01051324

Page 1

International Journal of Modern Research in Engineering and Technology (IJMRET) www.ijmret.org Volume 1 Issue 5 Ç December 2016. Numerical Solution of the Nonlocal Singularly Perturbed Problem

Musa ÇakÄąr, Derya Arslan Department of Mathematics, Faculty of Science, University of Yuzuncu Yil, Van, Turkey Ministry of National Education, Turkey Abstract: The study is concerned with a different perspective which the numerical solution of the singularly perturbed nonlinear boundary value problem with integral boundary condition using finite difference method on Bakhvalov mesh. So, we show some properties of the exact solution. We establish uniformly convergent finite difference scheme on Bakhvalov mesh. The error analysis for the difference scheme is performed. The numerical experiment implies that the method is the first order convergent in the discrete maximum norm, independently of đ?œ€- singular perturbation parameter with effective and efficient iterative algorithm. The numerical results are shown in table and graphs. Keywords: Singular perturbation, finite difference method, Bakhvalov mesh, uniformly convergence, integral condition. 1.

Introduction

This paper focused on the first-order nonlinear singularly perturbed differential equation. đ?œ€đ?‘˘â€˛ + đ?‘“ đ?‘Ľ, đ?‘˘ = 0,

đ?‘Ľ ∈ 0, đ??ż ,

đ??ż>0

(1)

subject to the integral boundary condition đ??ż

đ?‘˘ 0 = đ?›˝đ?‘˘ đ??ż +

đ?‘? đ?‘ đ?‘˘ đ?‘ đ?‘‘đ?‘ + đ?‘‘

(2)

0

where đ?œ€ is a small perturbation parameter 0 < đ?œ€ ≪ 1 . đ?›ź, đ?›˝, đ?‘‘ are constant, which are independent of đ?œ€. We assume that đ?‘? đ?‘Ľ , đ?‘“(đ?‘Ľ, đ?‘˘) are sufficiently smooth functions in the intervals (0, đ??ż] and (0, đ??ż]đ?‘Ľđ?‘…. Also, đ?œ•đ?‘“ ≼ đ?›ź > 0. The solution of (1) has in general a boundary layer with O(đ?œ€) near to đ?‘Ľ = 0. The first time, đ?œ•đ?‘˘ nonlocal problems were studied by Bitsadze and Samarskii [16]. Differential equations with conditions which connect the values of the unknown solution at the boundary with values in the interior are known as nonlocal boundary value problems [16]. Existence and uniqueness of nonlocal problems can be seen in [1-3]. Singular perturbation problems can also arise in fluid mechanics, quantum mechanics, hydromechanical problems, chemical-reactor theory, control theory, oceanography, meteorology, electrical networks and other physical models. The subject of nonlocal boundary value problems for singularly perturbed differential equations has been studied by many authors. For example, Cziegis [4] studied the numerical solution of singularly perturbed nonlocal problem. Cziegis [5] analyzed the difference schemes for problems with nonlocal conditions. Amiraliyev and ÇakÄąr [8] applied the difference method on a Shishkin mesh to the Singularly perturbed threepoint boundary value problem. Amiraliyev and ÇakÄąr [9] studied numerical solution of the singularly perturbed problem with nonlocal boundary condition. ÇakÄąr and Arslan [14] analyzed a numerical method for nonlinear singularly perturbed multi-point boundary value problem. Adzic and Ovcin [11] studied nonlinear spp with nonlocal boundary conditions and spectral approximation. Amiraliyev, Amiraliyeva and Kudu [12] applied a numerical treatment for singularly perturbed differential equations with integral boundary condition. Herceg [6] studied the numerical solution of a singularly perturbed nonlocal problem. Herceg [7] researched solving a nonlocal singularly perturbed problem by splines in tension. ÇakÄąr [10] studied uniform second–order difference method for a singularly perturbed three-point boundary value problem. Geng [15] applied a numerical algorithm for nonlinear multi-point boundary value problems. ÇakÄąr [13] obtained a numerical study on the difference solution of singularly perturbed semilinear problem with integral boundary condition. It is well known that standard discretization methods for solving nonlocal singular perturbation problem are unstable and these don’t give accurate results for đ?œ€. Therefore, it is very important to find suitable numerical methods to these problem. So, we use finite difference method in this paper. According to this method, we will analyze that this method for the numerical solution of the nonlocal problem (1)-(2) is uniformly convergent of first order on Bakhvalov mesh, in discrete maximum norm, indepedently of singular perturbation parameter đ?œ€. In section 2, some properties of the exact solution of problem described in (1)-(2) is investigated. According to the perturbation parameter, by the method of integral identities with the use exponential basisfunctions and interpolating quadrature rules with the weight and remainder terms in integral form uniformly convergent finite difference scheme on Bakhvalov mesh is established in section 3. The error analysis for the difference scheme is performed in section 4. In section5, numerical examples are presented to find the solution of approximation. In

www.ijmret.org

Page 13

International Journal of Modern Research in Engineering and Technology (IJMRET) www.ijmret.org Volume 1 Issue 5 Ç December 2016. this paper, we use đ?‘”(đ?‘Ľ) continuous function.

∞

for the continuous maximum norm on the corresponding interval. đ?‘” đ?‘Ľ is any

2. Properties of the Exact Solution The exact solution of (1)-(2) has some properties, which are needed in later sections for analysis of the numerical solution. Thus, we describe these properties with Lemma 2.1. Lemma 2.1. If

đ?œ•đ?‘“ (đ?‘Ľ,đ?‘˘) đ?œ•đ?‘˘

≼ đ?›ź > 0 and đ?‘“ đ?‘Ľ, đ?‘˘ ∈ (0, đ??ż]đ?‘Ľđ?‘…, the estimates đ?‘˘(đ?‘Ľ)

≤ đ??ś0

(3)

where đ??ś0 = đ?‘˘ 0 đ?‘’

−đ?›źđ?‘Ą đ?œ€

+ 1−đ?‘’

−đ?›źđ?‘‡ đ?œ€

đ?›ź −1 đ??š

∞,

đ??š đ?‘Ľ = −đ?‘“(đ?‘Ľ, 0)

and 1 −đ?›źđ?‘Ľ ≤đ??ś 1+ đ?‘’ đ?œ€ đ?œ€

�′ (�)

(4)

hold for the solution đ?‘˘(đ?‘Ľ), where đ??ś0 and đ??ś are constant independent of đ?œ€ and h. Proof. We use intermediate value theorem for đ?‘“ đ?‘Ľ, đ?‘˘ in Equation (1) đ?‘“ đ?‘Ľ, đ?‘˘ − đ?‘“(đ?‘Ľ, 0) đ?œ•đ?‘“ đ?‘Ľ, đ?‘˘ = , đ?‘˘ đ?œ•đ?‘˘

� = ��,

0<�<1

and đ?‘Ž đ?‘Ľ =

đ?œ•đ?‘“ đ?‘Ľ, đ?‘˘ . đ?œ•đ?‘˘

Thus, we obtain the following linear equation, đ?œ€đ?‘˘â€˛ + đ?‘Ž đ?‘Ľ đ?‘˘ đ?‘Ľ = đ??š đ?‘Ľ

(5)

We can write the solution of the Equation (5) as follows: −1 đ?‘Ľ 0 đ?‘Ž đ?œ? đ?‘‘đ?œ?

