A New Second-Order Difference Approximation for Nonlocal Boundary Value Problem with Boundary Layers

The aim of this paper is to present finite difference method for numerical solution of singularly perturbed linear differential equation with nonlocal boundary condition. Initially, the nature of the solution of the presented problem for the numerical solution is discussed. Subsequently, the difference scheme is established on Bakhvalov-Shishkin mesh. Uniform convergence in the second-order is proven with respect to the ε− perturbation parameter in the discrete maximum norm. Finally, an example is provided to demonstrate the success of the presented numerical method. Thus, it is shown that indicated numerical results support theoretical results.


Introduction
In the present study, linear singularly perturbed problem with nonlocal boundary condition is discussed as follows: Problems such as the nonlocal singular perturbation problem (1.1)- (1.3), are problems where the coefficients of the highest order derivative are a very small positive parameters such as 0 < ε << 1. Solving this problems with classical numerical methods may not be the right choice due to the difficulties that may arise from the small perturbation parameter [18,23,24,25,26,27]. These difficulties are quick and fairly irregular variations within thin transition layers. These lead to unlimited number of derivatives in the solution of singular perturbation problems. Therefore, it is important to choose the most suitable numerical methods for singularly perturbed problems. These include finite difference and finite element methods. Thus, in the present study, we wanted to demonstrate that these difficulties can be overcome with the finite difference method.
Studies conducted on singular perturbation problems commenced in the 1900s. These problems were known to be common in the fields of natural sciences, engineering, medical sciences, fluid mechanics, aerodynamics, magnetic dynamics, diffusion theory, reaction diffusion, light emitting waves, electron plasma waves, communication networks, plasma dynamics, refined gas dynamics, mass transport, plastics, chemical reactor theory, oceanography, meteorology, electricity current, ion acoustic waves plasma and several physical modelling techniques (see, [2,4,9,14,15,18,24,25,26,27]). Lately, singularly perturbed problems, particularly with the nonlocal boundary condition and boundary layers have been studied by several researchers (e.g., [1,7,8,10,11,12,16,17,19,20,23] and the references therein). Bakhvalov used a special transformation in numerical solution of boundary solid problems [5]. Bitsadze and Samarskii obtained several generalizations for linear elliptic boundary value problems [6].Čiegis, studied numerical solution of the singular perturbation problem with nonlinear boundary condition [13]. Different from the previous studies in the literature and for the first time, this problem is solved with the presented finite difference method on Bakhvalov-Shishkin mesh in order to demonstrate that the difference scheme has second-order convergence and a better result could be obtained. Especially, this method shows uniformly convergent provided only that ε ≤ CN −1 . Namely, Bakhvalov-Shishkin mesh gives a stronger error bound for ε ≤ CN −1 . Bakhvalov-Shishkin mesh is a modification of the Shishkin mesh described that incorporates idea by Bakhvalov. But the original Bakhvalov mesh requires the solution of a nonlinear equation to determine the transition point where the mesh switches from coarse to fine. Instead, the transition points are as in the Shishkin mesh [21]. There are many studies on the B-S (Bakhvalov-Shishkin) mesh: T. Linss has studied simple upwind difference scheme on a B-S mesh [21]. Analysis of a Galerkin finite element method on Bakhvalov-Shishkin mesh for a linear convection-diffusion problem is investiagated by Linss [22]. Uniform second-order hybrid schemes on Bakhvalov-Shishkin mesh are analyzed in [29]. Hybrid difference schemes with variable weights on Bakhvalov-Shishkin mesh are examined to the derivative for quasi-linear singularly perturbed convection-diffusion boundary value problems in [28]. Linear Galerkin finite element method on Bakhvalov-Shishkin mesh for singularly perturbed convection-diffusion problem is worked in [30].
The present study is structured as follows: Section 2 focused on the exact solution of the problem provided in (1.1)-(1.3) and several asymptotic evaluations on the fourth-order derivatives of the exact solution. In Section 3, the difference scheme is constructed as hybrid scheme. Subsequently, the structure of the Bakhvalov-Shishkin mesh is introduced. In Section 4, the second-order uniform convergence of the difference scheme is obtained according to ε. The present study is finalized with the conclusion section. Henceforth, C and C 0 are positive constants independent of ε and the mesh parameter in the following sections.

