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Journal of Modern Methods in Numerical Mathematics 10:1-2 (2019), 16–35
A class of difference schemes uniformly
convergent on a modified Bakhvalov mesh
Samir Karasulji´
ca,∗, Helena Zarinb, Enes Duvnjakovi´
ca
aDepartment of Mathematics, Faculty of Sciences, University of Tuzla, Univerzitetska 4, 75000 Tuzla, Bosnia and Herzegovina
bDepartment of Mathematics and Informatics, Faculty of Sciences, University of Novi Sad, Trg Dositeja Obradovi´
ca 4, 21 000
Novi Sad, Serbia
Abstract
In this paper we consider the numerical solution of a singularly perturbed one-dimensional semilinear
reaction-diffusion problem. We construct a class of finite-difference schemes to discretize the problem and
we prove that the discrete system has a unique solution. The central result of the paper is second-order
convergence uniform in the perturbation parameter, which we obtain for the discrete approximate solution
on a modified Bakhvalov mesh. Numerical experiments with two representatives of the class of difference
schemes show that our method is robust and confirm the theoretical results.
Keywords: Singular perturbation, nonlinear, boundary layer, Bakhvalov mesh, layer-adapted mesh,
uniform convergence.
2010 MSC: 65L10, 65L11, 65L50.
1. Introduction
We consider the boundary value problem
ε2y00(x)=f(x,y) on [0,1],(1.1)
y(0) =0,y(1) =0,(1.2)
where 0 <ε<1 is a perturbation parameter and fis a non-linear function. We assume that the nonlinear
function fis continuously differentiable, i.e. for k>2,f∈Ck([0,1] ×R),and that it has a strictly positive
derivative with respect to y
∂f
∂y=fy≥m>0 on [0,1]×R(m=const).(1.3)
∗Corresponding author
Email addresses: samir.karasuljic@untz.ba (Samir Karasulji ´
c), helena.zarin@dmi.uns.ac.rs (Helena Zarin),
enes.duvnjakovic@untz.ba (Enes Duvnjakovi´
c)
Received: 8 January 2019 Accepted: 1 July 2019
http://dx.doi.org/10.20454/jmmnm.2018.1513
2090-8296 c
2019 Modern Science Publishers. All rights reserved.
Karasulji´
c et al., Journal of Modern Methods in Numerical Mathematics 10:1-2 (2019), 16–35 17
The boundary value problem (1.1)–(1.2) under the condition (1.3) has a unique solution, see Lorenz [21].
Differential equations with the small parameter εmultiplying the highest order derivate terms are said to
be singularly perturbed.
Singularly perturbed equations occur frequently in mathematical models of various areas of physics,
chemistry, biology, engineering science, economics and even sociology. These equations appear in analysis
of practical applications, for example in fluid dynamics (aero and hydrodynamics), semiconductor theory,
advection-dominated heat and mass transfer, theory of plates, shellsand chemical kinetics, seismology,
geophysics, nonlinear mechanics and so on.
A common features of singularly perturbed equations is that their solutions have tiny boundary or interior
layers, in which there is a sudden change of the solution’s values of these equations. Such sudden changes
occur e.g. in physics when viscous gas flows at high speed and has contact with a solid surface, then in
chemical reaction, in which besides the reactants, a catalyst is also involved.
Using classical numerical methods such are finite difference methods and finite element methods, which do
not take into account the appearance of the boundary or inner layer, we get results which are unacceptable
from the standpoint of stability, the value of the error or the cost of calculation.
Our goal is to construct a numerical method to overcome the previously listed problems, i.e., to construct
an ε–uniformly convergent numerical method for problem (1.1) −(1.3).
The numerical method is said to be an ε–uniformly convergent in the maximum discrete norm of the order
r, if
y−y
∞6CN−r,
where yis the exact solution of the original continuous problem, yis the numerical solution of a given
continuous problem, Nis the number of mesh points, and Cis a constant which does not depend of Nnor
ε.
Many authors have analyzed and made a great contribution to the study of the problem (1.1)–(1.3) with
different assumptions about the function f; and as well as more general nonlinear problems.
There were many constructed ε–uniformly convergent difference schemes of order 2 and higher (Herceg
[6], Herceg and Surla [11], Herceg and Miloradovi´
c [10], Herceg and Herceg [7], Kopteva and Linß [15],
Kopteva and Stynes [17,18], Kopteva, Pickett and Purtill [16], Linß, Roos and Vulanovi´
c [20], Sun and
Stynes [30,31], Stynes and Kopteva [29], Surla and Uzelac [33], Vulanovi´
c [34,35,36,37,38,39], Kopteva
[14] etc.).
The numerical method which we are going to construct and analyze in this paper is a synthesis of the two
approaches in numerical solving of the problem (1.1)–(1.3), and in an adequate approximation of the given
boundary problem and the use of a layer–adaptive mesh. As mentioned above, the exact solutions of the
singular perturbation boundary value problems usually exhibits sharp boundary or interior layers.
The first approach in numerical solving the singular perturbation boundary value problem is a method of
fitted operators. Construction and analysis of these exponentially fitted differences schemes for solving
linear singular–perturbation problems can be seen in Roos [26], O’Riordan and Stynes [23] etc, while the
appropriate schemes for nonlinear problems can be seen in Niijima [22], O’Riordan and Stynes [24], Stynes
[28] and others. The above mentioned fitted exponential difference schemes are uniformly convergent. In
order to obtain an ε–uniformly convergent method, we need to use a appropriate layer-adapted mesh.
Shishkin mesh [27] and their modification [32,39,19] and others, Bakhvalov mesh [1] and their modification
[6,12,9,10,34,37] and others are the most used layer–adapted meshes.
The method, appropriate for our purpose, was first presented by Boglaev [2], where the discretisation of
the problem (1.1)–(1.3) on a modified Bakhvalov mesh was analysed and first order uniform convergence
with respect to εwas demonstrated.
Karasulji´
c et al., Journal of Modern Methods in Numerical Mathematics 10:1-2 (2019), 16–35 18
Using the method of [2], authors constructed new difference schemes in papers [3] and [4] for the problem
(1.1)–(1.3) and carried out numerical experiments.
In [5,13] authors constructed new difference schemes and proved the uniqueness of the numerical solu-
tion and an ε–uniform convergence on a modified Shishkin mesh, and at the end presented numerical
experiments.
In order to obtain better results, instead of Shishkin mesh, we will use a modification of Bakhvalov mesh.
We have decided to use the modification of Bakhvalov mesh constructed by Vulanovi´
c [37]. This mesh has
the features that we need in our analysis of the numerical method value of the error.
Shishkin mesh is much simpler than Bakhvalov mesh, but difference schemes applied to Bakhvalov mesh
show better results. In order to get better results we used a modification of Bakhvalov mesh.
This paper consist of six parts and it has the following structure. The first part is Introduction. Next,
in Section 2 a class of difference schemes are constructed, and it is proven the theorem of existence and
uniqueness of the numerical solution. Mesh construction is in Section 3. In Section 4, it is showed and
proven the theorem of an ε–uniform convergence. In Section 5 are numerical experiments which confirm
the theoretical results. The last two sections are Conclusion and Acknowledgments.
We use RN+1to denote the real (N+1)–dimensional linear space of all column vectors
u=(u0,u1,...,uN)T.
