• Previous Article
    An AIS-based optimal control framework for longevity and task achievement of multi-robot systems
  • NACO Home
  • This Issue
  • Next Article
    Global convergence of an SQP algorithm for nonlinear optimization with overdetermined constraints
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

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.
Citation: Xiao-Yu Zhang, Qing Fang. A sixth order numerical method for a class of nonlinear two-point boundary value problems. Numerical Algebra, Control & 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, 1381 (2004), 11. Google Scholar

[2]

U. M. Ascher, R. M. M. Mattheij and R. D. Russell, "Numerical Solution of Boundary Value Problems for Ordinary Differential Equations,", Prentice Hall, (1988). Google Scholar

[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. Google Scholar

[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. doi: doi:10.1007/BF01386325. Google Scholar

[5]

J. C. Butcher, "Numerical Methods for Ordinary Differential Equations,", 2nd edition, (2008). Google Scholar

[6]

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

[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. doi: doi:10.1093/imamat/24.1.35. Google Scholar

[8]

L. Collatz, "The Numerical Treatment of Differential Equations,", Springer, (1966). Google Scholar

[9]

Q. Fang, Convergence of Ascher-Mattheij-Russell finite difference method for a class of two-point boundary value problems,, Information, 9 (2006), 563. Google Scholar

[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. doi: doi:10.1016/S0377-0427(01)00392-2. Google Scholar

[11]

H. B. Keller, "Numerical Methods for Two-Point Boundary Value Problems,", Blaisdell, (1968). Google Scholar

[12]

M. Kumar, Higher order method for singular boundary-value problems by using spline function,, Appl. Math. Comput., 192 (2007), 175. doi: doi:10.1016/j.amc.2007.02.156. Google Scholar

[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. doi: doi:10.1016/j.cam.2004.11.045. Google Scholar

[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. doi: doi:10.1016/j.amc.2005.12.038. Google Scholar

[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. doi: doi:10.1016/j.amc.2005.01.134. Google Scholar

[16]

G. H. Shortley and R. Weller, The numerical solution of Laplace's equation,, J. Appl. Phys., 9 (1938), 334. doi: doi:10.1063/1.1710426. Google Scholar

[17]

J. Stoer and R. Bulirsch, "Introduction to Numerical Analysis,", 3rd edition, (2002). Google Scholar

[18]

T. Yamamoto, Harmonic relations between Green's functions and Green's matrices for boundary value problems,, RIMS Kokyuroku, 1169 (2000), 15. Google Scholar

[19]

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

[20]

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

[21]

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

[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. doi: doi:10.1007/BF03167497. Google Scholar

show all references

References:
[1]

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

[2]

U. M. Ascher, R. M. M. Mattheij and R. D. Russell, "Numerical Solution of Boundary Value Problems for Ordinary Differential Equations,", Prentice Hall, (1988). Google Scholar

[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. Google Scholar

[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. doi: doi:10.1007/BF01386325. Google Scholar

[5]

J. C. Butcher, "Numerical Methods for Ordinary Differential Equations,", 2nd edition, (2008). Google Scholar

[6]

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

[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. doi: doi:10.1093/imamat/24.1.35. Google Scholar

[8]

L. Collatz, "The Numerical Treatment of Differential Equations,", Springer, (1966). Google Scholar

[9]

Q. Fang, Convergence of Ascher-Mattheij-Russell finite difference method for a class of two-point boundary value problems,, Information, 9 (2006), 563. Google Scholar

[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. doi: doi:10.1016/S0377-0427(01)00392-2. Google Scholar

[11]

H. B. Keller, "Numerical Methods for Two-Point Boundary Value Problems,", Blaisdell, (1968). Google Scholar

[12]

M. Kumar, Higher order method for singular boundary-value problems by using spline function,, Appl. Math. Comput., 192 (2007), 175. doi: doi:10.1016/j.amc.2007.02.156. Google Scholar

[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. doi: doi:10.1016/j.cam.2004.11.045. Google Scholar

[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. doi: doi:10.1016/j.amc.2005.12.038. Google Scholar

[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. doi: doi:10.1016/j.amc.2005.01.134. Google Scholar

[16]

G. H. Shortley and R. Weller, The numerical solution of Laplace's equation,, J. Appl. Phys., 9 (1938), 334. doi: doi:10.1063/1.1710426. Google Scholar

[17]

J. Stoer and R. Bulirsch, "Introduction to Numerical Analysis,", 3rd edition, (2002). Google Scholar

[18]

T. Yamamoto, Harmonic relations between Green's functions and Green's matrices for boundary value problems,, RIMS Kokyuroku, 1169 (2000), 15. Google Scholar

[19]

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

[20]

