Hindawi Publishing Corporation
Advances in Numerical Analysis
Volume 2012, Article ID 349618, 21 pages
doi:10.1155/2012/349618
Research Article
A Note on Fourth Order Method for
Doubly Singular Boundary Value Problems
R. K. Pandey and G. K. Gupta
Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, India
Correspondence should be addressed to R. K. Pandey, rkp@maths.iitkgp.ernet.in
Received 31 May 2012; Accepted 5 September 2012
Academic Editor: Hassan Safouhi
Copyright q 2012 R. K. Pandey and G. K. Gupta. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
We present a fourth order finite difference method for doubly singular boundary value problem
pxy x qxfx, y, 0 < x ≤ 1 with boundary conditions y0 A or y 0 0 or limx → 0 pxy x 0 and αy1 βy 1 γ, where α> 0, β≥ 0, γ and A are finite
constants. Here p0 0 and qx is allowed to be discontinuous at the singular point x 0. The
method is based on uniform mesh. The accuracy of the method is established under quite general
conditions and also corroborated through one numerical example.
1. Introduction
Consider the following class of singular two point boundary value problems:
pxy
qxf x, y ,
0 < x ≤ 1,
1.1
with boundary conditions
or,
y0 A, αy1 βy 1 γ
or lim pxy x 0 , αy1 βy 1 γ,
y 0 0
x→0
1.2
1.3
where α > 0, β ≥ 0 and A, γ are finite constants. Here p0 0 and qx is unbounded near
x 0. The condition p0 0 says that the problem is singular and as qx is unbounded near
x 0, so the problem is doubly singular 1. Let px, qx, and fx, y satisfy the following
conditions:
2
Advances in Numerical Analysis
A-1
i p0 0, px > 0 in 0, 1,
ii px ∈ C0, 1 ∩ C1 0, 1,
iii px xb0 gx, 0 ≤ b0 < 1, gx > 0, and
iv G ∈ C4 0, 1 and Gv x exists on 0, 1,
where Gx 1/gx on 0, 1.
A-2
i qx > 0 in 0, 1, qx is unbounded near x 0,
ii qx ∈ L1 0, 1,
iii qx xa0 Hx, H0 > 0 with a0 > −1,
iv Hx ∈ C4 0, 1 and H v x exists on 0, 1, and
v 1 a0 − b0 ≥ 0.
A-3 fx, y is continuous on {0, 1 × R}, ∂f/∂y exists, and is continuous
and ∂f/∂y ≥ 0 for all 0 ≤ x ≤ 1 and for all real y.
Thomas 2 and Fermi 3 independently derived a boundary value problem for determining
the electrical potential in an atom. The analysis leads to the nonlinear singular second order
problem
y x−1/2 y3/2
1.4
with a set of boundary conditions. The following are of our interest:
i the neutral atom with Bohr radius b given by y0 1, by b − yb 0;
ii the ionized atom given by y0 1, yb 0.
Furthermore, Chan and Hon 4 have considered the generalized Thomas-Fermi equation
xb y cqxye ,
y0 1,
ya 0
1.5
for parameter values 0 ≤ b < 1, c > 0, d > −2, e > 1, and qx xbd .
Such singular problems have been the concern of several researchers 5–8. The
existence-uniqueness of the solution of the boundary value problem 1.1 with boundary
condition 1.2 or 1.3 is established in 1, 9–13. Bobisud 1 has mentioned that in case
limx → 0 qx/p x / 0, the condition y 0 0 is quite severe, that is, it is sufficient but not
necessary for forcing the solution to be differentiable at x 0. In fact if limx → 0 pxy x 0,
then
lim y x lim
x→0
x→0
qx
· lim f x, yx .
p x x → 0
1.6
Thus limx → 0 y x 0 if either limx → 0 qx/p x 0 or f0, y0 0. But
0 if qx has discontinuity at x 0; thus it is natural to consider the
limx → 0 qx/p x /
weaker boundary condition limx → 0 pxy x 0.
Advances in Numerical Analysis
3
There is a considerable literature on numerical methods for qx 1 but to the best of
our knowledge very few numerical methods are available to tackle doubly singular boundary
value problems. Reddien 14 has considered the linear form of 1.1 and derived numerical
methods for qx ∈ L2 0, 1 which is stronger assumption than A-2ii.
Some second order methods Chawla and Katti 15, Pandey and Singh 16, 17
as well as fourth order methods Chawla et al. 18–21, Pandey and Singh 22–24 have
been developed for qx 1, px. Most of the researchers have developed methods for the
function px xb0 , 0 ≤ b0 < 1 and the boundary conditions y0 A and y1 B.
Chawla 19 has given fourth order method for the problem 1.1-1.2 with qx 1
and px xb0 , 0 ≤ b0 < 1. The method is extended by Pandey and Singh 22 to a
class of function px xb0 gx satisfying A-1i–iii with Gx 1/gx analytic in the
neighborhood of the singular point.
In this work we extend the fourth order accuracy method developed in 22 to doubly
singular boundary value problems 1.1-1.2 and 1.1, 1.3 both. That is, we allow the data
function to be discontinuous at the singular point x 0. Further, we do not require analyticity
of the function Gx 1/gx. The convergence of the method is established under quite
general conditions on the functions px, qx, and fx, y. Fourth order convergence of the
method is corroborated through one example and maximum absolute errors are displayed in
Table 1.
The work is organized in the following manner. In Section 2, first the method is
described and then its construction is explained. In Section 3, convergence of the method
is established and in Section 4, the order of the method is corroborated through one example.
2. Finite Difference Method
This section is divided in to two parts: i description of the method and ii derivation of the
method. The coefficients not specified explicitly in this section are specified in the appendices.
2.1. Description of the Method
In this section we first state the method; detailed derivation is given in Section 2.2.