đ?‘˘ đ?‘Ľ =đ?‘˘ 0 đ?‘’đ?œ€

đ?‘Ľ

1 đ?œ€

+

−1 đ?‘Ľ đ?œ‰ đ?‘Ž đ?œ? đ?‘‘đ?œ?

đ??š(đ?œ‰)đ?‘’ đ?œ€

đ?‘‘đ?œ‰

(6)

0

In Equation (5), đ?‘Ľ = đ?‘‡ and đ?‘Ľ = đ?‘ , we obtain Eq (7)-Eq(9) −1 đ?‘‡ 0 đ?‘Ž đ?œ? đ?‘‘đ?œ?

đ?‘˘ đ?‘‡ =đ?‘˘ 0 đ?‘’đ?œ€

+

−1 đ?‘ 0 đ?‘Ž đ?œ đ?‘‘đ?œ

đ?‘˘ đ?‘ =đ?‘˘ 0 đ?‘’đ?œ€

1 đ?œ€

+

�

−1 đ?‘‡ đ?œ‰ đ?‘Ž đ?œ? đ?‘‘đ?œ?

đ??š(đ?œ‰)đ?‘’ đ?œ€

đ?‘‘đ?œ‰

(7)

đ?‘‘đ?œ‰

(8)

0 đ?‘

1 đ?œ€

−1 đ?‘ đ?œ‰ đ?‘Ž đ?œ đ?‘‘đ?œ

đ??š(đ?œ‰)đ?‘’ đ?œ€ 0

and � 0 =

đ?‘‘+

−1 đ?‘‡ đ?‘‡ đ?‘Ž đ?œ? đ?‘‘đ?œ? đ??š(đ?œ‰)đ?‘’ đ?œ€ đ?œ‰ 0 đ?œ€

đ?›˝

1

−1 đ?‘‡ đ?‘‡ đ?‘‡ đ?‘Ž đ?œ? đ?‘‘đ?œ? đ?‘?(đ?‘ ) [ 0 đ??š đ?œ‰ đ?‘’ đ?œ€ đ?œ‰ đ?‘‘đ?œ‰ ]đ?‘‘đ?‘ 0 đ?œ€ . −1 đ?‘‡ −1 đ?‘ đ?‘‡ − đ?›˝đ?‘’ đ?œ€ 0 đ?‘Ž đ?œ? đ?‘‘đ?œ? − 0 đ?‘?(đ?‘ )đ?‘’ đ?œ€ 0 đ?‘Ž đ?œ? đ?‘‘đ?œ? đ?‘‘đ?‘

đ?‘‘đ?œ‰ +

1

(9)

Now, we evaluate to Equation (9) for đ?‘Ž đ?œ? ≼ đ?›ź > 0 and đ?‘? ∗ = đ?‘šđ?‘Žđ?‘Ľ đ?‘?(đ?‘Ľ) . First, we prove the estimate the following inequality, −1 đ?‘‡ 0 đ?‘Ž đ?œ? đ?‘‘đ?œ?

1 − đ?›˝đ?‘’ đ?œ€

�

−

−1 đ?‘ 0 đ?‘Ž đ?œ? đ?‘‘đ?œ?

đ?‘? đ?‘ đ?‘’đ?œ€

đ?‘‘đ?‘ ≼ 1 − đ?›˝đ?‘’

−đ?›źđ?‘‡ đ?œ€

0

�

− đ?‘?∗

đ?‘’

−đ?›źđ?‘ đ?œ€

đ?‘‘đ?‘ ≼ 1 − đ?›˝đ?‘’

−đ?›źđ?‘‡ đ?œ€

− đ?‘?∗ 1 − đ?‘’

−đ?›źđ?‘‡ đ?œ€

≼ đ??ś0

0

>0 Second, we prove the estimate the following inequality,

www.ijmret.org

Page 14

International Journal of Modern Research in Engineering and Technology (IJMRET) www.ijmret.org Volume 1 Issue 5 Ç December 2016. đ?‘‘+ ≤ đ?‘‘ + đ?›˝

1 đ?œ€

�

đ?›˝ đ?œ€

−1 đ?‘‡ đ?œ‰ đ?‘Ž đ?œ? đ?‘‘đ?œ?

đ??š đ?œ‰ đ?‘’đ?œ€ 0 đ?‘‡

đ??š

−1 đ?‘‡ đ?œ‰ đ?‘Ž đ?œ? đ?‘‘đ?œ?

đ?‘’đ?œ€

∞

đ?‘‘đ??ƒ +

0

= � + � ≤ � + � = � + �

�

1 đ?œ€

đ?‘‘đ?œ‰ +

0

1 đ?œ€

�

1 đ?œ€

đ??š

1 đ?œ€

đ??š

đ?‘’

∞

�

đ?‘?(đ?‘ ) [

−1 đ?‘‡ đ?œ‰ đ?‘Ž đ?œ? đ?‘‘đ?œ?

đ??š đ?œ‰ đ?‘’đ?œ€

�

�

đ?‘? đ?‘

đ??š

≤ đ?‘‘ + đ?›˝ đ??š

∞

0

đ?‘‘đ??ƒ +

−đ?›źđ?‘‡ đ?œ€ 1−đ?‘’ đ?œ€ đ?›ź

+

−đ?›źđ?‘‡ 1 1−đ?‘’ đ?œ€ đ?›ź

∞

đ?‘‘đ?œ‰ đ?‘‘đ?‘

0

−đ?›ź đ?‘‡âˆ’đ?œ‰ đ?œ€

1 1 + đ??š đ?›ź đ?›ź

∞

−1 đ?‘‡ đ?œ‰ đ?‘Ž đ?œ? đ?‘‘đ?œ?

đ??š đ?œ‰ đ?‘’đ?œ€

0

∞

đ?‘‘đ?œ‰ ]đ?‘‘đ?‘

0

+ đ?‘?

