordinary differential equation, Singularly perturbed problems, Asymp- totic expansion, Boundary layer, Finite-difference scheme, Exponentially fitted finite difference scheme, Fitted operator method, Boundary value technique, Fitted mesh method.

1. INTRODUCTION

Singular perturbation problems appear in many branches of applied mathematics, and for more than two decades quite a good number of research works on the qualitative and quantitative analysis of these problems for both ordinary differential equations (ODES) and partial differential equations (PDEs) have been reported in the literature. Most of the papers connected with computational aspects are confined to second-order equations. Only a few results are reported for higher-order equations. Singularly perturbed higher-order problems are classified on the basis of how the order of the original differential equation is affected if one sets E = 0 (11. Here, E is a small positive parameter multiplying the highest derivative of the differential equation. We say We wish to thank the referees for their valuable suggestions in improving the paper.

08981221/03/$ - see front matter @ 2003 Elsevier Science Ltd. All rights reserved. ‘M-et by 44-W PII: 80898-1221(03)002244

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464 V. SHANTHI AND N. RAMANUJAM

that the singular perturbation problem (SPP) is of convection-diffusion type if the order of the DE is reduced by one, whereas it is called reaction-diffusion type if the order is reduced by two. We now present a brief summary of the works on analytical and numerical methods for higher- order equations. O’Malley [2,3], Howes [4,5], and Zhao [6] derived analytical results on third- order nonlinear SPPs. Fe&an [7] considered higher-order problems and his approach is based on the nonlinear analysis involving fixed-point theory, Leray-Schauder theory, etc. In [8] an iterative method is described. Niederdrenk and Yserentant [9] reported works on higher-order convection-diffusion type problems. The works of Gartland [lo] are connected with exponentially fitted higher-order differences with identity expansion (HODIE) method. Further, if the order of the equation is even, then a finite element method (FEM) based on standard Cm-’ splines on a Shishkin mesh is reported in [ll]. In [11,12] a FEM for convection-reaction type problems is described. Also, Semper [12] and Roos [S] considered fourth-order equations and applied a standard FEM. Roberts [13] suggested a numerical method of finding an approximate solution for third-order ODES.

In [13,14], Roberts introduced a technique called boundary value technique (BVT). In [15-171, the authors have applied the same technique to find numerical solution for singularly perturbed second-order boundary value problems. In [4,5], Howes considered higher-order equations (order

> 2).

Motivated by the works of [4,5,13-171 we in the present paper suggest computational methods for the boundary value problems (BVPs) of the form (1.1),(1.2) which effectively use an asymp- totic expansion approximation of the solution, a fitted operator method (FOM), fitted mesh method (FMM), and BVT.

Consider the BVP

-Qyz) + b(z)y”(X) - c(x)y(x) = -f(z), x E D, (1.1) Y(O) = P7 Y(l) = 4, y”(0) = -T, y”(1) = -s, (1.2)

where E > 0 is a small positive parameter, and b(z), c(x), and f(x) are sufficiently smooth functions satisfying the following conditions:

b(x) 2 P > 0, (1.3)

0 2 C(X) 2 -7, Y > 0, (1.4)

P-h>v>O, 8 > 1 is arbitrarily close to 1, for some 17, (1.5) D= (O,l), b= [O,l], and y E C4(D) n C2 (D) .

Motivation for considering the above BVP comes from the reference [l, pp. 97,991, which states that the deflection of an elastic beam with small flexural rigidity under tension subject to a speci- fied load and stability problems in fluid dynamics which lead to the Orr-Sommerfield equation [3] and (121 yield fourth-order singularly perturbed differential equations, whose order decreases by two when one sets E = 0. Moreover, the discussions presented in [l, pp. 25,261 motivated us to consider this type of boundary condition so as to establish a uniform stability result and other estimates. These boundary conditions enable us to transform a fourth-order differential equation into a system of two second-order ordinary differential equations with Dirichlet type boundary conditions so that the maximum principle theory can be extended, which is not available for higher-order equations as far as authors’ knowledge goes.

In Section 2, some analytical results are derived for the BVP (1.1) ,( 1.2). Section 3 deals with some analytical and numerical results for singularly perturbed second-order ordinary differential equation of reaction-diffusion type. Computational methods for solving the BVP (1.1),( 1.2) are described in Section 4. Among the methods suggested in this section, BVT and FMM give an excellent portrait of the solution, especially within the boundary layers. We not only give

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methods which are easy to implement, but also necessary theory (convergence, stability, and error estimates).

The error estimates for these methods are discussed in detail in Section 5. Section ,6 gives an al- ternate procedure to problem (1.1),(1.2) w h en condition (1.4) is not met. Numerical experiments

are conducted in Section 7. Conclusions are drawn in the last section.

In the following, the generic constant C is independent of mesh points, mesh size, and the small positive parameter E.

2. SOME

ANALYTICAL

RESULTS

In this section, we first establish a maximum principle for the following problem. Then, using this principle, a stability result for the same problem is derived. Further, an asymptotic expansion approximation is constructed for the solution.

The SPBVP (1.1),(1.2) can be transformed into an equivalent problem of the form

Ay = F H Ply := -y:‘(z) - y&) = 0,

P2y:= --EY[(X) + b(z)yz(z) + &)Yl(Z) = f(s), x E D, (2.1) RlY := Yl(O) = P, R2Y := Yl(l) = 4,

R3y := y2(0) = r, R4y := yz(1) = s, (24

where Y = (~1~~2)~.

REMARKS. Hereafter, only problem (2.1),(2.2) is considered subject to conditions (1.3)-( 1.5).