For positive integer N ≥ 2, we consider the mesh wh {xk }N
k0 over 0, 1 : 0 x0 <
x1 < x2 < · · · xN 1, h 1/N. For uniform mess xk kh, where b0 is mentioned in
condition A-1. Let rx : fx, yx yx is the solution, yk yxk , and so forth. Now
we approximate the differential equation 1.1 with boundary condition 1.2 on the grid wh
by the following difference equations:
yk−1
−
Jk−1
yk1
1
1
a0,k rk a1,k rk1 − a−1,k rk−1 ,
yk Jk Jk−1
Jk
k 11N − 1,
2.1
yN−1
−
JN−1
1
JN−1
α
βGN
γ
a0,N rN − a−1,N rN−1 a1,N rN−θ ,
βGN
yN 2.2
4
Advances in Numerical Analysis
for boundary value problem 1.1-1.2. In the case of boundary condition 1.3 we use the
following difference equation:
−y1 y2
a−1,1 r0 a0,1 r1 a1,1 r2 .
J1
J1
2.3
Here y yk denotes the approximate solution, yk ≈ yk , Gk Gxk , and
xk1
Jk pτ
−1
2.4
dτ.
xk
Note that the functions px and Gx are defined in assumption A-1. Thus the method for
boundary value problem 1.1-1.2 is given by difference equations 2.1-2.2. Further, for
boundary value problem 1.1 and 1.3 the method is given by difference equations 2.1
with k 21N − 1 and 2.2-2.3.
2.2. Construction of the Method
In this section we describe the derivation of the method. Local truncation errors are
mentioned without proof.
With zx pxy x the differential equation 1.1 becomes z x qxfx, yx.
Now integrating z x qxfx, yx twice, first from xk to x and then from xk to xk1 and
changing the order of integration we get
yk1 − yk zk Jk xk1 xk1
pτ
xk
−1
dτ qtrtdt,
2.5
t
where zk zxk , rx : ft, yx and Jk get
yk − yk−1 zk Jk−1 −
xk1
pτ−1 dτ. In an analogous manner, we
xk
xk t
pτ
−1
dτ qtrtdt.
2.6
xk−1
xk−1
Eliminating zk from 2.5 and 2.6 we obtain the Chawla’s identity
I−
yk1 − yk yk − yk−1 Ik
−
k ,
Jk
Jk−1
Jk Jk−1
k 11N − 1,
2.7
where
Ik Ik−
xk1 xk1
−1
−1
pτ
xk
xk
t
t
pτ
xk−1
xk−1
dτ qtrtdt,
dτ qtrtdt.
2.8
Advances in Numerical Analysis
5
Via Taylor expansion of Gx, rx and Hx about xk in Ik± and taking the following approximations for rk and rk
rk rk rk1 − rk−1 1 2 − h r ηk ,
2h
6
xk−1 < ηk < xk1 ,
rk1 − 2rk rk−1
1
− h2 r iv ξk ,
12
h2
2.9
xk−1 < ξk < xk1 ,
we get the approximation for the smooth solution yx as:
yk−1
−
Jk−1
yk1
1
1
a0,k rk a1,k rk1 − a−1,k rk−1 tk ,
yk Jk Jk−1
Jk
k 11N − 1,
2.10
where tk is local truncation error given by
iv
tk Hk G
k b03,k Hk G ξk b04,k fk Hk fk Hk fk Gk b12,k Gk b13,k
1
1
Hk fk Hk fk Hk fk Gk b21,k Gk b22,k
2
2
1 1 1 1
Hk fk Hk fk Hk fk Hk fk Gk b30,k Gk b31,k
6
2
2
6
1 iv 1 1 1 1
iv
Hk f ξk Hk fk Hk fk Hk fk H ηk fk Gk b40,k
24
6
4
6
24
2.11
1
− h2 f ηk b10,k Gk Hk b11,k Gk H k b20,k Gk Hk
6
−
1
b20,k h2 Gk Hk f iv ξk ,
24
k 11N − 1, xk−1 < ξk , ηk < xk1 .
The coefficients b03,k , b04,k , b10,k , b11,k , b12,k , b13,k , b20,k , b21,k , b22,k , b30,k , b31,k , b40,k , and so
forth, are specified in the appendices and the functions Gx, Hx are defined in the
assumptions A-1, A-2.
2.2.1. Discretization of the Boundary Condition at x 1
We write 2.6 for k N
I−
yN−1
yN
−
pN yN
− N .
JN−1 JN−1
JN−1
2.12
6
Advances in Numerical Analysis
Now, we use the boundary condition at x 1, Taylor expansion of rx and following appro
and rN
ximations for rN
1 1 θ
θ
1
h2 θ rN rN−θ −
r ηN , xN−1 < ηN < xN ,
rN−1 h
θ
θθ − 1
6
1 − θ
2 1
1
1
h1 θ rN−θ −
r ηN , xN−1 < ηN
rN−1 2 rN < xN ; 0 < θ < 1
θθ − 1
3
1 − θ
h θ
2.13
rN
rN
in 2.12 and get the discretization at x 1 as follows:
yN−1
−
JN−1
1
JN−1
α
βGN
γ
a0,N rN − a−1,N rN−1 a1,N rN−θ tN ,
βGN
yN 2.14
where the local truncation error tN is given by
tN 1
JN−1
− iv −
−
−
−
G
a
H
f
G
ξ
a
H
f
H
f
a
G
a
HN fN G
N
N
N
N
N 03,N
N
N
N 12,N
N 13,N
N
04,N
1
1 HN fN
HN
fN
HN
fN GN a−21,N GN a−22,N
2
2
1
1 1 1 HN fN
HN
fN HN fN HN
fN GN a−30,N GN a−31,N
6
2
2
6
1
1 1 1 1 iv
iv
HN fξN HN fN HN fN HN fN H ηN fN GN a−40,k
24
6
4
6
24
− −
θ
a10,N GN HN a−11,N GN HN a−20,N GN HN
− h2 f ηN
6
−
h1 θ −
−
−
a20,N h2 GN HN f ηN
, ηN
< xN .
−
, xN−1 < ξN
3
2.15
The coefficients a−ij,N are specified in the appendices.