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

[21]

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

[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. doi: doi:10.1007/BF03167497. Google Scholar

[1]

Feliz Minhós, A. I. Santos. Higher order two-point boundary value problems with asymmetric growth. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 127-137. doi: 10.3934/dcdss.2008.1.127

[2]

Jeremiah Birrell. A posteriori error bounds for two point boundary value problems: A green's function approach. Journal of Computational Dynamics, 2015, 2 (2) : 143-164. doi: 10.3934/jcd.2015001

[3]

Shao-Yuan Huang, Shin-Hwa Wang. On S-shaped bifurcation curves for a two-point boundary value problem arising in a theory of thermal explosion. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4839-4858. doi: 10.3934/dcds.2015.35.4839

[4]

Jerry L. Bona, Hongqiu Chen, Shu-Ming Sun, Bing-Yu Zhang. Comparison of quarter-plane and two-point boundary value problems: The KdV-equation. Discrete & Continuous Dynamical Systems - B, 2007, 7 (3) : 465-495. doi: 10.3934/dcdsb.2007.7.465

[5]

Jerry Bona, Hongqiu Chen, Shu Ming Sun, B.-Y. Zhang. Comparison of quarter-plane and two-point boundary value problems: the BBM-equation. Discrete & Continuous Dynamical Systems - A, 2005, 13 (4) : 921-940. doi: 10.3934/dcds.2005.13.921

[6]

Wenming Zou. Multiple solutions results for two-point boundary value problem with resonance. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 485-496. doi: 10.3934/dcds.1998.4.485

[7]

Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. II. Convergence of the method of finite differences. Inverse Problems & Imaging, 2016, 10 (4) : 869-898. doi: 10.3934/ipi.2016025

[8]

K. Q. Lan, G. C. Yang. Optimal constants for two point boundary value problems. Conference Publications, 2007, 2007 (Special) : 624-633. doi: 10.3934/proc.2007.2007.624

[9]

Lizhi Ruan, Changjiang Zhu. Boundary layer for nonlinear evolution equations with damping and diffusion. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 331-352. doi: 10.3934/dcds.2012.32.331

[10]

Xiaohai Wan, Zhilin Li. Some new finite difference methods for Helmholtz equations on irregular domains or with interfaces. Discrete & Continuous Dynamical Systems - B, 2012, 17 (4) : 1155-1174. doi: 10.3934/dcdsb.2012.17.1155

[11]

Messoud Efendiev, Alain Miranville. Finite dimensional attractors for reaction-diffusion equations in $R^n$ with a strong nonlinearity. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 399-424. doi: 10.3934/dcds.1999.5.399

[12]

Ciprian G. Gal, Mahamadi Warma. Reaction-diffusion equations with fractional diffusion on non-smooth domains with various boundary conditions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1279-1319. doi: 10.3934/dcds.2016.36.1279

[13]

Piermarco Cannarsa, Giuseppe Da Prato. Invariance for stochastic reaction-diffusion equations. Evolution Equations & Control Theory, 2012, 1 (1) : 43-56. doi: 10.3934/eect.2012.1.43

[14]

Martino Prizzi. A remark on reaction-diffusion equations in unbounded domains. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 281-286. doi: 10.3934/dcds.2003.9.281

[15]

Angelo Favini, Atsushi Yagi. Global existence for Laplace reaction-diffusion equations. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1-21. doi: 10.3934/dcdss.2020083

[16]

Matthias Büger. Planar and screw-shaped solutions for a system of two reaction-diffusion equations on the circle. Discrete & Continuous Dynamical Systems - A, 2006, 16 (4) : 745-756. doi: 10.3934/dcds.2006.16.745

[17]

Marek Fila, Hirokazu Ninomiya, Juan-Luis Vázquez. Dirichlet boundary conditions can prevent blow-up in reaction-diffusion equations and systems. Discrete & Continuous Dynamical Systems - A, 2006, 14 (1) : 63-74. doi: 10.3934/dcds.2006.14.63

[18]

Peter E. Kloeden, Thomas Lorenz, Meihua Yang. Reaction-diffusion equations with a switched--off reaction zone. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1907-1933. doi: 10.3934/cpaa.2014.13.1907

[19]

Navnit Jha. Nonpolynomial spline finite difference scheme for nonlinear singuiar boundary value problems with singular perturbation and its mechanization. Conference Publications, 2013, 2013 (special) : 355-363. doi: 10.3934/proc.2013.2013.355

[20]

Jong-Shenq Guo, Yoshihisa Morita. Entire solutions of reaction-diffusion equations and an application to discrete diffusive equations. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 193-212. doi: 10.3934/dcds.2005.12.193

 Impact Factor: 

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]