© Hindawi Publishing Corp. EXPONENTIALLY FITTED SPLINE APPROXIMATION METHOD FOR SOLVING SELFADJOINT SINGULAR PERTURBATION PROBLEMS

A numerical method based on cubic spline with exponential fitting factor is given for the selfadjoint singularly perturbed two-point boundary value problems. The scheme derived in this method is second-order accurate. Numerical examples are given to support the predicted theory.


Introduction. We consider the following selfadjoint singularly perturbed two-point boundary value problem:
Ly ≡ −ε a(x)y + b(x)y = f (x) on (0, 1), where η 0 , η 1 are given constants and ε is a small positive parameter.Further, the coefficients f (x), a(x), and b(x) are smooth functions and satisfy Under these conditions, the operator L admits a maximum principle [8].
The problems in which a small parameter multiplies to a highest derivative arise in various fields of science and engineering, for instance, fluid mechanics, fluid dynamics, elasticity, quantum mechanics, chemical reactor theory, hydrodynamics, and so forth.
Out of the three principal approaches to solve such problems numerically, namely, the finite-difference methods, the finite-element methods, and the spline approximation methods, the first two have been used by several authors.Niijima [6] gave uniformly second-order accurate difference schemes whereas Miller [5] gave sufficient conditions for the uniform first-order convergence of a general three-point difference scheme.Boglaev [3] and Schatz and Wahlbin [9] used finite-element techniques to solve such problems.It is known that the most classical methods fail when ε is small relative to the mesh width h that is used for discretization of the operator L.
In this paper, we have used the third approach, namely, the spline approximation method, to solve problems of type (1.1).There are two possibilities to obtain small truncation error inside the boundary layer(s).The first is to choose a fine mesh there, whereas the second one is to choose a difference formula reflecting the behaviour of the solution(s) inside the boundary layer(s).The present work deals with the second approach, whereas the first one is currently under the investigation of the authors.
We have reduced the original problem (i.e., problem (1.1)) to the normal form.In the normalized form, we replace the perturbation parameter ε affecting the highest derivative by a fitting factor σ (x,ε).Using cubic spline, this factor is determined in such a way that the truncation error of the corresponding scheme for the boundary layer function(s), in the case of constant coefficients, should be equal to zero.This procedure is known as the exponential fitting or the introducing of artificial viscosity [2,4].By making use of the continuity of the first-order derivative of the spline function, the resulting spline difference scheme gives a tridiagonal system which can be solved efficiently by the wellknown algorithms.
In Section 2, we give a brief description of the method.The derivation of the difference scheme has been given in Section 3. The fitting factor is determined in Section 4, whereas the second-order accuracy of the method is shown in Section 5. To demonstrate the applicability of the proposed method, four numerical examples have been solved in Section 6 and the results are presented along with their comparison with those obtained by other authors.Finally, the discussion on these numerical results, along with some comparisons with the results obtained earlier by others, is presented in Section 7.

Description of the method.
Rewrite (1.1) as where and transform (2.1) into the normal form, that is, where with Multiplying (2.4) throughout by −ε (where 0 < ε ≤ 1), we get where We define the fitting comparison problem associated with (2.7) by where σ (x,ε) is an exponential fitting factor which is to be determined subsequently.
The approximate solution of problem (2.9) is sought in the form of the cubic spline function S j (x), which is defined as follows: let For the values V (x 0 ), V (x 1 ),...,V (x n ), there exists an interpolating cubic spline with the following properties: (i) S j (x) coincides with a polynomial of degree 3 on each interval Hence, analogous to [1], the cubic spline can be given as where x ∈ x j−1 ,x j , h= x j − x j−1 , j = 1, 2,...,n, M j = S j x j , j = 0, 1,...,n. (2.12) Using this spline function, we will derive the difference scheme in Section 3, which will give us the approximate solution of V (x).Since U(x) is known, therefore the solution to the original problem will be obtained using (2.3).
Remark 3.1.The scheme without using fitting factor will be given by (3.9) 4. Determination of the fitting factor.In order to get a suitable fitting factor σ (x,ε), we will use the following lemma.Lemma 4.1 [4]. where q 0 and q 1 are bounded functions of ε independent of x and N is a constant independent of ε.
The matrix of the system (3.3) is inverse monotone if h 2 W i /6σ i ≤ 1, i = j, j ± 1. Thus, we take a fitting factor in the following way: where µ(ρ) (with ρ at x j given by ρ j = W j /ε) is to be determined.We require that the truncation error for the boundary layer functions should be equal to zero when W (x) = W = constant.
From the condition Rd j = 0 for W (x) = W = constant, we have The condition Re j = 0 for W (x) = W = constant will give the same µ(ρ).Therefore, we define Hence, the variable fitting factor σ j is defined as 5. Proof of the uniform convergence.Throughout the paper, M will denote a positive constant which may take different values in different equations (inequalities) but are always independent of h and ε.
The scheme (3.3), (3.8) can be written in the matrix form where A is a matrix of the system (3.3) and ν and Z are corresponding vectors.Now, the local truncation error τ j (φ) of the scheme (3.3) is defined by where φ(x) is an arbitrary sufficiently smooth function.Therefore, (5.3) In order to estimate the values |V j − ν j |, we will estimate the truncation error τ j (V ) and the norm of the matrix A −1 .From (4.7), it is obvious that where C is some positive constant.Now, for the case Ch 2 ≤ ε, we see that Hence, that is, σ j approximates ε with the error O(h 2 ).

