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Exponential B-Spline Method for Second Order Singularly Perturbed 

Boundary Value Problem with Negative Shift 

 

1Biswajit Kaushik , 2Myana Rajkumar , *3Diddi Kumara Swamy 

1,2,3Department of Mathematics, Indian Institute of Information Technology, Sonepat, India 

*Corresponding author: ksdiddi@iiitsonepat.ac.in 

 

Article History: 

Received: 12-04-2024 

Revised: 26-05-2024 

Accepted: 14-06-2024 

 

 

 

Abstract:  

In this paper, we present a numerical scheme based on exponential B-spline method for the 

solution of singularly perturbed delay differential equation.The equation is transformed to 

singularly perturbed boundary value problem by applying Taylor's series expansion. A three 

term recurrence relation is obtained and invariant embedded algorithm is applied to get the 

approximate solution .The convergence of the proposed scheme is discussed. The efficiency of 

the scheme is illustrated by presenting different examples .The results are compared with 

available literature and comparatively the method yields better results. 

Keywords: Exponential B-spline, Negative shift, Invariant embedded algorithm, Tridiagonal 

system, Boundary layer. 

 

1. Introduction 

The numerical treatment of a singularly perturbed delay differential equations generated lots of 

interest in the recent years due to applicability these families of equations in the process of 

transforming a real life situation into a mathematical form for numerous disciplines of science and 

technology. An ordinary differential equation having a delay term and multiplied the highest order 

derivative by a small positive parameter , known as a singularly perturbed delay differential 

equation. Like in [8] Lange, C.G. and Miura, R.M. had studied initial exit time problem with the 

modelling of neuronal variability's activation. In [1] Derstine, M.W studied problems related to 

variational and bistable devices problem in control theory. Kadalbajoo et al. [3, 4] proposed some 

numerical approximations for a singularly perturbed differential-difference equation containing a 

delay term. Various Numerical approaches were proposed for differential equations with negative 

shifts, Mixed shifts and some schemes with fitted parameters by D.Kumara Swamy et al [7,13, 

14,15,19].An integration method based on numerical approach was proposed by Y.N Reddy et 

al.[12] for a singularly perturbed differential equation having a negative shift. McCartin introduced 

the concept of Exponential B spline[9].He also described convergent rates and extremal features of 

the exponential spline approximation and also developed cardinal bases and B-spline bases for the 

space of exponential splines. Reza Mohammadi[10] proposed a method which is based on 

exponential B-spline to approximate the soluton of partial differential equation of convection-

diffusion kind having boundary conditions of Dirichlet’s type.  Von Neumann method was used to 

prove the stability of the method.  A numerical scheme for a family of  reaction-diffusion equations 

was proposed by A. S. V. Ravi Kanth et al.[6] based on Exponential B-spline for space derivative. In 

recent years, studies like wavelets theory, medical imaging, and image processing have all greatly 

benefited from the application of exponential splines. sinuk kang[5],the author came up with a 



Communications on Applied Nonlinear Analysis 

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technique for the cardinal exponential splines’s space by using exponential B-spline as a Reisz basis.

 W. K. Zahra et al. [18] developed a method that attempted to solve a singularly perturbed 

boundary value problem with a small unknown perturbation parameter using exponential splines and 

shishkin mesh discretization .Chandra Sekhara Rao , Mukesh Kumar[11] proposed a method for a 

self adjoint singularly perturbed boundary value problem based on the Exponential B-spline 

collocation approach. 

 We have implemented Exponential B-spline approach to approximate the solution of a 

differential equation of second order with negative shift having a boundary with a layer structure. We 

have discussed the problem in section 2 of this paper. The numerical method have been developed in 

Section 3 and the convergence analysis was performed in Section 4. The effectiveness of the 

proposed scheme have been covered in Section 5. 

 

2. Objectives 

The main objective of this paper is to implement Exponential B-spline method on a singularly 

perturbed delay differential equation to approximate its solution. We discuss the convergence 

analysis of the proposed method. 

