01052539

Page 1

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

Finite Difference Method for Nonlocal Singularly Perturbed Problem Musa ÇakĹr, Derya Arslan Department of Mathematics, Faculty of Science, University of Yuzuncu Yil, Van, Turkey Ministry of National Education, Turkey

Abstract: In this study, nonlinear singularly perturbed problems with nonlocal condition are evaluated by finite difference method. The exact solution đ?‘˘(đ?‘Ľ) has boundary layers at đ?‘Ľ = 0 and đ?‘Ľ = 1. We present some properties of the exact solution of the multi-point boundary value problem (1)-(3). According to the perturbation parameter, by the method of integral identities with the use exponential basis functions and interpolating quadrature rules with the weight and remainder terms in integral form uniformly convergent finite difference scheme on Bakhvalov mesh is established. The error analysis for the difference scheme is performed. đ?œ€ − uniform convergence for approximate solution in the discrete maximum norm is provided, which is the firstorder (O(h)). This theoretical process is applied on the sample. By Thomas Algorithm, it has been shown to be consistent with the theoretical results of numerical results. The results were embodied in table and graphs. The relationship between the approximate solution with the exact solution are obtained by Maple 10 computer program. Keywords: Singular perturbation, finite difference method, Bakhvalov mesh, nonlocal boundary condition, uniform convergence.

1. Introduction In this paper we shall consider nonlocal singularly perturbed nonlinear problem −đ?œ€ 2 đ?‘˘â€˛â€˛ + đ?‘“ đ?‘Ľ, đ?‘˘ = 0, 0 ≤ đ?‘Ľ ≤ 1, đ?‘˘ 0 =0 đ?‘š

đ?‘˜0 đ?‘˘ 1 =

(1) (2) 1

đ?‘˜đ?‘– đ?‘˘(đ?‘ đ?‘– ) + đ?‘˜đ?‘š +1 đ?‘–=1

� � �� + �

đ?œ•đ?‘“ đ?‘Ľ, đ?‘˘ ≼ đ?›ź > 0, đ?‘ đ?‘– ∈ 0,1 , đ?œ•đ?‘˘ where, 0 < đ?œ€ ≪ 1 is small perturbation parameter. đ?‘“ đ?‘Ľ, đ?‘˘ ∈ đ??ś 1 0,1 đ?‘Ľđ?‘… ,

(3)

0

đ?‘– = 1, ‌ , đ?‘š, đ?‘˜0 ≼ 0

Singularly perturbed differential equations are symbolized by the existence of a small parameter đ?œ€ − multiplying the highest order derivatives [4]. Like these problems, thin transition layers where the solution varies rapidly, while away from layers it behaves regularly and varies slowly. Classical numerical methods are inappropriate for the singularly perturbed. So, it is important to develop suitable numerical methods to these problems, whose accuracy does not depend on the parameter value đ?œ€; that is, methods that are convergence đ?œ€ uniformly [18]. Such problems arise in many areas of applied mathematics: fluid mechanics, chemical-reactor theory, control theory, electrical networks, and other physical models. In recent years, Singularly perturbed differential equations were studied by many authors in various fields of applied mathematics and engineering. For examples, Cziegis [1] studied the numerical solution of singularly perturbed nonlocal problem. Cziegis [2] analyzed the difference schemes for problems with nonlocal conditions. Bakhvalov [3] investigated on optimization of methods for solving boundary-value problems in the presence of a boundary layer. Amiraliyev and Cakir [5] studied numerical solution of a singularly perturbed three -point boundary value problem. Amiraliyev and ÇakÄąr [6] researched numerical solution of the singularly perturbed problem with nonlocal boundary condition. Amiraliyev and Duru [7] studied a note on a parameterized singular perturbation problem. Boglaev [8] applied uniform numerical methods on arbitrary meshes for singularly perturbed problems with discontinuous data. Adzic and Ovcin [9] obtained nonlinear spp with nonlocal boundary conditions and spectral approximation. Amiraliyev, Amiraliyeva and Kudu [13] analyzed a numerical treatment for singularly perturbed differential equations with integral boundary condition. Herceg [10] estimated on the numerical solution of a singularly perturbed nonlocal problem. Herceg [11] analyzed solving a nonlocal singularly perturbed problem

www.ijmret.org

Page 25

International Journal of Modern Research in Engineering and Technology (IJMRET) www.ijmret.org Volume 1 Issue 5 Ç December 2016. by splines in tension. ÇakÄąr [14] researched uniform second –order difference method for a singularly perturbed three-point boundary value problem. Geng [12] gave a numerical algorithm for nonlinear multi-point boundary value problems. ÇakÄąr and Arslan [15] applied a numerical method for nonlinear singularly perturbed multipoint boundary value problem. Bitsadze and Samarskii [16] studied on some simpler generalization of linear elliptic boundary value problems. ÇakÄąr [17] obtain a numerical study on the difference solution of singularly perturbed semilinear problem with integral boundary condition.ÇakÄąr [5] applied the difference method on a Shishkin mesh to the singularly perturbed three-point boundary value problem. Some methods for singularly perturbed problems have been studied in different ways [9-12]. In this study we present uniformly convergent difference scheme on an equidistant mesh for the numerical solution of the problem (1)–(3). The difference scheme is constructed by the method integral identities with the use exponential basis functions and interpolating quadrature rules with the weight and remainder terms in integral form [18]. In Section 2, we obtain some analytical results of the multi-point bounday value problem (1)-(3). The difference scheme constructed on Bakhvalov mesh for numerical solution (1)–(3) is described in Section 3. We prove that the method is first–order convergent in the discrete maximum norm. In the following Section, a numerical example is considered and constructed which is in agreement with the theoretical results. According to results, the finite difference method on Bakhvalov mesh is more powerful method than other methods for nonlinear singularly perturbed multi-point boundary value problem.

2. The continuous problem Here we give some properties of the solution of (1)-(3) with Lemma 2.1. Because, they are needed in later sections for the analysis appropriate numerical solution. We use đ?‘” ∞ for the continuous maximum norm on the [0,1], where đ?‘” đ?‘Ľ , is any continuous function. Lemma 2.1. đ?œ•đ?‘“ Let đ?‘“ đ?‘Ľ, đ?‘˘ ∈ đ??ś 1 0,1 đ?‘Ľđ?‘… and đ?œ•đ?‘˘ đ?‘Ľ, đ?‘˘ is uniformly bounded in đ?‘˘ đ?‘Ľ . We assume that đ?‘š

1

đ?‘˜đ?‘– đ?‘¤0 (đ?‘ đ?‘– ) + đ?‘˜đ?‘š +1

đ?‘˘ đ?‘Ľ đ?‘‘đ?‘Ľ < đ?‘˜0

(4)