The discretization 2.14 involves unknown yN−θ , which we approximate in the
following way:
yN−θ y N−θ −
h3 θ2 1 − θ yN ,
6
2.16
where
y N−θ
θ
yN−1 θ1 − θ
1 − θ
1 θ hα
θ
β
hγ
−
.
β
2.17
Advances in Numerical Analysis
7
Now, let r N−θ fxN−θ , y N−θ then replacing rN−θ by r N−θ in 2.14 we obtain the following
discretization for a smooth solution yx at k N
yN−1
−
JN−1
1
JN−1
α
βGN
γ
yN a0,N rN − a−1,N rN−1 a1,N r N−θ tN ,
βGN
2.18
where
h3 2
∂f
∗
θ 1 − θa1,N yN
,
xN−θ , yN−θ
6
∂y
∈ min yN−θ , y N−θ , max yN−θ , y N−θ .
1
tN t N −
∗
yN−θ
2.19
2.2.2. Discretization of the Boundary Condition at x 0
Integrating the differential equation 1.1 twice, first from x1 to x; then from x1 to x2 and
interchanging the order of integration we get
y2 − y1 J1
x1
qtrtdt x2 x2
0
pτ
x1
−1
dτ qtrtdt.
2.20
t
Via Taylor expansion of rx, Hx, and Gx about x x1 we get the following discretization
for k 1:
−y1 y2
a−1,1 r0 a0,1 r1 a1,1 r2 t1 ,
J1
J1
2.21
where the local truncation error t1 is given by
t1 1
1
1
1
H1 f1 H1 f1 H1 f1 H1 f1
6
2
2
6
6x14a0
−
ψ4
1
1
1
1
1
H1 f1iv H1 f1 H1 f1 H1 f1 H1iv f1
24
6
4
6
24
1
1
1 a G a Giv
H1 f1
J1
6 03,1 1 24 04,1 1
1 1 a G a G
H1 f1 H1 f1
2 12,1 1 6 13,1 1
24x14a0
ψ5
8
Advances in Numerical Analysis
1
1
1
H1 f1 H1 f1 H1 f1
a21,1 G1 a22,1 G1
2
2
2
1
1
1
1
H1 f1 H1 f1 H1 f1 H1 f1 a30,1 G1 a31,1 G1
6
2
2
6
1
1 1 1 1 iv
iv
H1 f1 H1 f1 H1 f1 H1 f1 H1 f1 a40,1 G1
24
6
4
6
24
2a0
3a
2x 0
h2 x
− f η1 − 1 H1 1 H1
6
ψ2
ψ3
1 a10,1 G1 a11,1 G1 H1 a20,1 G1 H1
J1
3a0
x1
1 h2 iv
H1 a G1 H1 ,
− f ξ1 12
ψ3
2J1 20,1
0 < ξ1 < x2 .
2.22
The coefficients ai,j are specified in the appendices and the functions Gx, Hx are defined
in the assumptions A-1 and A-2.
The discretization 2.22 involves y0 ; so for y0 we use the following approximation:
x12a0 −b0
H1 G1 f1
1 a0 2 a0 − b0 1
1
1
G1 H1 f1
x13a0 −b0
−
1 a0 2 − b0 2 a0 − b0 3 a0 − b0 2−b0 3a0 −b0 1
1
G1 H1 f1
−
1a0 2a0 3a0 −b0 y 0 y1 x12a0 −b0
1
1
f2 − f0 ,
−
2
1 a0 2 a0 3 a0 − b0 2.23
where
y0 y0 τ0 ,
τ0 c∗ h4a0 −b0 ,
c∗ is some constant.
2.24
Let r 0 fx0 , y 0 ; then in 2.21 replacing r0 by r 0 we obtain the following discretization for
k 1:
1
1
1
y1 − y2 b−1,1 r 0 b0,1 r1 b1,1 r2 t1 0,
J1
J1
2.25
Advances in Numerical Analysis
9
where t1 t1 b−1,1 τ0 ∂f/∂yx0 , y0∗ , y0∗ ∈ min{y0 , y 0 }, max{y0 , y 0 }, and coefficients bi,j ,
are specified in appendices.
3. Convergence of the Method
In this section we show that under suitable conditions the method 2.1-2.2 for the boundary
value problems 1.1-1.2 described in Section 2 is of fourth order accuracy.
Let Y y1 , . . . , yN T , RY r1 , . . . , rN T , Q q1 , . . . , qN T , and T t1 , . . . , tN T ,
then the finite difference method given by 2.1-2.2 for the boundary value problems 1.11.2 can be expressed in matrix form as
BY P R Y GY Q,
3.1
and Y y1 , . . . , yN T corresponding to smooth solution yx satisfies the perturbed system
BY P RY GY T Q,
3.2
where B bij and P pij are N × N tridiagonal matrices.
Let E Y −Y e1 , . . . , eN T , then from Mean Value Theorem RY −RY ME, M diag{U1 , . . . , UN }Uk ∂fk /∂yk ≥ 0, then from 3.1 and 3.2 we get the error equation as
B P M WE T.
3.3
We recall that by the notation Z ≥ 0 we mean that all components zi of the vector Z
satisfy zi ≥ 0. Similarly by B ≥ 0 we mean all the elements bij of the matrix B satisfy bij ≥ 0.
From Corollary of Theorems 7.2 and 7.4 in 25 an irreducible tridiagonal matrix B is
monotone if
bk,k1 ≤ 0,
bk,k−1 ≤ 0,
N
j1
bk,j
≥ 0,
> 0,
k 1, 2, . . . , N,
for at least one i.
3.4
Furthermore, from Theorem 7.3 in 25, B−1 exists and the elements of B−1 are nonnegative.
Now, if B P M W is monotone and B P M W ≥ B, then from Theorem 7.5 in 25
B P M W−1 ≤ B−1 .
3.5
Let vector norm E
and matrix norm B
be defined by
E
max |ek |,
1≤k≤N
B
max
1≤k≤N
N bkj ,
j1
3.6
10
Advances in Numerical Analysis
then from 3.3 we get
E
B P M W−1 |T |,
3.7
where |T | ≡ |t1 |, |t2 |, . . . , |tN |.
