On construction of a global numerical solution for a semilinear singularly--perturbed reaction diffusion boundary value problem

Abstract

A class of different schemes for the numerical solving of semilinear singularly--perturbed reaction--diffusion boundary--value problems was constructed. The stability of the difference schemes was proved, and the existence and uniqueness of a numerical solution were shown. After that, the uniform convergence with respect to a perturbation parameter ε\varepsilon on a modified Shishkin mesh of order 2 has been proven. For such a discrete solution, a global solution based on a linear spline was constructed, also the error of this solution is in expected boundaries. Numerical experiments at the end of the paper, confirm the theoretical results. The global solutions based on a natural cubic spline, and the experiments with Liseikin, Shishkin and modified Bakhvalov meshes are included in the numerical experiments as well.

Full text

arXiv:2009.06523v1 [math.NA] 14 Sep 2020
On construction of a global numerical solution for a semilinear
singularly–perturbed reaction diffusion boundary value problem
Samir Karasulji´c∗,†and Hidajeta Ljevakovi´c‡
Abstract
A class of different schemes for the numerical solving of semilinear singularly—perturbed reaction—diffusion
boundary–value problems was constructed. The stability of the difference schemes was proved, and the exis-
tence and uniqueness of a numerical solution were shown. After that, the uniform convergence with respect
to a perturbation parameter εon a modified Shishkin mesh of order 2 has been proven. For such a discrete
solution, a global solution based on a linear spline was constructed, also the error of this solution is in
expected boundaries. Numerical experiments at the end of the paper, confirm the theoretical results. The
global solutions based on a natural cubic spline, and the experiments with Liseikin, Shishkin and modified
Bakhvalov meshes are included in the numerical experiments as well.
1 Introduction
We consider the semilinear boundary–value singularly–perturbed problem
ε2y′′ −f(x, y) = 0, x ∈(0,1), y(0) = y(1) = 0,(1a)
with the condition ∂f (x, y)
∂y := fy>m > 0,(1b)
where 0 < ε << is a small perturbation parameter, and mis a positive constant, fis a nonlinear function,
f(x, y)∈Ck([0,1] ×R), k >2.The problem (1a) under the condition (1b) has a unique solution, (see Lorenz
[28]). It’s a well-known fact in theory that the exact solution to (1a)–(1b) has two exponential boundary layers,
i.e. near the end points x= 0 and x= 1.
Differential equations like (1a) and similar occur in mathematical modeling of many problems in physics,
chemistry, biology, engineering sciences, economics and even social sciences. Numerical solutions of singularly–
perturbed boundary–value problems obtained by some classical methods are usually useless. That is because
the exact solutions of the singularly–perturbed boundary–value problems depend on the perturbation parameter
ε, but classical methods don’t take in account the influence of the perturbation parameter. The singularly–
perturbed problems require special developed numerical methods in order to obtain the accuracy, which is
uniform respect to the parameter ε. Numerical methods that act uniformly well for all the values of the singular
perturbation parameter are called ε-uniformly convergent numerical methods.
Many authors have worked on the numerical solution of the problem (1a)–(1b) with different assumptions
about the function f, as well as more general nonlinear problems. There were many constructed ε–uniformly
convergent difference schemes of order 2 and higher (Herceg [8], Herceg, Surla and Rapa ji´c [9], Herceg and
Miloradovi´c [10], Herceg and Herceg [11], Kopteva and Linß [17], Kopteva and Stynes [18,19], Kopteva, Pickett
and Purtill [20], Linß, Roos and Vulanovi´c [22], Sun and Stynes [31], Stynes and Kopteva [32], Surla and Uzelac
[34], Vulanovi´c [35,36,37,38,40], etc.
In the paper [1] Boglaev introduced a new method for the numerical solving of the problem (1a)–(1b), using
the representation of the exact problem to (1a)–(1b) via the Green function. In this paper we use this method
to construct a new different scheme.
The author’s results in the numerical solving of the problem (1a)–(1b) and others results can be seen in [2],
[3], [5], [6], [7], [12], [4], [13], [16], [15], [14], [27], [26].
∗corresponding author
†University of Tuzla, Faculty of Sciences and Mathematics, Univerzitetska br. 4, 75 000 Tuzla, Bosnia and Herzegovina,
email:samir.karasuljic@untz.ba
‡University of Tuzla, Faculty of Sciences and Mathematics, Univerzitetska br. 4, 75 000 Tuzla, Bosnia and Herzegovina, email:
hidajeta1993@gmail.com
1

2 Theoretical background
The estimates of solution’s derivatives are a very important tool in the analysis of numerical methods considering
the singularly–perturbed boundary–value problems. The construction of layer–adapted meshes is based on these
estimates, also in the sequel they will be used in the analysis of the consistency. Bearing in mind the above,
we state the following theorem about a decomposition of the solution yto a layer component sand a regular
component rand the appropriate estimates.
Theorem 2.1. [35]The solution yto problem (1a)–(1b)can be represented in the following way:
y=r+s,
where for j= 0,1, ..., k + 2 and x∈[0,1] we have that
r(j)(x)≤C, (2)
and s(j)(x)≤Cε−je−x
ε√m+e−1−x
ε√m.(3)
2.1 Layer–adapted mesh
It’s a well–known fact that the exact solution to problems like (1a)–(1b) changes rapidly near the end points
x= 0 and x= 1.Many meshes have been constructed for the numerical solving problems that have a layer or
layers of an exponential type. In the present paper we shall use three different meshes. We will get these meshes
0 = x0< x1< . . . < xN= 1,by using appropriate generating functions, i.e. xi=ψ(i/N).The generating
function are constructed as follows.
Let N+ 1 be the number of mesh points, q∈(0,1/2) mesh parameter. Define the Shishkin mesh transition
point by
λ:= min 2εln N
√m,1
4.(4)
The first mesh we will use in the sequel is a modified Shishkin mesh proposed by Vulanovi´c [39]. The generating
function for this mesh is
ψ(t) = 




