

































 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                                                  OTOBONG J. TOM1 *  

https://ijojournals.com/                                                                                             Volume 08 || Issue 04 || April, 2025 || 

“SEMIGROUP APPROACH FOR THE SOLUTION OF BOUNDARY LAYER EQUATION WITH SINC FUNCTION TERM" 

 

 

SEMIGROUP APPROACH FOR THE SOLUTION OF BOUNDARY LAYER 

EQUATION WITH SINC FUNCTION TERM 

OTOBONG J. TOM1 &OTOBONG G. UDOAKA2 
 

1FEDERAL UNIVERSITY OF TECHNOLOGY, IKOT ABASI. 
2AKWA IBOM STATE UNIVERSITY, IKOT AKPADEN 

 
 

Declarations 
1. Funding: Not applicable 
2. Informed Consent Statement: Not applicable 
3. Data Availability: Not applicable 
4. Conflict of Interest Statement: No conflict of interest. 

 

Abstract 

Boundary layer equation is crucial in fluid dynamics for modeling viscous flow near surfaces. It 

provides insight into flow behaviour, drag reduction, heat transfer and stability, making it 

essential in both theoretical and applied fluid mechanics. This paper examines the semigroup 

approach for solving the boundary layer equation incorporating a Sinc function term.The 

addition of the Sinc function term necessitates specialized functional analysis techniques and its 

effects are analyzed. We establish well-posedness in a suitable function space by analyzing the 

existence, uniqueness of the mild solution. We demonstrate the semigroup method's 

effectiveness in capturing boundary layer dynamics, contributing to the study of semigroup 

methods in fluid mechanics and partial differential equations. The influences of the Sinc function 

term are analyzed and illustrative examples are included to validate the approach and to also 

highlight its applicability. 

Key words: Boundary layer equation, Banach Spaces, Sinc function, Strongly Continuous 

Semigroup, Mild solution. 

 

 

 

IJO JOURNALS

Volume 08 | Issue 04 | April 2025 |           https://ijojournals.com/index.php/m/index 22



 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                                                  OTOBONG J. TOM1 *  

https://ijojournals.com/                                                                                             Volume 08 || Issue 04 || April, 2025 || 

“SEMIGROUP APPROACH FOR THE SOLUTION OF BOUNDARY LAYER EQUATION WITH SINC FUNCTION TERM" 

 

 

1. INTRODUCTION 

The boundary layer equation plays a crucial role in fluid dynamics, describing the behavior of 

viscous fluid flow near solid surfaces. Prandtl [1] initiated the boundary layer theory, 

significantly advancing the study of viscous flows. This equation has been extensively studied 

due to its in aerodynamics, heat transfer, and hydrodynamics stability. The traditional method 

for solving the boundary layer equation include similarity transformations, perturbation 

methods, and numerical simulations. However, in recent years, the semigroup approach has 

emerged as a powerful tool for the existence, uniqueness, and stability of solutions to partial 

differential equations (PDEs). This method leverages the properties of operator semigroups to 

study the time evolution of solutions in appropriate functional spaces. Semigroup theory 

provides a robust framework for studying time-dependent PDEs. Hille and Phillips [2] laid the 

foundation of semigroup analysis, later extended by Pazy [3] to cover evolutionary equations in 

Banach spaces. The semigroup approach has been successfully applied to fluid dynamics 

problems, including the Navier-Stokes equations [4]and parabolic PDEs [5]. Henry [6] applied 

semigroup method for nonlinear equations where he showed that the mild solution method 

could be used to establish the well-posedness of these equations in various function spaces. 

Recently, the work of Pruss [7] extended the semigroup approach to treat boundary boundary 

value problems with non-homogeneous conditions, which arise in practical fluid dynamics 

applications. Husssian and Kato [8] explored the use of semigroup approach to solve boundary 

layer equation numerically. Their research showed that semigroup-based numerical methods 

offer significant advantages in terms of stability compared to other methods like finite 

difference schemes, particularly for complex flow configurations and high Renolds numbers.  