Furthermore, if B P M W ≥ B and B is also monotone matrix, then from 3.5 we
get
E
≤ B −1 |T |.
3.8
−1
Now from 22, B −1 bi,j
of matrix B is given by
−1
bi,j
⎧
P xi P xN − P xj βGN /α
⎪
⎪
⎪
, i ≤ j,
⎪
⎪
⎨
P xN βGN /α
⎪
⎪
P xj P xN − P xi βGN /α
⎪
⎪
⎪
, i ≥ j,
⎩
P xN βGN /α
3.9
where
P x x
pτ
−1
dτ.
3.10
0
Let xr iv , r i for i 0 1 3, and y be bounded on 0, 1. Now, as b0 is fixed in 0, 1 and a0 is
chosen such that 1 a0 − b0 > 0, for sufficiently small h we get
|tk | ≤ Ch5 xk−1a0 ,
4
tN ≤ Ch ,
3.11
for suitable constants C and C.
Now from 3.8–3.11 we get
⎡
∗ 5⎣
|ei | ≤ C h
P xi 1− P xN βGN /α
#
i
P xj xj−1a0
j1
⎫⎤ 3.12
⎧
N−1
⎬
⎨
βG
βG
P xi N
N
⎦,
xj−1a0 P xN − P xj α
hα ⎭
P xN βGN /α ⎩ji1
Advances in Numerical Analysis
11
where C∗ max{C, C}. It is easy to see that
xi
i
−1a0
h P xj xj
<
P xx−1a0 dx,
0
j1
h
3.13
xN βGN −1a0
βGN −1a0
xj
x
P xN − P xj P xN − P x <
dx.
α
α
xi
N−1
ji1
From 3.12-3.13 we obtain
⎧
⎪
1 − b0 xi1−b0
α 1 − b0 a0 1 − a0 β
⎪
a0
⎪
x
−
⎪
⎪
⎪
−a0 1 − b0 a0 i
α 1 − b0 β
⎪
⎪
⎪
⎪
C∗ h4 ⎨
×sup|Gx| 1 − b0 d1 ,
|ei | ≤
1 − b0 ⎪
0,1
⎪
⎪
⎪
⎪
⎪
⎪
2 − b0 β
1
⎪ 1−b0
⎪
⎪
x
sup|Gx| d2 ,
ln
⎩ i
xi
α 1 − b0 β 0,1
a0 /
0
3.14
a0 0,
where
α1 − b0 a0 1−b0 a0 /−a0
α 1 − b0 β 1 − b0 1−b0 /−a0
3 − b0 a0 1−b0 a0 /−a0
2 − b0 a0 1−b0 a0 /−a0
G x ×
sup
supG x
1−b0 /−a0
1−b0 /−a0
23 − b0 2 − b0 0,1
0,1
d1 4 − b0 a0 1−b0 a0 /−a0
5 − b0 a0 1−b0 a0 /−a0
v supG x sup
x
G
4!5 − b0 1−b0 /−a0 0,1
0,1
3!4 − b0 1−b0 /−a0
1
1
α
d2 e1/2−b0 supG x e2/3−b0 supG x
2
23 − b0 α 1 − b0 β e 2 − b0 0,1
0,1
1
1
v 3/4−b0 4/5−b0 e
e
sup G x supG x .
3!4 − b0 4!5 − b0 0,1
0,1
3.15
In view of the following inequalities
1−b0 a0 /−a0
α 1 − b0 β
α 1 − b0 a0 1 − a0 β
x1−b0 xa0 −
≤
,
α 1 − b0 β
α 1 − b0 a0 1 − a0 β
β
1
1
1−b0
x
e−α−1−b0 β/α1−b0 β
ln
≤
x
α 1 − b0 β
1 − b0 3.16
12
Advances in Numerical Analysis
for x ∈ 0, 1, B P M W−1 ≤ B−1 and from 3.14 it is easy to establish that
E
∞ O h4 .
3.17
Thus we have established the following result.
Theorem 3.1. Assume that px, qx, and fx, y satisfy assumptions given in A-1, A-2, and
A-3, respectively. Let rx : fx, yx, then the method is of fourth order accuracy for sufficiently
small mesh size h provided, xr iv x, r i , i 0 1 3, and y are bounded on 0, 1.
Remark 3.2. We have developed the numerical method for the boundary condition y 0 0
but could not establish the fourth order convergence although the order of the accuracy is
verified through one example in the next section.
4. Numerical Illustrations
To illustrate the convergence of the method and to corroborate their order of the accuracy,
we apply the method to following example. The maximum absolute errors are displayed in
Table 1.
Example 4.1. Consider
− b0 x1.9−b0 ey − 0.9 3.5x3.5 / 1 x3.5
1
−0.1 1.9
,
y
1.9 − b0 x
1 x3.5
4 x1.9−b0 1 x3.5 1
1
y0 ln
or y 0 0 ,
− 1.9 − b0 ,
y1 5y 1 ln
4
5
xb0
4.1
with exact solution yx ln1/4 x1.9−b0 .
The method is applied on the example for b0 0.1, 0.5. Maximum absolute errors and
order of accuracy for Example 4.1 are displayed in Table 1.
We use the following approximation of Jk in our numerical program:
⎞
1−b0
2−b0
2−b0
− xk1−b0
Gk xk1
−
x
x
⎟ ⎜
k
Gk ⎝ k1
−
⎠Gk
2 − b0
1 − b0
⎛
Jk 1−b0
− xk1−b0
xk1
1 − b0
1
Gk
2
1
G
6 k
3−b0
xk1
− xk3−b0
3 − b0
4−b0
xk1
− xk4−b0
4 − b0
xk2
1−b0
xk1
− xk1−b0
− 3xk
1 − b0
− 2xk
3−b0
xk1
− xk3−b0
3 − b0
2−b0
xk1
− xk2−b0
3xk2
4.2
2 − b0
2−b0
xk1
− xk2−b0
2 − b0
−
xk3
1−b0
xk1
− xk1−b0
1 − b0
.
