A Numerical Solution of a Convection-Dominated Equation Arising in Biology

The theory of singularly perturbed problems has become very important area of interest in the recent years. In these problems, a small parameter multiplies the highest order derivative and there exist boundary layers where the solutions change rapidly. It is a well known fact that the solution of singularly perturbed boundary-value problem exhibits a multiscale character (Doolan et al., 1980; Kevorkin and Cole, 1996; Miller et al., 1996). A reproducing kernel Hilbert space is a useful framework for constructing approximate solutions of differential equations (Akram and Rehman, 2011a, b). The following convection-dominated equation associated with biology (Lin et al., 2009) can be considered as:


INTRODUCTION
The theory of singularly perturbed problems has become very important area of interest in the recent years.In these problems, a small parameter multiplies the highest order derivative and there exist boundary layers where the solutions change rapidly.It is a well known fact that the solution of singularly perturbed boundary-value problem exhibits a multiscale character (Doolan et al., 1980;Kevorkin and Cole, 1996;Miller et al., 1996).A reproducing kernel Hilbert space is a useful framework for constructing approximate solutions of differential equations (Akram and Rehman, 2011a, b).
The following convection-dominated equation associated with biology (Lin et al., 2009) can be considered as: where, 0< ε<1,  : A small positive parameter f (x) : Continuous functions on [0,1] This problem arises in transport phenomena in chemistry and biology.
Let L be the differential operator and homogenization of the boundary conditions of system (1) can be transformed into the following form: The solution of system (2) provides the solution of the system (1).

METHODOLOGY
Reproducing kernel spaces: The reproducing kernel space W 3 2 [0,1] is defined by . The inner product and norm in W 3 2 [0,1] are given by: Theorem 1: The space W 3 2 [0, 1] is a reproducing kernel Hilbert space.That is ,∀ 0, 1 and each fixed x, y [0, 1], there exists R x (y) W 3 2 [0, 1] such that <u(y), R x (y)> = u(x) and R x (y) is called the reproducing kernel function of space 0,1 The reproducing kernel function R x (y) is given by: (5) where, The exact and approximate solutions: In the problem (2), the linear operator L: 0, 1 → 0, 1 is bounded.Using the adjoint operator L* of L and choose a countable dense subset T = {x 1 , x 2 , ..., x n , ...}⊂ [0, 1] and let: is a complete system of 0, 1 and | To orthonormalize the sequence Ψ x in the reproducing kernel space 0, 1 Gram-Schmidt process can be used, as: Theorem 2: For all u(x) 0, 1 the series is convergent in the norm of ||.|| W 3 2 On the other hand, if u(x) is the exact solution of the system (2) then: 0, 1 and can be expanded in the form of Fourier series about normal orthogonal system as: (9) Since the space 0, 1 is Hilbert space so the series is convergent in the norm of ||.|| W 3 2 From Eq. ( 7) and ( 9), it can be written as: is the exact solution of Eq. ( 2) and Lu = f(x), then: The approximate solution obtained by the n-term intercept of the exact solution u (x), given by:
The comparison of the errors in absolute values between the method developed in this study and Bspline method (Lin et al., 2009) is shown in Table 1.

CONCLUSION
In this study, the reproducing kernel space method (RKSM) is developed for the solution singularly perturbed boundary value problem.The results obtained from our method are compared with the results obtained from the B-spline method (Lin et al., 2009) and found that present method gives better results.The results revealed that the method is a powerful mathematical tool for the solution of singularly perturbed boundary value problem.Numerical example also shows the accuracy of the method.