The Sinc function and its interpolation techniques have been widely studied in numerical 

analysis [10, 11,12]. The function is well known for its applications in signal processing and 

numerical analysis, introducing oscillatory behavior into the equation, which affects the 

solution properties. Also, the function’s unique properties such as band-limited representation 

and rapid decay, makes it useful for solving differential equations. Lund and Bowers [10] have 

explored Sinc-base methods for approximating PDE solutions, but their interaction with 

semigroup methods remains an open area of research. This study aims to bridge this gap by 

developing a semigroup-theoretic approach for analyzing the well-posedness of boundary layer 

equation with Sinc function term. The integration of Sinc function term in the equation 

introduces additional complexities, requiring additional mathematical techniques for analysis 

and solution. Our approach focuses on establishing well-posedess of the equation with the 

inclusion of the Sinc function term. 

IJO JOURNALS

Volume 08 | Issue 04 | April 2025 |           https://ijojournals.com/index.php/m/index 23



 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                                                  OTOBONG J. TOM1 *  

https://ijojournals.com/                                                                                             Volume 08 || Issue 04 || April, 2025 || 

“SEMIGROUP APPROACH FOR THE SOLUTION OF BOUNDARY LAYER EQUATION WITH SINC FUNCTION TERM" 

 

 

2. PRELIMINARIES 

In this section, we introduce the fundamental mathematical concepts and notations required 

for our analysis of the boundary layer equation with a Sinc function term using the semigroup 

approach. 

2.1. Functional Space and Operators 

Let X  be a Banach space, and consider a differential operator A  on a dense subspace 

 D A X . We work within the frame work of semigroup theory, requiring the following 

definitions: 

2.2. Normed and Banach Spaces: A normed space  ,X   is a vector space if it is 

complete with respect to this norm. A normed space  ,X   is called Banach space 

if every Cauchy sequence in  ,X   converges. 

2.3. Hilbert Space: A special case of Banach space where the norm is induced by an inner 

product ,  . 

We primarily consider function spaces such as  pL Ω  and Sobolevspaces  kH Ω , which are 

crucial in studying PDEs [ 5, 13, 14, 21]. 

 

 

3. BOUNDARY LAYER EQUATION WITH SINC TERM 

The general form of the boundary layer equation with a Sinc function term can be written as: 

     , , 1
u

Au f x t S x t
t


  


 

were : 

  ,u x t  represents the velocity field, 

 A  is a differential operator capturing the viscous effects, 

  ,f x t  is an external forcing term, 

IJO JOURNALS

Volume 08 | Issue 04 | April 2025 |           https://ijojournals.com/index.php/m/index 24



 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                                                  OTOBONG J. TOM1 *  

https://ijojournals.com/                                                                                             Volume 08 || Issue 04 || April, 2025 || 

“SEMIGROUP APPROACH FOR THE SOLUTION OF BOUNDARY LAYER EQUATION WITH SINC FUNCTION TERM" 

 

 

  ,S x t is a Sinc function-modulated term affecting the solution structure. 

The boundary conditions and initial conditions depend on the physical context. 

4. SEMIGROUP THEORY 

Let X be a Banach space, and let  :A D A X X   be a densely defined, closed linear 

operator. A strongly continuous semigroup (or 0C Semigroup, or a semigroup of class 0C ) 

 
0t

T t


 on the Banach space X  is a family of bounded linear operators such that  

  

   
     

 
 

0

0 the identity operator

 for all , 0 2

lim  for all 
t

T I

T s t T s T t t s

T t x x x X


 


  
  


 

The generator of the semigroup, denoted by A , is defined as 

  
 

0
lim ,
t

T t x x
Ax

t


  

whenever the limit exists. The well-posedness of the equation depends on whether A  

generates a semigroup on X [3, 18, 19, 22, 24]. 

 

4.1. Spectral Properties of the Operators:  

The spectrum of A  denoted by  A , plays a crucial role in determining the stability of the 

solutions. The resolvent operator    
1

,R A I A 


   helps analyze whether A  an 

exponentially stable semigroup. 

4.2. Sinc Function and its Properties 

The Sinc function is define as 

   
 

   
sin

Sinc , 0, Sinc 0 1 3
x

x x
x




    

IJO JOURNALS

Volume 08 | Issue 04 | April 2025 |           https://ijojournals.com/index.php/m/index 25



 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                                                  OTOBONG J. TOM1 *  

https://ijojournals.com/                                                                                             Volume 08 || Issue 04 || April, 2025 || 

“SEMIGROUP APPROACH FOR THE SOLUTION OF BOUNDARY LAYER EQUATION WITH SINC FUNCTION TERM" 

 

 

It has useful properties such as rapid decay and interpolation capabilities, making it a valuable 

tool in numerical method and spectral analysis [12]. 