We equip space RN+1with usual maximum vector norm
kuk∞=max
06i6N|ui|.
The induced norm of a linear mapping A=(aij) : RN+1→RN+1is
kAk∞=max
06i6N
N
X
j=0aij.
Remark 1.1.Throughout this paper we let C, sometimes subscripted, denote a generic positive constant that
may take different values in different formulas, but it is always independent of Nand ε.
2. Construction of the scheme
We will use the well–known Green’s function for the operator Lεy:=ε2y00 −γy,for the construction of the
difference scheme, where γis a constant. The value of γwill be determined later in this section.
This method, as we mentioned in Introduction, first was introduced by Boglaev in his paper [2]. Detailed
construction of difference schemes done by this method, can be found in [5,13]. In [13] was obtained the
following equality
β
sinh(βhi−1)yi−1− β
tanh(βhi−1)+β
tanh(βhi)!yi+β
sinh(βhi)yi+1
=1
ε2
xi
Z
xi−1
uII
i−1(s)ψ(s,y(s))ds +
xi+1
Z
xi
uI
i(s)ψ(s,y(s))ds
,
y0=0,yN=0,i=1,2,··· ,N−1,
(2.1)
where
Karasulji´
c et al., Journal of Modern Methods in Numerical Mathematics 10:1-2 (2019), 16–35 19
ψ(x,y(x)) =f(x,y(x)) −γy(x),
0=x0<x1<x2<··· <xN=1,(2.2)
is an arbitrary mesh on [0,1],hi=xi+1−xi,
β=√γ
ε,(2.3)
functions uI
iand uII
iare the solutions of the next boundary value problem
Lεy=0 on (xi,xi+1),
ui(xi)=1,ui(xi+1)=0,
i=0,1, ..., N−1,
and
Lεy=0 on (xi,xi+1),
ui(xi)=0,ui(xi+1)=1,
i=0,1, ..., N−1.
and
uI
i(x)=sinh β(xi+1−x)
sinh βhi,uII
i(x)=sinh β(x−xi)
sinh βhi,x∈[xi,xi+1],
i=0,1,2, ..., N−1.
We cannot, in general, explicitly compute the integrals on the right-hand side of (2.1). In order to get a
simple enough difference scheme, we approximate the function ψon [xi−1,xi]∪[xi,xi+1] using
ψi=ψ(xi−1,yi−1)+qψ(xi,yi)+ψ(xi+1,yi+1)
q+2,(2.4)
where q∈R+,while yiare approximate values of the solution yof the problem (1.1)–(1.3) at mesh points
xi.
Finally, from (2.1), using (2.4), we get the following difference scheme
(q+1)ai+di+4di+1yi−1−yi−(q+1)ai+1+di+1+4diyi−yi+1
−f(xi−1,yi−1)+q f (xi,yi)+f(xi+1,yi+1)
γ(4di+4di+1)=0,
y0=0,yN=0,i=1,2, ..., N−1,
(2.5)
where ai=1
sinh(βhi−1),di=1
tanh(βhi−1),4di=di−ai.
Let us introduce the discrete problem of the problem (1.1)–(1.3), using (2.5) on the mesh (2.2) we can write
Fy =F0y,F1y,...,FNyT=0,(2.6)
where are
F0y:=y0=0,
Fiy:=γ
4di+4di+1n(q+1)ai+di+4di+1yi−1−yi
−(q+1)ai+1+di+1+4diyi−yi+1
−f(xi−1,yi−1)+q f (xi,yi)+f(xi+1,yi+1)
γ(4di+4di+1)},
i=1,2,...,N−1,
FNy:=yN=0.
(2.7)
Karasulji´
c et al., Journal of Modern Methods in Numerical Mathematics 10:1-2 (2019), 16–35 20
Theorem 2.1. The discrete problem (2.6) for γ≥fy,has the unique solution y,where
y=(y0,y1,y2, ..., yN−1,yN)T∈RN+1.Moreover, for any v,w∈RN+1, the following stabilty inequality holds
kw−vk∞61
mkFw −Fvk∞.(2.8)
Proof. We use a technique from [10,37], the proof of existence of the solution of Fy =0 is based on the proof
of the following relation:
F0y−1
∞≤C,where F0yis a Fr ´
echet derivative of F.
The Fr´
echet derivative H:=F0yis a tridiagonal matrix. Let H=[hij].The non-zero elements of this
tridiagonal matrix are
h0,0=hN,N=1,
hi,i=γ
4di+4di+1−q(ai+ai+1)−2(di+di+1)−q
γ·∂f(xi,yi)
∂yi(4di+4di+1)<0,
hi,i−1=γ
4di+4di+1(4di+4di+1)1−1
γ·∂f(xi−1,yi−1)
∂yi−1+(q+2)ai>0,
hi,i+1=γ
4di+4di+1(4di+1+4di)1−1
γ·∂f(xi+1,yi+1)
∂yi+1+(q+2)ai+1>0,
i=1,2,...,N−1.
(2.9)
Hence His an L–matrix. Let us show that His an M–matrix. Now, we have
|hi,i|−|hi,i−1|−|hi−1,i|
=γ
4di+4di+1
(4di+4di+1)∂f(xi−1,yi−1)
∂yi−1
+q∂f(xi,yi)
∂yi
+∂f(xi+1,yi+1)
∂yi
γ
>(q+2)m.(2.10)
Based on (2.10), we have proved that His an M–matrix. Since His an M–matrix, now we obtain
H−1
∞≤1
(q+2)m.(2.11)
Finally, by the Hadamard Theorem (5.3.10 from [25]), the first statement of our theorem follows.
The second part of the proof is based on the part of the proof of [6]. We have that
Fw −Fv =(F0u)(w−v),for some u=(u0,u1,...,uN)T∈RN+1.(2.12)
and
w−v=(F0u)−1(Fw −Fv).(2.13)
Now, based on (2.11), we have that
kw−vk∞=
(F0u)−1(Fw −Fv)
∞
61
(q+2)mkFw −Fvk∞61
mkFw −Fvk∞.(2.14)
Karasulji´
c et al., Journal of Modern Methods in Numerical Mathematics 10:1-2 (2019), 16–35 21
3. Mesh construction
The exact solution yof the problem (1.1)–(1.2) has boundary layers of exponential type near the points
x=0 and x=1.In order to achieve an ε–uniformly convergence of the numerical method, it is necessary
to use a layer-adapted mesh. In the construction of this mesh the occurrence of boundary or inner layers
needs to be taken in account. We will use the modified Bakhvalov mesh from [37], which has a sufficiently
smooth generating function, that is going to provide the necessary characteristics of the mesh that we need
for further analysis.
The mesh 4:x0<x1< ... < xNis generated by xi=ϕ(ti),ti=i/N=ih,h=1/N,i=0,1,...,N;N=2m,m∈
N\{1},with the mesh generating function
ϕ(t)=
κ(t) :=aεt
p−t,t∈[0, α],
π(t)=:ω(t−α)3+κ00(α)(t−α)2
2+κ0(α)(t−α)+κ(α),t∈[α, 1/2],
1−ϕ(1 −t),t∈[1/2,1],
(3.1)
here pis an arbitrary parameter from (ε?)1/3,1/2, ε ∈(0, ε?] and α=p−ε1/3>0,where we assume that
ε?<1
8.The coefficient ωis determined from π1
2=1
2,we get
ω=1
2−α−31
2−ap1
2−α2+p1
2−αε1/3+αε2/3,
and ais chosen such that ω>0,(such a,independent of ε, obviously exist).