0

đ?‘–=1

where đ?‘¤0 (đ?‘Ľ) ≼ đ?‘¤(đ?‘Ľ) , −đ?œ€ 2 đ?‘¤ ′′ + đ?‘Ž đ?‘Ľ đ?‘¤ đ?‘Ľ = 0 đ?‘¤ 0 = 0,

(5)

� 1 =1

(6)

the solution of this problem. So, the solution of the Eq. (1) satisfies the inequalities �(�)

đ??ś[0,1]

≤ đ??ś0

(7)

and đ?‘˘â€˛ (đ?‘Ľ) ≤ đ??ś 1 +

− 1 − đ?›źđ?‘Ľ đ?‘’ đ?œ€ +đ?‘’ đ?œ€

đ?›ź 1−đ?‘Ľ đ?œ€

,0 < đ?‘Ľ < 1

(8)

where, đ??ś0 and đ??ś are constants independent of đ?œ€ and the đ?‘•. Proof. We use intermediate value theorem for đ?‘“(đ?‘Ľ, đ?‘˘) in the Eq.(1) and we obtain the following the equalities, đ?‘“ đ?‘Ľ, đ?‘˘ − đ?‘“(đ?‘Ľ, 0) đ?œ•đ?‘“ = đ?‘Ľ, đ?œ— , đ?‘˘âˆ’0 đ?œ•đ?‘˘

www.ijmret.org

đ?œ— = đ?›žđ?‘˘,

0<�<1

Page 26

International Journal of Modern Research in Engineering and Technology (IJMRET) www.ijmret.org Volume 1 Issue 5 Ç December 2016. đ?‘“ đ?‘Ľ, đ?‘˘ = đ?‘“ đ?‘Ľ, 0 + đ?‘˘(đ?‘Ľ)

đ?œ•đ?‘“ đ?‘Ľ, đ?œ— đ?œ•đ?‘˘

= −đ??š(đ?‘Ľ) + đ?‘Ž(đ?‘Ľ)đ?‘˘(đ?‘Ľ) where đ?‘Ž đ?‘Ľ , đ??š đ?‘Ľ are sufficiently smooth on 0,1 and 0,1 đ?‘Ľđ?‘…. Also, đ?œ•đ?‘“ đ?‘Ľ, đ?œ— = đ?‘Ž đ?‘Ľ ≼ đ?›ź > 0, đ?œ•đ?‘˘

đ?œ— − đ?‘–đ?‘›đ?‘Ąđ?‘’đ?‘&#x;đ?‘šđ?‘’đ?‘‘đ?‘–đ?‘Žđ?‘Ąđ?‘’ đ?‘Łđ?‘Žđ?‘™đ?‘˘đ?‘’.

So, we obtain the following linear equation, −đ?œ€ 2 đ?‘˘â€˛â€˛ + đ?‘Ž đ?‘Ľ đ?‘˘ đ?‘Ľ = đ??š đ?‘Ľ ,

0 < đ?‘Ľ < 1.

(9)

Now, we can write the solution of the Eq.(9) as follows đ?‘˘ đ?‘Ľ = đ?‘Ł đ?‘Ľ + đ?œ†đ?‘¤(đ?‘Ľ)

(10)

−đ?œ€ 2 đ?‘Ł ′′ + đ?‘Ž đ?‘Ľ đ?‘Ł đ?‘Ľ = đ??š(đ?‘Ľ)

(11)

đ?‘Ł 0 = 0,

(12)

đ?‘Ł 1 =0

where đ?‘Ł đ?‘Ľ is solution of the Eq. (11)- the Eq. (12). First, we give the estimate đ?‘Ł đ?‘Ľ , đ?‘Ł(đ?‘Ľ) ≤ đ?‘Ł 0 + đ?‘Ł 1 + đ?›ź −1 đ??š

∞

≤ đ??ś1 .

(13)

Second, we give the estimate đ?‘¤ đ?‘Ľ , đ?‘¤(đ?‘Ľ) ≤ đ?‘¤ 0 + đ?‘¤ 1 + đ?›ź −1 0

∞

≤1

(14)

Then, from the Eq.(13) - the Eq.(14) we have the following inequality đ?‘˘(đ?‘Ľ) ≤ đ?‘Ł đ?‘Ľ + đ?œ† đ?‘¤ đ?‘Ľ

≤ đ??ś1 + đ?œ†

đ?‘˘(đ?‘Ľ) ≤ đ??ś0 .

(15)

we now prove the estimate the Eq. (8). if �′′ � is pulled from the Eq. (9), we obtain �′′ � =

−1 đ??š đ?‘Ľ − đ?‘Ž đ?‘Ľ đ?‘˘(đ?‘Ľ) đ?œ€2

(16)

and from the Eq. (16)

www.ijmret.org

Page 27

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

1 đ??š đ?‘Ľ −đ?‘Ž đ?‘Ľ đ?‘˘ đ?‘Ľ đ?œ€2

≤

≤

đ??ś . đ?œ€2

(17)

Now, we take derivative of the Eq. (9) and if it called đ?‘˘â€˛ (đ?‘Ľ) = đ?‘Ł0 (đ?‘Ľ), the Eq. (9) takes the form with boundary condition −đ?œ€ 2 đ?‘Ł0 ′′ + đ?‘Ž đ?‘Ľ đ?‘Ł0 đ?‘Ľ = ∅(đ?‘Ľ)

(18)

�′ 0 = �0 0 = 0, �′ 1 = �0 1 = 1

(19)

In a similar manner, we proceed with the estimation of ∅ đ?‘Ľ , đ?‘˘â€˛ 0 and đ?‘˘â€˛ 1 respectively, from the Eq. (7), we obtain the following equation ∅ đ?‘Ľ = đ??š ′ đ?‘Ľ − đ?‘Žâ€˛ đ?‘Ľ đ?‘˘ đ?‘Ľ ≤ đ??ś1 .

(20)

We use the relation for đ?‘” ∈ đ??ś 2 as follows: đ?‘” ′ (đ?‘Ľ) =

đ?‘” đ?›˝ −đ?‘” đ?›ź − đ?›˝âˆ’đ?›ź

� �

đ?›˝âˆ’đ?‘Ą − đ?‘‡0 đ?‘Ľ − đ?‘Ą đ?‘” ′′ đ?‘Ą đ?‘‘đ?‘Ą, đ?›ź < đ?‘Ľ < đ?›˝ , đ?›˝âˆ’đ?›ź

�<�

(21)

where đ?‘‡0 đ?‘Ľ − đ?‘Ą =

1, 0,

đ?‘Ľâˆ’đ?‘Ą>0 đ?‘Ľ − đ?‘Ą < 0.

the Eq. (21) with the values đ?‘” đ?‘Ľ = đ?‘˘ đ?‘Ľ , đ?›ź = 0, đ?›˝ = đ?œ€, đ?‘Ľ = 0 and from the Eq. (7)- the Eq. (17) đ?‘˘â€˛ 0 ≤

