Fourth Order Fitted Scheme for Second Order Singular Perturbation Boundary Value Problems

Fitted fourth order central difference scheme is presented for solving singularly perturbed two-point boundary value problems with the boundary layer at one end point. A fitting factor is introduced in a tri-diagonal finite difference scheme and is obtained from the theory of singular perturbations. Thomas Algorithm is used to solve the system and its stability is investigated. To demonstrate the applicability of the method, we have solved linear and nonlinear problems. From the results, it is observed that the present method approximates the exact solution very well.


Introduction
There are varieties of physical processes in which a boundary layer may arise in the solutions for certain parameter ranges. These types of problems may be generally characterized as singular perturbation problems, and the parameter is termed as the perturbation parameter. Detailed theory and analytical discussion on singular perturbation problems, can be referred in Bender and Orsazag [1], Kevorkian and Cole [3], Nayfeh [5][6], O'Mally [7] and Van Dyke [12] and for exponential fitted methods Miller,J.J.H., O'Riordan,E. and Shishkin, G.I. [4], Y.N.Reddy, P.Pramod. Chakravarthy [9] and Reddy Y.N., Awoke A. [10][11].
Fitted stable fourth order scheme is presented for solving singularly perturbed two-point boundary value problems with the boundary layer at one end point. A fitting factor is introduced in a tri-diagonal finite difference scheme and is obtained from the theory of singular perturbations. Thomas Algorithm is used to solve the system and its stability is investigated. To demonstrate the applicability of the method, we have solved several linear and nonlinear problems. From the results, it is observed that the present method approximates the exact solution very well.

Fourth Order Finite Difference Method
Consider a linear singularly perturbed two-point boundary value problem of the form: (2b) where ε is a small positive parameter (0<ε<<1) and α, β are known constants. We assume that a(x), b(x) and f(x) are sufficiently continuously differentiable functions in[0,1]. Further more, we assume that b(x) ≤ 0, a(x) ≥ M > 0 throughout the interval[0,1], where M is some positive constant.
A finite difference scheme is often a convenient choice for the numerical solution of two point boundary value problems, because of their simplicity. Let us divide the interval [0,1] Differentiating twice both sides of equation (5) with respect to x, we get (4) (4). For detail discussions of the method refer Joshua Y.Choo and David H.Schultz [2].

Fitted Fourth Order Scheme
A difference scheme with a fitting factor containing exponential functions is known as exponentially fitted difference scheme. From the theory of singular perturbations it is known that the solution of (1)- (2) By taking first terms of the Taylor's series expansion for a(x) and b(x) about the point '0', (10) becomes, 1] into N equal parts with constant mesh length h. Let 0=x 0 , x 1 , x 2 , …x N =1 be the mesh points. Then we have x i = ih; i=0, 1, 2, …, N.
From (11) we have Now, we consider the stable fourth order central difference scheme (8) and introduce the fitting factor ( ) σ ρ : 6 12 Rewriting (13), we have is a fitting factor which is to be determined in such a way that the solution of (14) converges uniformly to the solution of (1)- (2).
Multiplying (14) by h and taking the limit as h→ 0; we get ( ) By using (12) in (15) We have; (18) where the fitting factor σ is given by (18). The equivalent three term recurrence relation of equation (19) is given by: This gives us the tri-diagonal system which can be solved easily by Thomas Algorithm.

Thomas Algorithm
A brief discussion on solving the tri-diagonal system using Thomas algorithm is presented as follows: Consider the scheme given in (20): By comparing (24) and (22), we get the recurrence relations To solve these recurrence relations for i=0,1,2,3, ,N-1, we need the initial conditions for 0 W and 0 T . For this we take With these initial values, we compute i W and i T for i=1,2,3,….,N-1 from (25) in forward process, and then obtain i y in the backward process from (22)and (21b).

Stability Analysis
We will now show that the algorithm is computationally stable. By stability, we mean that the effect of an error made in one stage of the calculation is not propagated into larger errors at later stages of the calculations. Let us now examine the recurrence relation given by (25a). Suppose that a small error 1 i e − has been made in the calculation of From (26) and (25a), we have under the assumption that the error is small initially. From the assumptions made earlier that a(x)>0 , b(x)≤0 and its derivatives also non-positive, we have Form (25a) we have Successively, it follows that 2 1 Therefore the recurrence relation (25a) is stable. Similarly we can prove that the recurrence relation (25b) is also stable. Finally the convergence of the Thomas Algorithm is ensured by the condition i W <1, i=1, 2, 3,…., N-1.

Example
Consider the following homogeneous singular perturbation problem from Bender and Orszag[

Example
Let us consider the following non-homogeneous singular perturbation problem from fluid dynamics for fluid of small viscosity, Reinhardt

Non-linear Problems
Nonlinear singular perturbation problems were converted as a sequence of linear singular perturbation problems by using quasilinearization (Replacing the non-linear problem by a sequence of linear problems) method. The outer solution (the solution of the given problem by putting ε=0) is taken to be the initial approximation.

Example
Consider the following singular perturbation problem from Bender

Conclusions
We have presented fitted fourth order finite difference  Choo's method for solving singularly perturbed two-point boundary value problems The present fitted fourth order finite difference method for solving singularly perturbed two-point boundary value problems produce better approximation to the exact solution, specifically in the boundary layer region with step size h ε > , the perturbation parameter.