We consider the following linear differential equation of second order with negative shift to 

implement the proposed method, 

                                            휀𝑢′′(ȶ) + 𝑚(ȶ)𝑢′(ȶ − 𝛿) + 𝑛(ȶ)𝑢(ȶ) = 0 ,       0 ≤ ȶ ≤ 1,                 (2.1)    

with imposed boundary conditions, 

                                           𝓊(ȶ) = 𝜑  ,  −𝛿 ≤ ȶ ≤ 0   and    𝓊(1) = 𝜓                                               (2.2) 

where 휀 and 𝛿 are perturbation parameter and delay argument respectively such that, 

0 < 휀 ≪ 1 , 0 < 𝛿 < 1   and 𝛿 = 𝑜(휀). 

  Furthermore, 𝑚(ȶ) and 𝑛(ȶ) are functions such that they are 𝑐∞ in the open interval (0,1) and 𝜑 , 𝜓 

are constants. 

Let us assume 𝑚(ȶ) ≥ ռ > 0  throughout the interval [0, 1],where, ռ is a positive constant .The 

boundary layer will be in the neighborhood of ȶ = 0.Again assuming 𝑚(ȶ) ≤ ռ < 0 throughout the 

interval [0, 1], where, ռ is a negative constant.The boundary layer will be in the neighborhood of ȶ = 

1. 

Taylor's series expansion yields, 

                                                 𝓊′(ȶ − 𝛿) ≈ 𝓊′(ȶ) − 𝛿𝓊′′(ȶ)                                                         (2.3) 

Using equation (2.3) in equation (2.1), 

                                               −휀𝓊′′(ȶ) + ⨍(ȶ)𝓊′(ȶ) + 𝑔(ȶ)𝓊(ȶ) = 0                                          (2.4)    

Where, 

                        ⨍(ȶ) =
𝑚(ȶ)

𝜎𝑚(ȶ)−1
 , 𝑔(ȶ) =

𝑛(ȶ)

𝜎𝑚(ȶ)−1
 , 𝜎 =

𝛿
  

With reference to Elsgolt’s and Norkin [2] equation (2.4) thus obtained from (2.1) is valid since, 

0 < 𝛿 < 1 



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3. Methods 

Let us consider the equation (2.4), 

𝐿𝓊 ≡ −휀𝓊′′(ȶ) + ⨍(ȶ)𝓊′(ȶ) +  𝑔(ȶ)𝓊(ȶ) = 0  ,  ȶ ∈ [0,1]                                                        (3.1) 

Let   Ω: 0 = ȶ0 < ȶ1 < ⋯⋯ < ȶռ = 1 be partition on [0,1] with uniform step size ɦ =
1

ռ
 

Define the Exponential B-spline function  Ᏸ𝒿(ȶ) , defined as follows [9]ռ 

Ᏸ𝒿(ȶ) =

{
 
 
 
 

 
 
 
 ƙ5 [(ȶ𝒿−2 − ȶ) −

1

ƥ
sinh{ƥ(ȶ𝒿−2 − ȶ)}]    ,                       ȶ ∈ [ȶ𝒿−2, ȶ𝒿−1] ,

ƙ1 + ƙ2(ȶ𝒿 − ȶ) + ƙ3𝑒
ƥ(ȶ𝒿−ȶ) + ƙ4𝑒

−ƥ(ȶ𝒿−ȶ)  ,         ȶ ∈ [ȶ𝒿−1, ȶ𝒿]

ƙ1 + ƙ2(ȶ − ȶ𝒿) + ƙ3𝑒
ƥ(ȶ−ȶ𝒿) + ƙ4𝑒

−ƥ(ȶ−ȶ𝒿)  ,          ȶ ∈ [ȶ𝒿, ȶ𝒿+1],

ƙ5 [(ȶ − ȶ𝒿+2) −
1

ƥ
sinh{ƥ(ȶ − ȶ𝒿+2)}]    ,                      ȶ ∈ [ȶ𝒿−2, ȶ𝒿−1],

   0                                                                                otherwise
 

 

 Where , 

                                ƙ1 =
ƥɦ𝚌

ƥɦ𝚌−𝚜
 , 

                                ƙ2 =
ƥ

2
[
𝚌(𝚌−1)+𝚜2

(ƥɦ𝚌−𝚜)(1−𝚌)
], 

                                ƙ3 =
1

4
[
𝑒−ƥɦ(1−𝚌)+𝚜(𝑒−ƥɦ−1)

(ƥɦ𝚌−𝚜)(1−𝚌)
], 

                                ƙ4 =
1

4
[
𝑒ƥɦ(𝚌−1)+𝚜(𝑒ƥɦ−1)

(ƥɦ𝚌−𝚜)(1−𝚌)
], 

                                 ƙ5 =
ƥ

2(ƥɦ𝚌−𝚜)
 

where 𝚜 = sinh(ƥɦ),  𝚌 = cosh(ƥɦ) and ƥ is a non-negative parameter. 

