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2012, 2(1): 31-43. doi: 10.3934/naco.2012.2.31

A sixth order numerical method for a class of nonlinear two-point boundary value problems

Xiao-Yu Zhang 1, and  Qing Fang 1,

1. 

Department of Mathematical Sciences, Faculty of Science, Yamagata University, Yamagata 990-8560, Japan, Japan

Received  April 2011 Revised  June 2011 Published  March 2012

In this paper, we are concerned with the numerical solution of a class of nonlinear two-point boundary value problems with general boundary conditions. We propose a new numerical method of sixth order accuracy by integrating compact finite difference methods with the Green's function approach. It is the first sixth order accurate numerical scheme on non-uniform grids for the problem. We also give numerical results of some practical problems including reaction-diffusion equations. It is remarked that our numerical method is also efficient for layer equations.
Keywords: finite difference methods, Green's functions, Two-point boundary value problems, layer equations., reaction-diffusion equations.
Mathematics Subject Classification: Primary: 65L10; Secondary: 65L1.
Citation: Xiao-Yu Zhang, Qing Fang. A sixth order numerical method for a class of nonlinear two-point boundary value problems. Numerical Algebra, Control and Optimization, 2012, 2 (1) : 31-43. doi: 10.3934/naco.2012.2.31
References:
[1]

S. Aguchi and T. Yamamoto, Numerical methods with fourth order accuracy for two-point boundary value problems, RIMS Kokyuroku, Kyoto Univ., 1381 (2004), 11-20.

[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 finite-difference schemes for the solution of boundary value problems in second-order non-linear ordinary differential equations, using non-uniform grids, J. Comput. Methods Sci. Engineer., 8 (2008), 3-18.

[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), 313-327. doi: doi:10.1007/BF01386325.

[5]

J. C. Butcher, "Numerical Methods for Ordinary Differential Equations," 2nd edition, John Wiley & Sons, Chichester, 2008.

[6]

M. M. Chawla, A sixth order tridiagonal finite difference method for non-linear two-point boundary value problems, BIT, 17 (1977), 128-133. doi: doi:10.1007/BF01932284.

[7]

M. M. Chawla, A sixth-order tridiagonal finite difference method for general non-linear two-point boundary value problems, J. Inst. Math. Appl., 24 (1979), 35-42. 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 Ascher-Mattheij-Russell finite difference method for a class of two-point boundary value problems, Information, 9 (2006), 563-572.

[10]

Q. Fang, T. Tsuchiya and T. Yamamoto, Finite difference, finite element and finite volume methods applied to two-point boundary value problems, J. Comput. Appl. Math., 139 (2002), 9-19. doi: doi:10.1016/S0377-0427(01)00392-2.

[11]

H. B. Keller, "Numerical Methods for Two-Point Boundary Value Problems," Blaisdell, Waltham, 1968.

[12]

M. Kumar, Higher order method for singular boundary-value problems by using spline function, Appl. Math. Comput., 192 (2007), 175-179. 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 non-linear two point boundary value problems with singularity, J. Comput. Appl. Math., 182 (2005), 173-187. doi: doi:10.1016/j.cam.2004.11.045.

[14]

R. K. Mohanty and U. Arora, A TAGE iterative method for the solution of non-linear singular two point boundary value problems using a sixth order discretization, Appl. Math. Comput., 180 (2006), 538-548. 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 two-point non-linear boundary value problems, Appl. Math. Comput., 172 (2006), 148-162. 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), 334-348. doi: doi:10.1063/1.1710426.

[17]

J. Stoer and R. Bulirsch, "Introduction to Numerical Analysis," 3rd 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), 15-26.

[19]

T. Yamamoto, Harmonic relations between Green's functions and Green's matrices for boundary value problems II, RIMS Kokyuroku, Kyoto Univ., 1286 (2002), 27-33.

[20]

T. Yamamoto, Harmonic relations between Green's functions and Green's matrices for boundary value problems III, RIMS Kokyuroku, Kyoto Univ., 1381 (2004), 1-10.

[21]

T. Yamamoto, Discretization principles for linear two-point boundary value problems, Numer. Funct. Anal. and Optimiz., 28 (2007), 149-172. doi: doi:10.1080/01630560600791296.

[22]

T. Yamamoto and S. Oishi, A mathematical theory for numerical treatment of nonlinear two-point boundary value problems, Japan J. Indust. Appl. Math., 23 (2006), 31-62. doi: doi:10.1007/BF03167497.

show all references

References:
[1]

S. Aguchi and T. Yamamoto, Numerical methods with fourth order accuracy for two-point boundary value problems, RIMS Kokyuroku, Kyoto Univ., 1381 (2004), 11-20.

[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 finite-difference schemes for the solution of boundary value problems in second-order non-linear ordinary differential equations, using non-uniform grids, J. Comput. Methods Sci. Engineer., 8 (2008), 3-18.

[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), 313-327. doi: doi:10.1007/BF01386325.

[5]

J. C. Butcher, "Numerical Methods for Ordinary Differential Equations," 2nd edition, John Wiley & Sons, Chichester, 2008.

[6]

M. M. Chawla, A sixth order tridiagonal finite difference method for non-linear two-point boundary value problems, BIT, 17 (1977), 128-133. doi: doi:10.1007/BF01932284.

[7]

M. M. Chawla, A sixth-order tridiagonal finite difference method for general non-linear two-point boundary value problems, J. Inst. Math. Appl., 24 (1979), 35-42. 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 Ascher-Mattheij-Russell finite difference method for a class of two-point boundary value problems, Information, 9 (2006), 563-572.

[10]

Q. Fang, T. Tsuchiya and T. Yamamoto, Finite difference, finite element and finite volume methods applied to two-point boundary value problems, J. Comput. Appl. Math., 139 (2002), 9-19. doi: doi:10.1016/S0377-0427(01)00392-2.

[11]

H. B. Keller, "Numerical Methods for Two-Point Boundary Value Problems," Blaisdell, Waltham, 1968.

[12]

M. Kumar, Higher order method for singular boundary-value problems by using spline function, Appl. Math. Comput., 192 (2007), 175-179. 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 non-linear two point boundary value problems with singularity, J. Comput. Appl. Math., 182 (2005), 173-187. doi: doi:10.1016/j.cam.2004.11.045.

[14]

R. K. Mohanty and U. Arora, A TAGE iterative method for the solution of non-linear singular two point boundary value problems using a sixth order discretization, Appl. Math. Comput., 180 (2006), 538-548. 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 two-point non-linear boundary value problems, Appl. Math. Comput., 172 (2006), 148-162. 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), 334-348. doi: doi:10.1063/1.1710426.

[17]

J. Stoer and R. Bulirsch, "Introduction to Numerical Analysis," 3rd 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), 15-26.

[19]

T. Yamamoto, Harmonic relations between Green's functions and Green's matrices for boundary value problems II, RIMS Kokyuroku, Kyoto Univ., 1286 (2002), 27-33.

[20]

T. Yamamoto, Harmonic relations between Green's functions and Green's matrices for boundary value problems III, RIMS Kokyuroku, Kyoto Univ., 1381 (2004), 1-10.

[21]

T. Yamamoto, Discretization principles for linear two-point boundary value problems, Numer. Funct. Anal. and Optimiz., 28 (2007), 149-172. doi: doi:10.1080/01630560600791296.

[22]

T. Yamamoto and S. Oishi, A mathematical theory for numerical treatment of nonlinear two-point boundary value problems, Japan J. Indust. Appl. Math., 23 (2006), 31-62. doi: doi:10.1007/BF03167497.

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