đ?‘˘ đ?œ€ −đ?‘˘ 0 + đ?œ€

đ?œ€ 0

đ??ś đ?‘˘â€˛â€˛ đ?‘Ą đ?‘‘đ?‘Ą ≤ . đ?œ€

(22)

Similarly, the Eq. (21) with the values đ?‘” đ?‘Ľ = đ?‘˘ đ?‘Ľ , đ?›ź = 1 − đ?œ€, đ?›˝ = 1, đ?‘Ľ = 1 and from the Eq. (7)- the Eq. (17) đ?‘˘â€˛ 1 ≤

đ?‘˘ 1 −đ?‘˘ 1−đ?œ€ + đ?œ€

1 1−đ?œ€

đ??ś đ?‘˘â€˛â€˛ đ?‘Ą đ?‘‘đ?‘Ą ≤ . đ?œ€

(23)

We write the solution of the Eq. (18)- the Eq.(19) in the form, đ?‘Ł0 đ?‘Ľ = đ?‘Ł1 đ?‘Ľ + đ?‘Ł2 (đ?‘Ľ) where đ?‘Ł1 đ?‘Ľ , đ?‘Ł2 đ?‘Ľ are respectively the solution of the following problems,

www.ijmret.org

−đ?œ€ 2 đ?‘Ł1 ′′ (đ?‘Ľ) + đ?‘Ž đ?‘Ľ đ?‘Ł1 đ?‘Ľ = ∅(đ?‘Ľ)

(24)

�1 0 = 0, �′ 1 = �1 1 = 0

(25)

Page 28

International Journal of Modern Research in Engineering and Technology (IJMRET) www.ijmret.org Volume 1 Issue 5 Ç December 2016. −đ?œ€ 2 đ?‘Ł2 ′′ đ?‘Ľ + đ?‘Ž đ?‘Ľ đ?‘Ł2 đ?‘Ľ = 0,

(26)

đ?‘Ł2 0 = đ?‘Ł0 (0) , đ?‘˘â€˛ 1 = đ?‘Ł2 1 = đ?‘Ł0 (1) According to the maximum principle in the Eq.(24)- the Eq. (25),we can give the following Barrier function, đ?œ“1 đ?‘Ľ = ∓đ?‘Ł1 đ?‘Ľ + đ?›ź −1 ∅(đ?‘Ľ)

∞

This Barrier function provides the conditions of the maximum principle and đ?‘Ł1 đ?‘Ľ ≤ đ??ś. In a similar manner, according to the maximum principle in the Eq.(26), we can write đ?‘Ł2 đ?‘Ľ ≤ đ?›ł(đ?‘Ľ) where đ?›ł(đ?‘Ľ) is the solution of the following problem with constant coefficient, −đ?œ€ 2 đ?›ł ′′ (đ?‘Ľ) + đ?›źđ?›ł đ?‘Ľ = 0 đ?›ł 0 = đ?‘Ł0 (0) ≤

đ??ś , đ?œ€

(27)

� 1 = �0 (1) ≤

đ??ś đ?œ€

(28)

where � � ≼ � > 0 and the solution of � � as follows, �0 0 �

− đ?›ź 1−đ?‘Ľ đ?œ€

−đ?‘’

đ?›ź 1−đ?‘Ľ đ?œ€

� � =

�

+ đ?‘Ł0 (1) đ?‘’

−đ?‘’ đ?œ€ + đ?‘’

− đ?›źđ?‘Ľ đ?œ€

−đ?‘’

đ?›źđ?‘Ľ đ?œ€

(29)

− đ?›ź đ?œ€

after some arragement, we can obtain, đ??ś − đ?‘’ đ?œ€

� � ≤ Finally, from �′ � = �0 � , �1 �

đ?›źđ?‘Ľ đ?œ€

+đ?‘’

− đ?›ź 1−đ?‘Ľ đ?œ€

.

(30)

≤ đ??ś, đ?›ł đ?‘Ľ ≤ đ?‘Ł2 đ?‘Ľ , đ?‘Ł0 đ?‘Ľ = đ?‘Ł1 đ?‘Ľ + đ?‘Ł2 (đ?‘Ľ), we have the following

inequality, đ?‘˘â€˛ đ?‘Ľ ≤đ??ś+

= đ?‘Ł0 đ?‘Ľ đ??ś đ?‘’ đ?œ€

đ?›źđ?‘Ľ − đ?œ€

≤ �1 � + �2 � +�

− đ?›ź 1−đ?‘Ľ đ?œ€

≤đ??ś 1+

1 − đ?‘’ đ?œ€

đ?›źđ?‘Ľ đ?œ€

− đ?›ź 1−đ?‘Ľ đ?œ€

+ đ?‘’−

which leads to the Eq. (8).

3. Costruction of the Difference Scheme and Non-Unifom Mesh In what follows, we give by the following non-uniform mesh on 0,1 : đ?œ”đ?‘ = 0 = đ?‘Ľ0 < đ?‘Ľ1 < â‹Ż < đ?‘Ľđ?‘ −1 < đ?‘Ľđ?‘ = 1, đ?‘Ľđ?‘ đ?‘– = đ?‘ đ?‘– , đ?‘ đ?‘– =

đ?‘ đ?‘– . đ?‘ . 1

We present some properties of the mesh function đ?‘”(đ?‘Ľ) defined on đ?œ”đ?‘ , which is needed in this section for analysis of the numerical solution.

www.ijmret.org

Page 29

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

𝑔𝑥 ,𝑖 =

𝑔𝑖+1 − 𝑔𝑖 , 𝑕𝑖+1

𝑔𝑖 − 𝑔𝑖−1 , 𝑕𝑖

𝑔𝑥 ,𝑖 =

𝑔𝑖+1 − 𝑔𝑖 ћ𝑖

𝑔𝑥,𝑖 − 𝑔𝑥 ,𝑖 1 𝑔𝑖+1 − 𝑔𝑖 𝑔𝑖 − 𝑔𝑖−1 = − ћ𝑖 ћ𝑖 𝑕𝑖+1 𝑕𝑖

𝑔𝑥 𝑥 ,𝑖 =

𝑕𝑖 = 𝑥𝑖 − 𝑥𝑖−1 , ћ𝑖 = 𝑔

= 𝑔

∞

𝑕𝑖 + 𝑕𝑖+1 2 ∞,

𝜔𝑁

= max 𝑔𝑖 0≤𝑖≤𝑁

The difference scheme, we shall construct as following from the identity ћ𝑖 −1

𝑥 𝑖+1

𝑥 𝑖+1

−𝜀 2 𝑢′′ (𝑥)𝜑𝑖 (𝑥)𝑑𝑥 + ћ𝑖 −1

𝑥 𝑖−1

𝑓 𝑥, 𝑢 𝜑𝑖 𝑥 𝑑𝑥 = 0,

𝑖 = 1, 𝑁 − 1

(31)