 Let 𝒰(ȶ) approximate the exact solution 𝓊(ȶ) of the equation (3.1),then we have, 

 

                          𝒰(ȶ) =  ∑    ɤ𝒿
ռ+1
𝒿=−1 Ᏸ𝒿(ȶ)             [Mac-Cartin 1991]                                   (3.2) 

 Now by using the conditions (3.2) the values of the unknowns ɤ𝒿 can be found. 

The values of the 𝒰(ȶ) and its 1st order and 2nd order derivatives can be determined at the mesh 

points ȶ𝒿 as follows: 

 𝒰(ȶ𝒿) = ɤ𝒿−1Ᏸ𝒿−1(ȶ𝒿) + ɤ𝒿Ᏸ𝒿(ȶ𝒿) + ɤ𝒿+1Ᏸ𝒿+1(ȶ𝒿)  

            𝒰(ȶ𝒿) =
 𝚜−ƥɦ

2(ƥɦ𝚌−𝚜)
ɤ𝒿−1 + ɤ𝒿 +

 𝚜−ƥɦ

2(ƥɦ𝚌−𝚜)
ɤ𝒿+1                 (3.3) 

           𝒰′(ȶ𝒿) =
ƥ(1−𝚌)

2(ƥɦ𝚌−𝚜)
ɤ𝒿−1 −

ƥ(1−𝚌)

2(ƥɦ𝚌−𝚜)
ɤ𝒿+1                             (3.4)           



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             𝒰′′(ȶ𝒿) =
ƥ2 𝚜

2(ƥɦ𝚌−𝚜)
[ɤ𝒿−1 − 2ɤ𝒿 + ɤ𝒿+1] ,       0 ≤ 𝒿 ≤ ռ                                                  (3.5) 

Now, the approximation of the equation (3.1) at the mesh points can be described as 

 −휀 𝒰′′(ȶ𝒿) + ⨍(ȶ𝒿) 𝒰
′(ȶ𝒿) + 𝑔(ȶ𝒿)𝒰(ȶ𝒿) = 0 ,  ȶ𝒿 ∈ [0,1]  

             −휀 𝒰′′(ȶ𝒿) + ⨍𝒿 𝒰
′(ȶ𝒿) + 𝑔𝒿𝒰(ȶ𝒿) = 0  ,       where ⨍(ȶ𝒿) = ⨍𝒿, 𝑔(ȶ𝒿) = 𝑔𝒿              (3.6)     

Plugging (3.3), (3.4) and (3.5) into (3.6) we get 

 
− ƥ2 𝚜

2(ƥɦ𝚌−𝚜)
[ɤ𝒿−1 − 2ɤ𝒿 + ɤ𝒿+1] + ⨍𝒿 [

ƥ(1−𝚌)

2(ƥɦ𝚌−𝚜)
ɤ𝒿−1 −

ƥ(1−𝚌)

2(ƥɦ𝚌−𝚜)
ɤ𝒿+1] + 𝑔𝒿 [

 𝚜−ƥɦ

2(ƥɦ𝚌−𝚜)
ɤ𝒿−1 + ɤ𝒿 +

 𝚜−ƥɦ

2(ƥɦ𝚌−𝚜)
ɤ𝒿+1] = 0 

 By performing mathematical maneuver we obtain a three term recurrence relation as follows: 

                   𝒦𝒿ɤ𝒿−1 + 𝔏𝒿ɤ𝒿 +Ϻ𝒿ɤ𝒿+1 = 𝘙𝒿                                                                             (3.7)                     

 Where, 

                𝒦𝒿 = −휀ƥ
2 𝚜 + ⨍𝒿ƥ(1 − 𝚌) + 𝑔𝒿(𝚜 − ƥɦ)   

                            𝔏𝒿 = 2휀ƥ
2 𝚜 + 2𝑔𝒿(ƥɦ𝚌 − 𝚜) 