4λt, t ∈[0,1/4],
p(t−1/4)3+ 4λt, t ∈[1/4,1/2],
1−ψ(1 −t), t ∈[1/2,1],
(5)
where pis chosen so that ψ(1/2) = 1/2,i.e. p= 32(1−4λ).Note that ψ∈C1[0,1] with kψ′k∞6C, kψ′′k∞6C.
Therefore the mesh size hi=xi+1 −xi, i = 0,...N −1 satisfy (see [23])
hi=Z(i+1)N
i/N
ψ′(t) d t6CN −1,|hi+1 −hi|=Zi/N
(i−1)/N Zt+1/N
t
ψ′′(s) d s
6CN −2.(6)
The second mesh is the Shishkin mesh [30]. The generating function for this mesh is
ψ(t) = 




4λt, t ∈[0,1/4]
λ+ 2(1 −2λ)(t−1/4), t ∈[1/2,1/4],
1−ψ(1 −t), t ∈[1/2,1].
(7)
The third mesh is the modified Bakhvalov mesh also proposed by Vulanovi´c [35]. The generating function for
this mesh is
ψ(t) = 




µ(t) := aεt
q−t, t ∈[0, α],
µ(α) + µ′(α)(t−α), t ∈[α, 1/2],
1−ψ(1 −t), t ∈[1/2,1],
(8)
where aand qare constants, independent of ε, such that q∈(0,1/2), a ∈(0, q/ε),and additionally a√m>2.
The parameter αis the abscissa of the contact point of the tangent line from (1/2,1/2) to µ(t),and its value is
α=q−paqε(1 −2q+ 2aε)
1 + 2aε .
2

The fourth mesh proposed by Liseikin [24,25], and we will use its modification from [27]. The generating
function for this mesh is
ψ(t, ε, a, k) =















c1εk((1 −dt)−1/a −1) ,06t61/4,
c1hεkan/(1+na)−εk+d1
aεka(n−1)/(1+na)(t−1/4)+
1
2d21
a1
a+ 1εka(n−2)/(1+na)(t−1/4)2+c0(t−1/4)3i,1/46t61/2,
1−ψ(1 −t, ε, a, k),1/26t61,
(9)
where d= (1 −εka/(1+na))/(1/4), a is a positive constant subject to a≥m1>0, and a= 1, c0>0, n= 2,
k= 1, c0= 0,and 1
c1= 2 εkan/(1+na)−εk+d
4aεka(n−1)/(1+na)+d2
2
1
a1
a+ 1εka(n−2)/(1+na)(1/4)2+c0(1/4)3i
is chosen here.
3 Difference scheme
We will consider an arbitrary mesh with mesh points
0 = x0< x1< . . . < xN= 1,
and let it be hi=xi+1 −xi, i = 0,1,...,N −1.In constructing a new difference scheme for the problem
(1a)–(1b) we use the following scheme from Boglaev [1]
β
sinh(βhi−1)yi−1−β
tanh(βhi−1)+β
tanh(βhi)yi+β
sinh(βhi)yi+1
=1
ε2"Zxi
xi−1
uII
i−1ψ(s, y) d s+Zxi+1
xi
uI
iψ(s, y) d s#, i = 1,2,...,N −1, y0=yN= 0,(10)
where
ψ(s, y) = f(s, y)−γy, β =√γ
ε.(11)
We can’t calculate the integrals in (10) because we don’t know the exact solution yto the problem (1a)–(1b).
The next step is to approximate the function ψby a constant value. Approximations of the function ψare
ψ−
i= (1 −t)ψ(xi−1, y(xi−1)) + tψ(xi, y(xi)), x ∈[xi−1, xi],(12)
ψ+
i=tψ(xi, y(xi)) + (1 −t)ψ(xi+1 , y(xi+1)), x ∈[xi, xi+1], t ∈[0,1].(13)
By using the approximations (12), (13) into (10), after calculating the integrals and some computing, and taking
in account that
Zxi
xi−1
uII
i−1ds=cosh(βhi−1)−1
βsinh(βhi−1),
Zxi+1
xi
uI
ids=cosh(βhi)−1
βsinh(βhi),
we get the difference scheme
(1 −t) cosh(βhi−1) + t
sinh(βhi−1)(yi−1−yi)−(1 −t) cosh(βhi) + t
sinh(βhi)(yi−yi+1 )
−(1 −t)fi−1+tfi
γ·cosh(βhi−1)−1
sinh(βhi−1)−tfi+ (1 −t)fi+1
γ·cosh(βhi)−1
sinh(βhi)= 0,(14)
where fk=f(xk,yk), k ∈ {i−1, i, i + 1},and t∈[0,1].
The previous form of the difference scheme can be written in the following form
(1 −t) cosh(βhi−1) + 1 −t+ 2t−1
sinh(βhi−1)(yi−1−yi)−(1 −t) cosh(βhi) + 1 −t+ 2t−1
sinh(βhi)(yi−yi+1 )
−(1 −t)fi−1+ (1 −t+ 2t−1)fi
γ·cosh(βhi−1)−1
sinh(βhi−1)−(1 −t+ 2t−1)fi+ (1 −t)fi+1
γ·cosh(βhi)−1
sinh(βhi)= 0,
3