By this choice of αand πwe get
ϕ∈C2[0,1]\n1
2o,(3.2a)
ϕ0(t)6C,t∈[0,1],(3.2b)
and ϕ00(t)6C,t∈[0,1]\n1
2o.(3.2c)
Values of the mesh sizes hi,and values of differences hi+1−hi,will be given in the next lemma.
Lemma 3.1. The mesh sizes hi=xi+1−xi,defined by the generating function (3.1), satisfy
hi6CN−1,i=0,1,...,N−1,(3.3a)
and
|hi−hi−1|6CN−2,i=1,2,...,N−1.(3.3b)
Proof. Due to (3.2b),we have
hi=Z(i+1)/N
i/N
ϕ0(t) d t6CZ(i+1)/N
i/N
dt6CN−1.(3.4)
Let us divide the proof of (3.3b), because of (3.2a), into three parts.
Firstly, when i∈{1,...,N−1}\{N/2−1,N/2,N/2+1},based on (3.2c), we have
|hi−hi−1|=Z(i+1)/N
i/NZt
t−1/N
ϕ00(s) d sdt
6CZ(i+1)/N
i/NZt
t−1/N
dsdt
6CN−2.(3.5)
Secondly, for i=N/2,we get
Karasulji´
c et al., Journal of Modern Methods in Numerical Mathematics 10:1-2 (2019), 16–35 22
hN/2−hN/2−1=1−ϕ(1 −(N/2+1)/N)−1
2−1
2−ϕ((N/2−1)/N)
=ϕ((N/2−1)/N)−ϕ(1 −(N/2+1)/N)
=0,
(3.6)
and finally, for i=N/2−1 or i=N/2+1,we get
|hN/2−1−hN/2−2|=|hN/2+1−hN/2|66ω
N3+µ00(α)+(3 −6α)ω
N26C
N2.(3.7)
Now, using (3.4), (3.5), (3.6) and (3.7) the inequalities (3.3a) and (3.3b) are proven.
4. Uniform convergence
In this section we prove the theorem on ε–uniform convergence of the discrete problem (2.6). The proof of
the theorem is based on relation
y−y
∞6C
Fy −Fy
∞.
Stability of the difference sheme is proven in Theorem 2.1, and as Fy =0,it is enough to estimate the value
of the expression
Fy
∞.
The proof uses the decomposition of the solution yto the problem (1.1)–(1.2) to a layer sand a regular
component r, given in the following assertion.
Theorem 4.1. [34] The solution yto problem (1.1)–(1.2) can be represented in the following way:
y=r+s,
where for j=0,1,...,k+2 and x∈[0,1] we have
r(j)(x)6C,(4.1)
and s(j)(x)6Cε−je−x
ε√m+e−1−x
ε√m.(4.2)
Proof. See in Vulanovi´
c [34].
Note that e−x
ε√m>e−1−x
ε√m,∀x∈[0,1/2] and e−x
ε√m6e−1−x
ε√m,∀x∈[1/2,1].These inequalities and the
estimate (4.2) imply that the analysis of the error value can be done for
Fy
∞on the part of the mesh which
corresponds to [0,1/2] omitting the function e−1−x
ε√m,keeping in mind that on this part of the mesh we have
that hi−16hi.An analogous analysis holds for the part of the mesh which corresponds to x∈[1/2,1],but
with the omission of the function e−x
ε√mand using the inequality hi−1>hi.
In order to simplify our analysis, let us write Fiy,i=1,2,...,N−1,in the following form
Fiy=γyi−1−2yi+yi+1
+γ˜
q
ai(yi−1−yi)−ai+1(yi−yi+1)−f(xi,yi)
γ(∆di+ ∆di+1)
∆di+ ∆di+1
+2γai(yi−1−yi)−ai+1(yi−yi+1)
∆di+ ∆di+1
,
−f(xi−1,yi−1)+f(xi+1,yi+1)(∆di+ ∆di+1)
∆di+ ∆di+1
,
i=1, ..., N−1.
(4.3)
Karasulji´
c et al., Journal of Modern Methods in Numerical Mathematics 10:1-2 (2019), 16–35 23
Using marks Pi,Qi,Ri,we have
Fiy=Pi+Qi+Ri,i=1,2,...,N−1,(4.4)
where
Pi=γyi−1−2yi+yi+1,(4.5)
and using Taylor expansions for yi−1and yi+1,we get
Qi=γq y0
i
hisinh(βhi−1)−hi−1sinh(βhi)
sinh(βhi)(cosh(βhi−1)−1) +sinh(βhi−1)(cosh(βhi)−1)
+y00
i
2·h2
i−1sinh(βhi)+h2
isinh(βhi−1)
sinh(βhi)(cosh(βhi−1)−1) +sinh(βhi−1)(cosh(βhi)−1)
−ε2y00
i
γ·sinh(βhi)(cosh(βhi−1)−1) +sinh(βhi−1)(cosh(βhi)−1)
sinh(βhi)(cosh(βhi−1)−1) +sinh(βhi−1)(cosh(βhi)−1)
+y000
i
6·h3
isinh(βhi−1)−h3
i−1sinh(βhi)
sinh(βhi)(cosh(βhi−1)−1) +sinh(βhi−1)(cosh(βhi)−1)
+y(iv)(ζ−
i−1)h4
i−1sinh(βhi)+y(iv)(ζ+
i)h4
isinh(βhi−1)
24(sinh(βhi)(cosh(βhi−1)−1) +sinh(βhi−1)(cosh(βhi)−1))
,
(4.6)
and
Ri=2γy0
i
hisinh(βhi−1)−hi−1sinh(βhi)
sinh(βhi)(cosh(βhi−1)−1) +sinh(βhi−1)(cosh(βhi)−1)
+2γy00
i
1
2·h2
i−1sinh(βhi)+h2
isinh(βhi−1)
sinh(βhi)(cosh(βhi−1)−1) +sinh(βhi−1)(cosh(βhi)−1)
−1
β2·sinh(βhi)(cosh(βhi−1)−1) +sinh(βhi−1)(cosh(βhi)−1)
sinh(βhi)(cosh(βhi−1)−1) +sinh(βhi−1)(cosh(βhi)−1)!
+γy000
i
3·h3
isinh(βhi−1)−h3
i−1sinh(βhi)
sinh(βhi)(cosh(βhi−1)−1) +sinh(βhi−1)(cosh(βhi)−1)
+y000
iε2(hi−1−hi)
+2γ
y(iv)(ζ−
i−1)h4
i−1
24 sinh(βhi)+y(iv)(ζ+
i)h4
i
24 sinh(βhi−1)
sinh(βhi)(cosh(βhi−1)−1) +sinh(βhi−1)(cosh(βhi)−1)
−ε2
2hy(iv)(µ−
i−1)h2
i−1+y(iv)(µ+
i)h2
ii
(4.7)
i=1,2,...,N−1 and ζ−
i−1, µ−
i−1∈(xi−1,xi), ζ+
i, µ+
i∈(xi,xi+1).
