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Erratum to: B-spline collocation methods and their convergence for a class of nonlinear derivative dependent singular boundary value problems
Applied Mathematics and Computation 361 (2019) 198–201
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Erratum to: B-spline collocation methods and their convergence for a class of nonlinear derivative dependent singular boundary value problems V.M.K. Prasad Goura∗, Pradip Roul Department of Mathematics, Visvesvaraya National Institute of Technology, Nagpur 440010, India
a r t i c l e
i n f o
Keywords: Singular boundary value problem Uniform mesh Convergence analysis
a b s t r a c t In a recent paper [1], we presented two B-spline collocation methods and their convergence for a class of nonlinear derivative dependent singular boundary value problems. The first method is based on uniform mesh and the second method is based on non-uniform mesh. Convergent results for both methods were established. It is shown that the first method is of second order accuracy, while the second method is of fourth order accuracy. It is worth pointing out that there were some errors in the proof of Theorem 3 of [1], where the convergence proof of the first method is provided. In this erratum, we correct the mistakes occurred in the proof of Theorem 3. © 2019 Elsevier Inc. All rights reserved.
1. Introduction Recently, we proposed two numerical techniques [1] to approximate the solutions of a class of singular boundary value problems with derivative dependent source function:
( p(x )y ) = q(x ) f (x, y, y ), 0 ≤ x ≤ 1,
(1)
subject to the boundary conditions
y (0 ) = 0 (or lim p(x )y (0 ) = 0 ), x→0
μy ( 1 ) + η y ( 1 ) = ρ ,
(2)
with μ > 0, η ≥ 0, and ρ are finite constants. The first method was named as uniform mesh cubic B-spline collocation (UCS) approach, while the second method was named as non-uniform mesh cubic B-spline collocation (NCS) approach. We had established convergence results for both UCS and NCS methods, and it has been proved that UCS method yields O(h2 )convergent approximation and NCS method produces O(h4 )-convergent approximation for the solution of (1) and (2). In UCS approach, convergence result was proven by means of matrix analysis approach. However, we have committed some mistakes while proving the convergence of the method, as given in Theorem 3 of [1]. In this erratum, we provide the corrected proof for convergence of UCS method.
∗
Corresponding author. E-mail address: [email protected] (V.M.K.P. Goura).
https://doi.org/10.1016/j.amc.2019.05.022 0 096-30 03/© 2019 Elsevier Inc. All rights reserved.
V.M.K.P. Goura and P. Roul / Applied Mathematics and Computation 361 (2019) 198–201
199
2. Corrected proof of Theorem 3 of [1]: For completeness of the proof of Theorem 3 of [1], we first rewrite equation (27) of [1] in the following form:
i h h2 G Fi (−bi−1 + bi+1 ) − (bi−1 + 4bi + bi+1 ) 2 6 i + O(h4 ), i = 0, 1, 2, . . . , n. = − h2 H
i (bi−1 − 2bi + bi+1 ) − −R
(3)
Next, we eliminate the variables b−1 , bn+1 from the first and last equations of the system (3) respectively. Thus, after eliminating we obtain the following two equations:
0 − b0 2R
0 2 h2 G 3
2 0 − h G0 + b1 − 2R 3
0 + O(h4 ), = −h2 H
(4)
6(hμ + η )R n μh2 Fn ηh2 Gn 2μh2 Fn 6η R 2 η h2 G n n + − + bn + − hμ + 3η hμ + 3η hμ + 3η hμ + 3η hμ + 3η hμ + 3η 2 6hρ n + hFn + h Gn n + O(h4 ). = R − h2 H 2 6 hμ + 3η
bn−1 −
(5)
Eqs. (4) and (3) for i = 1, 2, . . . , n − 1 and (5) yield a system of (n + 1 ) equations in (n + 1 ) unknowns which can be written in the following matrix form:
P B = Q + R, where
⎛
p1,1 ⎜ p2,1 ⎜ ⎜ 0 ⎜ P=⎜ . ⎜ .. ⎜ ⎝ 0 0
(6)
p1,2 p2,2 p3,2 .. . 0 0
0 p2,3 p3,3 .. . 0 0
0 0 p3,4 .. . ··· ···
0 0 0 .. . ··· ···
··· ··· ··· .. . pn,n−1 0
··· ··· ··· .. . pn,n pn+1,n
⎞
0 0 0 .. . pn,n+1 pn+1,n+1
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
with
Q =
2 2 2 i + hFi − h Gi , pi+1, j+1 = 2R 0 − 2h G0 , p1,2 = −2R 0 − h G0 , pi+1, j = −R i p1,1 = 2 R 3 3 2 6 2 2 i 2 h2 G i − hFi − h Gi , 1 ≤ i, j ≤ n − 1, pn+1,n = − 6ηRn + μh Fn − , pi+1, j+2 = −R 3 2 6 hμ + 3η hμ + 3η n n
2 η h2 G η h2 G 6(hμ + η )R 2μh2 Fn n − , pn+1,n+1 = + − , B = b0 , b1 , · · · , bn−1 , hμ + 3η hμ + 3η hμ + 3η hμ + 3η
, −h2 H , · · · , −h2 H , −h2 H n + q† − h2 H 0 1 n−1
T
n + with q† = R
hFn 2
+
n h2 G 6
O ( h4 ).