𝑥 𝑖−1

where 𝜑𝑖 (𝑥) , 𝑖 = 1, 𝑁 − 1 are the basis functions and having the form

𝜑𝑖 𝑥 =

𝜑𝑖 (1 ) (x) and 𝜑𝑖

2

𝜑𝑖

1

𝜑𝑖

2

𝑥 − 𝑥𝑖−1 𝑕𝑖 𝑥𝑖+1 − 𝑥 𝑥 = 𝑕𝑖+1 0, 𝑥 =

𝑥𝑖−1 < 𝑥 < 𝑥𝑖 , 𝑥𝑖 < 𝑥 < 𝑥𝑖+1 , 𝑥 ∉ (𝑥𝑖−1 , 𝑥𝑖+1 )

𝑥 respectively, are the solutions of the following problems, −𝜀 2 𝜑′′ = 0

(32)

𝜑 𝑥𝑖−1 = 0, 𝜑 𝑥𝑖 = 1

(33)

−𝜀 2 𝜑′′ = 0

(34)

𝜑 𝑥𝑖 = 1, 𝜑 𝑥𝑖+1 = 0.

(35)

If we rearrange the Eq. (31) it gives −𝜀 2 ћ𝑖

𝑥 𝑖+1

−1

𝑢′′ (𝑥)𝜑𝑖 (𝑥)𝑑𝑥 + ћ𝑖 −1

𝑥 𝑖−1

𝑥 𝑖+1

𝑓 𝑥, 𝑢 𝜑𝑖 𝑥 𝑑𝑥 = 0,

𝑖 = 1, 𝑁 − 1

(36)

𝑥 𝑖−1

and after a sample calculation 𝜀 2 ћ𝑖

−1

𝑥 𝑖+1 𝑥 𝑖−1

𝑢′ 𝑥 𝜑𝑖′ 𝑥 𝑑𝑥 + 𝑓 𝑥𝑖 , 𝑢𝑖 + 𝑅𝑖 = 0,

𝑖 = 1, 𝑁 − 1

(37)

where 𝑅𝑖 = ћ𝑖 −1

𝑥 𝑖+1 𝑥 𝑖−1

www.ijmret.org

𝑥 𝑖+1

𝑑𝑥𝜑𝑖 𝑥 𝑥 𝑖−1

𝑑 𝑓 𝑥, 𝑢 𝐾0∗ 𝑥, 𝜉 𝑑𝜉 , 𝑑𝑥

𝑖 = 1, 𝑁 − 1

(38)

Page 30

International Journal of Modern Research in Engineering and Technology (IJMRET) www.ijmret.org Volume 1 Issue 5 Ç December 2016. and đ?‘Ľđ?‘–+1 − đ?‘Ľđ?‘–−1 đ?‘Ľđ?‘–+1 − đ?‘Ľđ?‘–−1 − đ?œ‰ + (đ?‘Ľđ?‘–+1 − đ?‘Ľđ?‘–−1 )−đ?&#x;? (đ?‘Ľđ?‘–+1 − đ?œ‰)0 −đ?‘Ľ . 2 2

đ??ž0∗ đ?‘Ľ, đ?œ‰ = đ?‘‡0 đ?‘Ľ − đ?œ‰ − đ?‘‡0

So, from the Eq. (37), we propose the following difference scheme for approximation the Eq. (1)-the Eq. (3): −đ?œ€ 2 đ?‘˘đ?‘Ľ đ?‘Ľ ,đ?‘– + đ?‘“ đ?‘Ľđ?‘– , đ?‘˘đ?‘– + đ?‘…đ?‘– = 0

đ?‘– = 1, đ?‘ − 1

(39)

It is now necessary to define an approximation for the second boundary condition of the Eq. (1). We accepted that đ?‘Ľđ?‘ đ?‘– is the mesh point nearest to đ?‘ đ?‘– . đ?‘š

1

đ?‘˜0 đ?‘˘đ?‘ =

đ?‘˜đ?‘– đ?‘˘đ?‘ đ?‘– + đ?‘˜đ?‘š +1 đ?‘–=1

� � �� + � 0

đ?‘š

đ?‘

=

đ?‘˜đ?‘– đ?‘˘đ?‘ đ?‘– + đ?‘˜đ?‘š +1 đ?‘–=1

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

(40)

đ?‘–=1

where remainder term đ?‘

đ?‘Ľđ?‘–

đ?‘&#x;đ?‘– =

đ?‘‘ đ?‘˘ đ?œ‰ đ?‘‘đ?œ‰. đ?‘‘đ?‘Ľ

đ?œ‰ − đ?‘Ľđ?‘–−1 đ?‘–=1

đ?‘Ľ đ?‘–−1

(41)

By neglecting đ?‘…đ?‘– and đ?‘&#x;đ?‘– in the Eq. (39) and the Eq. (40), we may propose the following difference scheme for approximating the Eq. (1)- the Eq. (3): −đ?œ€ 2 đ?‘Śđ?‘Ľ đ?‘Ľ ,đ?‘– + đ?‘“ đ?‘Ľđ?‘– , đ?‘Śđ?‘– = 0 ,

đ?‘– = 1, đ?‘

(42)

đ?‘Ś0 = 0

(43)

đ?‘š

đ?‘

đ?‘˜0 đ?‘Śđ?‘ =

đ?‘˜đ?‘– đ?‘Śđ?‘ đ?‘– + đ?‘˜đ?‘š +1 đ?‘–=1

đ?‘•đ?‘– đ?‘Śđ?‘– + đ?‘‘

(44)

đ?‘–=1

We will use the Bakhvalov mesh to be đ?œ€- uniform convergent of the difference scheme the Eq. (42)- the Eq. (44). Transition points of the Bakhvalov mes are defined as đ?œŽ = đ?‘šđ?‘–đ?‘›

1 ,− 4

�

−1

đ?œ€đ?‘™đ?‘›đ?œ€ .

and đ?‘Ľđ?‘– mesh points as follows: đ?œŽ< đ?œŽ=

1 4 1 4

, đ?‘Ľđ?‘– ∈ 0, đ?œŽ , đ?‘Ľđ?‘– = − , đ?‘Ľđ?‘– ∈ 0, đ?œŽ , đ?‘Ľđ?‘– = −

đ?›ź đ?›ź đ?‘

−1

−1

đ?‘Ľđ?‘– ∈ đ?œŽ, 1 − đ?œŽ , đ?‘Ľđ?‘– = đ?œŽ + đ?‘– − 4 đ?‘•

www.ijmret.org

4đ?‘–

đ?œ€ ln 1 − 1 − đ?œ€

đ?‘ −

, đ?‘–=

đ?‘ 4

+ 1, ‌ ,

4

đ?›ź 4đ?œ€

đ?œ€ ln 1 − 1 − đ?‘’

1

đ?‘

, � = 0,1, ‌ , .