Lemma 4.2. Let the mesh size hi=xi+1−xi,be defined by the generating function (3.1). It holds the estimate
yi−1−yi−(yi−yi+1)6Cy0
i(hi−hi−1)+y00(δ−
i−1)
2h2
i−1+y00(δ+
i)
2h2
i,
i=1,...,N/2,(4.8)
where are δ−
i−1∈(xi−1,xi), δ+
i∈(xi,xi+1).
Proof. The proof is trivial, using Taylor expansions for yi−1and yi+1we obtain (4.8).
Karasulji´
c et al., Journal of Modern Methods in Numerical Mathematics 10:1-2 (2019), 16–35 24
Lemma 4.3. Let the mesh size hi=xi+1−xi,be defined by the generating function (3.1) and βdefined by
(2.3). It holds estimate
y0
i|hisinh(βhi−1)−hi−1sinh(βhi)|
sinh(βhi)(cosh(βhi−1)−1)+sinh(βhi−1)(cosh(βhi)−1) 6Cy0
i(hi−hi−1),i=1,...,N/2.(4.9)
Proof. We have that
y0
ihisinh(βhi−1)−hi−1sinh(βhi)
sinh(βhi)(cosh(βhi−1)−1) +sinh(βhi−1)(cosh(βhi)−1)
=y0
i
βhi−1hiP+∞
n=1
β2n(h2n
i−h2n
i−1)
(2n+1)!
sinh(βhi)(cosh(βhi−1)−1) +sinh(βhi−1)(cosh(βhi)−1)
6y0
i
βhi−1hi(h2
i−h2
i−1)P+∞
n=1
β2nh2n−2
i
(2n)!
sinh(βhi)(cosh(βhi−1)−1) +sinh(βhi−1)(cosh(βhi)−1)
62y0
i
β2hi−1hi(hi−hi−1)P+∞
n=0
β2n+1h2n+1
i
(2n+2)!
4 sinh βhi−1
2sinh βhi
2sinh βhi−1+βhi
2
=2y0
i
β2hi−1hi(hi−hi−1)cosh(βhi)−1
βhi
4 sinh βhi−1
2sinh βhi
2sinh βhi−1+βhi
2
=y0
i
βhi−1(hi−hi−1) sinh2βhi
2
sinh βhi−1
2sinh βhi
2sinh βhi−1+βhi
2
6Cy0
i(hi−hi−1).
(4.10)
Remark 4.4.It is true that P+∞
n=0x2n+1
(2n+2)! =cosh x−1
x,cosh x−1=2 sinh2x
2
and sinh x(cosh y−1) +sinh y(cosh x−1) =4 sinh x
2sinh y
2sinh x+y
2.
Lemma 4.5. Let the mesh size hi=xi+1−xi,be defined by the generating function (3.1) and βdefined by
(2.3). We have the following estimate
y00
i
1
2[h2
i−1sinh(βhi)+h2
isinh(βhi−1)]−ε2
γ[sinh(βhi)(cosh(βhi−1)−1)+sinh(βhi−1)(cosh(βhi)−1)]
sinh(βhi)(cosh(βhi−1)−1)+sinh(βhi−1)(cosh(βhi)−1)
6Cy00
ih2
i,i=1,...,N/2.
(4.11)
Karasulji´
c et al., Journal of Modern Methods in Numerical Mathematics 10:1-2 (2019), 16–35 25
Proof. We have that
y00
i
1
2[h2
i−1sinh(βhi)+h2
isinh(βhi−1)]−ε2
γ[sinh(βhi)(cosh(βhi−1)−1)+sinh(βhi−1)(cosh(βhi)−1)]
sinh(βhi)(cosh(βhi−1)−1)+sinh(βhi−1)(cosh(βhi)−1)
=y00
i
sinh(βhi)h2
i−1
2−cosh(βhi−1)−1
β2+sinh(βhi−1)h2
i
2−cosh(βhi)−1
β2
sinh(βhi)(cosh(βhi−1)−1) +sinh(βhi−1)(cosh(βhi)−1)
=y00
i
h2
i−1sinh(βhi)β2h2
i−1
4! +β4h4
i−1
6! +···
sinh(βhi)(cosh(βhi−1)−1) +sinh(βhi−1)(cosh(βhi)−1)
+
h2
isinh(βhi−1)β2h2
i
4! +β4h4
i
6! +···
sinh(βhi)(cosh(βhi−1)−1) +sinh(βhi−1)(cosh(βhi)−1)
6y00
i
h2
i−1sinh(βhi)cosh(βhi−1)−1+h2
isinh(βhi−1)cosh(βhi)−1
sinh(βhi)(cosh(βhi−1)−1) +sinh(βhi−1)(cosh(βhi)−1)
6y00
ih2
i
h2
i−1
h2
i
sinh(βhi)cosh(βhi−1)−1+sinh(βhi−1)cosh(βhi)−1
sinh(βhi)(cosh(βhi−1)−1) +sinh(βhi−1)(cosh(βhi)−1)
6Cy00
ih2
i.
(4.12)
Remark 4.6.It is true that P+∞
n=0x2n+2
(2n+4)! =cosh x−1−x2
2
x2,0<
cosh x−1−x2
2
x2
cosh x−1<1,and
cosh x−1−x2
2
x26C(cosh x−1).
Lemma 4.7. Let the mesh size hi=xi+1−xi,be defined by the generating function (3.1) and βdefined by
(2.3). We have the following estimate
y000
i
h3
isinh(βhi−1)−h3
i−1sinh(βhi)
sinh(βhi)(cosh(βhi−1)−1) +sinh(βhi−1)(cosh(βhi)−1)
6Cy000
iε2+h2
i−1(hi−hi−1),i=1,...,N/2.(4.13)
Karasulji´
c et al., Journal of Modern Methods in Numerical Mathematics 10:1-2 (2019), 16–35 26
Proof. We have that
y000
i
h3
isinh(βhi−1)−h3
i−1sinh(βhi)
sinh(βhi)(cosh(βhi−1)−1) +sinh(βhi−1)(cosh(βhi)−1)
6y000
i
βhi−1hi(hi−1+hi)(hi−hi−1)
βhi
β2h2
i−1
2+βhi−1
β2h2
i
2
+β3h3
i−1h3
iP+∞
n=1
β2n(h2n
i−h2n
i−1)
(2n+3)!
sinh(βhi)(cosh(βhi−1)−1)+sinh(βhi−1)(cosh(βhi)−1)
=y000
i
2
γε2(hi−hi−1)
+β3h3
i−1h3
i(h2
i−h2
i−1)P+∞
n=0
β2n+2(n+1)h2n
i
(2n+5)!
sinh(βhi)(cosh(βhi−1)−1) +sinh(βhi−1)(cosh(βhi)−1)
6y000
i
2
γε2(hi−hi−1)+β3h3
i−1h3
i(hi−hi−1)(hi−1+hi)P+∞
n=0
β2n+2h2n
i
(2n+4)!
4 sinh βhi−1
2sinh βhi
2sinh β(hi−1+hi)
2
6y000
i
2
γε2(hi−hi−1)+2βh3
i−1(hi−hi−1)P+∞
n=0
β2n+4h2n+4
i
(2n+4)!