and finally
(1 −t)cosh(βhi−1) + 1
sinh(βhi−1)(yi−1−yi)−cosh(βhi) + 1
sinh(βhi)(yi−yi+1 )
−fi−1+fi
γ·cosh(βhi−1)−1
sinh(βhi−1)−fi+fi+1
γ·cosh(βhi−1)−1
sinh(βhi−1)
+ (2t−1) 1
sinh(βhi−1)(yi−1−yi)−1
sinh(βhi)(yi−yi+1 )
−fi
γ·cosh(βhi−1)−1
sinh(βhi−1)−fi
γ·cosh(βhi−1)−1
sinh(βhi−1)= 0, i = 1,...,N −1.(15)
4 Stability
The difference scheme (15) generates a nonlinear system. A goal of this section is to show that this system has
a unique solution. We are going to construct a discrete operator T, and show that the discrete operator Tis
inverse-monotone as well, which implies that our numerical method is stable, and the numerical solution exist
and it is a unique.
Let us set the discrete operator
T u = (T u0, T u 1, ...,TuN)T,(16)
where
T u0=−u0
T ui=γ
cosh(βhi−1)−1
sinh(βhi−1)+cosh(β hi)−1
sinh(βhi)
(1 −t)cosh(βhi−1) + 1
sinh(βhi−1)(yi−1−yi)−cosh(βhi) + 1
sinh(βhi)(yi−yi+1 )
−fi−1+fi
γ·cosh(βhi−1)−1
sinh(βhi−1)−fi+fi+1
γ·cosh(βhi−1)−1
sinh(βhi−1)
+ (2t−1) 1
sinh(βhi−1)(yi−1−yi)−1
sinh(βhi)(yi−yi+1 )
−fi
γ·cosh(βhi−1)−1
sinh(βhi−1)−fi
γ·cosh(βhi−1)−1
sinh(βhi−1)= 0, i = 1,...,N −1,(17)
T uN=−uN
Obviously, it is hold
Ty= 0,(18)
where y= (y0, y1,...,yN)Tthe numerical solution of the problem (1a)–(1b), obtained by using the difference
scheme (15). Now, we can state and prove the theorem of stability.
Theorem 4.1. The discrete problem (16)–(18)has a unique solution yfor γ>fy.Moreover, for every v, w ∈
RN+1 we have the following stability inequality
kv−wk6CkT v −T wk.(19)
Proof. We use a well known technique from [38] to prove the first statement of the theorem. The proof of
existence and uniqueness of the solution of the discrete problem T y = 0 is based on the proof of the relation:
kT yk6C, where T′is the Fr´echet derivative of T. The Fr´echet derivative H:= T′(y) is a tridiagonal matrix.
Let H= [hij ].The non-zero elements of this tridiagonal matrix are
h1,1=hN+1,N+1 =−1<0,
hi,i−1=γ
cosh(βhi−1)−1
sinh(βhi−1)+cosh(β hi)−1
sinh(βhi)"(1 −t) cosh(βhi−1) + t
sinh(βhi−1)−
∂f
∂yi−1
γ·(1 −t)(cosh(βhi−1)−1)
sinh(βhi−1)#,
hi,i+1 =γ
cosh(βhi−1)−1
sinh(βhi−1)+cosh(β hi)−1
sinh(βhi)"(1 −t) cosh(βhi) + t
sinh(βhi)−
∂f
∂yi+1
γ·(1 −t)(cosh(βhi)−1)
sinh(βhi)#,
4

hi,i =−γ
cosh(βhi−1)−1
sinh(βhi−1)+cosh(β hi)−1
sinh(βhi)(1 −t) cosh(βhi−1) + t
sinh(βhi−1)+(1 −t) cosh(βhi) + t
sinh(βhi)
+t
∂f
∂yi
γ·cosh(βhi−1)−1
sinh(βhi−1)+t
∂f
∂yi
γ·cosh(βhi)−1
sinh(βhi)#, i = 2,...,N. (20)
From (20), it’s obvious that
hi,i−1>0, hi,i+1 >0, hi,i <0,
and
|hi,i| − |hi,i−1| − |hi,i+1 |>m,
so we can conclude that His an M–matrix, and finally we obtain
kH−1k61
m.(21)
Using Hadamard’s theorem ([29, p 137]) we get that Thomeomorphism. Since clearly RN+1 is non–empty and
0 is the only image of the mapping T, we have that (18) has a unique solution.
The proof of second statement of the Theorem 19 is based on a part of the proof of Theorem 3 from [8]. We
have that T v−T w = (T′ξ)−1(v−w) for some ξ= (ξ0, ξ1,...,ξN)T∈RN+1 .Therefore v−w= (T′ξ)−1(T v −T w)
and finally due to inequality (21) we have that
kv−wk=k(T′ξ)−1(T v −T w )k61
mkT v −T wk.
5 Uniform convergence
The difference scheme (14) we can write in the following form
(1 −t)cosh(βhi−1)−1
sinh(βhi−1)(yi−1−yi)−cosh(βhi)−1
sinh(βhi)(yi−yi+1 )+yi−1−yi
sinh(βhi−1)−yi−yi+1
sinh(βhi)
−(1 −t)fi−1+tfi
γ·cosh(βhi−1)−1
sinh(βhi−1)−tfi+ (1 −t)fi+1
γ·cosh(βhi)−1
sinh(βhi)= 0, i = 1,...,N −1.(22)
In order to prove the Theorem of convergence, we need three estimates given in the next lemmas.
Lemma 5.1. [13]Assume that ε6C
N.In the part of the modified Shishkin mesh from Section 2.1 when
xi, xi±1∈[xN/4−1, λ]∪[λ, 1/2],we have the following estimate