3đ?‘ 4

−1

,đ?‘•

4đ?‘– đ?‘ 1

=

, � = 0,1, ‌ ,

đ?‘ 4

2 1−2đ?œŽ đ?‘

Page 31

International Journal of Modern Research in Engineering and Technology (IJMRET) www.ijmret.org Volume 1 Issue 5 Ç December 2016. đ?‘Ľđ?‘– ∈ 1 − đ?œŽ, 1 , đ?‘Ľđ?‘– = 1 − đ?œŽâˆ’

�

−1

đ?œ€ ln 1 − 1 − đ?œ€

3

1 − đ?œŽ = , đ?‘Ľđ?‘– ∈ 1 − đ?œŽ, 1 , đ?‘Ľđ?‘– = −

�

4

−1

3đ?‘ 4

4 đ?‘–−

3đ?‘

,đ?‘– =

đ?‘ −

đ?œ€ ln 1 − 1 − đ?‘’

đ?›ź 4đ?œ€

+ 1, ‌ , đ?‘

4

−1

4 đ?‘–−

3đ?‘ 4

đ?‘

3đ?‘ + 1, ‌ , đ?‘ 4

đ?‘–= where, N is even number, đ?œŽ ≪ 1.

4. Uniform Error Estimates Let đ?‘§ = đ?‘Ś − đ?‘˘, đ?‘Ľ ∈ đ?œ”đ?‘ , which is the error function of the difference scheme the Eq. (42)- the Eq. (44) and the solution of the discrete problem đ?‘™đ?‘§ ≥ −đ?œ€ 2 đ?‘§đ?‘Ľ đ?‘Ľ ,đ?‘– + đ?‘“ đ?‘Ľđ?‘– , đ?‘Śđ?‘– − đ?‘“ đ?‘Ľđ?‘– , đ?‘˘đ?‘–

= đ?‘…đ?‘– ,

đ?‘– = 1, đ?‘ − 1

(45)

�0 = 0

(46)

đ?‘š

đ?‘

đ?‘˜0 đ?‘§đ?‘ =

đ?‘˜đ?‘– đ?‘§đ?‘ đ?‘– + đ?‘˜đ?‘š+1 đ?‘–=1

đ?‘•đ?‘– đ?‘§đ?‘– = đ?‘&#x;đ?‘– , đ?‘– = 1, đ?‘

(47)

đ?‘–=1

where đ?‘…đ?‘– and đ?‘&#x;đ?‘– are defined in the Eq.(38) and the Eq.(41). Lemma 4. 2. The solution đ?‘§đ?‘– of problem the Eq. (45)- the Eq. (47 ) satisfies following đ?‘… đ?‘&#x;

∞,đ?œ” đ?‘

≤ đ??śđ?‘ −1

∞,đ?œ” đ?‘ đ?‘

(48)

≤ đ??śđ?‘ −1 .

(49)

Proof. We shall evaulate reminder term đ?‘…đ?‘– , 1

1) For đ?œŽ < 4 , đ?‘Ľđ?‘– ∈ 0, đ?œŽ For đ?‘Ľđ?‘– = −đ?›˝ −1 đ?œ€ ln 1 − 1 − đ?œ€

4đ?‘– đ?‘

đ?‘

, � = 0,1, ‌ , 4

we obtain đ?‘•đ?‘– = đ?‘Ľđ?‘– − đ?‘Ľđ?‘–−1 = −

�

−1

đ?œ€ ln 1 − 1 − đ?œ€

4đ?‘– + đ?‘

�

−1

đ?œ€ ln 1 − 1 − đ?œ€

4 đ?‘–−1 . đ?‘

According to đ?‘–, we use the intermediate value theorem as follows: đ?‘•đ?‘– =

�

−1

đ?œ€

4(1 − đ?œ€)đ?‘ −1 ≤4 1 − đ?‘–1 4(1 − đ?œ€)đ?‘ −1

�

−1

1 − đ?œ€ đ?‘ −1 ≤ đ??śđ?‘ −1

and we have ћ� =

www.ijmret.org

đ?‘•đ?‘– + đ?‘•đ?‘–+1 ≤ đ??śđ?‘ −1 . 2

Page 32

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

− 𝛼 𝑥 𝑖−1 𝜀

−𝑒

− 𝛼 𝑥 𝑖+1 𝜀

− 𝛼 −

𝛼

−1

𝜀 ln 1− 1−𝜀

≤𝑒

≤𝑒

4 𝑖−1 ln 1− 1−𝜀 𝑁

≤ 𝑒 ln

4 𝑖−1 𝑁

− 𝛼 −

− 1−𝜀

4 𝑖−1 𝑁

−𝑒

− 1−𝜀

−1

4 𝑖+1 𝑁

𝜀 ln 1− 1−𝜀

4 𝑖+1 𝑁

𝜀

4 𝑖+1 ln 1− 1−𝜀 𝑁

− 𝑒 ln

𝛼

−𝑒

𝜀

≤ 𝑒 ln

1− 1−𝜀

4 𝑖−1 𝑁

− 𝑒 ln

1− 1−𝜀

4 𝑖+1 𝑁

≤ 𝐶𝑁 −1

and 𝑒

− 𝛼 (1−𝑥 𝑖+1 ) 𝜀

− 𝑒

− 𝛼 (1−𝑥 𝑖−1 ) 𝜀

≤ 𝐶𝑁 −1

We use above inequalities and we can obtain 𝑅𝑖 ≤ 2𝐶𝑕𝑖 ≤ 𝐶𝑁 −1 . 1 , 𝑥𝑖 ∈ 0, 𝜎 4 We use the followig equation, 2) 𝜎 =

−𝛽

𝑕𝑖 = 𝑥𝑖 − 𝑥𝑖−1 = −𝛽 −1 𝜀 ln 1 − 1 − 𝑒 4𝜀

−𝛽 4𝑖 4𝑖 + 𝛽 −2 𝜀 ln 1 − 1 − 𝑒 4𝜀 𝑁 𝑁

we obtain, −𝛽

𝑕𝑖 = 𝛽 −1 𝜀

4(1 − 𝑒 4𝜀 )𝑁−1

≤ 4𝛽−1 𝑁 −1 ≤ 𝐶𝑁 −1

−𝛽

1 − 𝑖1 4(1 − 𝑒 4𝜀 )𝑁 −1

and ћ𝑖 =

𝑕𝑖 + 𝑕𝑖+1 ≤ 𝐶𝑁 −1 . 2

After doing some calculation, 𝑅𝑖 ≤ 𝐶𝑁 −1 . 𝑁

3) 𝑥𝑖 ∈ 𝜎, 1 − 𝜎 , 𝑥𝑖 = 𝜎 + 𝑖 − 4 𝑕

1

, 𝑖=

𝑁 4

+ 1, … ,

3𝑁 4

,𝑕

1

=

2 1−2𝜎 𝑁

, 𝑥𝑖 ∈ 𝜎, 1 − 𝜎 ,

1

Let 𝜎 = −𝛽−1 𝜀𝑙𝑛𝜀 < 4 , According to 𝑖, we use the intermediate value theorem and we obtain, 𝑕𝑖 ≤ 𝐶𝑁 −1 and ћ𝑖 = 𝑕(1) ≤ 𝐶𝑁 −1 . Consequently, 𝑅𝑖 ≤ 𝐶𝑁 −1 . 4) 𝑥𝑖 ∈ 𝜎, 1 − 𝜎 , 𝜎 =