4 sinh βhi−1
2sinh βhi
2sinh β(hi−1+hi)
2
6y000
i
2
γε2(hi−hi−1)+h2
i−1(hi−hi−1)
βhi−1
2
sinh βhi−1
2·cosh(βhi)−1−β2h2
i
2
sinh βhi
2sinh β(hi−1+hi)
2
6Cy000
iε2+h2
i−1|hi−hi−1|.
(4.14)
Remark 4.8.It is true that P+∞
n=0
β2n+4h2n+4
i
(2n+4)! =cosh(βhi)−1−β2h2
i
2and
0<cosh(βhi)−1−β2h2
i
2
sinh βhi
2sinh β(hi−1+hi)
2
<2.
Lemma 4.9. Let the mesh size hi=xi+1−xi,be defined by the generating function (3.1) and βdefined by
(2.3). We have the following estimate
y(iv)(ζ−
i−1)h4
i−1sinh(βhi)+y(iv)(ζ+
i)h4
isinh(βhi−1)
sinh(βhi)(cosh(βhi−1)−1) +sinh(βhi−1)(cosh(βhi)−1)
6Cε2y(iv)(ζ−
i−1)h2
i−1+y(iv)(ζ+
i)h2
i,i=1,...,N/2.(4.15)
Karasulji´
c et al., Journal of Modern Methods in Numerical Mathematics 10:1-2 (2019), 16–35 27
Proof. We have that
y(iv)(ζ−
i−1)h4
i−1sinh(βhi)+y(iv)(ζ+
i)h4
isinh(βhi−1)
sinh(βhi)(cosh(βhi−1)−1) +sinh(βhi−1)(cosh(βhi)−1)
6y(iv)(ζ−
i−1)h4
i−1sinh(βhi)
sinh(βhi)β2h2
i−1
2+sinh(βhi−1)β2h2
i
2
+y(iv)(ζ+
i)h4
isinh(βhi−1)
sinh(βhi)β2h2
i−1
2+sinh(βhi−1)β2h2
i
2
62
y(iv)(ζ−
i−1)h4
i−1sinh(βhi)
β2h2
i−1sinh(βhi)+sinh(βhi−1)+y(iv)(ζ+
i)h4
isinh(βhi−1)
β2sinh(βhi−1)h2
i−1+h2
i
6Cε2y(iv)(ζ−
i−1)h2
i−1+y(iv)(ζ+
i)h2
i.
(4.16)
Let us continue with the following lemma that will be further used in the proof of the ε–uniform convergence
theorem. In the lemma quite rough estimate of Fi, is given but it is fairly enough for our needs.
Lemma 4.10. Let the mesh size hi=xi+1−xi,be defined by the generating function (3.1) and βdefined by
(2.3). We have the following estimate
Fiy6C|si−1|ε2
h2
i−1
+ε2
h2
i
+4γ+q+2+1
N2,i=1,2,...,N/2.(4.17)
Proof. For Fiy,i=1,2,...,N/2 holds
Fiy=
γ
4di+4di+1(q+1)ai+di+4di+1(yi−1−yi)
−(q+1)ai+1+di+1+4di(yi−yi+1)
−f(xi−1,yi−1)+q f (xi,yi)+f(xi+1,yi+1)
γ(4di+4di+1)}
=
γ
4di+4di+1(q+1)ai+di+4di+1(yi−1−yi)
−(q+1)ai+1+di+1+4di(yi−yi+1)
−ε2y00
i−1+qy00
i+y00
i+1
γ(4di+4di+1)}
6γ(q+2)
ai(si−1−si)−ai+1(si−si+1)
4di+4di+1+γ|si−1−2si+si+1|+ε2s00
i−1+qs00
i+s00
i+1
+
γ
4di+4di+1(q+1)ai+di+4di+1(ri−1−ri)
−(q+1)ai+1+di+1+4di(ri−ri+1)−ε2r00
i−1+qr00
i+r00
i+1
γ(4di+4di+1)
(4.18)
For the layer component s,due to (4.2), we have
Karasulji´
c et al., Journal of Modern Methods in Numerical Mathematics 10:1-2 (2019), 16–35 28
γ(˜
q+2)
ai(si−1−si)−ai+1(si−si+1)
∆di+ ∆di+1
+γ|si−1−2si+si+1|+ε2s00
i−1+˜
qs00
i+s00
i+1
=γ(˜
q+2)
1
sinh(βhi−1)
cosh(βhi−1)−1
sinh(βhi−1)+cosh(βhi)−1
sinh(βhi)
(si−1−si)
−
1
sinh(βhi)
cosh(βhi−1)−1
sinh(βhi−1)+cosh(βhi)−1
sinh(βhi)
(si−si+1)
+γ|si−1−2si+si+1|+ε2s00
i−1+˜
qs00
i+s00
i+1
6γ(˜
q+2)
si−1−si
cosh(βhi−1)−1
+
si−si+1
cosh(βhi)−1!
+γ|si−1−2si+si+1|+ε2s00
i−1+˜
qs00
i+s00
i+1
6Ce−xi−1
ε√m
ε2
h2
i−1
+ε2
h2
i
+4γ+˜
q+2
.(4.19)
Now, for the regular component r,due to mesh sizes (3.3a), (3.3b), the estimate (4.1), expansions (4.4), (4.5)
(4.6), (4.7), and Lemma 4.2– Lemma 4.9, we have
γ
4di+4di+1(q+1)ai+di+4di+1(ri−1−ri)
−(q+1)ai+1+di+1+4di(ri−ri+1)−ε2ri−1+qri+ri+1
γ(4di+4di+1)o
6C
N2.(4.20)
Using (4.19) and (4.20) completes the proof of the lemma.
Now we can state and prove the main theorem on ε–uniform convergence.
Theorem 4.11. The discrete problem (2.7) on the Bakhvalov–type mesh from Section 3 is uniformly conver-
gent with respect to εand
max
06i6Ny(xi)−yi≤C
N2,
where yis a solution of the problem (1.1)–(1.3), yis the corresponding solution of (2.7) and C>0 is a
constant independent of Nand ε.
Proof. From (4.4),due to Lemma 4.2, Lemma 4.3, Lemma 4.5, Lemma 4.7 and Lemma 4.9 we have
Giy6Cy0
i(hi−hi−1)+y00(δ−
i−1)
2h2
i−1+y00(δ+
i)
2h2
i+y00
ih2
i
+y000
iε2+h2
i−1(hi−hi−1)+ε2y(iv)(ζ−
i−1)+y(iv)(µ−
i−1)h2
i−1
+ε2y(iv)(ζ+
i)+y(iv)(µ+
i)h2
i,i=1,2, . . . N/2.(4.21)
The statement of the theorem for the regular component due to (4.20) is proved.
For the layer component s,holds the estimate (4.2). On the observed part of mesh, we have that e−x
ε√m>
e−1−x
ε√m,now it is enough to estimate e−x
ε√m, on this part of mesh.
We use a technique from [1] and [6,34,37].