 

4.3. Mild Solution through Semigroup Approach: 

A mild solution to a differential equation is a solution that is defined through an integral 

equation involving a semigroup. For an evolution equation of the form 

   ,
du

Au f t
dt

   

the mild solution is obtain using Duhamel’s formula and is given by  

           0 0
4

t

u t T t u T t s f s ds    

where  T t  is the semigroup generated by the operator A  and 0u is the initial condition.The 

term mild is used because the solution is often less regular than classical solutions but still 

provides valuable information. 

Also, if  T t  is strongly continuous semigroup generated by A , then the mild solution can be 

expressed as 

              0 0
5

t

u t T t u T t s f s S s ds     

Equation  5 provides insight into the existence, uniqueness, and stability of the solution, See 

[23, 24]. 

4.4. Mathematical Formulation of the Problem 

We begin by representing the boundary layer equation as follows 

       , , , , 0 6
u

Au f x t S x t x t
t


    


Ω  

where  ,u x t  represent the unknown function (the state of the system) at time t  and position 

x , A  is a differential operator (often related to the Laplacian or similar operator that models 

IJO JOURNALS

Volume 08 | Issue 04 | April 2025 |           https://ijojournals.com/index.php/m/index 26



 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                                                  OTOBONG J. TOM1 *  

https://ijojournals.com/                                                                                             Volume 08 || Issue 04 || April, 2025 || 

“SEMIGROUP APPROACH FOR THE SOLUTION OF BOUNDARY LAYER EQUATION WITH SINC FUNCTION TERM" 

 

 

diffusion or advection in fluid dynamics),  ,f x t and  ,S x t  are external forcing terms. The 

initial condition for this equation is: 

   0,0 , ,u x u x x Ω where  0u x  represents the state of the system at time 0t  . 

5. Semigroup Representation and Solution Method 

The solution is sought in the layout of semigroup theory, which provides a powerful method to 

deal with evolution equations. We assume that A  generates a strongly continuous semigroup 

 
0t

T t


 on a Banachspace X . The semigroup approach allows us to express the solution in 

the form of equation  5 . Here, the term   0T t u  represents the evolution of initial condition, 

and the integral term capture the influence of the forcing    , ,f x t S x t over time, See [18, 

24]. 

5.1. Well-Posedness of the Problem 

To ensure the problem is well-posed, we need to establish that equation  5  exists, is unique, 

and depends continuously on the initial conditions. This is done through the following steps: 

 Existence: By the properties of semigroups, the integral equation for  u t  is well-

defined and yields a solution. The regularity of  ,f x t  and  ,S x t  ensure that the 

integral is finite and the solution ie well-behaved. 

 Uniqueness: If two solutions  1u t  and  2u t  exist, we show that    1 2u t u t  by 

employing the Banach fixed-point theorem [3, 25], this relying on the fact that the 

operator  T t  is strongly continuous and the forcing terms  ,f x t and  ,S x t  are 

assume be identical for both solutions. 

 Continuous Dependence: Since the solution depends continuously on the initial 

condition  0u x , small changes in the initial condition lead to small changes in the 

solution, which is a key property in proving stability. 

 

5.2. Influence of the SincFunctionTerm 

A distinctive feature of this research is the inclusion of Sinc function in the forcing term. The 

Sinc function introduce oscillations into the system, which can affect both the regularity and 

IJO JOURNALS

Volume 08 | Issue 04 | April 2025 |           https://ijojournals.com/index.php/m/index 27



 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                                                  OTOBONG J. TOM1 *  

https://ijojournals.com/                                                                                             Volume 08 || Issue 04 || April, 2025 || 

“SEMIGROUP APPROACH FOR THE SOLUTION OF BOUNDARY LAYER EQUATION WITH SINC FUNCTION TERM" 

 

 

stability of the solution. In the context of boundary layers, the Sinc function term might 

represent oscillatory external forces or disturbances that interact with the boundary layer 

dynamics. 