1 4

𝑕𝑖 ≤ 𝐶𝑁 −1 and ћ𝑖 = 𝑕(1) ≤ 𝐶𝑁 −1 . We have

www.ijmret.org

Page 33

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

5) đ?‘Ľđ?‘– ∈ 1 − đ?œŽ, 1 ,

đ?‘Ľđ?‘– = 1 − đ?œŽ + đ?›˝ đ?œ€ ln 1 − 1 − đ?œ€

4 đ?‘–−

3đ?‘ 4

,

đ?‘

đ?‘–=

3đ?‘ + 1, ‌ , đ?‘ 4

where, according to đ?‘–, we use the intermediate value theorem as follows: 4 đ?‘–−

−1

đ?‘•đ?‘– = đ?‘Ľđ?‘– − đ?‘Ľđ?‘–−1 = 1 − đ?œŽ + đ?›˝ đ?œ€ ln 1 − 1 − đ?œ€

3đ?‘ 4

đ?‘ 4 đ?‘–−

−2

− 1 − đ?œŽ + đ?›˝ đ?œ€ ln 1 − 1 − đ?œ€

3đ?‘ 4

đ?‘

Thus, we obtain ћ� =

đ?‘•đ?‘– + đ?‘•đ?‘–+1 ≤ đ??śđ?‘ −1 2

and đ?‘…đ?‘– ≤ đ??śđ?‘ −1 . 3 6) 1 − đ?œŽ = , 4

−1

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

−đ?›˝ 4đ?œ€

4 đ?‘–−

3đ?‘ 4

đ?‘

,đ?‘– =

3đ?‘ + 1, ‌ , đ?‘ 4

we use the intermediate value theorem and we obtain, ћ� =

đ?‘Ľ đ?‘–+1

đ?‘…đ?‘– = đ?œ†đ?‘– Ń›đ?‘– −1

đ?‘•đ?‘– + đ?‘•đ?‘–+1 ≤ đ??śđ?‘ −1 . 2 đ?‘Ľ đ?‘–+1

đ??śđ?‘•đ?‘– đ?œ‘đ?‘– đ?‘Ľ đ?‘‘đ?‘Ľ + đ?œ†đ?‘– Ń›đ?‘– −1

đ?‘Ľ đ?‘–−1

đ??śđ?‘•đ?‘– đ?œ‘đ?‘– đ?‘Ľ đ?‘‘đ?‘Ľ ≤ đ??śđ?‘ −1

đ?‘Ľ đ?‘–−1

After all these calculations, we have đ?‘… đ?‘

đ?‘ đ?‘Ľđ?‘–

đ?‘&#x;đ?‘– ≤

đ?‘Ľ − đ?‘Ľđ?‘–−1 đ?‘–=1

≤ đ??śđ?‘ −1 .

∞,đ?œ” đ?‘

đ?‘Ľ đ?‘–−1 3đ?‘

� � � �� ≤ �� 4

+

đ?‘•

đ?‘Ľ đ?‘–−1

đ?‘ 3

đ?‘Ľđ?‘–

1+ đ?‘Ľ đ?‘–−1

− 1 − đ?›źđ?‘Ľ đ?‘’ đ?œ€ +đ?‘’ đ?œ€

đ?‘Ľđ?‘–

1+ đ?‘Ľ đ?‘–−1

đ?‘–=3đ?‘ 4 +1

1

đ?‘–=1

1+

đ?‘–=đ?‘ 4 +1

đ?‘•

đ?‘•

đ?‘Ľđ?‘–

2

+

4

− 1 − đ?›źđ?‘Ľ đ?‘’ đ?œ€ +đ?‘’ đ?œ€

đ?›ź 1−đ?‘Ľ đ?œ€

− 1 − đ?›źđ?‘Ľ đ?‘’ đ?œ€ +đ?‘’ đ?œ€

đ?›ź 1−đ?‘Ľ đ?œ€

đ?›ź 1−đ?‘Ľ đ?œ€

đ?‘‘đ?‘Ľ

đ?‘‘đ?‘Ľ

đ?‘‘đ?‘Ľ

After a sample calculation for đ?‘&#x;đ?‘– , we obtain, đ?‘&#x;đ?‘– ≤ đ??śđ?‘ −1 . Lemma 4. 3. Let đ?‘§đ?‘– be solution of the Eq. (45)- the Eq. (47). Then there is the following inequality: đ?‘§

∞,đ?œ” đ?‘

≤đ??ś đ?‘…

∞,đ?œ” đ?‘

+ đ?‘&#x;

(50)

Ä°spat. If we rearrange the Eq. (45), it gives

www.ijmret.org

Page 34

International Journal of Modern Research in Engineering and Technology (IJMRET) www.ijmret.org Volume 1 Issue 5 Ç December 2016. đ?‘™đ?‘§ ≥ −đ?œ€ 2 đ?‘§đ?‘Ľ đ?‘Ľ ,đ?‘– + đ?‘Žđ?‘– đ?‘§đ?‘– = đ?‘…đ?‘– ,

đ?‘– = 1, đ?‘ − 1

where đ?œ•đ??š đ?‘Ą , đ?‘˘ + đ?›žđ?‘§đ?‘– , đ?œ•đ?‘˘ đ?‘– đ?‘–

đ?‘Žđ?‘– =

0<�<1

we can use the maximum principle, and so it is easy to obtain, �

∞,đ?‘¤ đ?‘

≤ đ?‘§đ?‘ + đ?›ź −1

đ?‘…

∞,đ?‘¤ đ?‘

đ?‘š

+ đ?‘&#x;

đ?‘

≤

đ?‘•đ?‘– đ?‘§đ?‘– + đ?‘‘ − đ?‘&#x;đ?‘– + đ?›ź −1

đ?‘˜đ?‘– đ?‘§đ?‘ đ?‘– + đ?‘˜đ?‘š +1 đ?‘–=1

đ?‘…

∞,đ?‘¤ đ?‘

+ đ?‘&#x;

đ?‘–=1

≤ đ?›ź −1

đ?‘…

≤đ??ś đ?‘…

∞,đ?‘¤ đ?‘

∞,đ?‘¤ đ?‘

+ đ?‘&#x;