Karasulji´
c et al., Journal of Modern Methods in Numerical Mathematics 10:1-2 (2019), 16–35 29
Case I
Let ti−1>τ, we have that
e−xi−1
ε√m6e−κ(τ)
ε√m=e−a√mp−3
√ε
3
√ε6e−C
3
√ε.(4.22)
Now for the layer component sdue to (4.21), holds
s0
i(hi−hi−1)+s00(δ−
i−1)
2h2
i−1+s00(δ+
i)
2h2
i+s00
ih2
i
+s000
iε2+h2
i−1(hi−hi−1)+ε2s(iv)(ζ−
i−1)+s(iv)(µ−
i−1)h2
i−1
+ε2s(iv)(ζ+
i)+s(iv)(µ+
i)h2
i
6C1
e−C
3
√ε
ε(hi−hi−1)+C2
e−C
3
√ε
ε2h2
i
+C3
e−C
3
√ε
ε3ε2+h2
i−1(hi−hi−1)6C
N2.
(4.23)
Case II
Let ti−1< τ iti−16p−3h,h=1/N.From ti−16p−3h⇔p−ti−1>3h⇔p−ti+1>hand p−ti−1=p−ti+1+2h,
we have that
p−ti+1>p−ti−1
3.(4.24)
Also, there holds the following estimate
e−xi−1
ε√m=e−a√mti−1
p−ti−16Ce−a√mp
p−ti−1.(4.25)
From the construction of the mesh (3.1), it implies
κ(k)(τ)=π(k)(τ),k∈{0,1,2}(4.26)
and
κ000(t)−π000(t)=6aεp
(p−t)4−6ω>6aεp
(p−τ)4−6ω=6 ap
3
√ε−ω!,t∈τ, p.(4.27)
Hence, for sufficiently small ε, we have ap
3
√ε−ω > 0,(4.28a)
κ000(t)−π000(t)>0,t∈τ, p,(4.28b)
and, due to (4.26) and (4.28b), we get
κ(k)(t)> π(k)(t),k∈{0,1,2},t∈τ, p.(4.28c)
Now, because of (3.1), (4.24) and (4.28c) we have
hi6Z(i+1)/N
i/N
κ0(t) d t6κ0(ti+1)
N=aεp
(p−ti+1)2·1
N69aεp
(p−ti−1)2·1
N(4.29a)
and
hi−hi−16Z(i+1)/N
i/NZt
t−1/N
κ00(s) d sdt
6κ00(ti+1)
N2=2aεp
(p−ti+1)3·1
N2654aεp
(p−ti−1)3·1
N2.(4.29b)
Karasulji´
c et al., Journal of Modern Methods in Numerical Mathematics 10:1-2 (2019), 16–35 30
Finally, for the layer component sfrom (4.21), due to (4.25), (4.29a) and (4.29b), we obtain the estimate
s0
i(hi−hi−1)+s00(δ−
i−1)
2h2
i−1+s00(δ+
i)
2h2
i+s00
ih2
i
+s000
iε2+h2
i−1(hi−hi−1)+ε2s(iv)(ζ−
i−1)+s(iv)(µ−
i−1)h2
i−1
+ε2s(iv)(ζ+
i)+s(iv)(µ+
i)h2
i
6C1
e−xi−1
ε√m
ε(hi−hi−1)+C2
e−xi−1
ε√m
ε2h2
i
+C3
e−xi−1
ε√m
ε3ε2+h2
i−1(hi−hi−1)+C4ε2e−xi−1
ε√m
ε4h2
i
6C5
e−a√mp
p−ti−1
ε·54aεp
(p−ti−1)3·1
N2+2e−a√mp
p−ti−1
ε2·81a2ε2p2
(p−ti−1)4·1
N2
+e−a√mp
p−ti−1
ε3· ε2+81a2ε2p2
(p−ti−1)4·1
N2!·54aεp
(p−ti−1)3·1
N2
+4ε2e−a√mp
p−ti−1
ε4·81a2ε2p2
(p−ti−1)4·1
N2
6C6
e−a√mp
p−ti−1
(p−ti−1)3+e−a√mp
p−ti−1
(p−ti−1)4+e−a√mp
p−ti−1
(p−ti−1)7·1
N2
·1
N2
6C
N2.(4.30)
Case III
At the end, let p−3h<ti−1< τ, there holds
e−xi−1
ε√m=e−a√m(p−3h)
3h6Ce−a√mp
3h.(4.31)
On the observed part of the mesh is hi−16hi,and for the mesh sizes hi−1,we have
hi−1=κ(ti)−κ(ti−1)=κ0(θi−1)h=aεph
(p−θi−1)2>aεph
(p−(p−3h))2=aεp
9h, θi−1∈[ti−1,ti].(4.32)
Now, due to (4.31), (4.32) and (4.17) we get
Giy6C1e−xi−1
ε√mε2
h2
i−1
+ε2
h2
i
+4γ+˜
q+2+1
N2
6C2e−xi−1
ε√m2h2+4γ+˜
q+2+1
N26C
N2.(4.33)
Case 3h63
√εis proved in Case I and Case II.
According to (4.20), (4.21), (4.23), (4.30) and (4.33), the proof of the theorem is complete.
Karasulji´
c et al., Journal of Modern Methods in Numerical Mathematics 10:1-2 (2019), 16–35 31
5. Numerical results
In this section we present the numerical results to confirm the uniform accuracy of the discrete problem
(2.6) using the mesh from Section 3. To demonstrate the efficiency of the method, we present two examples
having boundary layers.
Example 5.1. Consider the following problem from [37]
ε2y00 =y−1 for x∈[0,1],(5.1a)
y(0) =y(1) =0.(5.1b)
The exact solution of this problem (5.1a) −(5.1b) is given by
y(x)=1−e−x
ε+e−1−x
ε
1+e−1
ε
.(5.2)
The appropriate system was solved using initial guess y0=0,y01,...,y0N−1,0T,y0i=1,i=1,...,N−1,by
Newton’s method. The value of the constant γ=1 has been chosen so that the condition γ>fy(x,y),∀(x,y)∈
[0,1] ×[yL,yU]⊂[0,1] ×Ris fulfilled, where yLand yUare the lower and the upper solutions of the test
problem (5.1a)–(5.1b) and their values are yL=0 and yU=1.
Because of the fact that the exact solution is known, we define the computed error Enand the computed
rate of convergence Ord in the usual way
EN=
y−yN
∞(5.3)
and
Ord =ln EN−ln E2N
ln 2 .(5.4)
Other values of constants are m=1,q=4,a=1 and p=0.4.
The values of ENand Ord are given at the of the paper in Appendix (Table 1).
Example 5.2. Consider the following problem
ε2y00 =(y−1)(1 +(y−1)2) for x∈[0,1],(5.5a)
y(0) =y(1) =0,(5.5b)
The exact solution of the problem (5.5a) −(5.5b) is unknown. The system of nonlinear equations is solved
by Newton’s method with initial guess y0=0,y01,...,y0N−1,0T,y0i=1,i=1,...,N−1.The value of the
constant γhas been chosen so that local version of the condition (1.3) is fulfilled. In other words, because
fy(x,y)>m>0,∀(x,y)∈[0,1] ×[yL,yU],and in our example yL=0,yU=1, and 1 6fy(x,y)64,∀(x,y)∈
[0,1] ×[yL,yU],we get that γ=4.
Because the fact that we do not know the exact solution, we will replace yby ˆ
yin order to calculate ENand
Ord, where ˆ
yis the numerical solution of (5.5a) −(5.5b) obtained by using N=16384,(same procedure is
applied in [12], [8]).