 Oscillatory Behavior: The Sinc function’s oscillations may induce transient effects in the 

system, but under suitable condition (e.g., decaying forcing terms), these oscillations do 

not lead to unbounded growth of the solution. 

 Damping Effects: The decay of the Sinc function ensures that its influence fades over 

time, which implies that the system will eventually settle to a steady state, as shown by 

the exponential decay established earlier. 

 

5.3. Regularity of Solutions 

 This is another crucial aspect of the research. Regularity refers to the smoothness of the 

solution. If  u t  belongs to a function space with sufficient differentiability (e.g., 
1 2, ,C L etc.), 

it is said to be regular. The higher the regularity the smoother the solution, and this property 

often plays a crucial role in the stability and long-term behavior of solutions. The solution  u t  

is shown to belong to the class of 
1C under suitable regularity assumptions on the initial data 

0u  and the forcing term    , ,f x t S x t . Higher regularity ensures that the solution 

behaves smoothly over times and that its derivatives exist and are continuous. 

By applying standard semigroup theory, we can also show that higher derivatives of  u t  exist 

and are bounded, which is important for numerical simulations and further theoretical analysis, 

See [18, 19, 22]. 

6. RESULTS 

Well-Posedness of Boundary Layer Equation: We consider the initial-boundary value problem:  

 

 
   

   
 

0

, , , , 0
7

,0 ,

u
Au f x t S x t x t

t

u x u x x


    



 

Ω

Ω

 

where A is as assumed in section 4. 

IJO JOURNALS

Volume 08 | Issue 04 | April 2025 |           https://ijojournals.com/index.php/m/index 28



 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                                                  OTOBONG J. TOM1 *  

https://ijojournals.com/                                                                                             Volume 08 || Issue 04 || April, 2025 || 

“SEMIGROUP APPROACH FOR THE SOLUTION OF BOUNDARY LAYER EQUATION WITH SINC FUNCTION TERM" 

 

 

Theorem 5.1 (Existence and Uniqueness of Mild Solution): If A  is as assumed in section 5, and 

if  ,f x t  and  ,S x t  satisfy suitable regularity conditions, then equation  7  has a unique 

mild solution given by equation  5 . 

Proof: 

We apply Duhamel’s formula [3, 15] to construct a mild solution of equation  5 . Since the 

assumption of section 5 holds, it satisfies; 

  0 ,T I  

      T t s T t T s   

   tT t Me  for some , 0M    

The existence of the integral form follows from the properties of semigroups and the assumed 

regularity of  ,f x t and  ,S x t , ensuring that the function inside the integral is well-

defined. 

Next, we prove the uniqueness of the solution. Assume there exist two solutions  1u t  and 

 2u t  satisfying the integral equation 

 

                     1 2 1 2 1 2 1 20
0 0

t

u t u t T t u u T t s f s f s S s S s ds         

If    1 20 0u u and 1 2 1 2,f f S S  , then  

     
1 2 0

0 0
t t s

u t u t M e ds
 

    

Thus, 1 2u u , proving the uniqueness. 

Finally, we prove the continuity of the solution. By the strong continuity of  T t , we can show 

that  u t  is continuous in X  as follows: 

By assumption,  T t is strongly continuous semigroup, meaning that for all 0 ,u X  

IJO JOURNALS

Volume 08 | Issue 04 | April 2025 |           https://ijojournals.com/index.php/m/index 29



 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                                                  OTOBONG J. TOM1 *  

https://ijojournals.com/                                                                                             Volume 08 || Issue 04 || April, 2025 || 

“SEMIGROUP APPROACH FOR THE SOLUTION OF BOUNDARY LAYER EQUATION WITH SINC FUNCTION TERM" 

 

 

    0 0
0

lim
x

T t u u X


   

This directly implies that   0T t u  is continuous as function of t X . 

Now, we need to prove the continuity of integral term by analyzing the integral term: 

          
0

t

I t T t s f s S s ds    

To prove that  I t  is continuous in X , we consider a small perturbation h in time and 

examine the difference: 

 

 

                 
0 0

t h t

I t h I t T t h s f s S s ds T t s f s S s ds


           

Splitting the difference, we get 

               
0

t

I t h I t T t h s T t s f s S s ds          

           
t h

t
T t h s f s S s ds



    

For the first term: 

         
0

t

T t h s T t s f s S s ds      

Since  T t  is strongly continuous, for each fixed s,  

     
0

lim
h
T t h s T t s


    in X . 