6 hρ hμ+3η
bn
T
,
and R = [R0 , R1 , R2 , · · · , Rn ]T
with Ri = Let φ (x) be the unique cubic spline interpolant to the exact solution y(x) of the problem (15) and (16) of [1], as defined (x ) be the spline collocation approximation to the solution y(x) of the problem (15) and (16) of [1], by Eq. (18) of [1]. Let φ as follows n+1
(x ) = φ
bi i (x ).
i=−1
Our B-spline collocation method (6) consists in finding an approximation for the exact solution of problem (15) and (16) of [1] by solving the following system:
= Q, PB
(7)
= [ where B b0 , b1 , . . . , bn ]T . Subtracting (7) from (6), we get
) = R, P (B − B
(8)
= [ (b − where B − B b0 ), ( b1 − b1 ), . . . , ( bn − bn )]T . It is easy to see that for sufficiently small value of h, the off-diagonal 0 , lim p elements of the matrix P are non-zero, i.e. lim p1,2 = −2R 0 i+1, j = −Ri , lim pi+1, j+2 = −Ri , 1 ≤ i, j ≤ n − 1, lim pn+1,n = h→0
h→0
n . Thus, the matrix P is irreducible (See corollary of Theorem 7.2 of [2]). −2R
h→0
h→0
200
V.M.K.P. Goura and P. Roul / Applied Mathematics and Computation 361 (2019) 198–201
Now we determine row sums S¯0 , S¯1 , . . . , S¯n of the matrix P as follows
0 , S¯0 = p1,1 + p1,2 = −h2 G S¯i =
n−1
(9)
i , i = 1, 2, . . . , n − 1, ( pi+1, j + pi+1, j+1 + pi+1, j+2 ) = −h2 G
(10)
j=1
S¯n = pn+1,n + pn+1,n+1 =
n 3 η h2 G 6hμR 3μh2 Fn n + − . hμ + 3η hμ + 3η hμ + 3η
(11)
Thus, we can write
S¯ j =
n
d j,k , j = 0, 1, . . . , n,
(12)
k=0
(given in [1]), it follows from (9)-(11) where dj,k denotes the (j, k)th element of the matrix P. In view of definitions of G i that, for sufficiently small h, the matrix P is monotone (See Theorem 7.4 of [2]). Therefore, we find that P −1 exists. Let di,−1j denote the (i, j)th element of P −1 . We define the matrix and vector norms as
P−1 ∞ = max
0≤i≤n
n
|di,−1j |,
R∞ = max |Ri |.
j=0
0≤i≤n
From the theory of matrix, we have
P −1 P = I,
(13)
where I is the identity matrix. From Eq. (13), one can deduce that n n
di,−1j d j,k = 1, i = 0, 1, . . . , n.
(14)
k=0 j=0
Using Eq. (12) in Eq. (14) yields n
di,−1j S¯ j = 1, i = 0, 1, . . . , n.
(15)
j=0
Defining S¯k = min S¯ j , j = 0, 1, . . . , n, then from (15) we get n−1
di,−1j ≤
j=0
1 1 ≤ 2 , h m∗ S¯k
(16)
, −G , . . . , −G }. With the help of Taylor’s expansion, we can obtain the following where m∗ = min{−G 0 1 n−1 −1 di,n ≤
1 μ + 3η η ≤ + + O ( h ). S¯n 6 μR 2 h μ R n n
(17)
From (8), we have
B − B∞ = P−1 R∞ ≤ P −1 ∞ R∞ .