+ đ?‘&#x;

(51)

Theorem 4. 1. Let đ?‘˘ đ?‘Ľ be the solution of (1)-(3) and let đ?‘Ś be the solution of Eq. (45)- the Eq. (47). Then under the same assumptions of Lemma (4. 2) and Lemma (4.3) the following đ?œ€ −uniform error estimate đ?‘Śâˆ’đ?‘˘

≤ đ??śđ?‘ −1 đ?‘™đ?‘›đ?‘

∞,đ?‘¤ đ?‘

(52)

holds. 5) Algorithm and the Numerical Example We apply the scheme the Eq. (42)- the Eq. (44) to calculate the solution of approximation of the following problem: −đ?œ€ 2 đ?‘˘â€˛â€˛ + đ?‘˘2 đ?‘Ľ − đ?‘“ đ?‘Ľ = 0,

(53)

� 0 =0

(54)

� 1 = � 0.5 + �

(55)

which has the exact solution,

� � =

2đ?‘Ľ − 1

đ?‘’

1−đ?‘Ľ đ?œ€

đ?‘Ľ

− đ?‘’đ?œ€

1

+1.

đ?‘’đ?œ€ − 1 We solve the difference scheme the Eq. (42)- the Eq. (44) using the following iteration technique, đ?œ€2 2đ?œ€ 2 đ?œ•đ?‘“ (đ?‘›) đ?‘Śđ?‘–−1 − + đ?‘Ľ , đ?‘Ś đ?‘›âˆ’1 Ń›đ?‘– đ?‘•đ?‘– đ?‘•đ?‘– đ?‘•đ?‘–+1 đ?œ•đ?‘Ś đ?‘– đ?‘– = đ?‘“ đ?‘Ľđ?‘– + đ?‘“ đ?‘Ľđ?‘– , đ?‘Śđ?‘– đ?‘›âˆ’1

đ?‘Śđ?‘– đ?‘› +

đ?œ€2 đ?‘Ś đ?‘› 68 Ń›đ?‘– đ?‘•đ?‘–+1 đ?‘–+1

− đ?‘Śđ?‘– đ?‘›âˆ’1

đ?œ•đ?‘“ đ?‘Ľ , đ?‘Ś đ?‘›âˆ’1 , đ?‘– = 1, ‌ , đ?‘ − 1, đ?œ•đ?‘Ś đ?‘– đ?‘–

đ?‘š (đ?‘›) đ?‘Ś0

= 0,

(đ?‘›) đ?‘˜0 đ?‘Śđ?‘

đ?‘–=1

www.ijmret.org

đ?‘ (đ?‘›âˆ’1) đ?‘˜đ?‘– đ?‘Śđ?‘ đ?‘–

=

� = 1.2, ‌

(đ?‘›âˆ’1)

+ đ?‘˜đ?‘š +1

đ?‘•đ?‘– đ?‘Śđ?‘–

+đ?‘‘

(56)

đ?‘–=1

Page 35

International Journal of Modern Research in Engineering and Technology (IJMRET) www.ijmret.org Volume 1 Issue 5 Ç December 2016. đ?œ€2 2đ?œ€ 2 đ?œ•đ?‘“ (đ?‘›) đ?‘Śđ?‘–−1 − + đ?‘Ľ , đ?‘Ś đ?‘›âˆ’1 Ń›đ?‘– đ?‘•đ?‘– đ?‘•đ?‘– đ?‘•đ?‘–+1 đ?œ•đ?‘Ś đ?‘– đ?‘– = đ?‘“ đ?‘Ľđ?‘– + đ?‘“ đ?‘Ľđ?‘– , đ?‘Śđ?‘– đ?‘›âˆ’1 (đ?‘›)

đ?œ€2 đ?‘Śđ?‘› Ń›đ?‘– đ?‘•đ?‘–+1 đ?‘–+1

57

đ?œ•đ?‘“ đ?‘Ľ , đ?‘Ś đ?‘›âˆ’1 , đ?‘– = 1, ‌ , đ?‘ − 1, đ?œ•đ?‘Ś đ?‘– đ?‘–

− đ?‘Śđ?‘– đ?‘›âˆ’1 (đ?‘›)

đ?‘Śđ?‘ = đ?œ‡đ?‘›âˆ’1 ,

đ?‘Śđ?‘– đ?‘› +

đ?‘Śđ?‘ = đ?œ‡đ?‘›âˆ’1 − 1,

đ?œ‡0 = đ??ś0 ≼ 1

2

where 2

đ?œ€

đ?œ‡đ?‘› =

đ?‘•2

2

đ?œ€

đ?‘Śđ?‘ −1,đ?‘› +

đ?‘•2

2

2

đ?‘Śđ?‘ +1,đ?‘› − 1 + đ?‘Śđ?‘ +1,đ?‘›âˆ’1 2

đ?œ€

2

2

2

+ đ?‘Śđ?‘ ,đ?‘›âˆ’1

đ?‘•2

2

The system of the (57) is solved by the following procedure,

đ??´đ?‘– =

đ?œ€2 , Ń›đ?‘– đ?‘•đ?‘–

đ??ľđ?‘– =

đ?œ€2 , Ń›đ?‘– đ?‘•đ?‘–+1

đ??śđ?‘– =

đ??šđ?‘– = −đ?‘“ đ?‘Ľđ?‘– , đ?‘Śđ?‘– đ?‘›âˆ’1

+ đ?‘Śđ?‘– đ?‘›âˆ’1

�1 = 0,

đ?›˝đ?‘ +1 = đ?œ‡đ?‘›âˆ’1

2

(đ?‘›)

đ?‘Śđ?‘–

đ?œ•đ?‘“ đ?‘Ľ , đ?‘Ś đ?‘›âˆ’1 đ?œ•đ?‘Ś đ?‘– đ?‘–

đ?›˝1 = 0

đ?›źđ?‘ +1 = 0,

��+1 =

2đ?œ€ 2 đ?œ•đ?‘“ + đ?‘Ľ , đ?‘Ś đ?‘›âˆ’1 , đ?‘•đ?‘– đ?‘•đ?‘–+1 đ?œ•đ?‘Ś đ?‘– đ?‘–

2

đ??ľđ?‘– đ??šđ?‘– + đ??´đ?‘– đ?›˝đ?‘– , đ?›˝đ?‘–+1 = , đ??śđ?‘– − đ??´đ?‘– đ?›źđ?‘– đ??śđ?‘– − đ??´đ?‘– đ?›źđ?‘– (đ?‘›)

(0)

= ��+1 ��+1 + ��+1 ,

đ?‘Śđ?‘–

= 0.5 ,

đ?‘– = 1, ‌ , đ?‘ − 1

đ?‘– = đ?‘ − 1, ‌ ,2,1.