cosh(βhi−1)−1
sinh(βhi−1)(y(xi−1)−y(xi)) −cosh(βhi)−1
sinh(βhi)(y(xi)−y(xi+1))
cosh(βhi−1)−1
sinh(βhi−1)+cosh(β hi)−1
sinh(βhi)
6C
N2, i =N/4,...,N/2−1.
Lemma 5.2. [13]Assume that ε6C
N.In the part of the modified Shishkin mesh from Section 2.1 when
xi, xi±1∈[xN/4−1, λ]∪[λ, 1/2],we have the following estimate

y(xi−1)−y(xi)
sinh(βhi−1)−y(xi)−y(xi+1 )
sinh(βhi)
cosh(βhi−1)−1
sinh(βhi−1)+cosh(β hi)−1
sinh(βhi)
6C
N2, i =N/4,...,N/2−1.
Lemma 5.3. Assume that ε6C
N.In the part of the modified Shishkin mesh from Section 2.1 when xi, xi±1∈
[xN/4−1, λ]∪[λ, 1/2],we have the following estimate
γ
cosh(βhi−1)−1
sinh(βhi−1)+cosh(β hi)−1
sinh(βhi)
(1 −t)f(xi−1, y(xi−1)) + tf(xi, y(xi))
γ·cosh(βhi−1)−1
sinh(βhi−1)
−tf(xi, y(xi)) + (1 −t)f(xi+1 , y(xi+1 ))
γ·cosh(βhi)−1
sinh(βhi)
6C
N2, i =N/4,...,N/2−1.(23)
5

Proof. Taking into consideration the assumption ε6C
N,the equality ε2y′′(xi) = f(xi, y(xi)),and the Theorem
of decomposition, it is hold
γ
cosh(βhi−1)−1
sinh(βhi−1)+cosh(β hi)−1
sinh(βhi)
(1 −t)f(xi−1, y(xi−1)) + tf(xi, y(xi))
γ·cosh(βhi−1)−1
sinh(βhi−1)
−tf(xi, y(xi)) + (1 −t)f(xi+1 , y(xi+1))
γ·cosh(βhi)−1
sinh(βhi)
6|(1 −t)f(xi−1, y(xi−1)) + tf(xi, y(xi)) + tf(xi, y(xi)) + (1 −t)f(xi+1 , y(xi+1 ))|
6ε2[(1 −t) (|r′′(xi−1)|+|s′′(xi−1)|) + 2t(|s′′(xi)|+|r′′(xi)|) + (1 −t) (|s′′(xi+1 )|+|r′′(xi+1)|)]
6C1ε2"(1 −t) 1 + e−xi−1
ε√m
ε2!+ 2t 1 + e−xi
ε√m
ε2!+ (1 −t) 1 + e−xi+1
ε√m
ε2!#
6Cε2+1
N2, i =N/4,...,N/2−1.(24)
Theorem 5.1. The discrete problem (16)–(18)on the modified Shishkin mesh (5)from Section 2.1 is uniformly
convergent with respect εand
max
i|y(xi)−yi|6C










ln2N/N2, i = 0,...,N/4−1,
1/N2, i =N/4,...,3N/4,
ln2N/N2, i = 3N/4 + 1,...,N,
where y(xi)is the value of the exact solution, yiis the value of the numerical solution of the problem (1a)–(1b)
in the mesh point xi,respectively, and C > 0is a constant independent of Nand ε.
Proof.
Case 06i < N/4−1.Here it’s hold hi−1=hiand hi=O(εln N/N).We have
(T y)i=
γ
cosh(βhi−1)−1
sinh(βhi−1)+cosh(β hi)−1
sinh(βhi)(1 −t) cosh(βhi−1) + t
sinh(βhi−1)(y(xi−1)−y(xi)) −(1 −t) cosh(βhi) + t
sinh(βhi)(y(xi)−y(xi+1 ))
−(1 −t)f(xi−1, y(xi−1)) + tf(xi, y(xi))
γ·cosh(βhi−1)−1
sinh(βhi−1)
−tf(xi, y(xi)) + (1 −t)f(xi+1 , y(xi+1 ))
γ·cosh(βhi)−1
sinh(βhi)
=γ
2(cosh(βhi)−1) ty(xi−1)−2y(xi) + y(xi+1)−2f(xi, y(xi))
γ(cosh(βhi)−1)
+ (1 −t)cosh(βhi) (y(xi−1)−2y(xi) + y(xi+1)) −f(xi−1, y (xi−1)) + f(xi+1, y(xi+1))
γ·(cosh(βhi)−1)
=γ
2(cosh(βhi)−1) ty(xi−1)−2y(xi) + y(xi+1)−2ε2y′′(xi)
γ(cosh(βhi)−1)
+ (1 −t)cosh(βhi) (y(xi−1)−2y(xi) + y(xi+1)) −ε2y′′(xi−1) + y′′(xi+1)
γ·(cosh(βhi)−1).
Using Taylor’s expansions
y(xi−1)−2y(xi) + y(xi+1) = y′′(xi)h2
i+y(iv)(ξ−
i) + y(iv)(ξ+
i)
24 h4
i, ξ−
i∈(xi−1, xi), ξ+
i∈(xi, xi+1),
y′′(xi−1) + y′′(xi+1) = 2y′′(xi) + y(iv)(η−
i) + y(iv)(η+
i)
2h2
i, η−
i∈(xi−1, xi), η+
i∈(xi, xi+1),
cosh(βhi) = 1 + β2h2
i
2+Oβ4h4
i
we get
6

