An approximate solution of singularly perturbed problem on uniform mesh

ABSTRACT


Introduction
This study is concerned with the numerical solution the following singularly perturbed equation with integral boundary values for 0 ≤ ℓ 0 < ℓ 1 ≤ ℓ ε 2 u ′′ (t) + εa(t)u ′ (t) − g(t, u) = 0, 0 < t < ℓ, (1) Here, ε is the perturbation parameter and is defined as 0 < ε ≪ 1.A and B are fixed.a(t) and g(t, u) are continuous functions in the interval [0, ℓ] and [0, ℓ] × R, respectively.f 0 (t) and f 1 (t) are continuous functions on [ℓ 0 , ℓ 1 ].When ε = 0 in the Eq. ( 1), the new equation is an algebraic equation.The boundary conditions will be unnecessary for the solution of this equation.In this case, there will be two boundary layers t = 0 and t = 1 of the problem (1-(3).
Equations with a positive parameter ε in the coefficient of the highest order derivative are called singularly perturbed equations.Solutions of these problems have thin boundary layers.In these layers, the solution changes abruptly and rapidly, while in other parts of the definition region it changes slowly and regularly.This irregularity causes the solution of singularly perturbed problems to have unlimited derivatives.Thus, serious difficulties arise in the operation of such problems.These difficulties are also evident in numerical solution.Because the approximate solution diverges from the exact solution as the mesh steps get smaller.For this reason, it is very important to establish appropriate numerical methods for the solution of problems with singular perturbations.Known classical numerical methods cannot give numerical results suitable for the exact solution.Especially in this study, an efficient numerical method such as the finite difference method, which gives uniform convergence according to ε *Corresponding Author is preferred for the solution of such problems [1]- [15].Studies on problems with singular perturbations started in the 1900s.These problems are encountered in science, economics, sociology, engineering, medical science, fluids mechanics, aerodynamics, magnetic dynamics, emission theory, reaction diffusion, light emitting waves, communication lines, plasma dynamics, purified gas dynamics, motion of mass, plastics, chemical reactor theory, seismology, oceanography, meteorology, electric current, ion acoustic waves and some physical modeling [16]- [21].Also, Bakhvalov used a special transformation in the numerical solution of boundary layer problems [22].Bitsadze and Samarskii obtained some generalizations of linear elliptic boundary value problems [23].Herceg and Surla found the numerical solution of the singularly perturbed problem with non-local boundary values using spline tension [24].Gupta and Trofimchuk studied a sharper case for solving the second-order three-point boundary value problem [25].Amiraliyev and Cakir found a uniformly convergent approach for the zeroth order reduced equation and convective singularly perturbed problem [2].Cakir, for the three-point singularly perturbed problem, using the difference method, a second-order uniform convergence was obtained [3].Cakir and Amiraliyev evaluated the three-point singularly perturbed boundary value problem using a Shishkin mesh [4].In [1,[6][7][8]24] and [26,34] have been worked on the numerical solution of the singularly perturbed problem with nonlocal boundary and integral boundary conditions.The singularly perturbed problem has also been solved by different numerical methods [9,10,35].There are also studies on existence and uniqueness of these problems in the literature [36]- [38].
The study of the linear state of our problem (1)-( 3) is in [1], where the problem is solved with a layer-adapted mesh.The difference of this study from similar studies in the literature [6] is the use of a uniform mesh and the two integral boundary conditions of the problem (1)- (3).Although there are studies on singularly perturbed problems with two integral boundary conditions solved in Shishkin mesh as in [6], we have not come across any studies on singularly perturbed problems with two integral boundary conditions in uniform mesh.This gap, which is not in the literature, constitutes the motivation of our study.In this paper, the difference scheme is obtained using the integral rules from [30].In the second part, we investigated several important factors for the exact solution ( 1)-( 3).The difference procedure for the uniform mesh of the problem ( 1)-( 3) is given in part 3.In the fourth part, the convergence evaluation of the method is made.For the purpose of applying the theoretical procedure, an example whose exact solution is unknown is presented in Sect 5. Throughout this study, C and C 0 will be used as positive constants that do not depend on ε and h.The norm ∥.∥ is used to denote the maximum norm.