Now, we calculate the value of error ENand the the rate of convergence on the following way
EN=
ˆ
y−yN
∞(5.6)
and
Ord =ln EN−ln E2N
ln 2 .(5.7)
Other values of constants are m=1,q=4,a=1 and p=0.3.
The values of ENand Ord are given at the of the paper in Appendix (Table 2).
Karasulji´
c et al., Journal of Modern Methods in Numerical Mathematics 10:1-2 (2019), 16–35 32
6. Conclusion
In this paper we presented a discretization of an one–dimensional semilinear reaction–diffusion problem,
with suitable assumptions that have ensured the existence and uniqueness of the continuous problem. We
constructed a class of difference schemes, and we proved the existence and uniqueness of the numerical
solution, after which we proved the ε–uniform convergence using a suitable layer–adaptive mesh. Finally
we performed a numerical experiments to confirm the theoretical results.
7. Acknowledgment
This paper is the part of Project ”Numeriˇ
cko rjeˇ
savanje kvazilinearnog singularno–perturbacionog jednodi-
menzionalnog rubnog problema”. The paper has emanated from research conducted with the partial finan-
cial support of Ministry of education and sciences of Federation of Bosnia and Herzegovina and University
of Tuzla under grant 01/2-3995-V/17 of 18.12.2017.
Karasulji´
c et al., Journal of Modern Methods in Numerical Mathematics 10:1-2 (2019), 16–35 33
Appendix
N EnOrd EnOrd EnOrd
262.4133e−04 2.00 1.0062e−03 1.98 1.3294e−03 1.96
276.0436e−05 2.00 2.5429e−04 1.99 3.4128e−04 1.99
281.5095e−05 2.00 6.3802e−05 2.00 8.5934e−05 2.00
293.7691e−06 1.97 1.5961e−05 2.00 2.1523e−05 2.00
210 9.6433e−07 2.00 3.9909e−06 2.00 5.3833e−06 2.00
211 2.4108e−07 2.00 9.9777e−07 2.00 1.3460e−06 2.00
212 6.0271e−08 2.00 2.4945e−07 2.00 3.3651e−07 2.00
213 1.5068e−08 −6.2363e−08 −8.4127e−08 −
ε2−32−52−10
N EnOrd EnOrd EnOrd
261.3243e−03 1.96 1.3243e−03 1.96 1.3243e−03 1.96
273.3945e−04 1.99 3.3945e−04 1.99 3.3945e−04 1.99
288.5413e−05 2.00 8.5413e−05 2.00 8.5413e−05 2.00
292.1388e−05 2.00 2.1388e−05 2.00 2.1388e−05 2.00
210 5.3493e−06 2.00 5.3493e−06 2.00 5.3493e−06 2.00
211 1.3375e−06 2.00 1.3375e−06 2.00 1.3375e−06 2.00
212 3.3438e−07 2.00 3.3438e−07 2.00 3.3438e−07 2.00
213 8.3595e−08 −8.3595e−08 −8.3595e−08 −
ε2−15 2−25 2−30
N EnOrd EnOrd EnOrd
261.3243e−03 1.96 1.3243e−03 1.96 1.3243e−03 1.96
273.3945e−04 1.99 3.3945e−04 1.99 3.3945e−04 1.99
288.5413e−05 2.00 8.5413e−05 2.00 8.5413e−05 2.00
292.1388e−05 2.00 2.1388e−05 2.00 2.1388e−05 2.00
210 5.3493e−06 2.00 5.3493e−06 2.00 5.3493e−06 2.00
211 1.3375e−06 2.00 1.3375e−06 2.00 1.3375e−06 2.00
212 3.3438e−07 2.00 3.3438e−07 2.00 3.3438e−07 2.00
213 8.3595e−08 −8.3595e−08 −8.3595e−08 −
ε2−35 2−40 2−45
Table 1: Error ENand convergence rates Ord for approximate solution.
Karasulji´
c et al., Journal of Modern Methods in Numerical Mathematics 10:1-2 (2019), 16–35 34
N EnOrd EnOrd EnOrd
265.4721e−04 1.99 2.4570e−03 1.96 2.7149e−03 1.93
271.3773e−04 2.00 6.3358e−04 1.99 7.1008e−04 1.98
283.4477e−05 2.00 1.5962e−04 2.00 1.7971e−04 2.00
298.6159e−06 2.00 3.9985e−05 2.00 4.5044e−05 2.00
210 2.1478e−06 2.02 9.9654e−06 2.02 1.1237e−05 2.02
211 5.3066e−07 2.07 2.4624e−06 2.07 2.7767e−06 2.07
212 1.2635e−07 2.07 5.8629e−07 2.07 6.6115e−07 2.07
213 3.0091e−08 −1.3963e−07 −1.5746e−07 −
ε2−32−52−10
N EnOrd EnOrd EnOrd
261.2281e−03 1.80 1.2276e−03 1.80 1.2276e−03 1.80
273.5293e−04 1.94 3.5247e−04 1.94 3.5247e−04 1.94
289.1947e−05 1.98 9.1749e−05 1.98 9.1749e−05 1.98
292.3235e−05 2.00 2.3180e−05 2.00 2.3180e−05 2.00
210 5.8267e−06 2.00 5.8140e−06 2.00 5.8140e−06 2.00
211 1.4577e−06 2.00 1.4544e−06 2.00 1.4544e−06 2.00
212 3.6449e−07 2.00 3.6366e−07 2.00 3.6366e−07 2.00
213 9.1127e−08 −9.0921e−08 −9.0921e−08 −
ε2−15 2−25 2−30
N EnOrd EnOrd EnOrd
261.2276e−03 1.80 1.2276e−03 1.80 1.2276e−03 1.80
273.5247e−04 1.94 3.5247e−04 1.94 3.5247e−04 1.94
289.1749e−05 1.98 9.1749e−05 1.98 9.1749e−05 1.98
292.3180e−05 2.00 2.3180e−05 2.00 2.3180e−05 2.00
210 5.8140e−06 2.00 5.8140e−06 2.00 5.8140e−06 2.00
211 1.4544e−06 2.00 1.4544e−06 2.00 1.4544e−06 2.00
212 3.6367e−07 2.00 3.6367e−07 2.00 3.6367e−07 2.00
213 9.0921e−08 −9.0921e−08 −9.0921e−08 −
ε2−35 2−40 2−45
Table 2: Error ENand convergence rates Ord for approximate solution.
References
[1] N. S. Bakhvalov, Towards optimization of methods for solving boundary value problems in the presence of boundary layers, Zh. Vychisl.
Mat. i Mat. Fiz. 9(1969), 841–859 ((In Russian)).
[2] I. P. Boglaev, Approximate solution of a non-linear boundary value problem with a small parameter for the highest-order differential, Zh.
Vychisl. Mat. i Mat. Fiz. 24 (1984), no. 11, 1649 – 1656, In Russian.
[3] E. Duvnjakovi´
c and S. Karasulji´
c, Difference scheme for semilinear reaction-diffusion problem on a mesh of bakhvalov type, Mathe-
matica Balkanica 25, Fasc. 5 (2011), 499–504.
[4] E. Duvnjakovi´
c, S. Karasulji´
c, and N. Okiˇ
ci´
c, Difference Scheme for Semilinear Reaction-Diffusion Problem, (2010), 793–796.