(T y)i=γ·t
β2h2
i+ 2O(β4h4
i)y′′(xi)h2
i+y(iv)(ξ−
i) + y(iv)(ξ+
i)
24 h4
i−2ε2y′′(xi)
γβ2h2
i
2+O(β4h4
i)
+γ·(1 −t)
β2h2
i+ 2O(β4h4
i)1 + β2h2
i
2+O(β4h4
i)y′′(xi)h2
i+y(iv)(ξ−
i) + y(iv)(ξ+
i)
24 h4
i
−ε22y′′(xi) + y(iv)(η−
i)+y(iv)(η+
i)
2
γβ2h2
i
2+O(β4h4
i)

=γ·t
β2h2
i+ 2O(β4h4
i)y(iv)(ξ−
i) + y(iv)(ξ+
i)
24 h4
i−2ε2y′′(xi)
γO(β4h4
i)
+γ·(1 −t)
β2h2
i+ 2O(β4h4
i)y(iv)(ξ−
i) + y(iv)(ξ+
i)
24 h4
i+β2h2
i
2+O(β4h4
i)y′′(xi)h2
i+y(iv)(ξ−
i) + y(iv)(ξ+
i)
24 h4
i
−2ε2y′′(xi)
γ· O(β4h4
i) + ε2y(iv)(η−
i) + y(iv)(η+
i)
2γh2
iβ2h2
i
2+O(β4h4
i),(25)
and finally
|(T y)i|6Cln2N/N2, i = 0,1,...,N/4−1.(26)
Case N/46i < N/2.Due to (22) we have the next inequality
|(T y)i|6γ
cosh(βhi−1)−1
sinh(βhi−1)+cosh(βhi)−1
sinh(βhi)
·(1 −t)
cosh(βhi−1)−1
sinh(βhi−1)(y(xi−1)−y(xi)) −cosh(βhi)−1
sinh(βhi)(y(xi)−y(xi+1))
+
y(xi−1)−y(xi)
sinh(βhi−1)−y(xi)−y(xi+1)
sinh(βhi)
+
(1 −t)f(xi−1, y(xi−1)) + tf(xi, y(xi))
γ·cosh(βhi−1)−1
sinh(βhi−1)
+
tf(xi, y(xi)) + (1 −t)f(xi+1 , y(xi+1 ))
γ·cosh(βhi)−1
sinh(βhi),
and according Lemma 5.1, Lemma 5.2 and Lemma 5.3 we obtain
|(T y)i|6C/N2, i =N/4,...,N/2−1.(27)
Case i=N/2.This case is trivial, because hN/4−1=hN/4and the influence of the layer component sis
negligible.
Collecting (26), (27) and taking in account Case i=N/2,we have proven the theorem.
6 Global solution
In the paper [14] a global numerical solution was constructed using a spline in tension, and the authors proved
the uniform convergence of order 1 for this solution on the modified Shishkin mesh generated by (5). After
that they repaired the global numerical solution on [λ, 1−λ] and achieved the uniform convergence of order
2. That repaired global solution is composed of exponential and linear functions. In the sequel we avoid
exponential functions and give a global numerical solution composed by linear functions. We will also include
in the numerical experiments a global solution obtained by using a natural cubic spline, because this spline is
the lowest degree spline with a continuous second derivative.
Linear spline Let the global numerical solution to the problem (1a)–(1b) has the form
P(x) =



















p1(x), x ∈[x0, x1],
p2(x), x ∈[x1, x2],
.
.
.
pi(x), x ∈[xi−1, xi],
.
.
.
pN(x), x ∈[xN−1, xN],
(28)
7

where
pi(x) = 






yi−yi−1
xi−xi−1
(x−xi−1) + yi−1, x ∈[xi−1, xi],
0, x /∈[xi−1, xi],
(29)
and i= 1,2,...,N.
Theorem 6.1. The following estimate of the error holds
max
x∈[0,1] y(x)−P(x)6Cln2N/N2,(30)
where yis the exact solution to the problem (1a)–(1b)and Pis global numerical solution (28).
Proof. We divide this proof in three parts, [0, λ],[λ, xN/4+1] and [xN/4+ 1,1/2]. The proof is analogues on
[1/2,1].The proof is based on the inequality kP−yk∞6kP−Pk∞+kP−yk∞,and a theorem on the
interpolation error and its corollaries. For our purpose we use [21, Example 8.12]. By Pis designated a
piecewise polynomial obtained in the same way like P , but Ppasses trough the points with the coordinates
(xi−1, y(xi−1)),(xi, y(xi)), i = 1,2,...,N; instead of (xi−1, yi−1),(xi, yi+1), i = 1,2,...,N,
P(x) =



