Condition (1.3) means that the SPBVP (2.1),(2.2) is a nonturning-point problem. Condition (1.4) is imposed to ensure that system (2.1),(2.2) is quasi-monotone. Conditions (1.3),(1.4) are suf- ficient to establish the maximum principle for (2.1),(2.2). The last condition (1.5) and the maximum principle are used to derive a stability result for system (2.1),(2.2). Since the given BVP has Neumann boundary conditions, it has less severe boundary layers.

2.1. Maximum Principle and Stability Result

THEOREM 2.1. MAXIMUM PRINCIPLE. Consider BVP (2.1),(2.2) subject to conditions (1.3)-

(1.5). Assume that Plu 2 0, Pzu 1 0 in D, q(0) > 0, ui(l) 2 0, uz(0) 2 0, and up(l) 2 0. Then, u(z) 2 0, in [0, 11.

Here, u(x) = (2Ll(x), Us) for a-II 2 E b.

PROOF. Define s(z) = (si(z), ~~(2)) as

si(x)=(l+s) l-

( (9) and s&r) = 1,

where 0 < 6 <( 1.

Then s > 0, 4s = 6 > 0, and Pzs = b(z) + c(z)[(l + 6)(1 - (z2/2))] > ,8 - 8y > q > 0, 0 > 1 is arbitrarily close to 1. Further, we define

Assume that the theorem is not true. Then (Y > 0, and there exists a point xc such that z0 E D and either (-~i/si)(xc) = o or (-~z/sz)(xc) = Q. Consequently, (ui + CLQ)(S) 2 0, i = 1,2, 2 E D.

CASE 1. (--u~/s~)(x~) = cu; that is, (u~+cEs~)(~~) = 0. Therefore (~i+asi) attains its minimum

at z = 20. Therefore 0 < Pl(u + as)(xo) = -(ul + CXS~)“(XO) - (~2 + ask) 5 0. This is a contradiction.

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466 V. SH.~NTHI Aim N. RAMA~~~JAM

CASE 2. (-u2/52)(zo) = a; that is, (wJ+(Ys~)(zo) = 0. Therefore (U~+CYSZ) attains its minimum

at x = ZO. Therefore,

0 < P2(u + cys)(xo) = --E(u2 + QS2)“(ZO) + b(zo)(u2 + asz)(zo) + C(XO)(Ul + (YSl)(XO) L 0.

This is a contradiction.

Hence, we conclude that u(x) > 0, for all 2 E D,. a THEOREM 2.2. STABILITY RESULT. Consider the BVP (2.1),(2.2) subject to conditions (1.3)- (1.5). If y(z) is a smooth function, then

Ilu(x 5 Cm= Iy1W Iy2W Iv1(l)l, l~2~~~l,~~~l~~~l~~~~lpZ~l ,

1 >

for a~] x E 0, where ll~(z)ll = mdlyl(z)I, l~2b)l).

PROOF. Let

G = Cm= Iv1(O)L IY~(O)LIY~(~)I, I~2~~~l~~~~l~1~l~~~~IpZ~l .

{ >

Defining two functions w*(z) as

w~(z)=C1(l+S) 1-c ztYl(Z),

( > x E b, 6>0, O<S<<l,

wzf(x) = 9 f Yz(X),

we can prove that Plwk(z) > 0 and &W*(Z) 2 0, by a proper choice of C. Furthermore, w;(x) 2 0, atx=O, and x=1,

f&x, 2 0, at 2 = 0, and x=1,

by a proper choice of C. Therefore, by the maximum principle we get the required result. I 2.2. An Asymptotic Expansion Approximation of the Solution

Consider the BVP (2.1),(2.2). Using the standard perturbation method [18], we can construct an asymptotic expansion for the solution of (2.1),(2.2) as follows. Let

Y(x,~)=(Uo+Vo+Wo)+O(~)r

where uo = (~01, ~02) is the solution of the reduced problem of (2.1),(2.2), -u&(x) - qQ(x) = 0, Sol = P, UOl(l) = 4, b(X)~oz(X) +c(x)~ol(x) = f(s), 5 E 0,

vg is the left-layer correction given by vg = (wo~,wo~) with- UOl = 0,

(2.3) (2.4)

uo2 = {[r _ 21o2(())] - [s - uo2(q]e-JbTiSld ,[

1 - ,-(rn+rn)/fi -l

1 [

,-Qm~

1

, and wg is the right-layer correction given by wo = (1~01, ~02) with

WOl = 0, ~02 = -t [s - 7~02(1)] - [r - u02(0)]e- &mJz }[ l-e-(&-E+dm)Ifi -I

1 [

,-Cl-z,a/J;1, REMARK 2.1. If (uo1,u,-,2) is the solution of (2.3),(2.4), then ~01 is the solution of the BVP

C(X)

f(x)

-u&(x) + -UOl(X) = -

b(x)

b(x) ’

(2.5)

UOl(O)

=

P, UOl(l) = 4. (2.6) In the following it is assumed that the BVP (2.5),(2.6) can be solved exactly and closed form solution is available.

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THEOREM 2.3. The zero-order asymptotic expansion approximation yss = (ug + vg + wg) of the solution y(x) of (2.1),(2.2) satisfies the inequality

PROOF. Defining barrier functions 41 and $2 as

f#Jl = C(1+ S)

( 1 - ; > fik (Yl - Yla&),

42 = cfi* (Y2 - Y2&3.&), 4 = t&,42),

it is easy to verify that

p14 2 0, p24 2 0, 41(O) 2 0, 41(l) 2 0, d2P) 2 0, dJ2P) 2 0,

for a suitable selection of C. Then, by Theorem 2.2 we have the required result.