Since    f s S s  is integrable, we can use the Lebesgue Dominated Convergence Theorem 

[16, 17] to conclude that: 

           
00

lim 0.
t

h
T t h s T t s f s S s ds


       

IJO JOURNALS

Volume 08 | Issue 04 | April 2025 |           https://ijojournals.com/index.php/m/index 30



 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                                                  OTOBONG J. TOM1 *  

https://ijojournals.com/                                                                                             Volume 08 || Issue 04 || April, 2025 || 

“SEMIGROUP APPROACH FOR THE SOLUTION OF BOUNDARY LAYER EQUATION WITH SINC FUNCTION TERM" 

 

 

For the second term. 

        
0

lim
t h

th
T t h s f s S s ds




    

As 0h , the interval  ,t t h shrinks to zero, and since    f s S s  is integrable, this 

integral also vanishes. 

Finally, since both terms vanish as 0h , we have  

     
0

lim 0
h
I t h I t


    

Thus,  I t  is continuous in X , and their sum  u t  is also continuous in X . 

Therefore, the mild solution  u t  in equation  5 is continuous in X .     

Proposition 5.2. (Decay Rate of Solution): If   ,f x t  and  ,S x t  vanishes as ,t  then 

  0u t   exponentially fast. 

 

Proof: 

Since   ,f x t  and  ,S x t  tend to zero, the integral term in  

       0
0

t t stu t Me u M e f s S s ds
      

vanishes as t . Thus,  

    0 0tu t Me u   

This shows exponential decay of solution. 

Theorem 6.4. (Higher Regularity of Solutions): If  0u D A , and    , ,f x t S x t  are 

sufficiently smooth, then the solution satisfies 

IJO JOURNALS

Volume 08 | Issue 04 | April 2025 |           https://ijojournals.com/index.php/m/index 31



 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                                                  OTOBONG J. TOM1 *  

https://ijojournals.com/                                                                                             Volume 08 || Issue 04 || April, 2025 || 

“SEMIGROUP APPROACH FOR THE SOLUTION OF BOUNDARY LAYER EQUATION WITH SINC FUNCTION TERM" 

 

 

    1, 0, ,
du

Au f S u C T X
dt

    . 

Proof: 

First, since  0u D A , we apply A  to the mild solution formula 

           0 0

tdu
AT t u A T t s f s S s ds f S

dt
       

Using the properties of semigroups [3, 19], 

   AT t T t A , we write 

  
du

Au f S
dt

    

This implies that u  is differentiable with the stated regularity[6]. 

 

Example 6.1. Consider the boundary value problem: 

  

   

   
     

2

2

0

sinc , 0,1 , 0

,0 ,1 0, 0

0, sin .

u u
t x t

t x

u t u t t

u x u x x

  
     

  
  



V

 

This is a linear boundary layer equation with Sinc forcing. We interpret the sinc term as an 

external time-dependent forcing that is spartially uniform (i.e., acts identically across all spatial 

points). 

Let 

2

2

d
A

dx
V  with domain      2 1

00,1 0,1D A u H H   . Then A  generates a strongly 

continuous analytic semigroup   tAT t e . 

The mild solution is given by: 

IJO JOURNALS

Volume 08 | Issue 04 | April 2025 |           https://ijojournals.com/index.php/m/index 32



 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                                                  OTOBONG J. TOM1 *  

https://ijojournals.com/                                                                                             Volume 08 || Issue 04 || April, 2025 || 

“SEMIGROUP APPROACH FOR THE SOLUTION OF BOUNDARY LAYER EQUATION WITH SINC FUNCTION TERM" 

 

 

        0 0

t

u t T t u T t s f s ds   . 

The eigenfunctions of A  are: 

      
2

2sin ,n ne x n x n   Vλ . 

So the semigroup acts as: 

    0 01
,nt
n nn

T t u e u e e x



 λ

. 

Since      0 1

2
sin

2
u x x e x  , only the first mode is nonzero: 

    
2

1

2

2
T t e e x V

. 