p1(x), x ∈[x0, x1],
p2(x), x ∈[x1, x2],
.
.
.
pi(x), x ∈[xi−1, xi],
.
.
.
pN(x), x ∈[xN−1, xN],
(31)
where
pi(x) = 




yi−yi−1
xi−xi−1
(x−xi−1) + yi−1, x ∈[xi−1, xi],
0, x /∈[xi−1, xi],
(32)
and i= 1,2,...,N.
Taking in account the constructions (28), (31) and Theorem 5.1 holds
kP−Pk∞6Cln2N/N2, x ∈[0,1].(33)
The first part is on the subinterval [0, λ],this one corresponds with the mesh when i= 1,2,...,N/4.
Here, the mesh is equidistant i.e. hi−1=hi,and hi=O(εln N/N).Using Theorem 2.1, [21, Example 8.12],
hi=O(εln N/N) we have that
|y(x)−pi(x)|6h2
i
8max
ξ∈[xi−1,xi]|y′′(ξ)|6C1
ε2ln2N
N2max
ξ∈[xi−1,xi]|s′′(ξ) + r′′(ξ)|
6C2
ε2ln2N
N2max
ξ∈[xi−1,xi]ε−2e−ξ
ε√m+e−(ξ−1)
ε√m+r′′(ξ)
6C2
ε2ln2N
N2(ε−2+C3)6Cln2N
N2, i = 1,2,...,N/4.(34)
The remain of the proof, i.e. for x∈[λ, xN/4+1]∪[xN/4+1 ,1/2] which corresponds with the mesh for
i=N/4, N/4 + 1,...,N/2,we repeat from [14].
For i=N/4 + 1,...,N/2,the mesh isn’t equidistant but holds hi=O(1/N).According to the Theorem 2.1, to
the Theorem (5.1), [21, Example] and the features of the mesh we obtain
|y(x)−pi(x)|6h2
i
8max
ξ∈[xN/4+1,1/2] |y′′(ξ)|6C
N2.(35)
On [λ, xN/4+ 1],according to the Theorem 2.1 we obtain
y−pi(x) =y−yi−yi−1
xi−xi−1
(x−xi−1) + yi−1
8

=s−si−si−1
xi−xi−1
(x−xi−1) + si−1+r−ri−ri−1
xi−xi−1
(x−xi−1) + ri−1.
For the layer component s, based on the estimate (3), we have

s−si−si−1
xi−xi−1
(x−xi−1) + si−1
6|s|+|si+1 −si|+|si|6C1e−xi−1
ε√m+e−xi−1−1
ε√m6C
N2.(36)
For the regular component r, we apply again the estimate from [21, Example 8.12], the estimate (2), and we
have that 
r−ri−ri−1
xi−xi−1
(x−xi−1) + ri−1
6h2
i−1
8max
ξ∈[xi−1,xi]|r′′(ξ)|6C
N2.(37)
Collecting (33), (34), (35), (36) and (37), this theorem has been proven.
Cubic spline In the numerical experiments we will use a natural cubic spline as a global solution. We
construct it in the way as follows: design the natural cubic spline by C,
C(x) = Ci(x), x ∈[xi, xi+1], i = 0,1,...,N −1,(38)
where Ciare the cubic functions
Ci(x) = Mi
(xi+1 −x)3
6hi+1
+Mi+1
(x−xi)3
6hi+1
+yi+1 −yi
hi+1 −hi+1
6(Mi+1 −Mi)(x−xi) + yi−Mi
h2
i+1
6,(39)
the moments Mi:= C′′
i(xi), i = 1, N −1 we get from the system
hi
6Mi−1+hi+hi+1
3Mi+hi+1
6Mi+1 =yi+1 −yi
hi+1 −yi−yi−1
hi
, i = 1,2,...,N −1,(40)
and M0:= C′′
0(x0) = 0, MN:= C′′
N−1(xN) = 0.
7 Numerical experiments
In this section we conduct numerical experiments in order to confirm the theoretical results, i.e. to confirm the
accuracy of the different scheme (15) on the meshes (7), (5), (8) and (9).
Example 7.1. We consider the following boundary value problem
ε2y′′ =y+ cos2πx + 2ε2π2cos2πx on (0,1) ,(41)
y(0) = y(1) = 0.(42)
The exact solution of this problem is
y(x) = e−x
ε+ex
ε
1 + e−1
ε−cos2πx. (43)
The nonlinear system was solved using the initial condition y0=−0.5and the value of the constant γ= 1.
Because of the fact that the exact solution is known, we compute the error ENand the rate of convergence Ord
in the usual way
EN=ky−yNk∞,Ord = ln EN−ln E2N
ln(2k/(k+ 1)) ,(Shishkin),Ord = ln EN−ln E2N
ln 2 ,(Bakhvalov,Liseikin) (44)
where N= 2k, k = 4,5,...,12,and yis the exact solution of the problem (1a)–(43), while yNan appropriate
numerical solution of (16). The graphics of the numerical and exact solutions, for various values of the param-
eter εare on Figure 1(left), while fragments of these solutions are on Figure 1(right). The values of ENand
Ord are in Tables 1. The graphics of the exact and global solution obtained by using a linear spline, and the
corresponding error are shown on Figure 2, while the graphics of the exact and global solution obtained by using
a natural cubic spline, and the corresponding error are shown on Figure 3.
9