(18)
In view of (16), (17) and R∞ = O(h4 ), we can obtain that
B − B∞ = O(h2 ).
(19)
By using the first and second boundary conditions given in (16) of [1], it can be shown that
|b−1 − b˜ −1 | = O(h4 ) and |bn+1 − b˜ n+1 | = O(h4 ). Therefore, we find that
max
−1≤i≤n+1
| bi − b˜ i |= O(h2 ).
(20)
Using the definition of cubic B-spline basis functions given in Eq. (17) of [1], we can obtain that n+1 i=−1
|φi (x )| ≤ 10, 0 ≤ x ≤ 1.
(21)
V.M.K.P. Goura and P. Roul / Applied Mathematics and Computation 361 (2019) 198–201
201
With the help of Eqs. (20) and (21), we have
n+1 n+1 (x ) = φ ( x ) − φ bi i (x ) − bi i (x ) ∞ ≤
i=−1
i=−1
i=−1
i=−1
(22)
n+1 n+1 ( bi − bi ) i (x )
= O ( h2 ). (x ) = O(h2 ). From Theorem 2 of [1], we know that Hence, φ (x ) − φ ∞
y ( x ) − φ ( x )∞ = O ( h4 ).
(23)
Finally, using the triangle inequality and Eqs. (22) and (23) we can obtain
(x ) ≤ y(x ) − φ (x ) + φ (x ) − φ (x ) y ( x ) − φ ∞ ∞ ∞ 4 2 2 ≤ O ( h ) + O ( h ) = O ( h ).
(24)
This proves the second order convergence of UCS method. References [1] P. Roul, V.M.K. P. Goura, B-spline collocation methods and their convergence for a class of nonlinear derivative dependent singular boundary value problems, Appl. Math. Comput. 341 (2019) 428–450. [2] P. Henrici, Discrete Variable Methods in Ordinary Differential Equations, John Wiley and Sons, New York, NY, USA, 1962.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Erratum to: B-spline collocation methods and their convergence for a class of nonlinear derivative dependent singular boundary value problems V.M.K. Prasad Goura∗, Pradip Roul Department of Mathematics, Visvesvaraya National Institute of Technology, Nagpur 440010, India
a r t i c l e
i n f o
Keywords: Singular boundary value problem Uniform mesh Convergence analysis
a b s t r a c t In a recent paper [1], we presented two B-spline collocation methods and their convergence for a class of nonlinear derivative dependent singular boundary value problems. The first method is based on uniform mesh and the second method is based on non-uniform mesh. Convergent results for both methods were established. It is shown that the first method is of second order accuracy, while the second method is of fourth order accuracy. It is worth pointing out that there were some errors in the proof of Theorem 3 of [1], where the convergence proof of the first method is provided. In this erratum, we correct the mistakes occurred in the proof of Theorem 3. © 2019 Elsevier Inc. All rights reserved.
1. Introduction Recently, we proposed two numerical techniques [1] to approximate the solutions of a class of singular boundary value problems with derivative dependent source function:
( p(x )y ) = q(x ) f (x, y, y ), 0 ≤ x ≤ 1,
(1)
subject to the boundary conditions
y (0 ) = 0 (or lim p(x )y (0 ) = 0 ), x→0
μy ( 1 ) + η y ( 1 ) = ρ ,
(2)
with μ > 0, η ≥ 0, and ρ are finite constants. The first method was named as uniform mesh cubic B-spline collocation (UCS) approach, while the second method was named as non-uniform mesh cubic B-spline collocation (NCS) approach. We had established convergence results for both UCS and NCS methods, and it has been proved that UCS method yields O(h2 )convergent approximation and NCS method produces O(h4 )-convergent approximation for the solution of (1) and (2). In UCS approach, convergence result was proven by means of matrix analysis approach. However, we have committed some mistakes while proving the convergence of the method, as given in Theorem 3 of [1]. In this erratum, we provide the corrected proof for convergence of UCS method.