I

3. SOME

ANALYTICAL

AND

NUMERICAL

RESULTS

FOR SPBVP

FOR SECOND-ORDER

ODES

We state some results for the following SPBVP which are needed in the rest of the paper. Though the results are stated for the interval [0, 11, their forms are not changed when the equation is solved on any arbitrary closed and bounded interval. Consider the SPBVP

--EyZ+“(X) + b(z)yz*(z) = j(x) - C(Z)UOl (z), x E D, (3.1)

Y;(o) = T, Y;(l) = 3, (3.2)

where ZLO~(Z) is the solution of the BVP (2.5),(2.6).

3.1. Analytical Results

THEOREM 3.1. If (y1 , ~2) and y; are solutions of the BVPs (2.1), (2.2) and (3.1), (3.2), respectively then

IYZ(Xc) - YixS)I I CA

PROOF. Since (yl, yz) is the solution of (2.1),(2.2), then yz satisfies the BVP

--EY;(x) + bWY2(X) = f(x) - C(2)YlbL x E D, Y2@) = T, YZ(l) = 3.

Further, the function w = y2 - yz satisfies the BVP

--Ew”(X) + b(x)w(x) = -+)[Yl(X:) - UOl (z)l, x E D,

w(0) = 0, zu(1) = 0.

From the gtability result [19] we have

that is,

lY2k) - Yz*(Xc)l s c&. I

REMARK 3.1. If T and s are replaced by T + O(G) or/and s + O(fi), respectively, then the conclusion of the above theorem is still valid.

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468 V.SHANTHI AND N.RAMANUJAM

3.2. Numerical Results

Consider the SPBVP (3.1),(3.2), and its corresponding discrete problem defined in the more general form

-Eai(pi)62y;,;+ b(G)Y,*,j = f(G) - c(+ol(zi), (3.3)

Yrt,o = T, Y& = s, (3.4)

where 0 = 50 < 51 < . . . < 2, = 1, zi = &-i + hi, pi = hi/G, ai is a ‘fitting factor’,

D+y; i = Y&+1 - Y& hi+1 '

D-y; = Yrt,i - G,i-1

,z hi

62y2*

= (D+ - D-) yz,i

,a

Ki

h, = hi+1 + hi * 2 ’

and i = l(l)n - 1.

In the following, we define some special cases of the more general form (3.3),(3.4).

(i) Classical finite-difference scheme (CFDS) [20]. H ere, h= hi = hi+1 = l/n and ai = 1. (ii) Exponentially fitted finite-difference scheme (EFFDS) [19]. Here, h = hi = hi+1 = l/n,

p = pi = h/G and ai = p2b(zi)/(4sinh2(pm/2)).

(iii) Fitted mesh finite-difference scheme (FMFDS) [21]. Here,

ai = 19 r=min{i,/@lnn}, h=4(:),

H = 2(l - 27)/n, the set of mesh points is given by 02:: = {z~}~=~, where

h, i = l(l);,

hi = H, i = (a + 1) (1) (;) ,

h i= ( z + 1 (l)n, > n = 2~, r 2 3.

Error estimates for the above schemes are given in the following theorems. The results stated in the following theorems are valid in D.

THEOREM 3.2. The error in using the CFDS to solve problem (3.1),(3.2) at the inner grid points {si,i=1,2 ,..., n-1)satisfies

1YXd - Y&I I

Ch [l + &-l12e(zi, p)] , ifh < 4, C h + e-xi-la + e-(l-zi+~)JWT

1

, ifh > fi,

-

where e(zi,p) = e -4Tm + ,-(1-qPiq

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THEOREM 3.3. The error in using the EFFDS to solve problem (3.1),(3.2) at the inner grid points {zi, i = 1,2,. . . , n - 1) satisfies

fither, if b’(0) = b’(1) = 0, then

/Y;(G) - y;,i( I Ch’.

PROOF. See [19]. I

THEOREM 3.4. The error in using the FMFDS to solve problem (3.1),(3.2) at the inner grid points {zi, i = 1,2, . . . , n - 1) satisfies

lyz*(x;) - Y~,~I I: Cn-l Inn.

PROOF. See [21]. I

4. COMPUTATIONAL

METHODS

Description

Consider the BVP (2.1),(2.2). Let ~si(z) be the solution of the BVP (2.5),(2.6). The first step in the computational methods is to take usi (as we have said earlier it is assumed that the closed form solution is available) as an approximation for yr . From Theorem 2.3 it is obvious that ]yr (z) - ~ci(s)] I Cd. In the second step, we find the numerical solution for y2 by applying the following methods one-by-one to the BVP (3.1),(3.2).

(i) Fitted operator method [19,21]. (ii) Fitted mesh method [21].

(iii) Boundary value technique [14-17,22-241.

Description of the methods (i)-(iii)

(i) FITTED OPERATOR METHOD (FOM). The BVP (3.1),(3.2) is solved by the EFFDS on the entire interval [O, 11.

(ii) FITTED MESH METHOD (FMM). The BVP (3.1),(3.2) is solved by the FMFDS on the entire interval [0, 11.

(iii) BOUNDARY VALUE TECHNIQUE (BVT). In general, classical schemes fail to yield good ap- proximations inside the boundary layers, in which case one can observe that exponentially fitted schemes yield good approximations. According to Farrell [25], the exponentially fit- ted finite-difference schemes are more effective inside the layers. Hence, we are motivated to use the BVT, which applies an exponentially fitted finitedifference scheme inside the layers and a classical scheme which is sufficient in the region away from the layers. In [14-161, authors have applied the ,same BVT to solve second-order SPBVPs numerically. Here, we apply the same to solve the BVP (3.1),(3.2) numerically. The technique is described here.