0.0 0.2 0.4 0.6 0.8 1.0
x
- axis
−1.0
−0.8
−0.6
−0.4
−0.2
0.0
y
- axis
Exact
ε
= 2
−4
Exact
ε
= 2−6
Exact
ε
= 2−10
Numerical
ε
= 2−4
Numerical
ε
= 2−6
Numerical
ε
= 2−10
x
- axis
y
- axis
Exact
ε
= 2
−10
Exact
ε
= 2−10
Exact
ε
= 2−10
Exact
ε
= 2−10
mod. Bakhvalov
ε
= 2−10
mod. Shishkin
ε
= 2−10
Shishkin
ε
= 2−10
Liseikin
ε
= 2−10
Figure 1: Exact and numerical solutions (left), layer near x= 0 (right)
0.0 0.2 0.4 0.6 0.8 1.0
x
- axis
−1.0
−0.8
−0.6
−0.4
−0.2
0.0
y
- axis
Exact
ε
= 2
−7
Discrete numerical
ε
= 2−7,
N
= 16
Global numerical
ε
= 2−7,
N
= 16
0.0 0.2 0.4 0.6 0.8 1.0
modified Bakhvalov
0.0
0.1
0.0 0.2 0.4 0.6 0.8 1.0
modified Shishkin
0.0
0.2
0.0 0.2 0.4 0.6 0.8 1.0
Shishkin
0.0
0.1
0.0 0.2 0.4 0.6 0.8 1.0
Liseikin
0.00
0.05
0.0 0.2 0.4 0.6 0.8 1.0
modified Bakhvalov
0.00
0.02
0.0 0.2 0.4 0.6 0.8 1.0
modified Shishkin
0.0
0.1
0.0 0.2 0.4 0.6 0.8 1.0
Shishkin
0.000
0.025
0.0 0.2 0.4 0.6 0.8 1.0
Liseikin
0.00
0.02
0.0 0.2 0.4 0.6 0.8 1.0
modified Bakhvalov
0.000
0.005
0.0 0.2 0.4 0.6 0.8 1.0
modified Shishkin
0.000
0.025
0.0 0.2 0.4 0.6 0.8 1.0
Shishkin
0.00
0.01
0.0 0.2 0.4 0.6 0.8 1.0
Liseikin
0.000
0.005
Figure 2: Exact, discrete and global numerical solutions (left up), error (right up–N= 16, left down–N= 32,
right down–N= 64)
10

0.0 0.2 0.4 0.6 0.8 1.0
x
- axis
−1.0
−0.8
−0.6
−0.4
−0.2
0.0
y
- axis
Exact
ε
= 2
−7
Discrete numerical
ε
= 2−7,
N
= 16
Global numerical
ε
= 2−7,
N
= 16
0.0 0.2 0.4 0.6 0.8 1.0
modified Bakhvalov
0.00
0.05
0.0 0.2 0.4 0.6 0.8 1.0
modified Shishkin
0.0
0.2
0.0 0.2 0.4 0.6 0.8 1.0
Shishkin
0.0
0.1
0.0 0.2 0.4 0.6 0.8 1.0
Liseikin
0.00
0.05
0.0 0.2 0.4 0.6 0.8 1.0
modified Bakhvalov
0.00
0.02
0.0 0.2 0.4 0.6 0.8 1.0
modified Shishkin
0.00
0.05
0.0 0.2 0.4 0.6 0.8 1.0
Shishkin
0.00
0.02
0.0 0.2 0.4 0.6 0.8 1.0
Liseikin
0.00
0.01
0.0 0.2 0.4 0.6 0.8 1.0
modified Bakhvalov
0.000
0.002
0.0 0.2 0.4 0.6 0.8 1.0
modified Shishkin
0.00
0.02
0.0 0.2 0.4 0.6 0.8 1.0
Shishkin
0.00
0.01
0.0 0.2 0.4 0.6 0.8 1.0
Liseikin
0.000
0.002
Figure 3: Exact, discrete and global numerical solutions (left up), error (right up–N= 16, left down–N= 32,
right down–N= 64)
8 Conclusion
In the present paper we performed the construction of a numerical solution for the one–dimensional singularly–
perturbed reaction–diffusion boundary–value problem. The class of different schemes was constructed, and we
proved the existence and uniqueness of the discrete numerical solution. After that, we proved ε–uniformly
convergence of the constructed class of different schemes on the modified Shishkin mesh of order 2. A global
numerical solution was constructed based on a linear spline and proved that the order of the error value is
Oln2N/N2.The numerical experiments at the end of the paper confirm the theoretical results. The results
obtained by using a global numerical solution based on a natural cubic spline and the Shishkin, the modified
Bakhvalov and last but not least the Liseikin mesh are included in the numerical experiments. Although, the
theoretical analysis for these meshes wasn’t done, the results suggest that the order of convergence is 2 for all
of them. Especially, good results have been achieved by using the Liseikin mesh.
11