1

�

1 đ??š đ?›ź

1 đ??š đ?›ź

đ?‘? đ?‘ 1−đ?‘’

∞

đ?‘‘đ?‘

0 �

đ?‘? đ?‘ 1−đ?‘’

∞

−đ?›źđ?‘ đ?œ€

đ?‘‘đ?‘

0 �

1 đ??š đ?›ź

−đ?›źđ?‘ đ?œ€

đ?‘? đ?‘ 1−đ?‘’

∞

−đ?›źđ?‘ đ?œ€

đ?‘‘đ?‘

0

≤ đ??ś1

where �

đ?‘? đ?‘ đ?‘‘đ?‘ = đ?‘? 1 . 0

Consequently, we obtain the following inequality: đ?‘˘ 0 ≤đ??ś and

� �

≤ � 0 �

−đ?›źđ?‘Ľ đ?œ€

+ 1−đ?‘’

−đ?›źđ?‘‡ đ?œ€

đ?›ź −1 đ??š

≤ đ??ś0 −1 đ?‘‘ + đ??ś0 −1 đ?›˝ + đ?‘? ≤ [đ??ś0 −1 đ?‘‘ + đ??ś0 −1 đ?›˝ + đ?‘?

1 1

∞

đ?›ź −1 đ??š đ?›ź −1 đ??š

∞ ∞

đ?‘’

−đ?›źđ?‘Ľ đ?œ€

+ 1−đ?‘’

−đ?›źđ?‘‡ đ?œ€

đ?›ź −1 đ??š

∞

≤ đ??ś0 .

We now prove the estimate the Equation (4) : đ?œ•đ?‘“ đ?œ•đ?‘Ľ

≤ đ??ś, đ?‘˘(đ?‘Ľ) ≤ đ??ś0.

If the above inequalities replace the Equation (6), we have the following inequality � � where � 0

≤ đ??ś,

đ?›ź −1 đ??š

∞

≤ � 0 �

−đ?›źđ?‘Ľ đ?œ€

+ 1−đ?‘’

−đ?›źđ?‘‡ đ?œ€

đ?›ź −1 đ??š

∞

(10)

≤ đ??ś.

If we take derivative of the Equation (10) we obtain as following: đ??śđ?›ź −đ?›źđ?‘Ľ đ?›ź −đ?›źđ?‘Ľ 1 −đ?›źđ?‘Ľ đ?‘’ đ?œ€ + đ?‘’ đ?œ€ ≤ đ??ś 1+ đ?‘’ đ?œ€ đ?œ€ đ?œ€ đ?œ€ thus, the proving of the Lemma 1 is completed. đ?‘˘â€˛ đ?‘Ľ

3.

≤ � 0

The Establishment of Difference Scheme

Let us consider the following any non-uniform mesh on [0, đ??ż], đ?œ”đ?‘ = đ?‘Ľ0 < đ?‘Ľ1 < â‹Ż < đ?‘Ľđ?‘ −1 < đ?‘Ľđ?‘ ,

đ?‘•đ?‘– = đ?‘Ľđ?‘– − đ?‘Ľđ?‘–−1

đ?œ”đ?‘ = đ?œ”đ?‘ âˆŞ đ?‘Ľ = 0, đ?‘Ľ = đ??ż . First, we will construct the difference scheme for the Equation (1). First, we integrate the Equation (1) over (đ?‘Ľđ?‘–−1 , đ?‘Ľđ?‘– )

www.ijmret.org

Page 15

International Journal of Modern Research in Engineering and Technology (IJMRET) www.ijmret.org Volume 1 Issue 5 Ç December 2016. đ?‘•đ?‘– −1

đ?‘Ľđ?‘–

đ?‘Ľđ?‘–

đ?œ€đ?‘˘â€˛ (đ?‘Ľ))đ?‘‘đ?‘Ľ + đ?‘•đ?‘– −1

đ?‘Ľ đ?‘–−1

� �, � �� = 0,

đ?‘– = 1, đ?‘ − 1

(11)

đ?‘Ľ đ?‘–−1

and we obtain the following inequality đ?‘•đ?‘– −1 đ?œ€ đ?‘˘ đ?‘Ľđ?‘– − đ?‘˘ đ?‘Ľđ?‘–−1

đ?‘Ľđ?‘–

+ đ?‘•đ?‘– −1

đ?‘Ľđ?‘–

đ?‘“ đ?‘Ľ, đ?‘˘ đ?‘‘đ?‘Ľ = đ?œ€đ?‘˘đ?‘Ľ ,đ?‘– + đ?‘•đ?‘– −1

� �, � �� = 0

đ?‘Ľ đ?‘–−1

đ?‘Ľ đ?‘–−1

where đ?‘…đ?‘– = −đ?‘•đ?‘–−1

đ?‘Ľđ?‘–

đ?‘Ľđ?‘– − đ?‘Ľđ?‘–−1 đ?‘“ ′ (đ?‘Ľ, đ?‘˘)đ?‘‘đ?‘Ľ .

(12)

đ?‘Ľ đ?‘–−1

So, from the Equation (12) the difference scheme is defined by đ?œ€đ?‘˘đ?‘Ľ ,đ?‘– + đ?‘“ đ?‘Ľđ?‘– , đ?‘˘đ?‘– + đ?‘…đ?‘– = 0,

đ?‘– = 1, đ?‘

(13)

Second, we define an approximation for the boundary condition of the Equation (1). đ?‘

đ?‘Ľđ?‘–

đ?‘˘0 = đ?›˝đ?‘˘đ?‘ +

đ?‘? đ?‘ đ?‘˘ đ?‘ đ?‘‘đ?‘ + đ?‘‘

(14)

đ?‘Ľ đ?‘–−1

đ?‘–=1

where đ?‘

đ?‘

đ?‘Ľđ?‘–

đ?‘Ľđ?‘–

đ?‘? đ?‘ đ?‘˘ đ?‘ đ?‘‘đ?‘ = đ?‘–=1 đ?‘Ľ đ?‘–−1

đ?‘

[đ?‘? đ?‘ đ?‘˘ đ?‘ − đ?‘?đ?‘– đ?‘˘đ?‘– ]đ?‘‘đ?‘ đ?‘–=1

+

đ?‘Ľ đ?‘–−1 đ?‘

đ?‘?đ?‘– đ?‘˘đ?‘– đ?‘•đ?‘– = đ?‘&#x; + đ?‘–=1

đ?‘?đ?‘– đ?‘˘đ?‘– đ?‘•đ?‘–

(15)

đ?‘–=1

and remainder term đ?‘

đ?‘Ľđ?‘–

đ?‘&#x;=

đ?‘‘ đ?‘? đ?‘Ą đ?‘˘ đ?‘Ą đ?‘‘đ?’• . đ?‘‘đ?‘Ą

đ?‘Ľ − đ?‘Ľđ?‘–−1 đ?‘–=1

đ?‘Ľ đ?‘–−1

(16)

đ?‘

đ?‘˘0 = đ?›˝đ?‘˘đ?‘ +

đ?‘•đ?‘– đ?‘?đ?‘– đ?‘˘đ?‘– + đ?‘‘ + đ?‘&#x;.