Let k > 0 be such that ke < 1. Divide the interval [0, l] into three subintervals [0, ICE], [ka, 1 - kc], and [l - ke, 11. From this problem we derive three BVPs, namely, outer region problem, left inner region problem, and right inner region problem on the intervals [ks, 1 - ICE],

[0, kc], and [l - ka, 11, respectively.

To obtain boundary conditions for these problems at the transition points x = ke and x = 1-ke,

we use the zero-order asymptotic expansion of ys(x).

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470 V. SHANTHI AND N. RAMANUJAM

To solve the BVP (3.1),(3.2) by the BVT we consider the following problems. (1) Left inner region problem

-Y;“(x) + b(X)Y2*(X) = f(x) - Cool, x E (f&J=), (4.1)

Y;(o) = T, yz*(ke) =-Tl. (4.2)

To solve (4.1),(4.2), the EFFDS (Section 3) is used on the interval [O,ke] by taking h =

hl = Ice/n, T = T, and s = ~1.

(2) Outer region problem

--EY2*“(X) -i- qgY;l(X) = f(x) - C(2bOl b), x E (kc, 1 - kc), (4.3) y;(ke) = TI, y;(l - ka) = sl. (4.4) The CFDS (Section 3) is used to solve (4.3),(4.4) on the interval [ks, 1 - ICE] by taking

h = hz = (1 - 2kE)/ n, T = ~1, and s = sr.

(3) Right inner region problem

--EY;“(x) + @)Y3 (x> = f(x) - C(Z)UOl (xc), x E (1 - ke, l), (4.5)

y;(l -kc) = ~1, y;(l) = s. (4.6) Let h = hl = k&/n, T = sr, and s = s. Then EFFDS (Section 3) is applied to

solve (4.5),(4.6) on the interval [1 - kc, 11.

The above linear systems are solved by the self-correcting LU-decomposition procedure [26]. It is distinguished by the fact that it works even if the coefficient matrix is nearly singular, in which case the usual LU-decomposition algorithm fails. After solving the inner region problems and the outer region problem we combine their solutions to obtain an approximate solution to problem (3.1),(3.2) on the whole interval [0, l]. We repeat this process by increasing the value of k (thus widening the inner regions) until the solution profiles do not differ much from iteration to iteration. For computational purposes we use an absolute error criterion,

IlY(4m+’ - Ywq( I A

where ]( . ]I is the maximum norm, y(x)” is the mth iterative value of y, and A is a prescribed error tolerance.

REMARKS. It may be true that one can take the reduced problem solution as an approximation

in the outer region, but our numerical experiments show that one can get better approximation by numerical integration.

For the BVT, we have used a uniform mesh because the aim of this paper is not only to provide a numerical method but also a relevant theory, which is not available in a general nonuniform mesh method.

Regarding the value of k, since we do not have the exact layer width, we initially assume the value 1 for k in ks (layer width) and then increase the value of k to 10, 20, 30 in order to widen the layer. A table illustrating the BVT iterations is presented. One can even set k = O(lnN) analogous to the Shishkin mesh.

Since the problems defined on the subintervals mentioned above are independent of each other, the present method is suitable for parallel computing.

5. ERROR

ESTIMATES

5.1. FOM

THEOREM 5.1. Let (yi, yz) be the solution of (2.1),(2.2). Further, Jet y& be its numerical

solution obtained by the FOM. Then,

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PROOF.

IY2txi) - Y2*,il 5 IY2Czi) - Yz*(“i)l + )Yz*(xi) - Y2*,il

Using Theorems 3.1 and 3.3 in inequality (5.1), the theorem is proved.

5.2. FMM

(5.1) I

THEOREM 5.2. Let (yl,yz) be the solution of (2.1),(2.2). Further, let y& be its numerical

solution obtained by the FMM. Then,

Iy2(zi) - Y;,~( I C [n-l Inn + 4 .

PROOF. The result of the present theorem follows from inequality (5.1) and Theorems 3.1

and 3.5. I

5.3. BVT

THEOREM 5.3. Let (yr, yz) be the solution of (2.1),(2.2). Further, let y$ be its numerical

solution obtained by the BVT. Then,

ly2(ii) - y&l 5 C [hl + &I , xi E [Oy k&l, Or xi E [l - k&y 11, i = O(l)n,

and lYz”(xi) - Y2*,il I i C [h2 + h2e-1/2e(xi, P) + Jz] , if h2 5 &, C [ h2 + e-W~&% + e-(l-Zi+l)fi + JF] , if h2 > JZ,

for i = O(l)n, zi E [kE, 1 - ICE].

PROOF. Inequality (5.1), Theorems 3.1-3.3, and Remark 3.1 can be used to prove this theo-

rem. I

REMARK 5.1. So far, it has been assumed that the exact solution 2~01 is available. If not, one has to obtain a numerical solution for uai by a CFDS at the appropriate grid points (except at the transition points) depending on the method described above. At the transition points, the mesh size is taken as the maximum of meshes used in inner and outer regions. The values of ‘1~01 at the abovementioned grid points will be taken as an approximation for yr. If a second-order CFDS is applied to the BVP (2.5),(2.6), then the error will be of the order O(h2 + JE), where h = max{hr, hz}, for the method BVT and O(h*2 + J) E , w h ere h* = max{h, H}, for the method FMM. As done earlier, in the second equation the values of yi at the above grid points will be taken as uai,i. Then the results of Theorems 5.1-5.3 will remain the same for yz.