2−32−52−10 2−20 2−30 2−40
N EnOrd EnOrd EnOrd EnOrd EnOrd
245.277e-2 3.09 1.015e-1 2.08 1.255e-1 2.67 1.278e-1 2.65 1.278e-1 2.65 1.278e-1 2.65
251.234e-2 2.89 3.811e-2 2.83 3.570e-2 2.65 3.666e-2 2.64 3.666e-2 2.64 3.666e-2 2.64
262.819e-3 2.75 8.956e-3 2.87 9.194e-3 2.20 9.515e-3 2.26 9.515e-3 2.26 9.515e-3 2.26
276.374e-4 2.67 1.900e-3 2.74 2.804e-3 1.99 2.803e-3 1.99 2.803e-3 1.99 2.804e-3 1.99
281.429e-4 2.61 4.099e-4 2.61 9.198e-4 2.00 9.196e-4 2.00 9.196e-4 2.00 9.196e-4 2.00
293.175e-5 2.57 9.104e-5 2.52 2.911e-4 2.00 2.911e-4 2.00 2.911e-4 2.00 2.911e-4 2.00
210 6.988e-6 2.54 2.069e-5 2.42 8.987e-5 2.00 8.986e-5 2.00 8.986e-5 2.00 8.986e-5 2.00
211 1-522e-6 2.52 4.851e-6 2.33 2.719e-5 2.00 2.719e-5 2.00 2.719e-5 2.00 2.719e-5 2.00
212 3.286e-7 - 1.180e-6 - 8.091e-6 - 8.090e-6 - 8.090e-6 - 8.090e-6 -
mesh (7)
242.012e-1 2.88 1.944e-1 2.29 2.356e-1 1.80 2.408e-1 1.76 2.408e-1 1.766 2.408e-1 1.76
255.184e-2 3.06 6.598e-2 3.06 1.010e-1 2.20 1.049e-1 2.17 1.050e-1 2.17 1.050e-1 2.17
261.082e-2 3.10 1.377e-2 3.13 3.276e-2 2.36 3.450e-2 2.34 3.450e-2 2.34 3.450e-2 2.34
272.129e-3 2.96 2.547e-3 2.76 9.157e-3 2.40 9.768e-3 2.37 9.769e-3 2.37 9.769e-3 2.37
284.048e-4 2.94 5.413e-4 2.57 2.381e-3 2.42 2.581e-3 2.36 2.581e-3 2.36 2.581e-3 2.36
297.453e-5 2.93 1.226e-4 2.50 5.907e-4 2.51 6.619e-4 2.33 6.620e-4 2.33 6.620e-4 2.33
210 1.327e-5 2.93 2.818e-5 2.45 1.343e-4 2.82 1.674e-4 2.30 1.674e-4 2.30 1.674e-4 2.30
211 2.295e-6 2.91 6.478e-6 2.43 2.487e-5 2.92 4.211e-5 2.28 4.211e-5 2.28 4.212e-5 2.28
212 3.929e-7 - 1.484e-6 - 4.233e-6 - 1.055e-5 - 1.055e-5 - 1.055e-5 -
mesh (5)
245.240e-3 2.03 3.038e-2 1.97 5.847e-2 1.89 6.790e-2 1.87 6.822e-2 1.86 6.823e-2 1.86
251.282e-3 2.00 7.750e-3 1.94 1.577e-2 1.98 1.857e-2 1.97 1.867e-2 1.97 1.867e-2 1.96
263.186e-4 2.00 2.017e-3 1.96 4.009e-3 1.89 4.754e-3 1.99 4.779e-3 1.99 4.780e-3 1.99
277.954e-5 2.00 5.163e-4 1.99 1.076e-3 1.68 1.195e-3 2.00 1.202e-3 2.00 1.202e-3 2.00
281.987e-5 2.00 1.295e-4 2.00 3.355e-4 2.08 2.993e-4 2.00 3.009e-4 2.00 3.010e-4 2.00
294.969e-6 2.00 3.246e-5 2.00 7.912e-5 2.54 7.487e-5 2.00 7.527e-5 2.00 7.528e-5 2.00
210 1.242e-6 2.00 8.117e-6 2.00 1.357e-5 2.00 1.872e-5 1.99 1.882e-5 2.00 1.882e-5 2.00
211 3.105e-7 2.00 2.029e-6 2.00 3.397e-6 2.00 4.704e-6 1.85 4.705e-6 2.00 4.706e-6 2.00
212 7.764e-8 - 5.073e-7 - 8.494e-7 - 1.300e-6 - 1.176e-6 - 1.176e-6 -
mesh (8)
246.452e-3 2.01 1.209e-2 2.43 3.055e-2 1.96 3.593e-2 1.95 3.654e-2 1.94 3.660e-2 1.94
251.593e-3 2.00 2.234e-3 2.18 7.873e-3 1.69 9.332e-3 1.97 9.496e-3 1.85 9.513e-3 1.97
263.968e-4 2.00 4.897e-4 2.05 2.444e-3 1.78 2.355e-3 2.00 2.397e-2 2.00 2.401e-3 2.00
279.913e-5 2.00 1.177e-4 2.01 7.102e-4 1.96 5.902e-4 2.00 6.000e-4 2.00 6.017e-4 2.00
282.477e-5 2.00 2.913e-5 2.00 1.819e-4 2.48 1.476e-4 2.00 1.502e-4 2.00 1.505e-4 2.00
296.194e-6 2.00 7.264e-6 2.00 3.255e-5 3.27 4.469e-5 2.00 3.757e-5 2.00 3.764e-5 2.00
210 1.548e-6 2.00 1.814e-6 2.00 3.355e-6 1.99 1.354e-5 2.00 9.393e-6 2.00 9.410e-6 2.00
211 3.871e-7 2.00 4.536e-7 2.00 8.462e-7 1.54 3.867e-6 2.00 2.348e-6 2.00 2.352e-6 2.00
212 9.678e-8 - 1.134e-7 - 2.905e-7 - 1.196e-6 - 6.604e-7 - 5.881e-7 -
mesh (9)
Table 1: Values of ENand Ord
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