Advances in Numerical Analysis
13
Table 1: Maximum absolute errors and order of the methods for Example 4.1.
b0 0.1
Order
b0 0.5
y0 ln1/4 and θ 0.01
8.23 −7a
1.10 −6
5.05 −8
4.03
6.80 −8
3.13 −9
4.22 −9
1.94 −10
4.01
2.64 −10
y 0 0 and θ 0.01
3.37 −7
4.03 −7
2.04 −8
4.05
2.46 −8
1.25 −9
1.52 −9
7.74 −11
4.02
9.56 −11
N
32
64
128
256
32
64
128
266
a
Order
4.02
4.00
4.03
3.99
b0 0.1
Order
b0 0.5
y0 ln1/4 and θ 0.98
6.11 −7
8.11 −7
3.50 −8
4.13
4.60 −8
2.09 −9
2.72 −9
1.27 −10
4.04
1.67 −10
y 0 0 and θ 0.98
1.72 −7
1.95 −7
9.13 −9
4.24
9.80 −9
5.20 −10
5.39 −10
3.09 −11
4.08
3.23 −11
Order
4.14
4.03
4.31
4.06
−7
8.23 −7 8.23 × 10 .
4.1. Numerical Results for Uniform Mesh Case
Remark 4.2. In the discretization 2.18, θ may take any value in 0, 1 but from the Table 1,
maximum absolute errors are less for the value of θ close to one. So it is better to take value
of θ close to one.
Appendices
The coefficients involved in the method and their approximations are mentioned below.
A. Coefficients for Method
1
1
1
± b10,k Gk Hk b11,k Gk Hk b20,k Gk Hk b20,k Gk HK ,
2h
h
1
1
b00,k − 2 b20,k Gk Hk b01,k Hk Gk b02,k Gk Hk
2
h
a±1,k 1
a0,k
1
a−1,N
1
a0,N
1
b10,k Gk Hk b11,k Gk Hk b20,k Gk Hk ,
2
1
θ
a−10,N GN HN a−11,N GN HN a−20,N GN HN
JN−1 h1 − θ
1
2
a−20,N GN HN ,
h 1 − θ
1
1
a−01,N GN HN a−02,N GN HN a−11,N GN HN
a− GN HN a−10,N GN HN
JN−1 00,N
2
1
1 θ −
a10,N GN HN a−11,N GN HN a−20,N GN HN
a−20,N GN HN
2
hθ
14
Advances in Numerical Analysis
1 −
a
G
H
,
N
N
θh2 20,N
1
1
a−10,N GN HN a−11,N GN HN a−20,N GN HN
JN−1 hθ1 − θ
1
a−20,N GN HN ,
2
h θ1 − θ
1
a1,N
A.1
where
⎡ a±00,k a±01,k
1 ⎢
⎣−
1 − b0 2a0 −b0
xk±1
− xk2a0 −b0
φ2
⎤
xk±1 1a0
⎥
x
− xk1a0 ⎦,
ψ1 k±1
2a0 −b0
xk − xk±1 xk±1
1 3a0 −b0
1
xk±1
− xk3a0 −b0
φ2
φ3
1 − b0 1−b0 2−b0
xk±1
xk±1 − xk xk±1
1a0
1a0
xk±1
xk±1
− xk1a0 −
− xk1a0
ψ1
ψ1ϕ2
1
3a0 −b0
3a0 −b0
,
x
− xk
ϕ23 a0 − b0 k±1
a±02,k
2a0 −b0
xk±1 − xk 2 xk±1
1 1
3a0 −b0
2xk±1 − xk xk±1
−
φ2
φ3
1 − b0 xk±1 − xk 2 x1−b0 2 4a0 −b0
4a0 −b0
1a0
k±1
xk±1
−
xk±1
− xk
− xk1a0
φ4
1 a0 2φ2 4a0 −b0
2xk±1 − xk 3a0 −b0
xk±1
xk±1
−
− xk4a0 −b0
ϕ23 a0 − b0 ϕ2φ4
2xk±1 − xk 2−b0 1a0
2
1a0
xk±1 xk±1 − xk1a0 xk±1
− xk1a0
ϕ2ψ1
ϕ3ψ1
2
4a0 −b0
4a0 −b0
,
x
−
− xk
ϕ34 a0 − b0 k±1
−
a±03,k
h3
h3 2a0 −b0 3h2 3a0 −b0
1
1−b0
1a0
±
x
x
xk±1
− xk1a0 ∓
xk±1
1 − b0 1 a0 φ2 k±1
φ3 k±1
∓
6h 4a0 −b0
6 5a0 −b0
xk±1
xk±1
− xk5a0 −b0
φ4
φ5
−
3h2
2−b0
1a0
xk±1
− xk1a0
xk±1
1 a0 2 − b0 Advances in Numerical Analysis
h2
3
h
x4a0 −b0
x3a0 −b0 ∓
2 − b0 3 a0 − b0 k±1
3 a0 − b0 4 a0 − b0 k±1
15
±
2φ2 5a0 −b0
xk±1
− xk5a0 −b0
φ5
6
6
3−b0
1a0
xk±1
xk±1
− xk1a0 −
2 − b0 3 − b0 1 a0 2 − b0 3 − b0 ±
−
1
h
4a0 −b0
5a0 −b0
xk±1
xk±1
−
−xk5a0 −b0
4a0 −b0 4a0 −b0 5a0 − b0 6
4−b0
1a0
xk±1
xk±1
− xk1a0
ϕ41 a0 6
5a0 −b0
5a0 −b0
,
− xk
x
ϕ45 a0 − b0 k±1
a±04,k
h4 1−b0 1a0
h4 2a0 −b0 4h3 3a0 −b0
1
xk±1 xk±1 − xk1a0 −
x
x
±
1 − b0 ψ1
φ2 k±1
φ3 k±1
−
12h2 4a0 −b0 24h 5a0 −b0
24 6a0 −b0
xk±1
xk±1
xk±1
±
− xk6a0 −b0
φ4
φ5
φ6
4h3
2−b0
1a0
xk±1
xk±1
− xk1a0
ϕ2ψ1
3h2 φ2 4a0 −b0
4
h3
3a0 −b0
x
∓
xk±1
−
φ4 k±1
2 − b0 3 a0 − b0 ∓
#
6hφ2 5a0 −b0 6φ2 6a0 −b0
xk±1
xk±1
±
−
− xk6a0 −b0
φ5
φ6
12h2
3−b0
1a0
xk±1
xk±1
− xk1a0
ϕ2ψ1
2hφ3 5a0 −b0
h2
12
x
x4a0 −b0 ∓
−
ϕ3 4 a0 − b0 k±1
φ5 k±1
#
2φ3 6a0 −b0
xk±1
− xk6a0 −b0
φ5
24h
4−b0
1a0
xk±1
xk±1
− xk1a0
ϕ4ψ1
h
24
±
x5a0 −b0
ϕ4
5 a0 − b0 k±1
∓
−
1
6a0 −b0
− xk6a0 −b0
xk±1
5 a0 − b0 6 a0 − b0 16
Advances in Numerical Analysis
24
5−b0
1a0
xk±1
xk±1
− xk1a0
ϕ5ψ1
24
6a0 −b0
6a0 −b0
,
x
− xk
−
ϕ66 a0 − b0 k±1
1
1 1−b0 2a0 2a0 xk±1 − xk 2a0 −b0 xk±1 − xk 2a0 −b0
x
xk±1 xk±1
x
−
−
−xk
φ1
ψ1
ψ2 k±1 k±1
1 − b0 1 3a0 −b0
3a0 −b0
x
,
− xk
φ3 k±1
2a0 −b0
xk±1 − xk 2 xk±1
1
xk±1 − xk 2 2a0 −b0
xk±1
−
φ2
ψ1
1 − b0 a±10,k a±11,k
−
xk±1 − xk 1−b0 1−b0
xk±1 xk±1 − xk1−b0
ψ2
2xk±1 − xk 3a0 −b0
2 4a0 −b0
xk±1
xk±1
−
− xk4a0 −b0
φ3
φ4
1 2−b0
1
xk±1 − xk 3a0 −b0 xk±1 − xk 3a0 −b0
xk±1 x
xk±1 −
ψ1
ψ2 k±1
2 − b0 3 a0 − b0 ×
a±12,k
2a0
xk±1
−xk2a0
φ2 4a0 −b0
xk±1
−
− xk4a0 −b0
φ4
,
1
1 2a0
h 1a0
2a0
2 1−b0
x
x
−
− xk
h xk±1 ±
1 − b0
ψ1 k±1
ψ2 k±1
∓
h3 2a0 −b0 3h2 3a0 −b0
6h 4a0 −b0
x
x
x
∓
φ2 k±1
φ3 k±1
φ4 k±1
6 5a0 −b0
xk±1
− xk5a0 −b0
φ5
2h 2−b0
h 1a0
1 2a0
2a0
x
x
x
∓
−
− xk
ϕ2 k±1 ψ1 k±1
ψ2 k±1
h2
2
3a−0−b0 2hφ2 4a−0−b0
x
xk±1
∓
ϕ2 3 a0 − b0 φ4 k±1
#
2φ2 5a0 −b0 5a0 −b0 xk±1 −xk
φ5
2
1 2a0
h 1a0
3−b0
xk±1
xk±1 −
xk±1 − xk2a0
±
ϕ3
ψ1
ψ2
∓
h
1
4a0 −b0
5a0 −b0
xk±1
xk±1
− xk5a0 −b0 ,
4 a0 − b0 4 a0 − b0 5 a0 − b0 Advances in Numerical Analysis
1
1 2a0
h 1a0
1−b0
xk±1 −
xk±1 − xk2a0
±h3 xk±1
±
a±13,k 1 − b0
ψ1
ψ2
−
−
−
±
−
−
a±20,k
h4 2a0 −b0 4h3 3a0 −b0 12h2 4a0 −b0 24h 5a0 −b0
x
x
x
x
±
−
±
φ2 k±1
φ3 k±1
φ4 k±1
φ5 k±1
24 6a0 −b0
xk±1
− xk6a0 −b0
φ6
1 2a0
3h2 2−b0
h 1a0
xk±1 ±
xk±1 −
xk±1 − xk2a0
ϕ2
ψ1
ψ2
3φ2
h3 3a0 −b0 3h2 4a0 −b0
6h 5a0 −b0
±
x
x
x
−
±
ϕ2
φ3 k±1
φ4 k±1
φ5 k±1
#
6 6a0 −b0
6a0 −b0
x
−
− xk
φ6 k±1
6h 3−b0
1 2a0
h 1a0
2a0
x
x
x
−
− xk
±
ϕ3 k±1
ψ1 k±a ψ2 k±1
#
6φ3 h2 4a0 −b0
2h 5a0 −b0
2 6a0 −b0
6a0 −b0
x
x
x
∓
− xk
ϕ3 φ4 k±1
φ5 k±1
φ6 k±1
6
h
1 2a0
h 1a0
4−b0
xk±1
xk±1 −
xk±1 − xk2a0
∓
x5a0 −b0
±
ϕ4
ψ1
ψ2
5 a0 − b0 k±1
1
6a0 −b0
6a0 −b0
,
x
−xk
5 a0 − b0 6 a0 − b0 k±1
1
xk±1 − xk 2 2a0 −b0 2xk±1 − xk 3a0 −b0 xk±1 − xk 2 2a0 −b0
xk±1
xk±1
xk±1
−
φ2
φ3
ψ1
1 − b0 2x − x 2 4a0 −b0
k±1
k
3a0 −b0
xk±1
xk±1
− xk4a0 −b0 −
φ4
ψ2
2 1−b0 3a0
3a0
x
x
,
− xk
ψ3 k±1 k±1
−
a±21,k
17
#
h2 1a0
1
2 2a0
2 3a0
1−b0
3a0
±hxk±1
x
x
x
∓
− xk
1 − b0
ψ1 k±1
ψ2 k±1
ψ3 k±1
∓
h3 2a0 −b0 3h2 3a0 −b0