6. ADJOINT

SYSTEM

[27,28]

Consider the BVP (2.1),(2.2). Suppose that condition (1.4) is not met. Then we adjoin the following system to (2.1),(2.2):

fij42H

-9:‘(x) - &(5) = 0,

-&g(x) + b(x)&(x) - c+(x)&(x) + c-(x)&(x) = -f(x), x E D, -g(x) - &l(x) = 0, --E??;(x) +@)94(x) - c+(x)Ql(x)+c-(x)g3(z) = f(x), x E D, ih(O) = -p, Cl(l) = -q, $2(O) = -T, $2(l) = -s, @3(O) = P, 63(l) = Q1 $4(O) = T, $4(l) = s, (6.1) (6.2)

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V. SHANTHI AND N. RAMANUJAM FOM METHOD 472 Nodes 0.000000 0.003333 0.053333 0.103333 0.153333 0.203333 0.303333 0.403333 0.503333 0.603333 0.753333 0.803333 0.853333 0.903333 0.953333 1.000000

Table 1. Example 1. Table 2. Example 2.

Error Yl Y2 0.0000000E+00 0.0000000E+00 0.15205723-04 0.25221783-05 0.31249273-04 0.78113893-05 0.31250003-04 0.78125003-05 0.31250003-04 0.78125003-05 0.31250003-04 0.78125003-05 0.31250003-04 0.78125003-05 0.31250003-04 0.78125003-05 0.31250003-04 0.78125003-05 0.3125OOOE-04 0.78125003-05 0.31250OOE-04 0.78125003-05 0.31250003-04 0.78125003-05 0.31250003-04 0.78125003-05 0.31250003-04 0.78125003-05 0.31247243-04 0.78090333-05 0.0060000E+00 O.OOOOOOOE+O9 Nodes 0.000000 0.003333 0.053333 0.103333 0.153333 0.203333 0.303333 0.403333 0.503333 0.603333 0.753333 0.803333 0.853333 0.903333 0.953333 1.000000 Error Yl Y2 0.0000000E+00 0.0000000E+00 0.15205723-04 0.25014583-05 0.31249273-04 0.78113743-05 0.31250003-04 0.78125003-05 0.31250003-04 0.78125003-05 0.31250003-04 0.78125003-05 0.31250003-04 0.78125003-05 0.31250003-04 0.78125003-05 0.31250003-04 0.78125003-05 0.31250003-04 0.78125003-05 0.31250003-04 0.78125003-05 0.31250003-04 0.78125003-05 0.31250003-04 0.78125003-05 0.31250003-04 0.78125003-05 0.31247243-04 0.78083663-05 09000000E+00 0.00OOOOOE+00 yr represents the solution and ys represents the second derivative of the solution.

BVT METHOD

Table 3. Example 1. Table 4. Example 2.

Nodes Error Yl Y2 0.000000 0.0000000E+00 0.0000000E+00 0.000005 0.31234383-07 0.82811293-10 0.000100 0.61879143-06 0.16547463-08 0.000150 0.92357713-06 0.15548933-09 0.000200 0.12253303-05 0.33471843-08 0.000250 0.15240813-05 0.98763413-10 0.000300 0.18198583-05 0.27755583-16 0.200180 0.31250003-04 0.78125003-05 0.400060 0.31250003-04 0.78125003-05 0.599940 0.31250003-04 0.78125003-05 0.799820 0.31250003-04 0.78125003-05 0.999700 0.35667723-04 0.78125003-05 0.999750 0.15239733-05 0.5551115E-16 0.999800 0.12251713-05 0.23820623-05 0.999850 0.92372803-06 0.48038663-05 0.999900 0.61889313-06 0.71947363-05 0.999995 0.31276783-07 0.48369033-05 1 .oooooo 0.0000000E+00 0.0000000E+00

Nodes 0.000000 0.000005 0.000100 0.000150 0.000200 0.000250 0.000300 0.200180 0.400060 0.599940 0.799820 0.999700 0.999750 0.999800 0.999850 0.999900 0.999995 1.000000 Error Yl Y2 0.0000000E+00 0.0000000E+00 0.31234383-07 0.92041123-10 0.61879143-06 0.18186713-08 0.92357713-06 0.51726183-10 0.12253303-05 0.35515433-08 0.15240813-05 0.41505053-10 0.18198583-05 0.27675443- 16 0.31250003-04 0.78125003-05 0.31250003-04 0.78125003-05 0.31250003-04 0.78125003-05 0.31250003-04 0.78125003-05 0X1667723-04 0.55511153-16 0.15239733-05 0.23822023-05 0.12251713-05 0.48040703-05 0.92372803-06 0.71945293-05 0.61889313-06 0.48367403-05 0.31276783-07 0.17343603-05 0.0000000E+00 0.0000000E+00 E = 0.1000000E-03. Transition points = (0.003000, 1.0-0.003000),

(11)

where c+(z):=c(z) if c(z) 2 0 and zerootherwise, c-(z):=c(z) -c+(z) andy = (&,&,&,64). The results derived in the earlier sections can be extended to the above system. Because of the fact that if y = (yr, yz) is a solution of (2.1),(2.2) then 3 = (-yr, -ys, yi, ~2) is a solution of the above problem (6.1),(6.2), the results derived earlier for the BVP (2.1),(2.2) still hold good even if condition (1.4) is not met.

REMARK. In the adjoint system, the number of equations is doubled and hence occupies more memory spaces. Still we need this, in order to have the maximum principle and the stability result. To avoid the possibility of c+ and c- loosing smoothness and only belonging to C(D), we can assume that c(z) is positive throughout the interval.

FMM METHOD Table 5. Example 1.