Some properties of the Exact Solution
Here we will give a Lemma and its proof, which will be needed for later parts of the study.
Lemma 1.Let u(t) be the solution of the ( 1) hold, where 0 ⩽ t ⩽ ℓ and Proof.Using the mean value theorem for g(t, u) in (1), we have Let's rewrite the (1)-( 3) problem as follows to get the proof of (4) Here, using the maximum principle [4,39] and ( 6)-( 8) we arrive at the following inequality: Now, using the boundary values ( 7) and ( 8), let's obtain the inequality ( 4) If we write the inequalities (10) and (11) in the inequality (9), we get the following result: Thus, the proof of ( 4) is completed.Also, the proof of ( 5) is almost the same to that of [39].□

Uniform Mesh and Construction of the difference scheme
In this part, we will obtain the difference scheme for the (1)-( 3) problem.For this we will work on the uniform mesh.
where N is the number of discretization points.Let's give some notations for grid functions, where y i is the approximate value for u(t) at grid points t i .
Let's start constructing the difference scheme with the following equation for 1 where the functions {φ i (t)} N −1 i=1 have the form i (t) and φ (2) i (t), respectively, are the solution of the following problems: and the coefficient ξ −1 i in (12) as If we rearrange (12), we get the following system If we benefit the formulas from [5], we have the following system for 1 + h 2 ut t,i are substituted in the above equation, the following expression is obtained: So, from the above equations, the difference scheme is defined for 1 ⩽ i ⩽ N − 1: here Now, the approximations for the first and second boundary conditions need to be determined.Let t N 0 and t N 1 be the grid points nearest to ℓ 0 and ℓ 1 , respectively. and where Thus, we get the difference approximation corresponding to the first boundary value as: where (21) Now we get the difference approtimation corresponding to the second boundary value as follows: where If the error values in the ( 14), ( 21) and ( 24) are neglected, the following difference chart is found for 1 ⩽ i ⩽ N − 1:

Stability of the Difference Scheme
Here, we give the stability of the finite difference method with the Theorem 1 and the evaluation of the error functions with the Lemma 2.
The error term z is z = y − u for 1 < i < N .
Lemma 2. The estimates are valid for the terms R i , r 0 and r 1 to be obtained with the help of results of Section 1 and Lemma 1 Proof.We have from the expression (14) for R i on an arbitrary mesh as follows This inequality and (5) enable us to write the inequality as The mesh is uniform with h = ℓN −1 for 1 ≤ i ≤ N .So, from the above inequality, we get Let us evaluate (32) using the expression (21) for r 0 as The proof of ( 33) is similar to the proof of the inequality (32).All these complete the proof of Lemma 2. □ Lemma 3. If z i is the solution of ( 28)- (30) and Then there is the following the estimate Proof.Using Lemma 2, we easily obtain (36).□ Theorem 1.If u be the solution of ( 1)-(3) and y be the solution of ( 25)- (27).Then, the following estimate is satisfied.
This theorem gives the result of the convergence of the proposed method with the help of Lemma 2 and Lemma 3.

Numerical Illustrations
Here we provide some numerical results that exemplify the current method.By using the quasilinearization technique, the scheme ( 25)-( 27) can be arranged as: − ∂g ∂y t i , y where y (0) i for 1 ≤ i ≤ N and n ≥ 1 is the initial guess.Example 1.We apply the scheme ( 37)-( 39) to the following singularly perturbed problem with integral boundary conditions: The exact solution of the problem is unknown.For this reason, we have to use the double mesh as: .
The rates of convergence are defined as .
The e N is the maximum errors as  Uniform convergence rates p and error values are given in Table 1.p values are around one.
In Figure 1, the approximate solution curves have been plotted for each of the values N = 16, 32, 64, 128, 256 for t = 0.As N values increase, the approximate solution curves approach the coordinate axes around t = 0 and t = 1.Here it can be seen that the theoretical process is accurate and reliable.

Conclusion
In the study, the finite difference method is used to solve the problem with nonlocal conditions.The difference scheme has been established with the help of some integral forms on the uniform mesh.The difference problem was solved by the Gauss elimination method.Convergence analysis was performed.Uniform convergence was obtained from the first-order.The proposed method has been applied to a test problem.Numerical results show that the approaches described here contribute greatly to the understanding of singularly perturbed problem (see Table 1 and Figure 1).With the motivation given by this study, it is aimed to apply to nonlocal boundary condition and fuzzy problems with delay parameter with singularly perturbation feature.

Table 1 .
Convergence rates and maximum errors for ε and N .