(17)

đ?‘–=1

Thus, by neglecting đ?‘…đ?‘– and đ?‘&#x; in the Equation (13) and the Equation (17), we suggest the following difference scheme for approximating the Equations (1)-(2) : đ?œ€đ?‘Śđ?‘Ľ ,đ?‘– + đ?‘“ đ?‘Ľđ?‘– , đ?‘Śđ?‘– = 0 , đ?‘– = 1, đ?‘ (18) đ?‘

đ?‘Ś0 = đ?›˝đ?‘Śđ?‘ +

đ?‘•đ?‘– đ?‘?đ?‘– đ?‘Śđ?‘– + đ?‘‘

(19)

đ?‘–=1

4. Bakhvalov Mesh We use Bakhvalov mesh, because the difference scheme (18)-(19) must be đ?œ€ −uniform convergent. According to the transition point following as: đ??ż đ?œŽ = đ?‘šđ?‘–đ?‘› , −đ?›ź −1 đ?œ€đ?‘™đ?‘›đ?œ€ 2 the piecewise uniform mesh divides each of the interval 0, đ?œŽ and đ?œŽ, đ??ż into The mesh point đ?‘Ľđ?‘– is defined as follows: đ?œŽ< đ?œŽ=

đ??ż 2 đ??ż 2

đ?‘–đ?‘ đ?‘’, đ?‘Ľđ?‘– ∈ 0, đ?œŽ , đ?‘Ľđ?‘– = −đ?›ź −1 đ?œ€ ln 1 − 1 − đ?œ€ đ?‘–đ?‘ đ?‘’ , đ?‘Ľđ?‘– ∈ 0, đ?œŽ , đ?‘Ľđ?‘– = −đ?›ź −1 đ?œ€ ln 1 − 1 − đ?‘’

đ?‘Ľđ?‘– ∈ đ?œŽ, đ??ż , đ?‘Ľđ?‘– = đ?œŽ + đ?‘– −

đ?‘ 2

đ?‘•, đ?‘– =

www.ijmret.org

đ?‘ 2

2đ?‘– đ?‘

−đ?›źđ??ż 2đ?œ€

+ 1, ‌ , đ?‘ , đ?‘• =

, � = 0,1, ‌ , 2�

đ?‘

2

equidistant subintervals.

đ?‘ 2

, � = 0,1, ‌ ,

đ?‘ 2 đ??żâˆ’đ?œŽ

đ?‘

đ?‘ 2

.

Page 16

International Journal of Modern Research in Engineering and Technology (IJMRET) www.ijmret.org Volume 1 Issue 5 Ç December 2016. 5. Uniform Error Estimates In this section, we obtain the convergence of the method. First, we give the error function đ?‘§đ?‘– = đ?‘Śđ?‘– − đ?‘˘đ?‘– , đ?‘– = 0, đ?‘ , đ?‘§đ?‘– is the solution of the discrete problem, đ?œ€đ?‘§đ?‘Ľ ,đ?‘– + đ?‘“ đ?‘Ľđ?‘– , đ?‘Śđ?‘– − đ?‘“ đ?‘Ľđ?‘– , đ?‘˘đ?‘– = đ?‘…đ?‘– , đ?‘– = 1, đ?‘

(20)

đ?‘

đ?‘§0 = đ?›˝đ?‘§đ?‘ +

đ?‘•đ?‘– đ?‘?đ?‘– đ?‘§đ?‘– − đ?‘&#x;

(21)

đ?‘–=1

where đ?‘…đ?‘– and r are defined by Equation (12) and Equation (16). Lemma 5.1. Let đ?‘§đ?‘– be the solution to Equation (20), đ?‘?∗ = maxđ?‘– đ?‘?đ?‘– , đ?œŒđ?‘˜ =

đ?‘•đ?‘˜ đ?œ€

, đ?‘˜ = 1,2 and đ?›ź = maxđ?‘– đ?‘Žđ?‘–

and 1−đ?›˝

1 1 + đ?›źđ?œŒđ?‘–

đ?‘

đ?‘

− đ?‘?∗

1 1 + đ?›źđ?œŒđ?‘˜

đ?‘•đ?‘– đ?‘–=1

đ?‘–

≠0.

Then the estimate �

∞,đ?œ” đ?‘

≤đ??ś

đ?‘…

∞,đ?œ” đ?‘

+ đ?‘&#x;

(22)

holds. Proof. We use intermediate value theorem to rewrite the Equation (20) as follows: đ?‘“ đ?‘Ľđ?‘– , đ?‘Śđ?‘– − đ?‘“ đ?‘Ľđ?‘– , đ?‘˘đ?‘– = đ?‘“đ?‘˘ đ?‘Śđ?‘– − đ?‘˘đ?‘– = đ?‘“đ?‘˘ đ?‘§đ?‘– Here we assumed that đ?‘“đ?‘˘ = đ?‘Žđ?‘– . Thus, we can write đ?œ€đ?‘§đ?‘Ľ ,đ?‘– + đ?‘Žđ?‘– đ?‘§đ?‘– = đ?‘…đ?‘– , đ?‘– = 1, đ?‘ − 1

(23)

đ?‘

đ?‘§0 = đ?›˝đ?‘§đ?‘ +

đ?‘•đ?‘– đ?‘?đ?‘– đ?‘§đ?‘– − đ?‘&#x;.

(24)

đ?‘–=1

Rearranging Equation (23) gives �� =

đ?œ€ đ?‘•đ?‘– đ?‘…đ?‘– đ?‘§đ?‘–−1 + đ?œ€ + đ?‘Žđ?‘– đ?‘•đ?‘– đ?œ€ + đ?‘Žđ?‘– đ?‘•đ?‘–

and đ?‘–

�� = �0 [

đ?‘–

đ?‘žđ?‘˜ ] + đ?‘˜=1

đ?‘–

đ?œ‘đ?‘˜ [ đ?‘˜ =1

đ?‘žđ?‘— ] đ?‘— =đ?‘˜+1

where đ?‘žđ?‘˜ =

đ?œ€ đ?œ€ + đ?‘Žđ?‘˜ đ?‘•đ?‘˜

đ?œ‘đ?‘˜ =

đ?‘•đ?‘˜ đ?‘…đ?‘˜ . đ?œ€ + đ?‘Žđ?‘˜ đ?‘•đ?‘˜

and

According to the maximum principle for Equations (23)-(24), we have the following inequality đ?‘§ ∞,đ?œ” đ?‘ ≤ đ?‘§0 + đ?›ź −1 đ?‘… ∞,đ?œ” đ?‘ .

www.ijmret.org

(25)

Page 17

International Journal of Modern Research in Engineering and Technology (IJMRET) www.ijmret.org Volume 1 Issue 5 ǁ December 2016. Now, we evaluate the Equation (24). After a simple calculation we obtain 𝑁

𝑧0 ≤ 𝛽 𝑧𝑁 + 𝑏∗

𝑕𝑖 𝑧𝑖 + 𝑟

(26)

𝑖=1

where 𝑏∗ = max 𝑏𝑖 𝑖

and 𝑁

𝑧𝑁 = 𝑧0 ∅𝑁 +

𝜑𝑘 ∅𝑁−𝑘 .