Now consider: 

 

       

 

1

1

0

, sinc 1 sinc 1, ,

2 2
, odd,

1, 2 sin

0, even.

n nn

n

f s x s s e e x

n
e n x dx n

n

 




  




  






 

Hence: 

          sinc
1

odd

2 2
n t s s

n n n
n

T t s f s e e e
n

  
   λ

. 

Integrating: 

         1
0 0

odd

2 2
sincn

t t t s
n n
n

T t s f s ds e s ds e x
n



    
   

λ
. 

Combining, we have the mild solution: 

IJO JOURNALS

Volume 08 | Issue 04 | April 2025 |           https://ijojournals.com/index.php/m/index 33



 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                                                  OTOBONG J. TOM1 *  

https://ijojournals.com/                                                                                             Volume 08 || Issue 04 || April, 2025 || 

“SEMIGROUP APPROACH FOR THE SOLUTION OF BOUNDARY LAYER EQUATION WITH SINC FUNCTION TERM" 

 

 

           
22

11
0

odd

2 2 2
, sinc

2

t n t st
n n
n

u t x e e x e s ds e x
n




 


     
   

VV
. 

Truncating to first odd mode  1n  , we have: 

        
22

0

2 2 2
, sinc sin

2

t t stu t x e e s ds x
 


 

 
  
 


VV

. 

This is a practical, computable low-mode mild solution. 

Example 6.2. Consideranother case given by: 

 

   

       
   

2

2
sinc ,

,0 ,1 0, 0, sin ,

1 .

u u u
u x t

t x x

u t u t u x x

x x x







   
     
  

  



V

 

This is a semilinear case with Sinc forcing. In this case, the mild solution is obtained as follows: 

Let         sincxF u s u s u s s    , then: 

         0 0

t

u t T t u T t s F u s ds   . 

As in example 5.1, 

   
2

0

2
sin

2
tT t u e x  V

. 

For the nonlinear term 

Assume      , sinu t x t x  . Then: 

    cosxu t x     

      2 sin cosxu u t x x     . 

IJO JOURNALS

Volume 08 | Issue 04 | April 2025 |           https://ijojournals.com/index.php/m/index 34



 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                                                  OTOBONG J. TOM1 *  

https://ijojournals.com/                                                                                             Volume 08 || Issue 04 || April, 2025 || 

“SEMIGROUP APPROACH FOR THE SOLUTION OF BOUNDARY LAYER EQUATION WITH SINC FUNCTION TERM" 

 

 

We project xu u  onto  

    1 2 sine x x ; 

      
1

2 2
1 0

, 2 sin cosxu u e t x x dx       

    
1

2

0

1 1 1
sin cos

2 2
x x dx 

  
    

So: 

    2 2
1

1 2
, 2

2 2
xu u e t t  


    . 

Projection of forcing term: 

    
1

1 30

4
, 2 1 sin 2e x x x dx 


    . 

Putting everything together, the scalar integral equation for  t  is: 

  
 

   
22

2

30

2 2 4 2
sinc

2 2

t st t

t e e s s ds


 


   
    

 


VV

. 

This a nonlinearVolterra equation of the second kind. 

Remark: 

 The nonlinear term creates a coupling effect where past values of  s  impact 

present  t . 

 The integral equation can be approximated numerically (e.g., trapezoidal rule or fixed 

point iteration). 

 The result is a reduced-order approximation to the original PDE using the semigroup 

eigenfunction method. 

 

IJO JOURNALS

Volume 08 | Issue 04 | April 2025 |           https://ijojournals.com/index.php/m/index 35



 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                                                  OTOBONG J. TOM1 *  

https://ijojournals.com/                                                                                             Volume 08 || Issue 04 || April, 2025 || 

“SEMIGROUP APPROACH FOR THE SOLUTION OF BOUNDARY LAYER EQUATION WITH SINC FUNCTION TERM" 

 

 

7. CONCLUSION 

We have successfully establish a pattern for solving boundary layer equations incoperatingSinc 

function in the forcing term using semigroup theory. The major results include the well-

posedness, regularity and the decay rate of the solutions, along with detailed analysis of the 

role of the Sinc function term in these equations. This layout paves way for further exploration 

into numerical methods, such as the use of Sinc functions in discretization schemes, and also 

have applications in modeling physical systems with oscillatory boundary conditions. Illustrative 

examples have been shown to validate the approach and applicability. 