     Ϻ𝒿 = −휀ƥ2 𝚜 − ⨍𝒿ƥ(1 − 𝚌) + 𝑔𝒿(𝚜 − ƥɦ) 

       𝘙𝒿 = 0 

 System (3.7) consists of (ռ + 1) equations with (ռ + 3) unknowns, say   

 ɤ−1, ɤ0, ɤ1……… . . ɤռ+1 . Plugging the boundary conditions (2.2) in order to solve the system 

(3.7), we get two more additional equations, 

                                            {  
ɤ−1 =

2(ƥɦ𝚌−𝚜)

𝚜−ƥɦ
(𝜑 − ɤ0) − ɤ1

   ɤռ+1 =
2(ƥɦ𝚌−𝚜)

𝚜−ƥɦ
(𝜓 − ɤռ) − ɤռ−1

                                            (3.8)       

Now (3.8) can be used to eliminate ɤ−1 and    ɤռ+1 from (3.7) which yields a tridiagonal system in 

the unknowns   ɤ0, ɤ1……… . . ɤռ of the form, 

                                                 𝒜ɤ = 𝔇                                                                                         (3.9)    

Where, 

                           ɤ = (ɤ0, ɤ1……… . . ɤռ)
′  

                            𝔇 = (−2(ƥɦ𝚌 − 𝚜)𝒦0𝜑 ,0,0, ………… . , −2(ƥɦ𝚌 − 𝚜)Ϻռ𝜓)
′ 



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𝒜 =

(

 
 
 
 
 

𝔏0(𝚜 − 𝓅ɦ) − 2(𝓅ɦ𝚌 − 𝚜)𝒦0 (Ϻ0 − 𝒦0)(𝚜 − 𝓅ɦ)           0
 𝒦1  𝔏1            Ϻ1

                  
              

………………… . .………………… . .
………………… . .
…… . . ………… . .
……………………

                                                         (𝒦ռ −Ϻռ)(𝚜 − 𝓅ɦ)    𝔏ռ(𝚜 − 𝓅ɦ) − 2(𝓅ɦ𝚌 − 𝚜)Ϻռ)

 
 
 
 
 

 

The tridiagonal system (3.7) can be solved using invariant embedded algorithm.  

4. Results 

Convergence Analysis: The followings results are required to discuss the convergent analysis of the 

proposed scheme, 

Lemma 1. If   { Ᏸ−1,Ᏸ0, ……………… .Ᏸռ+1} is exponential spline basis, then   see.[11]     

 ∑ |Ᏸ𝒿(ȶ)| ≤
5

2

ռ+1
𝒿=−1     ,   0 ≤ ȶ ≤ 1                                                                   

Theorem 1 . Let exact solution 𝓊(ȶ) of the problem (2.1) can be interpolated using exponential B-

spline to a unique 𝒰(ȶ). And if 𝓊(ȶ) ∈ 𝑐4[0, ռ]  and  ⨍, 𝑔 ∈ 𝑐2[0, ռ], then ∃ a constant ϰ𝒿, 

independent of ɦ such that see.[11]  

                                      ‖𝐷𝒿(𝓊(ȶ) − 𝒰(ȶ))‖∞ ≤ ϰ𝒿ɦ
𝒿−1    ,   𝒿 = 0,1,2…. 

Theorem  2. Let exact solution 𝓊(ȶ) of the problem (3.1) can be approximated  using exponential B-

spline by 𝒰(ȶ). If 𝓊(ȶ) ∈ 𝑐4[0, ռ]  and  ⨍, 𝑔 ∈ 𝑐2[0, ռ], then ∃ a constant Ӄ, such that  

‖𝓊(ȶ) − 𝒰(ȶ)‖∞ ≤ Ӄɦ2  , 

  For sufficiently small  ɦ and  Ӄ is a positive constant. 

Proof . Let us consider the following for the problem (2.1). 

𝓊(ȶ)  be the exact solution,�̃�(ȶ) = ∑   ɤ̃𝒿
ռ+1
𝒿=−1 Ᏸ𝒿(ȶ) be an unique exponential B-spline interpolating 

the solution 𝓊(ȶ) and  𝒰(ȶ) = ∑   ɤ𝒿
ռ+1
𝒿=−1 Ᏸ𝒿(ȶ) be the approximate solution of the equation.                   