∗
Corresponding author. E-mail address: [email protected] (V.M.K.P. Goura).
https://doi.org/10.1016/j.amc.2019.05.022 0 096-30 03/© 2019 Elsevier Inc. All rights reserved.
V.M.K.P. Goura and P. Roul / Applied Mathematics and Computation 361 (2019) 198–201
199
2. Corrected proof of Theorem 3 of [1]: For completeness of the proof of Theorem 3 of [1], we first rewrite equation (27) of [1] in the following form:
i h h2 G Fi (−bi−1 + bi+1 ) − (bi−1 + 4bi + bi+1 ) 2 6 i + O(h4 ), i = 0, 1, 2, . . . , n. = − h2 H
i (bi−1 − 2bi + bi+1 ) − −R
(3)
Next, we eliminate the variables b−1 , bn+1 from the first and last equations of the system (3) respectively. Thus, after eliminating we obtain the following two equations:
0 − b0 2R
0 2 h2 G 3
2 0 − h G0 + b1 − 2R 3
0 + O(h4 ), = −h2 H
(4)
6(hμ + η )R n μh2 Fn ηh2 Gn 2μh2 Fn 6η R 2 η h2 G n n + − + bn + − hμ + 3η hμ + 3η hμ + 3η hμ + 3η hμ + 3η hμ + 3η 2 6hρ n + hFn + h Gn n + O(h4 ). = R − h2 H 2 6 hμ + 3η
bn−1 −
(5)
Eqs. (4) and (3) for i = 1, 2, . . . , n − 1 and (5) yield a system of (n + 1 ) equations in (n + 1 ) unknowns which can be written in the following matrix form:
P B = Q + R, where
⎛
p1,1 ⎜ p2,1 ⎜ ⎜ 0 ⎜ P=⎜ . ⎜ .. ⎜ ⎝ 0 0
(6)
p1,2 p2,2 p3,2 .. . 0 0
0 p2,3 p3,3 .. . 0 0
0 0 p3,4 .. . ··· ···
0 0 0 .. . ··· ···
··· ··· ··· .. . pn,n−1 0
··· ··· ··· .. . pn,n pn+1,n
⎞
0 0 0 .. . pn,n+1 pn+1,n+1
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
with
Q =
2 2 2 i + hFi − h Gi , pi+1, j+1 = 2R 0 − 2h G0 , p1,2 = −2R 0 − h G0 , pi+1, j = −R i p1,1 = 2 R 3 3 2 6 2 2 i 2 h2 G i − hFi − h Gi , 1 ≤ i, j ≤ n − 1, pn+1,n = − 6ηRn + μh Fn − , pi+1, j+2 = −R 3 2 6 hμ + 3η hμ + 3η n n
2 η h2 G η h2 G 6(hμ + η )R 2μh2 Fn n − , pn+1,n+1 = + − , B = b0 , b1 , · · · , bn−1 , hμ + 3η hμ + 3η hμ + 3η hμ + 3η
, −h2 H , · · · , −h2 H , −h2 H n + q† − h2 H 0 1 n−1
T
n + with q† = R
hFn 2
+
n h2 G 6
O ( h4 ).
6 hρ hμ+3η
bn
T
,
and R = [R0 , R1 , R2 , · · · , Rn ]T
with Ri = Let φ (x) be the unique cubic spline interpolant to the exact solution y(x) of the problem (15) and (16) of [1], as defined (x ) be the spline collocation approximation to the solution y(x) of the problem (15) and (16) of [1], by Eq. (18) of [1]. Let φ as follows n+1
(x ) = φ
bi i (x ).