Nodes Solution Second

Derivative 0.0000000E+00 0.43321703-03 0.38989533-02 0.73646893-02 O.l083042E-01 0.14296163-01 O.l776190E-01 0.21227633-01 0.24693373-01 0.35105173-01 0.94139433-01 0.15317373+00 0.27124423+00 0.38931083+00 0.50737933+00 0.62544783+00 0.74351633+00 0.8615849E+OO 0.9206191E+OO 0.97270733+00 0.9761731E-t00 0.97963883+00 0.9831045E-tOO 0.98657033+00 0.99003603+00 0.99350173+00 0.9969675E+OO 0.1000000E+01 0.0000000E+00 0.0000000E+00 0.25936243-05 0.28096693-04 0.16921813-04 0.10997673-03 0.24085913-04 0.58580213-04 0.27667953-04 0.59638423-04 0.29458983-04 0.25763533-03 0.30354493-04 0.61132053-03 0.30802243-04 0.12859033-02 0.31026123-04 0.26126733-02 0.31222103-04 0.10927953-02 0.31250003-04 0.78044573-05 0.31250003-04 0.78125003-05 0.31250003-04 0.78125003-05 0.31250003-04 0.78125003-05 0.31250003-04 0.78125003-05 0.31250003-04 0.78125003-05 0.31250003-04 0.78125003-05 0.31250003-04 0.78124993-05 0.31250003-04 0.76575493-05 0.31116883-04 0.44126993-02 0.30983763-04 0.21915903-02 0.30717523-04 0.10730713-02 0.30185053-04 0.50121433-03 0.29120093-04 0.19811773-03 0.26990203-04 0.24204753-04 0.22730453-04 0.80123613-04 0.14210773-04 0.10528703-03 O.OOOOOOOE+OO O.OOO9OOOE+OO E = 0.0001. Transition parameters = (0.0277258873, 1.0-0.0277258873).

7. NUMERICAL

RESULTS

In this section, two examples are given to illustrate the computational methods discussed in this paper.

(12)

474 V. SHANTHI AND N.RAMANUJAM FMM METHOD

Table 6. Example 2.

Nodes Solution Second

Derivative 0.0OOOOOOE+00 0.0000000E+00 o.oooooo(mtoo

0.43321703-03 0.25936243-05 0.27420173-04 0.38989533-02 0.16921813-04 0.10430673-03 0.73646893-02 0.24085913-04 0.49171263-04 0.10830423-01 0.27667953-04 0.71540053-04 O.l429616E-01 0.29458983-04 0.27109903-03 O.l77619OE-01 0.30354493-Ci4 0.62573143-03 0.21227633-01 0.30802243-04 0.13008873-02 0.24693373-01 0.31026123-04 0.26280253-02 0.35105173-01 0.31222103-04 0.11084073-02 0.94139433-01 0.31250003-04 0.78205423-05 0.15317373+00 0.31250003-04 0.78125003-05 0.27124223+00 0.31250003-04 0.78125003-05 0.38931083+00 0.31250003-04 0.78125003-05 0.50737933+00 0.31250003-04 0.78125003-05 0.62544783+00 0.31250003-04 0.78125003-05 0.74351633+00 0.31250003-04 0.78125003-05 0.86158493+00 0.31250003-04 0.78125013-05 0.9206191E+OO 0.31250003-04 0.79674483-05 0.97270733+00 0.31116883-04 0.44282703-02 0.97617313+00 0.30983763-04 0.22068623-02 0.97963883+00 0.30717523-04 0.10879363-02 0.98310453+00 0.30185053-04 0.51543153-03 0.98657033+00 0.29120093-04 0.21125973-03 0.99003603+00 0.26990203-04 0.35583943-04 0.99350173+00 0.22730453-04 0.71522983-04 0.9969675EfOO 0.14210773-04 0.10076313-03 0.1000000E+01 0.0000000E+00 . 0.00OOOOOE+00 E = 0.0001. Transition parameters = (0.0277258873,

1.0.0277258873). EXAMPLE 1. Consider the BVP

-&y”yr) + 4y”(Z) + y(z) = --f(x),

Y(O) = 1, Y(l) = 1, y”(0) = -1, y”(1) = -1,

where

EXAMPLE 2. Consider the BVP

-&YiV(2) + 4y”(Z) - y(z) = --f(5),

(13)

where

5E ,-WJE _ ,-w+x)/JE + e-2(1-x)/& _ ,-2(2-x)/&

f(x) = -2 - ?ip) + $ - 16

1 - e-4/&

Since this problem does not satisfy the condition that

the procedure given in Section 6 is used to solve it.

Tables l-6 present numerical results in the form of errors for the solution and its second derivative of the problems considered in Examples 1 and 2. Also, in Table 7 the BVT iterations are illustrated. In all the examples presented, one can easily find the closed form solution of the reduced problem.

8. SUMMARY

AND

CONCLUSIONS

In this paper, we presented computational methods to solve fourth-order SPBVPs for ODES subject to a particular type of boundary conditions. The boundary conditions not only help us to reduce the given fourth-order differential equation into a system of two second-order equations subject to suitable boundary conditions, but also to establish the maximum principle, a uniform stability result, and other necessary estimates [l, pp. 25,261. As mentioned in the introduction, the second-order singularly perturbed differential equations have been extensively studied, but only few results on higher-order equations are reported in the literature. The idea of an adjoint system presented in Section 6 is a new approach for solving a weakly coupled system of differential equations. The nonlinear BVPs of the following form can be solved by linearizing them by the Newton’s method of quasi linearization as done in (191:

&Y”W = F(x, Y,

Y”), x E D,

Y(O) = PY Y(l) = 97 y”(0) = -T, y”(1) = -s,

where

ql+,

Y, Y”) L P > 0, x E D, y E R (set of reals), 0 2 Fy(xc, Y, Y”) 2 -7, Y>O, P-@r>77>0,

B > 1 is arbitrarily close to 1.