Certain properties of the continuous problem
Proof. Once maximum principle for (1.1) is used, we obtain that Next, from boundary condition (1.3), we attain 3), the following inequality is found By setting (2.5) in inequality (2.4), we get Then by setting (2.6) in inequality (2.3), we have and this prove the inequality (2.1). The proof of inequality (2.2) is almost identical to that of [1,9] for k = 1 as Now, we obtain inequality (2.2) for k = 2. The proof of (2.2) for k = 3 is obtained in the same way. Let us begin by taking the derivative of equation (1.1) two times, and also, from (1.1) we get where v 1 (x) and v 2 (x) are the solutions of the following problems: We can give the solution of (2.11) in the form where the functions p 0 (x), q 0 (x) and R ε (x) are, respectively, the solutions of (2.21), (2.22) and (2.23) from [1]. Also, we see that these solutions have the following estimations: and |R ε (x)| ≤ C. From (2.7), (2.9), (2.12) and (2.13) the following inequality clearly leads to (2.2) for k = 2.
All these estimations conclude our proof.

The construction of difference scheme and mesh
In this section, the discretization of the problem (1.1)-(1.3) using finite difference method on Bakhvalov-Shishkin mesh is presented.

Bakhvalov-Shishkin mesh
The interval [0, 1] is divided into the three subintervals [0, Here σ 1 and σ 2 are referred as the transition points and are written as follows: Assumption 1: We shall assume throughout the paper that ε ≤ CN −1 as is generally the case in practice, where, N is a positive even integer.
are introduced through a set of the equalities:

The construction of the difference scheme
Here the following finite differences for any mesh function g i = g(x i ) are presented onω N as: , g0 Now, the difference scheme for the problem (1.1) should be constructed.
The following exact relation is obtained through the use of the interpolating quadrature formulas on subintervals [ If the above equality is arranged, it gives From here, the following equality is obtained by implementing partial integration and then by using the formula (2.2) of [3]: Finally, we propose the following difference scheme for approximating (1.1)-(1.3): , ..., N − 1, and the reminder term where the functions ϕ i (x) are in the form: Here, applying partial integration in the first expression of the left integral, we get and from here it follows that we use the formula (2.2) of [3] in (3.2) and propose the following difference scheme: where reminder term R 2 and the functions ψ i (x) take the form Here, it is necessary to define an approximation for the second boundary condition (1.3). The following equation is written using the interpolation quadrature formula with respect to x N0 and x N0+1 : where, reminder term r 0 Once reminder terms R 1 i and R 2 i and r 0 are neglected from (3.1), (3.3) and (3.4), it is possible to propose the following difference schemes for the problem (1.1)-(1.3): where x N0 is the mesh point nearest to l 1 .

Uniform error estimates
With respect to the examination of the presented method for the problem (1.1)-(1.3), this section provides the following discrete problem and its solution: where, R 1 i , R 2 i and r 0 are given by (3.1), (3.3) and (3.4) respectively. Lemma 2. If z i is the solution to (4.1)-(4.2), then the estimate becomes:
We can state the convergence result of this study the following Theorem 1. Proof. This follows immediately by mixing previous lemmas.

Algorithm and numerical results
This section focuses on the demonstration of the following procedure for the difference scheme (3.5)-(3.6). Moreover, the effectiveness of the presented method is confirmed by applying it to a linear problem (1.1)-(1.3). Initially, the algorithm for the solution of the difference scheme (3.5)-(3.6) is provided:

This algorithm is stable due to
Subsequently, the following problem is taken into consideration in order to prove that the presented method is working: The exact solution of the problem is where d = 5 + 2 cos(πx) + cos(πx) 2 + 4 sin(πx/2). The ε-uniform convergence rates are calculated using the following expression: The error estimates are also denoted by As presented in Table 1, when the ε is small, the solution changes fastly in the boundary layer regions. When N = 32, 64, . . . , 1024 takes increasing values, it is observed in table that the convergence rate p N is of the second-order. The exact solution and approximate solution curves are determined to be almost same, as presented in Figure 1. Therefore, it is possible to conclude that convergence is achieved. As indicated in Figure 2, errors in boundary layer regions with respect to the examination of the presented method for the problem (1.1)-(1.3), are maximum due to the irregularity caused by the sudden and rapid change of the solution in these regions around x = 0 and x = 1 for the different values of ε. Therefore, the numerical results indicated that the proposed scheme is working effectively.

Conclusions
In this study, we offered an effective finite difference method for solving secondorder linear singularly perturbed nonlocal boundary value problem. Uniform convergence in the second-order was proven with respect to the ε− perturbation parameter in the discrete maximum norm of the difference scheme. As a result, it was possible to conclude that the finite difference method, taken into consideration for the solution of problems that are not easy to solve with every numerical method and that have both nonlocal and singular perturbation properties, was very effective and convenient on nonuniform meshes (Shishkin, Bakhvalov, Bakhvalov-Shishkin etc.). The present study findings demonstrate that it would be possible to conduct further studies on delayed and partial differential equations, which contain more complex nonlocal conditions. Furthermore, it could be suggested that a study on the increase in the convergence rate to three or higher orders would be possible.