i=−1
Our B-spline collocation method (6) consists in finding an approximation for the exact solution of problem (15) and (16) of [1] by solving the following system:
= Q, PB
(7)
= [ where B b0 , b1 , . . . , bn ]T . Subtracting (7) from (6), we get
) = R, P (B − B
(8)
= [ (b − where B − B b0 ), ( b1 − b1 ), . . . , ( bn − bn )]T . It is easy to see that for sufficiently small value of h, the off-diagonal 0 , lim p elements of the matrix P are non-zero, i.e. lim p1,2 = −2R 0 i+1, j = −Ri , lim pi+1, j+2 = −Ri , 1 ≤ i, j ≤ n − 1, lim pn+1,n = h→0
h→0
n . Thus, the matrix P is irreducible (See corollary of Theorem 7.2 of [2]). −2R
h→0
h→0
200
V.M.K.P. Goura and P. Roul / Applied Mathematics and Computation 361 (2019) 198–201
Now we determine row sums S¯0 , S¯1 , . . . , S¯n of the matrix P as follows
0 , S¯0 = p1,1 + p1,2 = −h2 G S¯i =
n−1
(9)
i , i = 1, 2, . . . , n − 1, ( pi+1, j + pi+1, j+1 + pi+1, j+2 ) = −h2 G
(10)
j=1
S¯n = pn+1,n + pn+1,n+1 =
n 3 η h2 G 6hμR 3μh2 Fn n + − . hμ + 3η hμ + 3η hμ + 3η
(11)
Thus, we can write
S¯ j =
n
d j,k , j = 0, 1, . . . , n,
(12)
k=0
(given in [1]), it follows from (9)-(11) where dj,k denotes the (j, k)th element of the matrix P. In view of definitions of G i that, for sufficiently small h, the matrix P is monotone (See Theorem 7.4 of [2]). Therefore, we find that P −1 exists. Let di,−1j denote the (i, j)th element of P −1 . We define the matrix and vector norms as
P−1 ∞ = max
0≤i≤n
n
|di,−1j |,
R∞ = max |Ri |.
j=0
0≤i≤n
From the theory of matrix, we have
P −1 P = I,
(13)
where I is the identity matrix. From Eq. (13), one can deduce that n n
di,−1j d j,k = 1, i = 0, 1, . . . , n.
(14)
k=0 j=0
Using Eq. (12) in Eq. (14) yields n
di,−1j S¯ j = 1, i = 0, 1, . . . , n.
(15)
j=0
Defining S¯k = min S¯ j , j = 0, 1, . . . , n, then from (15) we get n−1
di,−1j ≤
j=0
1 1 ≤ 2 , h m∗ S¯k
(16)
, −G , . . . , −G }. With the help of Taylor’s expansion, we can obtain the following where m∗ = min{−G 0 1 n−1 −1 di,n ≤
1 μ + 3η η ≤ + + O ( h ). S¯n 6 μR 2 h μ R n n
(17)
From (8), we have
B − B∞ = P−1 R∞ ≤ P −1 ∞ R∞ .
(18)
In view of (16), (17) and R∞ = O(h4 ), we can obtain that
B − B∞ = O(h2 ).
(19)
By using the first and second boundary conditions given in (16) of [1], it can be shown that
|b−1 − b˜ −1 | = O(h4 ) and |bn+1 − b˜ n+1 | = O(h4 ). Therefore, we find that
max
−1≤i≤n+1
| bi − b˜ i |= O(h2 ).
(20)
Using the definition of cubic B-spline basis functions given in Eq. (17) of [1], we can obtain that n+1 i=−1
|φi (x )| ≤ 10, 0 ≤ x ≤ 1.
(21)
V.M.K.P. Goura and P. Roul / Applied Mathematics and Computation 361 (2019) 198–201
201
With the help of Eqs. (20) and (21), we have
n+1 n+1 (x ) = φ ( x ) − φ bi i (x ) − bi i (x ) ∞ ≤
i=−1
i=−1
i=−1
i=−1
(22)
n+1 n+1 ( bi − bi ) i (x )
= O ( h2 ). (x ) = O(h2 ). From Theorem 2 of [1], we know that Hence, φ (x ) − φ ∞
y ( x ) − φ ( x )∞ = O ( h4 ).
(23)
Finally, using the triangle inequality and Eqs. (22) and (23) we can obtain
(x ) ≤ y(x ) − φ (x ) + φ (x ) − φ (x ) y ( x ) − φ ∞ ∞ ∞ 4 2 2 ≤ O ( h ) + O ( h ) = O ( h ).
(24)
This proves the second order convergence of UCS method. References [1] P. Roul, V.M.K. P. Goura, B-spline collocation methods and their convergence for a class of nonlinear derivative dependent singular boundary value problems, Appl. Math. Comput. 341 (2019) 428–450. [2] P. Henrici, Discrete Variable Methods in Ordinary Differential Equations, John Wiley and Sons, New York, NY, USA, 1962.