Estimation of truncation error and the norm of
(5.7) We will estimate separately the parts of τ j (V ).First, we consider the case in which Ch 2 ≤ ε.
We will start with d(x).We calculate (5.9) Now, from Lemma 4.1, we have (5.10) implies ) ) (5.13) Putting all these expressions into (5.8) and (5.9), and since we get (5.15) From (5.6), σ j = ε +O(h 2 ), and using the above expressions for d j−1 and d j+1 , we have (5.16) But the expression for d(x) involves q 0 in the numerator, which is a bounded function of ε independent of x.Therefore, we get and the similar construction as was for d(x) will give us (5.24) Expanding g j−1 , g j+1 , and their derivatives in terms of g j and its derivatives, and using (5.6), we get τ j (g) ≤ Mh 3 g (iv) j .

Test examples and numerical results.
In this section, we present some numerical results which illustrate Theorem 5.1.Example 6.1 [11].Consider problem (1.1) with Its exact solution is given by Example 6.2 [7].Consider problem (1.1) with Its exact solution is given by Example 6.3 [4].Consider problem (1.1) with Its exact solution is not available.
Example 6.4 [9].Consider problem (1.1) with Its exact solution is given by   for different n and ε, where ν(x j ) is the approximate solution of (1.1) obtained via (2.7) and (2.3).Table 6.6 contains maximum errors based on the double-mesh principle (Doolan et al. [4]) (as for Example 6.3, the exact solution is not available): max 0≤j≤n ν n j − ν 2n 2j , n= 8, 16, 32, 64, 128, 256.(6.9) Tables 6.3 and 6.7 contain the numerical rate of uniform convergence for Examples 6.1 and 6.3, respectively, which is determined as in [4]: where , k= 0, 1, 2,..., (6.11) and ν h/2 k j denotes the value of ν j for the mesh length h/2 k .7. Discussion.We have described a numerical method for solving selfadjoint singular perturbation problem using cubic spline with exponential fitting.It is a practical method and can easily be implemented on a computer to solve such problems.The method has been analyzed for convergence.Four examples have been solved to demonstrate the applicability of the proposed method.
For Examples 6.1 and 6.3, we have computed the rate of convergence, see Tables 6.3 and 6.7 which show the uniform second-order convergence as predicted in the theory.The same can be seen for the other examples also.
As is seen from Tables 6.1 and 6.2, the results obtained using fitting factor are better than those without using fitting factor.Example 6.2 has been solved earlier by O'Riordan and Stynes [7].We obtain better results than those in [7].Using finite-element techniques, Schatz and Wahlbin [9] have solved Example 6.4.Table 6.8 shows (quite graphically) how badly standard methods can perform.
To further corroborate the applicability of the proposed method, graphs have been plotted for Examples 6.1 and 6.2 for values of x ∈ [0, 1] versus the computed (termed as approximate) solution obtained at different values of x for a fixed ε.For each plot, we took n = 20 and 40 for Examples 6.1 and 6.2, respectively.are graphs without using fitting factor for Example 6.2 for ε = 0.001 and ε = 0.0001, respectively, whereas Figures 7.6 and 7.8 are graphs which are plotted using fitting factor for the same value of n and ε = 0.001 and ε = 0.0001, respectively.It can be seen from Figures 7.1, 7.3, 7.5, and 7.7 that the exact and approximate solutions without using fitting factor deviate from each other in the boundary layer regions for smaller ε.To control these fluctuations, we used fitting-factor technique and the resulting behaviour of these two solutions can be seen from Figures 7.2  Finally, we would like to remark that we have replaced ε by σ (x,ε) in the normalized form and not in the original selfadjoint problem (i.e., problem (1.1)), with a(x) ≠ constant because in that case ε is a multiple of both the second and first derivative terms which will cause implicit expressions whereas in normalized form, ε is multiplied with the second derivative term only and hence, the fitting-factor technique on the normalized form can easily be implemented.This shows the importance of reducing the original selfadjoint problem to normal form.

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.
However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning
, 7.4, 7.6, and 7.8.The similar observation can be made for the other examples also.

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Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation