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Global convergence of an SQP algorithm for nonlinear optimization with overdetermined constraints
A sixth order numerical method for a class of nonlinear twopoint boundary value problems
1.  Department of Mathematical Sciences, Faculty of Science, Yamagata University, Yamagata 9908560, Japan, Japan 
References:
[1] 
S. Aguchi and T. Yamamoto, Numerical methods with fourth order accuracy for twopoint boundary value problems, RIMS Kokyuroku, Kyoto Univ., 1381 (2004), 1120. 
[2] 
U. M. Ascher, R. M. M. Mattheij and R. D. Russell, "Numerical Solution of Boundary Value Problems for Ordinary Differential Equations," Prentice Hall, Englewood Cliffs, NJ, 1988. 
[3] 
L. K. Bieniasz, Two new compact finitedifference schemes for the solution of boundary value problems in secondorder nonlinear ordinary differential equations, using nonuniform grids, J. Comput. Methods Sci. Engineer., 8 (2008), 318. 
[4] 
J. H. Bramble and B. E. Hubbard, On the formulation of finite difference analogue of the Dirichlet problem for Poisson's equation, Numer. Math., 4 (1962), 313327. doi: doi:10.1007/BF01386325. 
[5] 
J. C. Butcher, "Numerical Methods for Ordinary Differential Equations," 2^{nd} edition, John Wiley & Sons, Chichester, 2008. 
[6] 
M. M. Chawla, A sixth order tridiagonal finite difference method for nonlinear twopoint boundary value problems, BIT, 17 (1977), 128133. doi: doi:10.1007/BF01932284. 
[7] 
M. M. Chawla, A sixthorder tridiagonal finite difference method for general nonlinear twopoint boundary value problems, J. Inst. Math. Appl., 24 (1979), 3542. doi: doi:10.1093/imamat/24.1.35. 
[8] 
L. Collatz, "The Numerical Treatment of Differential Equations," Springer, Berlin, 1966. 
[9] 
Q. Fang, Convergence of AscherMattheijRussell finite difference method for a class of twopoint boundary value problems, Information, 9 (2006), 563572. 
[10] 
Q. Fang, T. Tsuchiya and T. Yamamoto, Finite difference, finite element and finite volume methods applied to twopoint boundary value problems, J. Comput. Appl. Math., 139 (2002), 919. doi: doi:10.1016/S03770427(01)003922. 
[11] 
H. B. Keller, "Numerical Methods for TwoPoint Boundary Value Problems," Blaisdell, Waltham, 1968. 
[12] 
M. Kumar, Higher order method for singular boundaryvalue problems by using spline function, Appl. Math. Comput., 192 (2007), 175179. doi: doi:10.1016/j.amc.2007.02.156. 
[13] 
R. K. Mohanty, A family of variable mesh methods for the estimates of $(du)/(dr)$ and solution of nonlinear two point boundary value problems with singularity, J. Comput. Appl. Math., 182 (2005), 173187. doi: doi:10.1016/j.cam.2004.11.045. 
[14] 
R. K. Mohanty and U. Arora, A TAGE iterative method for the solution of nonlinear singular two point boundary value problems using a sixth order discretization, Appl. Math. Comput., 180 (2006), 538548. doi: doi:10.1016/j.amc.2005.12.038. 
[15] 
R. K. Mohanty and N. Khosla, Application of TAGE iterative algorithms to an efficient third order arithmetic average variable mesh discretization for twopoint nonlinear boundary value problems, Appl. Math. Comput., 172 (2006), 148162. doi: doi:10.1016/j.amc.2005.01.134. 
[16] 
G. H. Shortley and R. Weller, The numerical solution of Laplace's equation, J. Appl. Phys., 9 (1938), 334348. doi: doi:10.1063/1.1710426. 
[17] 
J. Stoer and R. Bulirsch, "Introduction to Numerical Analysis," 3^{rd} edition, Springer, New York, 2002. 
[18] 
T. Yamamoto, Harmonic relations between Green's functions and Green's matrices for boundary value problems, RIMS Kokyuroku, Kyoto Univ., 1169 (2000), 1526. 
[19] 
T. Yamamoto, Harmonic relations between Green's functions and Green's matrices for boundary value problems II, RIMS Kokyuroku, Kyoto Univ., 1286 (2002), 2733. 
[20] 
T. Yamamoto, Harmonic relations between Green's functions and Green's matrices for boundary value problems III, RIMS Kokyuroku, Kyoto Univ., 1381 (2004), 110. 
[21] 
T. Yamamoto, Discretization principles for linear twopoint boundary value problems, Numer. Funct. Anal. and Optimiz., 28 (2007), 149172. doi: doi:10.1080/01630560600791296. 