The methods presented in this paper are easy to implement for one type of boundary value problem for fourth-order singularly perturbed ordinary differential equations. A complete theory for the methods is also given. The purpose of presenting the BVT is to obtain better approxima- tions in the boundary layer regions as the EFFDS is more effective in the layer regions. Among the methods described in this paper, the FMM requires less time and less space. But, it is ob- served that the BVT yields better approximations than the FMM inside the boundary layers. The important point to be noted in this paper is that the decoupling of (2.1),(2.2) reduces mem- ory space and time. Further, the present methods may be extended to systems of equations of the form

-Y:’ + f(x, Yl, Y2) = 0, YlKu = P, YlP) = 4, -t-y; + g(x, Yl, Y2> = 0, Y2(0) = -T, y1(1) = -s.

(14)

Table 7. Example 1. Solution iteration. NOCleS K=l K= 10 K = 20 K = 30 Exact Error O.OOOOOO0E+OO 0.10OOOOOE+01 0.1OOOOOOE+01 0.lOOOOOOE+01 O.lOOOOOOE+01 0.1000000E+01 0.0000000E+00 0.2OOOOOOE-03 0.1000025E+01 0.1000025E+01 0.1000025E+01 0.1000025E+01 0.1000013E+01 0.12475033-04 0.4OOOOOOE-03 0.1000050E+01 0.1000050E+01 0.1000050E+01 0.1000050E+01 0.1000025E+01 0.24900273-04 0.6OOOOOOE-03 0.1000075E+01 0.1000075E+01 0.1000075E+01 0.1000075E+01 0.1000038E+01 0.37275903-04 0.8000000E-03 0.1000100E+01 0.1000100E+01 0.1000100E+01 0.1000100E+01 0.1000050E+01 0.49602123-04 0.1OOOOOOE-02 0.1000000E+01 0.1000125E+01 0.1000125E+01 0.1000125E+01 0.1000063E+01 0.61879153-04 0.2OOOOOOE-02 0.1000250E+01 0.1000250E+01 0.1000250E+01 0.1000127E+01 0.12253303-03 03OOOOOOE-02 0.10003743+01 0.10003743+01 O.l000374E+Ol 0.1000192E+01 0.18198583-03 0.4OOOOOOE-02 0.10004983+01 0.10004983+01 O.l000498E+Ol 0.1000258E+01 0.24026143-03 0.5CMXMOOE-02 0.10006223+01 0.10006223+01 0.10006223+01 0.10003243+01 0.29738313-03 0.6OOOOOOE-02 0.10007463+01 0.10007463+01 0.10007463+01 0.1&03923+01 0.35337363-03 O.rOOWQOE-02 0.10008693+01 O.l000869E+Ol 0.10008693+01 0.1000461E+01 0.40825553-03 O.gOOOOOOE-02 O.l000992E+Ol 0.10009923+01 0.10009923+01 0.1000530E+01 0.46205073-03 OAWOOOOOE-02 0.1001115E+01 0.1001115E+01 0.1001115E+01 O.lOOO6OOE+01 0.51478063-03 0.KhlOOOOE-01 0.1000000E+01 0.10012373+01 0.10012373+01 0.1000671E+01 0.56646643-03 O.l5OWOOE-01 0.1001847E+01 0.10001847+01 0.1001037E+01 0.80994303-03 0.2OOOOOOE-01 0.KlOOOOOE+01 O.l00245OE+Ol 0.1001420E+01 0.10302503-02 0.25OOOOOE-01 0.1003@47EI+01 0.1001817E+01 0.12295923102 0.3OOOOOOE-01 0.lOOOOOOE+01 O.lOOOOO0E+01 0.14099643-02 O.l24OOOOE+OO 0.1011330E+01 ~0.10120423+01 0.10128183+01 @.10135783+01 0.1010715E+01 0.28633023-02 O.218OOOOE+OO O.l020045E+Ol 0.10204463+01 0.10208823+01 0.10213093+01 0.10182243+01 0.30850673-02 0.312OOOOE+OO 0.10262703+01 0.10264483+01 0.10266423+01 0.10268323+01 0.10237133+01 0.31189033-02 0.4060000E+00 0.1030005E+01 0.1030050E+01 0.10300983+01 0.1030145E+01 0.10270213+01 0.31240493-02 E = 0.10OOOOOE-01. ?tansition points = ,(0.3OOOOO, 0.97).

(15)

Nodes

0.5000000E+00 0.59400003+00 0.68800003+00 0.78200003+00 0.876OOOOE+OO 0.97OOOOOE+00 0.97500003+00 0.98OOOOOE+OO 0.985OOOOE+OO 0.9900000E+00 o.991OOOOE+OO 0.99200003+00 0.99300003+00 0.994OOOOE+OO 0.99500003+00 0.99600003+00 0.99700003+00 0.998OOOOE+00 0.99900003+00 0.9992000E+OO 0.9994000E+OO 0.9996OOOE-tOO 0.9998OOOE+OO 0.1000000E+01