27

𝑘 =1

The Equation (27) write in the Equation (26), we have 𝑁

𝑁

𝑧0 ≤ 𝛽 𝑧0 ∅𝑁 +

𝜑𝑘 ∅𝑁−𝑘 + 𝑏∗ 𝑘=1

𝑖

𝑕𝑖 𝑧0 ∅𝑖 + 𝑖=1

𝜑𝑘 ∅𝑖−𝑘 + 𝑟 𝑘=1

After doing some arragement, 𝑁 𝑘=1 𝜑𝑘 ∅𝑁−𝑘

𝑖 + 𝑏∗ 𝑁 𝑖=1 𝑕𝑖 𝑘=1 𝜑𝑘 ∅𝑖−𝑘 + 𝑟 𝑁 1 − 𝛽 ∅𝑁 − 𝑏∗ 𝑖=1 𝑕𝑖 ∅𝑖

𝛽

𝑧0 ≤

(28)

it follows that 𝛽 𝑧0 ≤

𝜌 𝑘 𝑅𝑘 𝑁 𝑘=1 1+𝛼𝜌 𝑘

𝑁

1

1−𝛽 ≤ 𝐶( 𝑅

𝑁

1

− 𝑏∗

1+𝛼𝜌 𝑖

∞,𝜔 𝑁

𝜌 𝑘 𝑅𝑘 𝑖 𝑘=1 1+𝛼𝜌 𝑘

𝑁 𝑖=1 𝑕𝑖

+ 𝑏∗

1+𝛼𝜌 𝑘

𝑁 𝑖=1 𝑕𝑖

1

𝑖

1 1+𝛼𝜌 𝑘

+ 𝑟

𝑖

1+𝛼𝜌 𝑘

+ 𝑟 )

(29)

where 𝜌𝑘 =

𝑕𝑘 𝜀

, 𝑘 = 1,2 and 𝛼 = max𝑖 𝑎𝑖 .

If the Equation(29) write in the Equation (25), we have 𝑧

≤𝐶

𝑅

∞,𝜔 𝑁

+ 𝑟

≤𝐶

𝑅

∞,𝜔 𝑁

+ 𝑟 .

∞,𝜔 𝑁

+ 𝛼 −1 𝑅

∞,𝜔 𝑁

𝜕𝑓

Lemma 5. 2. Under the assumptions of Lemma 1 and 𝜕𝑢 ≥ 𝛼 > 0, 𝑓 𝑥, 𝑢 ∈ 𝐶 2 0, 𝐿 , the solution of (1)-(2) satisfies the following estimates: 𝑅 ∞,𝜔 𝑁 ≤ 𝐶𝑁 −1 (30) 𝑟 ≤ 𝐶𝑁 −1

(31)

Proof. First, we consider error function 𝑅𝑖 , 𝑅𝑖 ≤ 𝑕𝑖−1

𝑥𝑖

𝑥𝑖 − 𝑥𝑖−1 𝑓 ′ (𝑥, 𝑢) 𝑑𝑥

𝑥 𝑖−1

= 𝑕𝑖−1 ≤

www.ijmret.org

𝐶𝑕𝑖−1

𝑥𝑖

𝑥𝑖 − 𝑥𝑖−1 𝑥 𝑖−1 𝑥𝑖 𝑥 𝑖−1

𝜕𝑓 𝑑𝑥 𝜕𝑓 𝑥, 𝑢 𝑥 + 𝜕𝑢 𝑑𝑥 𝜕𝑢

𝑥𝑖 − 𝑥𝑖−1 𝑑𝑥 + 𝐶𝑕𝑖−1

𝑥𝑖

𝑑𝑢 𝑑𝑥 𝑑𝑥

𝑥𝑖 − 𝑥𝑖−1 𝑢′ 𝑥 𝑑𝑥

𝑥 𝑖−1

Page 18

International Journal of Modern Research in Engineering and Technology (IJMRET) www.ijmret.org Volume 1 Issue 5 ǁ December 2016. ≤ 𝐶𝑕𝑖 + 𝐶𝑕𝑖−1

𝑥𝑖

1 −𝛼𝑥 𝑥𝑖 − 𝑥𝑖−1 1 + 𝑒 𝜀 𝑑𝑥 𝜀

𝑥 𝑖−1

≤ 𝐶 𝑕𝑖 + 𝑕𝑖−1 𝜀 −1

𝑥𝑖

𝑥𝑖 − 𝑥𝑖−1 𝑒

−𝛼𝑥 𝜀

𝑑𝑥 , 1 ≤ 𝑖 ≤ 𝑁.

(32)

𝑥 𝑖−1

In the first case, 𝑥𝑖 ∈ 0, 𝜎 , 1

𝑎) 𝜎 < 2 , 𝜎 = −𝛼 −1 𝜀𝑙𝑛𝜀: 𝑕𝑖 = 𝑥𝑖 − 𝑥𝑖−1 = 𝛼 −1 𝜀 ln 1 − 1 − 𝜀

2𝑖 2(𝑖 − 1) − ln 1 − 1 − 𝜀 𝑁 𝑁

where, according to 𝑖, we use intermediate value theorem as follows: 𝑕𝑖 = 𝛼 −1 𝜀

2(1 − 𝜀)𝑁 −1 ≤ 2𝛼 −1 1 − 𝜀 𝑁 −1 ≤ 𝐶𝑁 −1 1 − 𝑖1 2(1 − 𝜀)𝑁 −1

(33)