 

References 

[1] Prandtl, L. On the Motion of Fluids with very Little Viscosity, In Proceedings of the Third 
International Congress of Mathematicians, Heidelberg, Germany, (1964), 484-491.  

[2] Hille, E. and Phillips, R. S, Functional Analysis analysis and Semi-groups. AMS Colloquium 
Publication, vol. 32 (1957). 

[3] Pazy A., Semigroups of Linear Operators and Applications to Partial Differential 
Equations, Springer-Verlag, Applied Math. Sciences, Vol. 44, 1983. 

[4] Temam, R., Infinite-Dimensional Dynamical System in Mechanics and Physics. Springer-
Verlag 1997. 

[5] L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, Vol. 19, 
AMS, Providence, Rhode Island, 2002. 

[6] Henry, D., Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in 
Mathematics vol. 840, Pringer-Verlag, 1981. 

[7] Pr�̈ss, J., Evolutional Integral Equations and Applications. Birkhauser, (2015). 
[8] Hussain, S., and Kato, D., Semigroup-Based Numerical Approximations for Boundary 

Layer Equations. Journal of Computational Fluid Dynamics, 40(3) (2022) 785-810. 
[10] Lund, J., and Bowers, D.,Sinc Method for Quadrature and Differential Equations. SIAM 

(1992).  
[11] Stenger, F.  Numerical Methods Based on Sinc and Analytic Functions. Springer, Berlin, 

New York, (1993). 
[12] John, E. D. &Ogbonna ,N. A double Exponential Sinc Collocation Method for Volterra-

Fredholm Integral Equations of the Second Kind, J. Math. Soc. 35 (2016) 408-423. 
[13] S. Kesavan, Topics in Functional Analysis and Applications, New age International (P) 

Limited, New Delhi, (2003). 
[14] Haim Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, 

Universitext. Springer, New York, (2011). 

IJO JOURNALS

Volume 08 | Issue 04 | April 2025 |           https://ijojournals.com/index.php/m/index 36



 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                                                  OTOBONG J. TOM1 *  

https://ijojournals.com/                                                                                             Volume 08 || Issue 04 || April, 2025 || 

“SEMIGROUP APPROACH FOR THE SOLUTION OF BOUNDARY LAYER EQUATION WITH SINC FUNCTION TERM" 

 

 

[15] Michal Rozanski, BeataSikora, AndrianSmuda, and Roman Witula. On Theoretical and 
Practical aspect of Duhamel’s Integral. Archives of Control Sciences, Vol. 31(LXVII),  No. 
4, (2021) 815-847. 

[16] Duchesne, G. W., Lessard, JP. &Takayasu. A. A rigorous Integrator and Global Existence 
for Higher-Dimensional Semilinear Parabolic PDEs via Semigroup Theory. J SciComput. 
102(62) (2025). 

[17] Royden, H. L., and Fitzpatrick, P. M. Real Analysis. 4th ed., Pearson, 2010. 
[18] Klaus-JochenEngel and RainerNagel. One-Parameter Semigroups for Linear Evolution 

Equations. Springer(2000). 
[19] Lunardi A. Analytic Semigroups and Optimal Regurity in Parabolic Problems. Birkhauser, 

(1995). 
[20] Kato, T. Perturbation Theory for Linear Operators. Springer-Verlag 1980. 
[21] de Branges, L. Hilbert Spaces of Entire Functions and Applications in Fluid Mechanics. 

Cambridge University Press, (2015). 
[22] Triebel, H. Interpolation Theory, Function Spaces, Differential Operators. North-Holland, 

(1978). 
[23] Trefethen, L. N., and Weideman, J. A. The Exponentially convergent Sinc Method for 

Integral Equations. SIAM Review, 56(3) (2014), 385-458. 
[24] Tucsnak, M and Weiss, G. Observation and Control for Operator Semigroups. Birkhauser, 

(2009). 
[25] MannanMd. A., Rahman Md. R., Akter H., Nahar N., andMondal S. A Study of Banach 

Fixed-Point Theorem and It’s Applications. American Journal of Computational 
Mathematics, 11 (2021), 157-174. 

IJO JOURNALS

Volume 08 | Issue 04 | April 2025 |           https://ijojournals.com/index.php/m/index 37

https://link.springer.com/book/10.1007/b97696