It is easy to verify that đ??´đ?‘– > 0, đ??ľđ?‘– > 0 , đ??śđ?‘– > đ??´đ?‘– + đ??ľđ?‘– ,

đ?‘– = 1, ‌ , đ?‘ .

For this reason, the described procedure above is stable. Also, the Eq. (42)- the Eq. (44) has only one solution. In the computations in this section, we will take đ?‘‘ = −1. (0)

The initial guess in the iteration procedure is đ?‘Śđ?‘–

= 0. 5. The stopping criterion is taken as

(đ?‘›+1)

max đ?‘Śđ?‘– đ?‘–

(đ?‘›)

− đ?‘Śđ?‘–

≤ 10−5 .

The error estimates are denoted by đ?‘’đ?œ€đ?‘ = đ?‘Ś − đ?‘˘

∞,đ?œ” đ?‘

and

www.ijmret.org

Page 36

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

The corresponding đ?œ€ −uniform convergence rates are computed using the formula đ?‘ƒđ?œ€đ?‘ = log 2 The by

numerical comparison,

results the

obtained

error

and

from

đ?‘’đ?‘ . đ?‘’ 2đ?‘

the

uniform

rates

problem of

of

the

convergence

difference were

scheme

obtained

and

these are presentedin Table 1. Table 1. The computed maximum pointwise errors đ?‘’ đ?‘ and đ?‘’ 2đ?‘ , the numerical rate of convergence đ?‘ƒđ?‘ on the Bakhvalov mesh đ?œ”đ?‘ for different values of N and đ?œ€. đ?œ€/đ?‘

8

16

32

64

128

256

512

1024

2−10

0.1844643708

0.1031176337

0.0525233947

0.0264140654

0.0131517954

0.0064588555

0.0031701248

0.0012507942

p=0.826

p=0.973

p=0.991

p=1.006

p=1.025

p=1.026

p=1.341

0.2186063415

0.1221878445

0.0639603910

0.0335687579

0.0176559863

0.0092080361

0.0047118041

p=0.839

p=0.933

p=0.930

p=0.926

p=0.939

p=0.966

p=0.999

0.2468061329

0.1374469602

0.0731044491

0.0393749598

0.0214435825

0.0116955952

0.0063238248

p=0.844

p=0.910

p=0.892

p=0.876

p=0.874

p=0.887

p=0.913

2−16

0.2701179896

0.1497045164

0.0803967387

0.0439999693

0.0244759852

0.0137180219

0.0063238249

p=0.851

p=0.896

p=0.869

p=0.846

p=0.835

p=1.117

p=0.953

2−18

0.2895653366

00.1596652402

0.0862725672

0.0477070838

0.0268973496

0.0153331458

0.0087696300

p=0.858

p=0.888

p=0.854

p=0.826

p=0.810

p=0.806

p=0.809

0.3059909186

0.1679963817

0.0911603753

0.0506828286

0.0288403106

0.0166239446

0.0096388095

p=0.865

p=0.881

p=0.846

p=0.813

p=0.794

p=0.786

p=0.785

2

−12

2−14

2

−20

0.0023568060

0.0033579638

0.0032650757

0.0050041978

0.0055920974

The results show that the convergence rate of the considered scheme is essentially in accord with theoretical analysis.

Figure 1. Comparison of Approximate solution and Exact solution for đ?œ€ = 2−14

www.ijmret.org

Page 37

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?œ€ = 2−14 , 2−16 , 2−18 using N=256 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. Approximate solution compared with exact solution in Figure 1.

References [1] [2] [3]

[4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

[14]

[15] [16]

Cziegis, R., 1988. The Numerical Solution of Singularly Perturbed Nonlocal Problem. Lietuvas Matem.Rink.(in Russian), 28: 144-152. Cziegis, R., 1991.The Difference Schemes for Problems with Nonlocal Conditions. Informatica (Lietuva), 2: 155-170. Bakhvalov, N. S., 1969. On Optimization of Methods for Solving Boundary-Value Problems in the presence of a Boundary Layer. The Use of Special Transformations in the Numerical Solution of Bounary-Layer Problems. Zh. Vychisl. Mat. Mat.Fiz., 9, 4, 841-859. A. H. Nayfeh, 1993. Introduction to Perturbation Techniques.Wiley& Sons, New York, USA. G.M. Amiraliyev, M. Cakir,2007. Numerical Solution of a Singularly Perturbed Three -Point Boundary Value Problem, Int. J. Appl. Math. Vol.84, No.10, 1465–1481. Amiraliyev, G. M., ÇakĹr, M., 2002. Numerical Solution of the Singularly Perturbed Problem with Nonlocal Boundary Condition. Appl. Math. Mech.(English Ed.)23: 755 764. Amiraliyev, G. M., Duru, H., 2005. A Note on a Parameterized Singular Perturbation Problem. J. Comput. Appl. Math.182: 233-242. Boglaev, I., 2004. Uniform Numerical Methods on Arbitrary Meshes for Singularly Perturbed Problems with Discontinuous Data. Appl. Math. Comput.154: 815-833. Adzic, N., Ovcin, Z. 2001. Nonlinear spp with nonlocal boundary conditions and spectral approximation. Novi. Sad J. Math. Vol.31, No. 1, 85-91. D. Herceg, 1990. On The Numerical Solution of a Singularly Perturbed Nonlocal Problem, Univ. uNovom Sadu Zb. Rad. Prirod.-Mat. Fak. Ser. Mat. 20, 1-10. D. Herceg, 1991. Solving a Nonlocal Singularly Perturbed Problem by Splines in Tension, Univ. u Novom Sadu Zb. Rad. Prirod.-Mat. Fak. Ser. Mat.21, 2, 119-132. F.Z. Geng, 2012. A Numerical Algorithm for Nonlinear Multi-Point Boundary Value Problems, J. Comput. Appl. Math. 236, 1789–1794. Amiraliyev, G.M., Amiraliyeva, I. G., Kudu, M. 2007. A Numerical Treatment for Singularly Perturbed Differential Equations with İntegral Boundary Condition.Applied Mathematics and Computations.185, 574-582. ÇakĹr, M., 2010. Uniform Second –Order Difference Method for a Singularly Perturbed Three-Point Boundary Value Problem. Hindawi Publising Corporation Advances in Difference Equations Volume 2010, 13 pages. ÇakĹr, 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. Bitsadze, A. V. and Samarskii, A. A., 1969. On Some Simpler Generalization of LinearElliptic Boundary Value Problems. Doklady Akademii Nauk SSSR. 185:739-740.

www.ijmret.org

Page 38

International Journal of Modern Research in Engineering and Technology (IJMRET) www.ijmret.org Volume 1 Issue 5 ǁ December 2016. [17 Çakır, 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. [18] Amiraliyev, G. M., Duru, H., 2002. Nümerik Analiz. Pegema, Ankara

www.ijmret.org

Page 39

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.