Thus, we obtain 𝑅𝑖 ≤ 𝐶 𝑕𝑖 + 𝑕𝑖−1 𝜀 −1

𝑥𝑖

𝑥𝑖 − 𝑥𝑖−1 𝑒 𝑥 𝑖−1

≤ 𝐶 𝑕𝑖 + 𝜀 −1 −1

= 𝐶 𝑕𝑖 + 𝛼 𝑒

𝑥𝑖

𝑒

−𝛼𝑥 𝜀

−𝛼𝑥 𝜀

𝑑𝑥 ≤ 𝐶 𝑕𝑖 + 𝑕𝑖−1 𝜀 −1 𝑕𝑖

𝑑𝑥 = 𝐶 𝑕𝑖 + 𝛼 −1 𝑒

𝑥 𝑖−1 2𝑖 −𝛼 𝛼 −1 𝜀 ln 1− 1−𝜀 𝑁 𝜀

𝑒

−𝑒

−𝛼 𝑥 𝑖 𝜀

𝑥𝑖

−𝛼𝑥 𝜀

𝑒

𝑑𝑥

𝑥 𝑖−1

−𝑒

𝛼 −1 𝜀 ln 1− 1−𝜀

−𝛼 𝑥 𝑖−1 𝜀

2 𝑖−1 𝑁

= 𝐶 𝑕𝑖 + 2𝛼 −1 1 − 𝜀 𝑁 −1

≤ 𝐶 𝐶𝑁 −1 + 2𝛼 −1 1 − 𝜀 𝑁 −1 ≤ 𝐶𝑁 −1 . 1

1

𝑏) 𝜎 = 2 , 𝜎 = 2 < −𝛼 −1 𝜀𝑙𝑛𝜀 : −𝛼

𝑕𝑖 = 𝑥𝑖 − 𝑥𝑖−1 = 𝛼 −1 𝜀 ln 1 − 1 − 𝑒 2𝜀

−𝛼 2(𝑖 − 1) 2𝑖 − ln 1 − 1 − 𝑒 2𝜀 𝑁 𝑁

where, according to 𝑖, we use intermediate value theorem as follows: −𝛼

2(1 − 𝑒 2𝜀 )𝑁 −1

−1

𝑕𝑖 = 𝛼 𝜀

1 − 𝑖1 2(1 − 𝑒

−𝛼 2𝜀

)𝑁 −1

≤ 2𝛼 −1 𝑁 −1 ≤ 𝐶𝑁 −1 .

(34)

Hence 𝑅𝑖 ≤ 𝐶 𝑕𝑖 + 𝛼

−1

𝑒

−𝛼 𝑥 𝑖 𝜀

−𝑒

−𝛼 𝑥 𝑖−1 𝜀

−𝛼

−1

= 𝐶 𝑕𝑖 + 𝛼 𝑒

−𝛼 𝜀

𝑒

𝛼 −1 𝜀 ln 1− 1−𝑒 2𝜀

2𝑖 𝑁

−𝛼

−𝑒

𝛼 −1 𝜀 ln 1− 1−𝑒 2𝜀

2 𝑖−1 𝑁

≤ 𝐶𝑁 −1 In the second case, 𝑥𝑖 ∈ 𝜎, 1 , we obtain 𝑅𝑖 ≤ 𝐶 𝑕𝑖 + 𝜀 −1

𝑥𝑖

𝑒

−𝛼𝑥 𝜀

𝑥 𝑖−1

𝑑𝑥 ≤ 𝐶

1 + 𝜀 −1 𝑕𝑖 ≤ 𝐶 𝑁 −1 + 𝜀 −1 𝑕−1 ≤ 𝐶 𝑁 −1 + 𝜀 −1 𝑁 −1 ≤ 𝐶𝑁 −1 . 𝑁

Consquently, we have 𝑅

∞,𝜔 𝑁

≤ 𝐶𝑁 −1 .

(35)

Now, we evaluate error function𝑟 as follows: 𝑥 − 𝑥𝑖−1 ≤ 𝑕𝑖 , 𝑏 ∗ = 𝑚𝑎𝑥 𝑏 𝑥

≤ 𝐶 , 𝑏 ′ 𝑥 ≤ 𝐶 ve 𝑢′ (𝑥) ≤ 𝐶0

under the above assumptions,

www.ijmret.org

Page 19

International Journal of Modern Research in Engineering and Technology (IJMRET) www.ijmret.org Volume 1 Issue 5 ǁ December 2016. 𝑁

𝑥𝑖

𝑟 ≤

𝑏 ′ 𝑥 𝑢 𝑥 + 𝑏 𝑥 𝑢′ 𝑥 𝑑𝑥

𝑥 − 𝑥𝑖−1 𝑖=1

𝑥 𝑖−1 𝑁

≤

𝑥𝑖

1 𝛼𝑥 𝐶 + 𝐶 1 + 𝑒− 𝜀 𝜀

𝑕𝑖 𝑥 𝑖−1

𝑖=1 𝑁

𝑥𝑖

≤𝐶

𝑕𝑖 𝑥 𝑖−1

𝑖=1

𝑁 2

1 𝛼𝑥 1 + 𝑒 − 𝜀 𝑑𝑥 𝜀

𝑥𝑖

≤𝐶

𝑕𝑖 𝑥 𝑖−1

𝑖=1 𝜎

= 𝐶 𝑕𝑖 0

𝑑𝑥

𝑁

1 𝛼𝑥 1 + 𝑒 − 𝜀 𝑑𝑥 + 𝜀

𝑥𝑖

𝑕𝑖 𝑁 2

𝑖= +1

1 𝛼𝑥 1 + 𝑒 − 𝜀 𝑑𝑥 + 𝑕𝑖 𝜀

1 𝜎

𝑥 𝑖−1

1 𝛼𝑥 1 + 𝑒 − 𝜀 𝑑𝑥 𝜀

1 𝛼𝑥 1 + 𝑒 − 𝜀 𝑑𝑥 ≤ 𝐶 𝑕𝑖 𝜀

(36)

In the first case, 𝑥𝑖 ∈ 0, 𝜎 : 1 𝑎) 𝜎 < , 𝜎 = −𝛼 −1 𝜀𝑙𝑛𝜀, 2

𝑥𝑖−1 = 𝛼 −1 𝜀 ln 1 − 1 − 𝜀

2(𝑖 − 1) 𝑁

and 𝑕𝑖 = 𝑥𝑖 − 𝑥𝑖−1 = 𝛼 −1 𝜀 ln 1 − 1 − 𝜀

2𝑖 2(𝑖 − 1) − ln 1 − 1 − 𝜀 𝑁 𝑁

where, according to 𝑖, we use intermediate value theorem as follows: 𝑕𝑖 = 𝛼 −1 𝜀

2(1 − 𝜀)𝑁 −1 ≤ 2𝛼 −1 1 − 𝜀 𝑁 −1 ≤ 𝐶𝑁 −1 1 − 𝑖1 2(1 − 𝜀)𝑁 −1

(37)

Thus, we obtain 𝑟 ≤ 𝐶 𝑕𝑖 + 𝑕𝑖 ≤ 𝐶𝑁 −1 . b) 𝜎 =

1 1 , 𝜎 = < −𝛼 −1 𝜀𝑙𝑛𝜀, 2 2

After a simple calculation we obtain, −𝛼

𝑕𝑖 = 𝛼 −1 𝜀

2(1 − 𝑒 2𝜀 )𝑁 −1 −𝛼

1 − 𝑖1 2(1 − 𝑒 2𝜀 )𝑁 −1

≤ 2𝛼 −1 𝑁 −1 ≤ 𝐶𝑁 −1 .