Now, from lemma 1 we have        

∑|Ᏸ𝒿(ȶ)| ≤
5

2

ռ+1

𝒿=−1

    ,   0 ≤ ȶ ≤ 1 

Again, using Theorem 1 

|𝐿𝓊(ȶ𝒿) − 𝐿𝒰(ȶ𝒿)| ≤ |−휀(𝓊′′(ȶ𝒿) − 𝒰(ȶ𝒿)) +  ⨍(ȶ)(𝓊
′(ȶ𝒿) − 𝒰(ȶ𝒿)) +  𝑔(ȶ)(𝓊(ȶ𝒿) − 𝒰(ȶ𝒿))| 

≤ (휀ϰ2 + ‖⨍‖∞ϰ1ɦ +
‖𝑔‖∞ϰ0ɦ

2)ɦ2= ϰɦ2 

      Where,     ϰ = 휀ϰ2 + ‖⨍‖∞ϰ1ɦ +
‖𝑔‖∞ϰ0ɦ

2 

Thus we obtain,                  

      |𝐿𝒰(ȶ𝒿) − 𝐿�̃�(ȶ𝒿)| = |0 − 𝐿�̃�(ȶ𝒿)| = |𝐿𝓊(ȶ𝒿) − 𝐿�̃�(ȶ𝒿)| ≤  ϰɦ2                              (4.1) 



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As 𝐿𝒰(ȶ𝒿) = 0  , 0 ≤ 𝒿 ≤ ռ with the boundary conditions given in (2.2) gives a system of linear 

equations, 

                             𝒜ɤ = 𝔇 

Now, suppose  𝐿�̃�(ȶ𝒿) = ϣ(ȶ𝒿), 0 ≤ 𝒿 ≤ ռ , 

 with boundary conditions �̃�(ȶ0) = 𝜑 and �̃�(ȶռ) =  𝜓 leads to a linear system 

                                     𝒜ɤ̅ = �̅� 

Where, 

        ɤ̅ = (ɤ̅0, ɤ̅1……… . . ɤ̅ռ)
′ 

         �̅� = (�̅�0(𝚜 − ƥɦ) − 2(ƥɦ𝚌 − 𝚜)𝒦0𝜑 , �̅�1, �̅�2, ………… . , �̅�ռ(𝚜 − ƥɦ) − 2(ƥɦ𝚌 − 𝚜)Ϻռ𝜓)
′ 

Where,   �̅�𝒿 = 2(ƥɦ𝚌 − 𝚜)ϣ(ȶ𝒿) ,  0 ≤ 𝒿 ≤ ռ 

Then it follows 

                          𝒜(ɤ − ɤ̅) = (𝔇 − �̅�)                                                                                            (4.2)                         

Where, 

(ɤ − ɤ̅) = [ ɤ0 − ɤ̅0, ɤ1 − ɤ̅1, ɤ2 − ɤ̅2, …………ɤռ − ɤ̅ռ]
′ 

(𝔇 − �̅�) = [−2(ƥɦ𝚌 − 𝚜)(𝚜 − ƥɦ)ϣ(ȶ0),−2(ƥɦ𝚌 − 𝚜)ϣ(ȶ1), …,−2(ƥɦ𝚌 − 𝚜)(𝚜 − ƥɦ)ϣ(ȶռ)]
′ 

Using (4.1) we have,        

     ‖𝔇 − �̅�‖∞ = max
 0≤𝒿≤ռ

|𝔇𝒿 − �̅�𝒿| ≤ ϰɦ2(ƥɦ)3                                                                   (4.3)                

 

As  0 < 휀 ≪ 1 and with sufficiently small ɦ it can be verified that the coefficient matrix 𝒜 is 

Irreducible and monotone [21, 22]. Hence, 𝒜−1 exists. 

