An effective approximation for singularly perturbed problem with multi-point boundary value

This study deals with the singularly perturbed multi-point boundary value problem and an effective numerical method. The method analysis singularly perturbed problem with multi-point boundary value as theoretically and experimentally. It is shown that the presented method has first-order approximation in the discrete maximum norm. The numerical results are presented in table and graphs, and these results come out the validity of the theoretical analysis of our method.


Introduction
Consider the following second-order linear singularly perturbed multi-point boundary value problem where 0 < ε 1 is a small perturbation parameter; b, d, m and c i are given constants, 0 < s i < 1, i = 1, 2, ...m; and b (x) ≥ b 2 > 0 and f (x) are assumed to be continuous functions in [0, 1] , and moreover It is a well known fact that differential equations with a small parameter ε multiplying the highest-order derivative terms are called singularly perturbed differential equations. Standard numerical methods for solving singularly perturbed problems are fail to give accurate results and unstable due to the perturbation parameter ε. Therefore, there are some fitted numerical methods to solve equations like these, such finite difference methods, finite element methods etc. So, we prefer to use finite difference method for solving this problem in this paper.
In the above aforementioned papers, related studies to singularly perturbed problems are related only with the ordinary cases. In addition, available studies for the numerical solution of singularly perturbed problems with multi-point boundary conditions have not widespread yet. It can be seen in [5,10] that various difference schemes exist for multi-point and integral boundary conditions.
In this present paper, we use finite difference method on a Shishkin mesh for the numerical solution of the nonlocal problem (1)-(3). This method is shown uniformly convergent of first-order on Shishkin mesh, in discrete maximum norm. Some properties of the exact solution of the problem described in (1)-(3) is investigated in Section 2. Finite difference schemes on Shishkin mesh for the problem (1)-(3) are constructed in Section 3. The error analysis for the difference scheme is performed in Section 4. Finally, We formulate the iterative algorithm for solving the discrete problem and a numerical example present to find the solution of approximation in Section 5.
Henceforth, C, C 0 and C 1 will mean positive constants independent of ε and the mesh parameter.

Some properties of the continuous problem
Here we establish very important asymptotic properties of the exact solution of the problem (1)-(3) that will be used to anaylze appropriate finite difference problem.
where w 0 (x) ≥ |w (x)| is the solution of the following problem Then, the solution of the problem (1)-(3) satisfies the following inequalities: where Proof. Let us take u (1) = λ and the solution of the problem (1)-(3) as , and the function v (x) is the solution of the following problem Now, we use the maximum principle for the evaluation of the functions v (x) and w (x), and so we have and Finally, from (7) and (8), we obtain which proves (5).
Next, we will examine the inequality (6). Differentiating the Equation (1), we get the relation where After doing some calculation in the Equation (9), we obtain (see in [8]). Eventually, we have the inequality (6). And so, the proof of Lemma 1 is completed.

The Establishment of difference scheme
In here, we discretize the problem (1)-(3) using a finite difference method on a piecewise uniform mesh of Shishkin type. The Shishkin mesh is introduced for this study as follows.

Shishkin mesh
The approximation to the solution u of the problem (1) We introduce a set of the mesh pointsω

Construction of the difference scheme on Shishkin mesh
We introduce an any non-uniform mesh on the interval [0, 1] Before describing our numerical method, we introduce some notations for the mesh functions. We define the following finite difference for any mesh function g i = g(x i ) given onω N : Now, we construct the difference scheme for the Equation (1). Firstly, we will integrate the Equation (1) over i (x) and ϕ (2) i (x), respectively, are the solution of the problems as: After doing some arrangements in the Equation (11), we have where and Using the interpolating quadrature rules (2.1) and (2.2) from [4] with weight functions ϕ i (x) on subintervals (x i−1 , x i+1 ) from (12), we obtain the following precise relation: where If we neglect R i in the Equation (14), we can suggest the following difference scheme for the problem (1)-(3): where x N i is the mesh point nearest to s i , and θ i is given by (15).

Uniform error estimates
In this part, we will investigate the convergence of the method for the problem (1) and (3). We will give the error function where R i and θ i are defined by (13) and (15), respectively.

Lemma 2. Let z i be the solution (19)-(21) and
Then the estimate holds.
Proof. Let z (x) = z 1 (x) + λ z 2 (x) be the solution of the discrete problem (19)- (21), where z 1 (x) and z 2 (x) are the solution of the following problems, respectively: According to the maximum principle for z 1 (x) and z 1 (x), we have the following evaluations: and Next, we have from (23) and (24) which proves Lemma 2.
Lemma 3. Under the assumptions of section 1 and Lemma 2 the solution of the problem (1)-(3) satisfies the following estimates for the remainder term R i : Proof. The reminder term R i can be rewritten with (2) and using mean value theorem as In the first case, for 1 4 > b −1 ε ln n = σ and the interval [0, σ ] : In the second case, for 1 4 > b −1 ε ln n = σ and the interval [σ , 1 − σ ] : In the third case, for 1 4 > b −1 ε ln n = σ and the interval [1 − σ , 1] : and then, we evaluate R i for i = N 4 and i = 3N 4 , respectively: According to all these situations, we have This completes the proof of Lemma 3.
We can state the convergence result of this study the following Theorem 4.

Algorithm and numerical results
Here an effective algorithm has been given for the solution of the difference scheme (16)- (18) and numerical results have also been displayed in table and graphs.
(a) We give the algorithm for the solution of the difference scheme (16)- (18): (b) We examine the following problem to see how the method works: We have the exact solution of this problem as The corresponding ε− uniform convergence rates are computed using the formula The error estimates are denoted by e N = max ε e N ε , e N ε = y − u ∞,ω N .

Conclusion
In this study, we have offered an effective finite difference method for solving second-order linear singularly perturbed multi-point boundary value problem. The method has display uniform convergence with respect to the perturbation    parameter ε. Also, the method is first order convergent in the discrete maximum norm. Numerical example shows that recommended method has a good approximation characteristic as: in table and graphics, when N takes increasing values, it is seen that the convergence rate of the smooth convergence speed p N is first order. The exact solution and approximate solution curves are almost identical as shown in Figure 1. In Figure 2, as ε values decrease, the graph approaches more towards the coordinate axes in the boundary layer region around x = 0 and x = 1. In Figure 3, the errors in these regions are maximum because of the irregularity caused by the sudden and rapid change of solution in the boundary layer region around x = 0 and x = 1 for different values ε. Thus, numerical results prove that the proposed scheme is working very well.