(38)

Hence 𝑟 ≤ 𝐶 𝑕𝑖 + 𝑕𝑖 ≤ 𝐶𝑁 −1 . In the second case, 𝑥𝑖 ∈ 𝜎, 1 : 1

1

a) 𝜎 < 2 , −𝛼 −1 𝜀𝑙𝑛𝜀 < 2, we use the following equation, 𝑕 = 𝑕𝑖 = 𝑕2 =

2 1−𝜎 2 ≤ , 𝑁 𝑁

𝑖=

𝑁 + 1, 𝑁 2

we obtain, 𝑟 ≤ 𝐶 𝑕𝑖 + 𝑕𝑖 ≤ 𝐶 𝑕1 + 𝑕2 ≤ 𝐶 𝑁 −1 + 2𝑁 −1 ≤ 𝐶𝑁 −1 and

www.ijmret.org

Page 20

International Journal of Modern Research in Engineering and Technology (IJMRET) www.ijmret.org Volume 1 Issue 5 Ç December 2016. đ?‘&#x; ≤ đ??śđ?‘ −1 .

(39)

Teorem 5.1. Under the assumptions of Lemma 5.1. and Lemma 5.2. Let đ?‘˘(đ?‘Ľ), be the solution of (1)-(2) and đ?‘Śđ?‘– , be the solution of (18)-(19). Then, the following uniform error estimate satisfies đ?‘Śâˆ’đ?‘˘

6.

≤ đ??śđ?‘ −1 .

∞,đ?œ” đ?‘

(40)

Algorithm and Numerical Results

We give some numerical results to solve the nonlinear problem (18)-(19) on Bakhvalov mesh using quasilinearization technique. 6.1. Algorithm We present the following quasilinearization technique for the difference scheme (18)-(19): đ?œ€ đ?‘› đ?œ€ đ?œ•đ?‘“ đ?‘Ľđ?‘– , đ?‘Śđ?‘– đ?œ•đ?‘“ đ?‘Ľđ?‘– , đ?‘Śđ?‘– đ?‘Śđ?‘–−1 − + đ?‘Śđ?‘– đ?‘› = đ?‘“ đ?‘Ľđ?‘– , đ?‘Śđ?‘– − đ?‘Śđ?‘– đ?‘›âˆ’1 , đ?‘– = 1, ‌ , đ?‘ đ?‘•đ?‘– đ?‘•đ?‘– đ?œ•đ?‘Ś đ?œ•đ?‘Ś đ?‘Śđ?‘–

đ?‘›

=

đ?‘“ đ?‘Ľđ?‘– , đ?‘Śđ?‘– − đ?‘Śđ?‘–

đ?œ€

−

đ?œ•đ?‘“ đ?‘Ľ đ?‘– ,đ?‘Ś đ?‘–

đ?‘›âˆ’1

đ?‘•đ?‘–

+

đ?œ•đ?‘Ś đ?œ•đ?‘“ đ?‘Ľ đ?‘– ,đ?‘Ś đ?‘–

đ?œ€

đ?‘›

− đ?‘• đ?‘Śđ?‘–−1 đ?‘–

,

đ?‘– = 1, . . , đ?‘

đ?œ•đ?‘Ś

đ?‘ (đ?‘›âˆ’1)

đ?‘Ś0 đ?‘› = đ?›˝đ?‘Śđ?‘

(đ?‘›âˆ’1)

+

đ?‘•đ?‘– đ?‘?đ?‘– đ?‘Śđ?‘–

+ đ?‘‘,

� = 1,2, ‌

đ?‘–=1

6.2. Numerical Results We give some examples to see the effectiveness of the presented method Example 1 đ?œ€đ?‘˘â€˛ + 2đ?‘˘ − đ?‘’ −đ?‘˘ + đ?‘Ľ 2 = 0, 0 < đ?‘Ľ < 1 1 1 đ?‘˘ 0 = đ?‘˘ 1 + 2 4

(41)

1

đ?‘’ −đ?‘ đ?‘˘ đ?‘ đ?‘‘đ?‘ + 1

(42)

0

this problem has not the exact solution. Thus, we use đ?œ” −mesh to calculate the errors and convergence rates, respectively. 2đ?‘ đ?‘’ đ?‘ = max đ?‘’đ?œ€đ?‘ , đ?‘’đ?œ€đ?‘ = max đ?‘Śđ?‘–đ?‘ − đ?‘Ś2đ?‘– , đ?œ€

đ?‘–

đ?‘ƒđ?œ€đ?‘ = log 2

đ?‘’đ?‘ đ?‘’ 2đ?‘

2đ?‘ where, đ?‘Śđ?‘–đ?‘ and đ?‘Ś2đ?‘– are the numerical solutions for đ?‘ and 2đ?‘ .

According to (6.1), we give the algorithm as follows: đ?œ€ (đ?‘›) đ?œ€ đ?‘› −1 đ?‘Śđ?‘–−1 − + 2 + đ?‘’ −đ?‘Śđ?‘– đ?‘•đ?‘– đ?‘•đ?‘– đ?œ€

đ?‘Śđ?‘–

đ?‘›

=

đ?‘•đ?‘–

đ?‘›

đ?‘Śđ?‘–−1 − (2đ?‘Śđ?‘–

đ?‘›âˆ’1

đ?‘Śđ?‘– đ?‘› = 2đ?‘Śđ?‘– đ?‘›âˆ’1 + đ?‘Ľđ?‘–2 − đ?‘Śđ?‘– đ?‘›âˆ’1 + đ?‘Ľđ?‘–2 − đ?‘Śđ?‘– đ?œ€ đ?‘•đ?‘–

www.ijmret.org

đ?‘›âˆ’1

2 + đ?‘’ −đ?‘Śđ?‘–

+ 2 + đ?‘’ −đ?‘Śđ?‘–

đ?‘› −1

đ?‘› −1

2 + đ?‘’ −đ?‘Śđ?‘–

− đ?‘’ −đ?‘Śđ?‘–

đ?‘› −1

đ?‘› −1

− đ?‘’ −đ?‘Śđ?‘–

đ?‘› −1

) , đ?‘– = 1, . . , đ?‘

Page 21

International Journal of Modern Research in Engineering and Technology (IJMRET) www.ijmret.org Volume 1 Issue 5 Ç December 2016. đ?‘Ś0