[22] 
T. Yamamoto and S. Oishi, A mathematical theory for numerical treatment of nonlinear twopoint boundary value problems, Japan J. Indust. Appl. Math., 23 (2006), 3162. doi: doi:10.1007/BF03167497. 
show all references
References:
[1] 
S. Aguchi and T. Yamamoto, Numerical methods with fourth order accuracy for twopoint boundary value problems, RIMS Kokyuroku, Kyoto Univ., 1381 (2004), 1120. 
[2] 
U. M. Ascher, R. M. M. Mattheij and R. D. Russell, "Numerical Solution of Boundary Value Problems for Ordinary Differential Equations," Prentice Hall, Englewood Cliffs, NJ, 1988. 
[3] 
L. K. Bieniasz, Two new compact finitedifference schemes for the solution of boundary value problems in secondorder nonlinear ordinary differential equations, using nonuniform grids, J. Comput. Methods Sci. Engineer., 8 (2008), 318. 
[4] 
J. H. Bramble and B. E. Hubbard, On the formulation of finite difference analogue of the Dirichlet problem for Poisson's equation, Numer. Math., 4 (1962), 313327. doi: doi:10.1007/BF01386325. 
[5] 
J. C. Butcher, "Numerical Methods for Ordinary Differential Equations," 2^{nd} edition, John Wiley & Sons, Chichester, 2008. 
[6] 
M. M. Chawla, A sixth order tridiagonal finite difference method for nonlinear twopoint boundary value problems, BIT, 17 (1977), 128133. doi: doi:10.1007/BF01932284. 
[7] 
M. M. Chawla, A sixthorder tridiagonal finite difference method for general nonlinear twopoint boundary value problems, J. Inst. Math. Appl., 24 (1979), 3542. doi: doi:10.1093/imamat/24.1.35. 
[8] 
L. Collatz, "The Numerical Treatment of Differential Equations," Springer, Berlin, 1966. 
[9] 
Q. Fang, Convergence of AscherMattheijRussell finite difference method for a class of twopoint boundary value problems, Information, 9 (2006), 563572. 
[10] 
Q. Fang, T. Tsuchiya and T. Yamamoto, Finite difference, finite element and finite volume methods applied to twopoint boundary value problems, J. Comput. Appl. Math., 139 (2002), 919. doi: doi:10.1016/S03770427(01)003922. 
[11] 
H. B. Keller, "Numerical Methods for TwoPoint Boundary Value Problems," Blaisdell, Waltham, 1968. 
[12] 
M. Kumar, Higher order method for singular boundaryvalue problems by using spline function, Appl. Math. Comput., 192 (2007), 175179. doi: doi:10.1016/j.amc.2007.02.156. 
[13] 
R. K. Mohanty, A family of variable mesh methods for the estimates of $(du)/(dr)$ and solution of nonlinear two point boundary value problems with singularity, J. Comput. Appl. Math., 182 (2005), 173187. doi: doi:10.1016/j.cam.2004.11.045. 
[14] 
R. K. Mohanty and U. Arora, A TAGE iterative method for the solution of nonlinear singular two point boundary value problems using a sixth order discretization, Appl. Math. Comput., 180 (2006), 538548. doi: doi:10.1016/j.amc.2005.12.038. 
[15] 
R. K. Mohanty and N. Khosla, Application of TAGE iterative algorithms to an efficient third order arithmetic average variable mesh discretization for twopoint nonlinear boundary value problems, Appl. Math. Comput., 172 (2006), 148162. doi: doi:10.1016/j.amc.2005.01.134. 
[16] 
G. H. Shortley and R. Weller, The numerical solution of Laplace's equation, J. Appl. Phys., 9 (1938), 334348. doi: doi:10.1063/1.1710426. 
[17] 
J. Stoer and R. Bulirsch, "Introduction to Numerical Analysis," 3^{rd} edition, Springer, New York, 2002. 
[18] 
T. Yamamoto, Harmonic relations between Green's functions and Green's matrices for boundary value problems, RIMS Kokyuroku, Kyoto Univ., 1169 (2000), 1526. 
[19] 
T. Yamamoto, Harmonic relations between Green's functions and Green's matrices for boundary value problems II, RIMS Kokyuroku, Kyoto Univ., 1286 (2002), 2733. 
[20] 
T. Yamamoto, Harmonic relations between Green's functions and Green's matrices for boundary value problems III, RIMS Kokyuroku, Kyoto Univ., 1381 (2004), 110. 
[21] 
T. Yamamoto, Discretization principles for linear twopoint boundary value problems, Numer. Funct. Anal. and Optimiz., 28 (2007), 149172. doi: doi:10.1080/01630560600791296. 
[22] 
T. Yamamoto and S. Oishi, A mathematical theory for numerical treatment of nonlinear twopoint boundary value problems, Japan J. Indust. Appl. Math., 23 (2006), 3162. doi: doi:10.1007/BF03167497. 
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