Therefore, from (4.2) we must have 

           ‖ɤ − ɤ̅‖∞ ≤ ‖𝒜−1‖∞‖𝔇 − �̅�‖∞                                                                                         (4.4)                           

    Let 𝜌𝒿 ,  0 ≤ 𝒿 ≤ ռ be the row sum of the matrix 𝒜 such that 

                               𝜌0 = 𝔏0(𝚜 − ƥɦ) − 2(ƥɦ𝚌 − 𝚜)𝒦0 + (Ϻ0 − 𝒦0)(𝚜 − ƥɦ) 

                               𝜌𝒿 = 𝒦𝒿 + 𝔏𝒿 +Ϻ𝒿                       𝒿 = 1,2, ………… . , ռ − 1 

                               𝜌ռ = (𝒦ռ −Ϻռ)(𝚜 − ƥɦ) + 𝔏ռ(𝚜 − ƥɦ) − 2(ƥɦ𝚌 − 𝚜)Ϻռ 

Now from theory of matrices we have, 

                                    ‖𝒜−1‖∞ ≤
1

𝜌
≤

1

|𝜌|
≤

1

(ƥɦ)3
                                                                            (4.5)                                                       

                               Where,   ρ=min{𝜌0, 𝜌1, 𝜌3, ………𝜌ռ}   

Now, substituting the values of (4.3) and (4.5) in (4.4) gives 



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                       ‖ɤ − ɤ̅‖∞ ≤  ϰɦ2(ƥɦ)3
1

|𝜌|
≤ ϰɦ2(ƥɦ)3

1

(ƥɦ)3
= ϰɦ2                                                 (4.6)                                                        

Using Lemma 1 and equation (4.6) ,we obtained 

‖𝒰(ȶ) − �̃�(ȶ)‖
∞
≤ ∑ ( ɤ𝒿 − ɤ̅𝒿)|Ᏸ𝒿(ȶ)|

ռ+1
𝒿=−1   

                                                      

                            ≤ ∑ |Ᏸ𝒿(ȶ)|‖ɤ − ɤ̅‖∞ ≤
ռ+1
𝒿=−1

5ϰɦ2

2
                                                                         (4.7) 

                                       

   Applying the Theorem 1 we have 

                                    ‖𝓊(ȶ) − �̃�(ȶ)‖
∞
≤ ϰ0ɦ

4                                                                            (4.8) 

Now combining the results (4.7) and (4.8) , 

                                            ‖𝓊(ȶ) − 𝒰(ȶ)‖∞ ≤ Ӄɦ2 

 Where,  Ӄ = ϰ0ɦ
2 +

5ϰ

2
 

Hence, the Theorem is proved. 

Numerical Experiments: The efficiency of the proposed method is demonstrated by the four model 

problems. Our proposed solutions are compared with the exact solution those are available in the 

literature for various values of 휀 and  δ . We are using double mesh principle given by  

 𝐸ռ =  max
0≤𝑖≤ռ

  |𝓊𝑖
ռ −𝓊2𝑖

2ռ| to  calculate the absolute error wherein the exact solutions are not 

available.  

Example 5.1: Let us take a  differential equation containing a negative shift with left end layer 

휀𝓊′′(ȶ) + 𝓊′(ȶ − 𝛿) − 𝓊(ȶ) = 0 , 0 ≤ ȶ ≤ 1, with the boundary conditions 

𝓊(0) = 1 𝑎𝑛𝑑 𝓊(1) = 1 

We have the exact solution as 

                                               𝓊(ȶ) =
(1−e𝓌2)e𝓌1ȶ+(e𝓌1−1)e𝓌2ȶ)

(e𝓌1−e𝓌2)
 

Where,      

               𝓌1 =
− 1 − √1 + 4 (ε−δ)

2(ε−δ)
   and 𝓌1 =

− 1 + √ 1 + 4(ε−δ)

2(ε−δ)
 

The comparisons of the absolute error are presented in the Table 1 and table 2 with 

left end layer for various values of ε and δ  .And the impact of the parameter are shown in graph 

presented in the Figure 1 and Figure 2. 

Example  5.2: Let us take a variable coefficient  differential equation with negative shift 

휀𝓊′′(ȶ) + 𝑒−0.5ȶ𝓊′(ȶ − 𝛿) − 𝓊(ȶ) = 0 , with   𝓊(0) = 1  𝑎𝑛𝑑 𝓊(1) = 1 



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We are using double mesh principle to calculate the maximum absolute error and presented in the 

Table 3.  for  various values of ε and δ. The graph of the computed solution is presented in the 

Figure 3 . 