[5] E. Duvnjakovi´
c, S. Karasulji´
c, V. Paˇ
si´
c, and H. Zarin, A uniformly convergent difference scheme on a modified shishkin mesh for the
singularly perturbed reaction-diffusion boundary value problem, Journal of Modern Methods in Numerical Mathematics 6(2015),
no. 1, 28–43.
[6] D. Herceg, Uniform fourth order difference scheme for a singular perturbation problem, Numer. Math. 56 (1989), no. 7, 675–693.
[7] D. Herceg and Dj. Herceg, On a fourth-order finite difference method for nonlinear two-point boundary value problems, Novi Sad J.
Math 33 (2003), no. 2, 173–180.
[8] D. Herceg and H. Maliˇ
ci´
c, The mesh chopping algorithm for singularly perturbed boundary value problem, Novi Sad J.Math 30
(2000), no. 1, 155–164.
[9] D. Herceg, H. Maliˇ
ci´
c, and I. Liki´
c, On a finite difference analogue of fourth order for boundary value problem, Novi Sad J. Math. 30
(2000), no. 1, 197–203.
Karasulji´
c et al., Journal of Modern Methods in Numerical Mathematics 10:1-2 (2019), 16–35 35
[10] D. Herceg and M. Miloradovi´
c, On numerical solution of semilinear singular perturbation problems by using the hermite scheme on
a new bakhvalov-type mesh, Novi Sad J. Math 33 (2003), no. 1, 145–162.
[11] D. Herceg and K. Surla, Solving a nonlocal singularly perturbed problem by spline in tension, Review of research Faculty of
Sciences-University of Novi Sad Vol.21, No.2 (1991), 119–132.
[12] D. Herceg, K. Surla, and S. Rapaji´
c, Cubic spline difference scheme on a mesh of a bakhvalov type, Novi Sad J. Math. 28 (1998), no. 3,
41–49.
[13] S. Karasulji´
c, E. Duvnjakovi´
c, and H. Zarin, Uniformly convergent difference scheme for a semilinear reaction-diffusion problem,
Advances in Mathematics: Scientific Journal 4(2015), no. 2, 139–159.
[14] N. Kopteva, Maximum norm error analysis of a 2d singularly perturbed semilinear reaction-diffusion problem, Mathematics of
Computation 76 (2007), no. 258, 631–646.
[15] N. Kopteva and T. Linß, Uniform second-order pointwise convergence of a central difference approximation for a quasilinear convection-
diffusion problem, J. Comput. Appl. Math. 137 (2001), no. 2, 257–267.
[16] N. Kopteva, M. Pickett, and H. Purtill, A robust overlapping schwarz method for a singularly perturbed semilinear reaction-diffusion
problem with multiple solutions, Int. J. Numer. Anal. Model 6(2009), 680–695.
[17] N. Kopteva and M. Stynes, A robust adaptive method for a quasi-linear one-dimensional convection-diffusion problem, SIAM Journal
on Numerical Analysis 39 (2001), no. 4, 1446–1467.
[18] , Numerical analysis of a singularly perturbed nonlinear reaction–diffusion problem with multiple solutions, Appl. Numer.
Math. 51 (2004), no. 2, 273–288.
[19] T. Linß, G. Radojev, and H. Zarin, Approximation of singularly perturbed reaction-diffusion problems by quadratic c1-splines,
Numerical Algorithms 61 (2012), no. 1, 35–55.
[20] T. Linß, H.G. Roos, and R. Vulanovi ´
c, Uniform pointwise convergence on shishkin-type meshes for quasi-linear convection-diffusion
problems, SIAM J. Numer. Anal. 38 (2000), no. 3, 897–912.
[21] J. Lorenz, Stability and monotonicity properties of stiffquasilinear boundary problems, Zb.rad. Prir. Mat. Fak. Univ. Novom Sadu,
Ser. Mat. 12 (1982), 151–176, MR 85e:34046.
[22] K. Niijima, A uniformly convergent difference scheme for a semilinear singular perturbation problem, Numer.Math. 43 (1984), no. 2,
175–198.
[23] E. O’Riordan and M. Stynes, A uniformly accurate finite-element method for a singularly perturbed one-dimensional reaction-diffusion
problem, Mathematics of Computation 47 (1986), no. 176, 555–570.
[24] , l1and l∞uniform convergence of a difference scheme for a semilinear singular perturbation problem., Numer. Math. 50
(1986/87), 519–532.
[25] J. M. Ortega and W. C. Rheinboldt, Iterative solution of nonlinear equations in several variables, SIAM, Philadelphia, USA, 2000.
[26] H.G. Roos, Global uniformly convergent schemes for a singularly perturbed boundary-value problem using patched base spline-functions,
Journal of Computational and Applied Mathematics 29 (1990), no. 1, 69–77.
[27] G. I. Shishkin, Grid approximation of singularly perturbed parabolic equations with internal layers, Sov. J. Numer. Anal. M.Russian
Journal of Numerical Analysis and Mathematical Modelling 3(1988), no. 5, 393–408 (In Russian).
[28] M. Stynes, An adaptive uniformly convergent numerical method for a semilinear singular perturbation problem, SIAM Journal on
Numerical Analysis 26 (1989), no. 2, 442–455.
[29] M. Stynes and N. Kopteva, Numerical analysis of singularly perturbed nonlinear reaction-diffusion problems with multiple solutions,
Computers and Mathematics with Applications 51 (2006), no. 5, 857–864.
[30] G. Sun and M. Stynes, An almost fourth order uniformly convergent difference scheme for a semilinear singularly perturbed reaction-
diffusion problem, Numer. Math. 70 (1995), no. 4, 487–500.
[31] , A uniformly convergent method for a singularly perturbed semilinear reaction-diffusion problem with multiple solutions, Math.
Comput. 65 (1996), no. 215, 1085–1109.
[32] K. Surla and Z. Uzelac, A uniformly accurate difference scheme for singular perturbation problem, Indian J. pure appl. Math. 27
(1996), no. 10, 1005–1016.
[33] K. Surla and Z. Uzelac, On stability of spline difference scheme for reaction-diffusion time-dependent singularly perturbed problem,
Novi Sad J. Math. 33 (2003), no. 2, 89–94.
[34] R. Vulanovi´
c, On a numerical solution of a type of singularly perturbed boundary value problem by using a special discretization mesh,
Novi Sad J. Math. 13 (1983), 187–201.
[35] R. Vulanovi´
c, Mesh generation methods for numerical solution of quasilinear singular perturbation problems, Univ. u Novom Sadu
Zb. Rad, Prirod-Mat. Fak. Ser. Mat 19 (1989), no. 2, 171–193.
[36] R. Vulanovi´
c, A second order numerical method for non-linear singular perturbation problems without turning points, USSR Comp.
Math. Math+.31 (1991), no. 4, 522–532.
[37] R. Vulanovi´
c, On numerical solution of semilinear singular perturbation problems by using the hermite scheme, Univ. u Novom Sadu
Zb. Rad, Prirod-Mat. Fak. Ser. Mat 23 (1993), no. 2, 363–379.
[38] , Fourth order algorithms for a semilinear singular perturbation problem, Numer. Algor. 16 (1997), no. 2, 117–128.
[39] , An almost sixth-order finite-difference method for semilinear singular perturbation problems, Computational methods in
applied mathematics 4(2004), no. 3, 368–383.