6h 4a0 −b0
x
x
x
∓
φ2 k±1
φ3 k±1
φ4 k±1
6 5a0 −b0
xk±1
− xk5a0 −b0
φ5
#
h2 1a0
1
2h 2a0
2 3a0
2−b0
3a0
x
x
x
x
−
∓
− xk
ϕ2 k±1 ψ1 k±1
ψ2 k±1
ψ3 k±1
18
Advances in Numerical Analysis
φ2 −
h2 3a0 −b0
x
φ3 k±1
##
2h 4a0 −b0
2 5a0 −b0
xk±1
xk±1
±
−
− xk5a0 −b0
φ4
φ5
a±22,k
,
h2 1a0
1
∓hx 2x2
2h 2a0
1−b0
2
xk±1 h xk±1 ∓
x
1 − b0
ψ1
ψ2 k±1
2 − b0 #
1 3a0
xk±1 − xk3a0
ψ3
−
h4 2a0 −b0 4h3 3a0 −b0 12h2 4a0 −b0 24h 5a0 −b0
x
x
x
x
±
∓
φ2 k±1
φ3 k±1
φ4 k±1
φ5 k±1
24 6a0 −b0
xk±1
− xk6a0 −b0
φ6
2φ2
h3 3a0 −b0 3h2 4a0 −b0
x
x
±
−
φ3 k±1
φ4 k±1
2 − b0 #
6h 5a0 −b0
6 6a0 −b0
xk±1
xk±1
∓
−
− xk6a0 −b0
φ5
φ6
#
2ϕ1
h2 4a0 −b0
h 5a0 −b0
1 6a0 −b0 6a0 −b0 φ3
x
x
x
,
−
∓
−xk
ϕ3
φ4 k±1
φ5 k±1
φ6 k±1
a±30,k
h3 1a0
1
3h2 2a0
6h 3a0
1−b0
xk±1 ±
xk±1 −
x
x
±
1 − b0
ψ1
ψ2 k±1
ψ3 k±1
#
1 4a0
xk±1 − xk4a0
−
ψ4
h3 2a0 −b0 3h2 3a0 −b0
6h 4a0 −b0
xk±1
xk±1
x
∓
φ2
φ3
φ4 k±1
6 5a0 −b0
5a0 −b0
x
,
− xk
φ5 k±1
∓
a±31,k 1
1−b0
1−b0
−
±hxk±1
2
1
x2−b0
− b0 k±1
h3 1a0 3h2 2a0 6h 3a0
±
x
x
x
−
±
ψ1 k±1 ψ2 k±1 ψ3 k±1
#
6 4a0
xk±1 − xk4a0
−
ψ4
−
h4 2a0 −b0 4h3 3a0 −b0 12h2 4a0 −b0 24h 5a0 −b0
x
x
x
x
±
−
±
φ2 k±1
φ3 k±1
φ4 k±1
φ5 k±1
Advances in Numerical Analysis
19
24 6a0 −b0
xk±1
− xk6a0 −b0
φ6
φ2
h3 3a0 −b0 3h2 4a0 −b0
x
x
±
−
φ3 k±1
φ4 k±1
2 − b0 −
#
6h 5a0 −b0
6 6a0 −b0 6a0 −b0 xk±1 −
x
±
−xk
φ5
φ6 k±1
a±40,k
,
h4 1a0
1
4h3 2a0 12h2 3a0 24h 4a0
1−b0
xk±1
xk±1 ∓
x
x
x
∓
1 − b0
ψ1
ψ2 k±1
ψ3 k±1
ψ4 k±1
#
24 5a0
xk±1 − xk5a0
ψ5
h4 2a0 −b0 4h3 3a0 −b0 12h2 4a0 −b0
x
x
x
±
−
φ2 k±1
φ3 k±1
φ4 k±1
24 6a0 −b0
24h
6a0 −b0
,
−
x
±
− xk
φ5 φ6 k±1
−
bij,k φi i
1
j a0 − b0 ,
aij,k
Jk
ψi j2
a−ij,k
Jk−1
,
i
1
j a0 ,
ϕi i
1
j − b0 .
j2
j1
A.2
B. Approximations of the Coefficients
For uniform mesh xk kh, k 01N and h 1/N then for fixed xk as h → 0, we get the
following approximations for the coefficients:
b00,k ∼
hxka0
Gk
b11,k ∼
,
b01,k ∼ a0 − 2b0 h3 xka0
4Gk
b12,k ∼ 3a0 − 4b0 ,
b20,k ∼
h5 xka0 −1
9Gk
,
h3 xka0 −1
4Gk
h3 xka0
6Gk
b13,k ∼
b30,k ∼ 6a0 − 7b0 ,
b02,k ∼
h3 xka0
2Gk
b03,k ∼ 2a0 − 3b0 ,
h5 xka0
3Gk
h5 xka0 −1
30Gk
,
,
,
b10,k ∼ 2a0 − 3b0 h3 xka0 −1
3Gk
b21,k ∼ 5a0 − 6b0 b31,k ∼
h5 xka0
6Gk
,
,
18Gk
b40,k ∼
,
2h5 xka0
15Gk
12Gk
2h5 xka0
b04,k ∼
h5 xka0 −1
h3 xka0 −1
3Gk
b22,k ∼
,
,
,
2h5 xka0
9Gk
,
20
Advances in Numerical Analysis
2h3
h4
h3
h4
h4
2
− h
, a−02,N < , a−10,N < , a−11,N < , a−20,N < ,
a00,N < , a−01,N <
2
3
6
3
4
6
2h5
h6
h5
h6
2h5
h6
− , a−04,N < , a−12,N < , a−13,N < , a−21,N <
, a−22,N < ,
a03,N <
5
3
5
6
15
9
h5
h6
h6
− , a−31,N <
, a−40,N <
.
a30,N <
10
12
15
B.1
Acknowledgments
The authors are thankful to the referee for the valuable suggestions. This work is supported
by Department of Science and Technology, New Delhi, India.