Example 5.3: Let us take a  differential equation containing a negative shift with right end layer 

휀𝓊′′(ȶ) − 𝓊′(ȶ − 𝛿) − 𝓊(ȶ) = 0 , with the boundary conditions 

𝓊(0) = 1 𝑎𝑛𝑑 𝓊(1) = −1 

We have the exact solution as 

                                  𝓊(ȶ) =
((1+e𝓌2)e𝓌1ȶ−(e𝓌1+1)e𝓌2ȶ)

(e𝓌2−e𝓌1)
 

                              Where 𝓌1 =
 1 − √1 + 4 (ε−δ)

2(ε+δ)
   and 𝓌1 =

 1 + √ 1 + 4(ε−δ)

2(ε+δ)
 

The comparison of the absolute errors are presented in the Table 4 and table 5 with 

left end layer for various values of ε and δ  .And the impact of the parameter are shown in graph 

presented in the Figure 4. and Figure 5. 

Example  5.4: Let us take a variable coefficient  differential equation with a delay term 

                         휀𝓊′′(ȶ) − 𝑒ȶ𝓊′(ȶ − 𝛿) − ȶ𝓊(ȶ) = 0 , with   𝓊(0) = 1  𝑎𝑛𝑑 𝓊(1) = 1      

The maximum absolute errors are  calculated by double mesh principle is presented in the Table 6.  

for various values of ε and δ. The graph of the computed solution is presented in the Figure 7 . 

Table 1:Maximum Absolute Error with 휀 = 0.1 for various values of δ and ռ  (Example 5.1) 

 

Table 2: Absolute Error with 휀 = 0.02, δ = 0.001and ɦ = 0.01    ( Example 5.1 ) 

ռ → 102 103 104 

δ ↓ Proposed 

method 

Previous 

result[13] 

Proposed 

method 

Previous 

result[13] 

Proposed 

method 

Previous 

result[13] 

0.01 2.5733e-05 1.3798e-04 2.5763e-07 1.3907e-05 2.7092e-09 1.3887e-06 

0.03 2.9690e-05 1.0765e-04 2.9705e-07 1.0849e-05 2.9838e-09 1.0600e-06 

0.06 6.1468e-05 6.1798e-05 6.2550e-07 6.2273e-06 6.2161e-09 6.3164e-07 

0.08 1.4035e-05 3.0995e-05 1.4099e-06 3.1233e-06 1.400e-08 3.3537e-07 

ȶ Solution by Proposed 

Method 

Exact 

Solution 

Solution by 

Method [15] 

Absolute Error by  

Proposed method 

Absolute Error 

by method[15] 

0.0 1.00000000 1.00000000 1.00000000 0.00000000 0.00000000 

0.02 0.59650758 0.59611217 0.40030832 3.95409e-04 1.95804e-01 

0.04 0.46326487 0.46292257 0.38470877 3.42308e-04 7.82138e-02 

0.06 0.42273389 0.42247446 0.39155769 2.59429e-04 3.09168e-02 

0.08 0.41407601 0.41386729 0.39941373 2.08724e-04 1.44536e-02 

0.1 0.41644444 0.41626092 0.40746127 1.83512e-04 8.79965e-03 

0.2 0.45613474 0.45597317 0.45020470 1.61571e-04 5.76847e-03 

0.3 0.50314705 0.50299127 0.49743206 1.55780e-04 5.55921e-03 



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Table 3: Maximum absolute Error with 휀 = 0.1 for various values of δ and ռ using double mesh 

principle (for Example 5.2) 

ռ → 102 103 104 

δ ↓ Proposed method Proposed method Proposed method 

0.001 0.0265796 0.0032995 3.3605207e-04 

0.003 0.0271238 0.0033683 3.4306667e-04 

0.006 0.0279825 0.0034768 3.5414669e-04 

0.008 0.0285853 0.0035531 3.6193372e-04 

 

Table 4: Maximum absolute Error with 휀 = 0.1 for various values of δ and ռ(for Example 5.3) 

ռ → 102 103 104 

δ ↓  Proposed 

Method 

Results[13] Proposed 

method 

Results[13] Proposed 

method 

Results[13] 

0.01 1.8402e-05 4.9650e-05 1.8410e-07 4.9729e-06 1.6235e-09 4.9586e-07 

0.03 1.4093e-05 5.8439e-05 1.4097e-07 5.8534e-06 1.1188e-09 5.8693e-07 

0.06 9.7089e-06 7.1489e-05 9.7114e-08 7.1607e-06 1.0594e-09 7.2219e-07 

0.08 7.5492e-06 8.0100e-05 7.5512e-08 8.0235e06 6.7766e-10 8.0780e-07 

 