Table 7. (cont.) K=l K = 10 K = 20 K = 30 Exact 0.10312503+01 0.10312503+01 0.10312503+01 0.10312503+01 0.10281253+01 0.31247163-02 0.1030005E+01 0.1030050E+01 0.1030098E+01 0.10301463+01 0.10270213+01 0.31240493-02 0.10262703+01 0.1026448E-tOl 0.10266423+01 0.10268323+01 0.10237133+01 0.31189033-02 0.10200453+01 0.10204453+01 0.1020882EfOl 0.10213093+01 0.10182243+01 0.3085667E-02 O.l011330E+Ol 0.10120423+01 0.10128183+01 0.10135783+01 0.1010715E+01 0.28633023-02 0.10036373+01 O.l002228E+Ol 0.22275343-02 O.l003047E+Ol 0.1001817E+01 0.12295913-02 0.1000000E+01 0.10024503+01 0.1001420E+01 0.10302493-02 0.1001847E+01 O.l001847E+Ol 0.1001037E+01 0.80994243-03 0.1000000E+01 0.10012373+01 0.10012373+01 0.1000671E+01 0.56646593-03 0.1001115E+01 0.1001115E+01 0.1001115E+01 0.1000600E+01 0.51478083-03 0.10009923+01 0.1000992E+01 0.1000992E-tOl 0.1000530E+01 0.46205153-03 0.10008693+01 0.10008693+01 0.1000869E+Ol 0.1000461E+01 0.40825713-03 0.10007453+01 0.1000745E+01 O.l000745E+Ol 0.10003923+01 0.35337273-03 0.10006223+01 0.1000622E+01 0.10006223+01 O.l000324E+Ol 0.29738283-03 0.10004983+01 O.l000498E+Ol 0.10004983+01 0.10002583+01 0.24026193-03 0.1000374E+01 0.10003743+01 0.10003743+01 0.1000192E+01 0.18198713-03 0.10002493+01 0.10002493+01 O.l000249E+Ol 0.1000127E+01 0.12253153-03 0.10000OOE+01 0.1000125E+01 0.1000125E+01 0.1000125E+01 0.1000063E+01 0.61878363-04 0.1000100E+01 0.1000100E+01 0.1000100E+01 0.1000100E+01 0.1000050E+01 0.49602963-04 0.1000075E+Ol 0.1000075E+01 0.1000075E+01 0.1000075E+01 0.1000038E+01 0.37274683-04 0.1000050E+01 0.1000050E+01 0.1000050E+01 0.1000050E+01 0.1000025E+01 0.2490069E-04 O.l000025E+01 0.1000025E+01 O.l000025E+01 0.1000025E+01 0.1000013E+01 0.12473393-04 0.1000000E+01 0.10000OOE+01 O.1OCOOOOE+O1 0.1000000E+01 0.1000000E+01 0.0000000E+00 E = 0.1000000E-01. Transition points = (0.300000,0.97).

(16)

478 V.SHANTH~ ANDN.RAMANUJAM

REFERENCES

1. H.G. Roes, M. Stynes and L. Tobiska, Numerical Methods for Singularly Perturbed Diflerential Equations- Convection-Diflusion and Flov Problems, Springer, (1996).

2. R.E. O’Malley, Introduction to Singular Per&rbatiOnS, Academic Press, New York, (1974).

3. R.E. O’Malley, Singular Perturbation Methods for Ch-dinary Diflmential Equations, Springer-Verlag, New York, (1990).

4. F.A. Howe+ Differential inequalities of higher order and the asymptotic solution of nonlinear boundary value problems, SIAM J. Math. Anal. 13 (l), 61-80, (1982).

5. F.A. Howes, The asymptotic solution of a class of third-order boundary value problem arising in the theory of thin film flow, SIAM J. Appl. Math. 43 (5), 993-1004, (1983).

6. W. Zhao, Singular perturbations of boundary value problems for a class of third order nonlinear ordinary differential equations, Journal of Difierential Equations 88 (2), 265-278, (1990).

7. M. Fe&an, Singularly perturbid higher order boundary value problems, Jownal of Diflerential Equations III (l), 79-102, (1994).

8. H.G. Roos and M. Stynee, A uniformly convergent discretization method for a fourth order singular pertur- bation problem, Banner Math. Schtijten 228, 30-40, (1991).

9. K. Niederdrenk and H. Yserentant, The uniform stability of singularly perturbed discrete and continuous boundary value problems, Namer. Math. 41, 223-253, (1983).

10. E.C. Gartland, Graded-mesh difference schemes for singularly perturbed two-point boundary value problems, Math. Comput. 51, 631-657, (1988).

11. G. Sun and M. Stynes,, Finite element methods for singularly perturbed higher order elliptic two point boundary value problems I: Reaction-diffusion type, IMA J. Numer. Anal. 15, 11’7-139, (1995).

12. B. Semper, Locking in finite element approximation of long thin extensible beams, IMA J. Numer. Anal. 14, 97-109, (1994).

13. S.M. Roberts, Further examples of the boundary value technique in singular perturbation problems, Journal of Mathematical Analysis and Applications 133, 411-436, (1988).

14. S.M. Roberts, A boundary value technique for singular perturbation problems, J. of Mathematical Analysis and Applications 87, 489-508, (1982).

15. J. Jayakumar and N. Ramanujam, A computational method for solving singular perturbation problems, Applied Mathematics and Computation 65, 31-48, (1993).

16. J. Jayakumar and N. Ramanujam, A numerical method for singular perturbation problems arising in chemical reactor theory, Computers Math. Applic. 27 (5), 83-99, (1994).

17. S. Natesan and N. Ramanujam, A computational method for solving singularly perturbed turning point problems exhibiting twin boundary layers, Appl. Math. Computation 93, 259-275, (1998).

18. A.H. Nayfeh, Introduction to Pertwbation Methods, John Wiley and Sons, New York, (1981).

19. E.P. Doolan, J.J.H. Miller and W.H.A. Schilders, Uniform Numerical Methods for Problems wilh Initidl and Boundary Layers, Boole Press, Dublin, (1980).

20. M.K. Jain, Numerical Solution of Diflerential Equations, Wiley Eastern Limited, New Delhi, India, (1979). 21. J.J.H. Miller, E. O’Riordan and G.I. Shishkin, Fitted Numerical Methods for Singular Perturbation Problems.

Ewor Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions, World Scientific, Singapore, (1996).

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References

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