References
1 L. E. Bobisud, “Existence of solutions for nonlinear singular boundary value problems,” Applicable
Analysis, vol. 35, no. 1–4, pp. 43–57, 1990.
2 L. H. Thomas, “The calculation of atomic fields,” Mathematical Proceedings of the Cambridge Philosophical
Society, vol. 23, no. 5, pp. 542–548, 1927.
3 E. Fermi, “Un methodo statistico per la determinazione di alcune proprieta dell’atomo,” Atti
dell’Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali, vol. 6, pp. 602–607,
1927.
4 C. Y. Chan and Y. C. Hon, “A constructive solution for a generalized Thomas-Fermi theory of ionized
atoms,” Quarterly of Applied Mathematics, vol. 45, no. 3, pp. 591–599, 1987.
5 L. E. Bobisud, D. O’Regan, and W. D. Royalty, “Singular boundary value problems,” Applicable
Analysis, vol. 23, no. 3, pp. 233–243, 1986.
6 L. E. Bobisud, D. O’Regan, and W. D. Royalty, “Solvability of some nonlinear boundary value
problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 12, no. 9, pp. 855–869, 1988.
7 E. Hille, “Some aspects of the Thomas-Fermi equation,” Journal d’Analyse Mathématique, vol. 23, no. 1,
pp. 147–170, 1970.
8 A. Granas, R. B. Guenther, and J. Lee, “Nonlinear boundary value problems for ordinary differential
equations,” Dissertationes Mathematicae, vol. 244, p. 128, 1985.
9 D. R. Dunninger and J. C. Kurtz, “Existence of solutions for some nonlinear singular boundary value
problems,” Journal of Mathematical Analysis and Applications, vol. 115, no. 2, pp. 396–405, 1986.
10 R. K. Pandey and A. K. Verma, “Existence-uniqueness results for a class of singular boundary value
problems arising in physiology,” Nonlinear Analysis: Real World Applications, vol. 9, no. 1, pp. 40–52,
2008.
11 R. K. Pandey and A. K. Verma, “A note on existence-uniqueness results for a class of doubly singular
boundary value problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 7-8, pp.
3477–3487, 2009.
12 R. K. Pandey and A. K. Verma, “On solvability of derivative dependent doubly singular boundary
value problems,” Journal of Applied Mathematics and Computing, vol. 33, no. 1-2, pp. 489–511, 2010.
13 Y. Zhang, “A note on the solvability of singular boundary value problems,” Nonlinear Analysis: Theory,
Methods & Applications, vol. 26, no. 10, pp. 1605–1609, 1996.
14 G. W. Reddien, “Projection methods and singular two point boundary value problems,” Numerische
Mathematik, vol. 21, pp. 193–205, 1973.
15 M. M. Chawla and C. P. Katti, “Finite difference methods and their convergence for a class of singular
two-point boundary value problems,” Numerische Mathematik, vol. 39, no. 3, pp. 341–350, 1982.
16 R. K. Pandey and A. K. Singh, “On the convergence of a finite difference method for a class of singular
boundary value problems arising in physiology,” Journal of Computational and Applied Mathematics, vol.
166, no. 2, pp. 553–564, 2004.
Advances in Numerical Analysis
21
17 R. K. Pandey and A. K. Singh, “On the convergence of finite difference methods for weakly regular
singular boundary value problems,” Journal of Computational and Applied Mathematics, vol. 205, no. 1,
pp. 469–478, 2007.
18 M. M. Chawla and C. P. Katti, “A uniform mesh finite difference method for a class of singular twopoint boundary value problems,” SIAM Journal on Numerical Analysis, vol. 22, no. 3, pp. 561–565, 1985.
19 M. M. Chawla, “A fourth-order finite difference method based on uniform mesh for singular twopoint boundary-value problems,” Journal of Computational and Applied Mathematics, vol. 17, no. 3, pp.
359–364, 1987.
20 M. M. Chawla, R. Subramanian, and H. L. Sathi, “A fourth-order spline method for singular twopoint boundary value problems,” Journal of Computational and Applied Mathematics, vol. 21, no. 2, pp.
189–202, 1988.
21 M. M. Chawla, R. Subramanian, and H. L. Sathi, “A fourth order method for a singular two-point
boundary value problem,” BIT Numerical Mathematics, vol. 28, no. 1, pp. 88–97, 1988.
22 R. K. Pandey and A. K. Singh, “On the convergence of fourth-order finite difference method for
weakly regular singular boundary value problems,” International Journal of Computer Mathematics, vol.
81, no. 2, pp. 227–238, 2004.
23 R. K. Pandey and A. K. Singh, “A new high-accuracy difference method for a class of weakly nonlinear
singular boundary-value problems,” International Journal of Computer Mathematics, vol. 83, no. 11, pp.
809–817, 2006.
24 R. K. Pandey and A. K. Singh, “On the convergence of a fourth-order method for a class of singular
boundary value problems,” Journal of Computational and Applied Mathematics, vol. 224, no. 2, pp. 734–
742, 2009.
25 P. Henrici, Discrete Variable Methods in Ordinary Differential Equations, John Wiley & Sons, New York,
NY, USA, 1962.
Advances in
Operations Research
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Advances in
Decision Sciences
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Mathematical Problems
in Engineering
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Algebra
Hindawi Publishing Corporation
http://www.hindawi.com
Probability and Statistics
Volume 2014
The Scientific
World Journal
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International Journal of
Differential Equations
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at
http://www.hindawi.com
International Journal of
Advances in
Combinatorics
Hindawi Publishing Corporation
http://www.hindawi.com
Mathematical Physics
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International
Journal of
Mathematics and
Mathematical
Sciences
Journal of
Hindawi Publishing Corporation
http://www.hindawi.com
Stochastic Analysis
Abstract and
Applied Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
International Journal of
Mathematics
Volume 2014
Volume 2014
Discrete Dynamics in
Nature and Society
Volume 2014
Volume 2014
Journal of
Journal of
Discrete Mathematics
Journal of
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Applied Mathematics
Journal of
Function Spaces
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Optimization
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014