Table 5: Absolute Error with   휀 = 0.001, δ = 0.003 and ɦ = 0.01 (for Example 5.3) 

0.4 0.55502160 0.55487431 0.54961366 1.47294e-04 5.26065e-03 

0.5 0.61224452 0.61210912 0.60726922 1.35403e-04 4.83990e-03 

0.6 0.67536714 0.67524765 0.67097295 1.19493e-04 4.27470e-03 

0.7 0.74499772 0.74489886 0.74135933 9.88619e-05 3.53953e-03 

0.8 0.82180724 0.82173453 0.81912938 7.27047e-05 2.60515e-03 

0.9 0.90653585 0.90649574 0.90505767 4.01012e-05 1.43807e-03 

1.0 1.00000000 1.00000000 1.00000000 0.00000000 0.00000000 

ȶ Solution by Proposed 

Method 

Exact 

Solution 

Solution by 

Method [15] 

Absolute Error by  

Proposed method 

Absolute Error 

by method[15] 

0.0 1.00000000 1.00000000 1.00000000 0.00000000 0.00000000 

0.1 0.90536618 0.90519656 0.90493550 1.69625e-04 4.30683e-04 

0.2 0.81968792 0.81938081 0.81890826 3.07117e-04 7.79665e-04 

0.3 0.74211773 0.74170069 0.74105916 4.17042e-04 1.05857e-03 

0.4 0.67188829 0.67138491 0.67061074 5.03387e-04 1.27755e-03 

0.5 0.60830494 0.60773531 0.60685947 5.69633e-04 1.44547e-03 

0.6 0.55073872 0.55011991 0.54916868 6.18814e-04 1.57004e-03 

0.7 0.49862021 0.49796665 0.49696223 6.53568e-04 1.65798e-03 

0.8 0.45143388 0.45075769 0.44971877 6.76186e-04 1.71511e-03 

0.9 0.40871297 0.40802431 0.40696648 6.88656e-04 1.74649e-03 

0.92 0.40066672 0.39997663 0.39891664 6.90087e-04 1.75008e-03 

0.94 0.39277848 0.39208729 0.39102493 6.91195e-04 1.75355e-03 

0.96 0.38498654 0.38429459 0.38317158 6.91948e-04 1.81496e-03 

0.98 0.36841582 0.36772893 0.36289935 6.86889e-04 5.51647e-03 

1.0 1.00000000 1.00000000 1.00000000 0.00000000 0.00000000 



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Table 6: Maximum absolute Error with 휀 = 0.1 for various values of δ and ռ using  double mesh 

principle (Example 5.4) 

 

 

 

 

 

 

 

Figure 1. Graphical representation with 휀 = 0.1 for various values of δ (example5.1) 

 

 

Figure 2.  Graphical representation with 휀 = 0.01 for various values of δ (example5.1) 

 

 

 

 

ռ → 102 103 104 

δ ↓ Proposed method Proposed method Proposed method 

0.000 0.02230547 0.00305640 3.11355660e-04 

0.003 0.02081696 0.00282670 2.87617565e-04 

0.007 0.01910969 0.00256855 2.61006577e-04 

0.015 0.01639981 0.00217005 2.20066455e-04 



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Figure 3. Graphical representation with 휀 = 0.1 for various values of δ (example5.2) 

 

Figure 4. Graphical representation with 휀 = 0.1 for different values of δ (example5.3) 

 

Figure 5. Graphical representation with 휀 = 0.01 for various values of δ (example5.3) 



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Figure  6. Graphical representation with 휀 = 0.1 for various values of δ (example5.4) 

 

5. Discussion 

In an effort to solve delay differential equations with boundary layers, we employed the exponential 

B-spline method in this study. With this approach, we first applied Taylor's series to transform a 

second order singularly perturbed differential equation with a delay term problem into a neutral type 

singularly perturbed differential equation. The suggested approach is applied to four model 

problems, and the results are contrasted to the precise solutions that were found. In cases where exact 

solutions were not found, the absolute error was calculated using the double mesh concept. We have 

finally drawn the graphs showing the answers for various values of 휀 and δ. 

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