đ?‘›

1 = đ?‘Śđ?‘ đ?‘›âˆ’1 + 2

đ?‘

đ?‘•đ?‘– đ?‘?đ?‘– đ?‘Śđ?‘– đ?‘›âˆ’1 + 1,

� = 1,2, ‌

đ?‘–=1

the initial guess in the iteration procedure is đ?‘Śđ?‘– 0 = 0.5, (đ?‘›) (đ?‘›âˆ’1) The stopping criterion is taken as đ?‘Śđ?‘– − đ?‘Śđ?‘– ≤ 10−5 . The numerical results obtained from the problem of the difference scheme by comparison, the error and uniform rates of convergence were found and these are shown in Table 1. Consequently, numerical results show that the proposed scheme is working very well. Table 1. The computed maximum pointwise errors đ?‘’đ?‘ and đ?‘’2đ?‘ , the numerical rate of convergence đ?‘?đ?‘ on the Bakhvalow mesh đ?œ”đ?‘ for different values of đ?‘ and đ?œ€. 8 16 32 64 đ?œ€/đ?‘ 2−1

0.0364705764 p=0.76

0.0213939341 p=0.88

0.0115893393 p=1.05

0.0051560120

2−2

0.0457353908 p=0.79

0.0263063297 p=0.89

0.0141504831 p=0.94

0.0073451227

2−3

0.0547795035 p=0.79

0.0316355902 p=0.87

0.0172632555 p=0.93

0.0090159047

2−4

0.0592489265 p=0.74

0.0354285745 p=0.87

0.0192899598 p=0.93

0.0100776157

2−5

0.0617440905 p=0.71

0.0376291853 p=0.87

0.0204860025 p=0.93

0.0107447933

2−6

0.0632177961 p=0.70

0.0389122636 p=0.87

0.0212015157 p=0.92

0.0111341923

2−7

0.0640959677 p=0.69

0.0396529864 p=0.87

0.0216490391 p=0.93

0.0113563269

2−8

0.0646130925 p=0.68

0.0400746536

0.0219009225 p=1.00

0.010926500

P=0.87

Figure 1. Exact solution distribution for đ?‘ = 16 and đ?œ€ = 2−4 , 2−6 , 2−8

www.ijmret.org

Page 22

International Journal of Modern Research in Engineering and Technology (IJMRET) www.ijmret.org Volume 1 Issue 5 Ç December 2016.

Figure 2. Error distribution for đ?‘ = 16 and đ?œ€ = 2−2 , 2−4 , 2−6 , 2−8 From the graps it is show that the error is maximum near the boundary layer and it is almost zero in outer region in the Figure 2. When đ?œ€ −values are small, graphic curves leaned more towards the coordinate axes in Figure 1. Consequently, the purpose of this study was to give uniform finite difference method for numerical solution of nonlinear singularly perturbed problem with nonlocal boundary conditions. The numerical method was constructed on Bakhvalov mesh. The method was pointed out to be convergent, uniformly in the Îľparameter, of first order in the discrete maximum norm. The numerical example illustrated in practice the result of convergence proved theoretically. References [1]

N. Adzic. (2000) Spectral Approximation and Nonlocal Boundary Value Problems. Novi Sad J. Math., 30:1-10. [2] M. Benchohra and S. K. Ntouyas. (2000) Existence of Solutions of Nonlinear Differential Equations with Nonlocal Conditions. J. Math. Anal. Appl., 252:477-483. [3] Jankowski, T. (2004) Existence of Solutions of Differential Equations with Nonlinear Multi-point Boundary Conditions. Comput. Math. Appl., 47:1095-1103. [4] Cziegis, R. (1988) The Numerical Solution of Singularly Pertßrbed Nonlocal Problem. Lietuvos Matematikos Rinkinys, 28, 144-152. (In Russian) [5] Cziegis, R. (1991) The Difference Schemes for Problems with Nonlocal Conditions. Informatica (Lietuva), 2, 155-170. [6] Herceg, D. (1990) On the Numerical Solution of a Singularly Perturbed Nonlocal Problem. Univ. u Novom Sadu Zb. Rad. Prirod.-Mat. Fak. Ser. Mat., 20, 1-10. [7] Herceg, D. (1991) Solving a Nonlocal Singularly Perturbed Problem by Splines in Tension. Univ. u Novom Sadu Zb. Rad. Prirod.-Mat. Fak. Ser. Mat, 21, 119-132. [8] Amiraliyev, G.M. and Cakir, M. (2007) Numerical Solution of a Singularly Perturbed Three-Point Boundary Value Problem. International Journal of Applied Mathematics, 84, 1465-1481. [9] Amiraliyev, G. M. and ÇakĹr, M. (2002) Numerical Solution of the Singularly Perturbed Problem with Nonlocal Boundary Condition. Applied Mathematics and Mechanics (English Edition), 23, 755-764. [10] ÇakĹr, M. (2010) Uniform Second-Order Difference Method for a Singularly Perturbed Three-Point Boundary Value Problem. Hindawi Publising Corporation Advances in Difference Equations, Vol. 2010, 13 p. [11] Adzic, N. and Ovcin, Z. (2001) Nonlinear Spp with Nonlocal Boundary Conditions and Spectral Approximation. Novi Sad Journal of Mathematics, 31, 85-91. [12] Amiraliyev, G.M., Amiraliyeva, I.G. and Kudu, M. (2007) A Numerical Treatment for Singularly Perturbed Differential Equations with İntegral Boundary Condition. Applied Mathematics and Computations, 185, 574-582. [13] ÇakĹr, M. (2016) A Numerical Study on the Difference Solution of Singularly Perturbed Semilinear Problem with Integral Boundary Condition. Mathematical Modelling and Analysis Vol. 21, 644-658.

www.ijmret.org

Page 23

International Journal of Modern Research in Engineering and Technology (IJMRET) www.ijmret.org Volume 1 Issue 5 ǁ December 2016. [14] Çakır, M. and Arslan, D. (2016) A Numerical Method for Nonlinear Singularly Perturbed Multi-Point Boundary Value Problem. Journal of Applied Mathematics and Physics, 4, 1143-1156. [15] Geng, F.Z. (2012) A Numerical Algorithm for Nonlinear Multi-Point Boundary Value Problems. Journal of Computational and Applied Mathematics, 236, 1789-1794. [16] Bitsadze, A. V. and Samarskii, A. A., (1969) On Some Simpler Generalization of Linear Elliptic Boundary Value Problems. Doklady Akademii Nauk SSSR. 185: 739-740.

www.ijmret.org

Page 24

Turn static files into dynamic content formats.

Create a flipbook
Issuu converts static files into: digital portfolios, online yearbooks, online catalogs, digital photo albums and more. Sign up and create your flipbook.