Skip to main content
SpringerLink
Log in

A new sixth-order approximation for nonlinear two-point boundary value problems: application of single-step alternating group explicit iteration method to engineering problems

  • Original Article
  • Published:
Engineering with Computers Aims and scope Submit manuscript

Abstract

We report a new three-point compact sixth-order approximation for the solution of nonlinear two-point boundary value problems \( {-}w^{\prime\prime} + f\left( {t,w} \right) = 0 \), subject to natural boundary conditions prescribed at two end points. We also discuss the application of single-step alternating group explicit (SAGE) iteration method to the nonlinear difference equation as a computational tool. The error analysis of the SAGE iteration method is discussed briefly. We have compared the results obtained by using the proposed SAGE iteration method with the results obtained by corresponding two-step alternating group explicit iteration method to demonstrate experimentally the superiority of the proposed method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. Keller HB (1968) Numerical methods for two point boundary value problem. Blaisdell Pub. Co, Waltham

    MATH  Google Scholar 

  2. Evans DJ (1985) Group explicit methods for solving large linear systems. Int J Comput Math 17:81–108

    Article  Google Scholar 

  3. Evans DJ (1987) The alternating group explicit (AGE) matrix iterative method. Appl Math Model 11:256–263

    Article  MathSciNet  Google Scholar 

  4. Yousif WS, Evans DJ (1987) The modified alternating group explicit (MAGE) algorithm for solving tridiagonal linear equations. Comput Math Appl 14:169–172

    Article  MathSciNet  Google Scholar 

  5. Evans DJ, Yousif WS (1988) The modified alternating group explicit (MAGE) method. Appl Math Model 12:262–267

    Article  Google Scholar 

  6. Evans DJ, Ahmad AR (1996) Comparison of SOR and AGE methods for solution of the two-point boundary value problem. Adv Eng Softw 26:101–110

    Article  Google Scholar 

  7. Sukon KS, Evans DJ (1996) Two parameter AGE method for the solution of tridiagonal linear system of equations. Int J Comput Math 60:265–278

    Article  Google Scholar 

  8. Sukon KS (1996) On two parameter alternating group explicit (TAGE) method for singular perturbation problems. Parallel Algorithm Appl 10:71–77

    Article  Google Scholar 

  9. Evans DJ, Mohanty RK (2002) Alternating group explicit method for the numerical solution of non-linear singular two point boundary value problems using a fourth order finite difference method. Int J Comput Math 79:1121–1133

    Article  MathSciNet  Google Scholar 

  10. Mohanty RK, Sachdev PL, Jha N (2003) TAGE method for non-linear singular two point boundary value problem using a fourth order difference scheme. Neural Parallel Sci Comput. 11:281–297

    MathSciNet  MATH  Google Scholar 

  11. Evans DJ, Mohanty RK (2005) On the application of the SMAGE parallel algorithms on a non-uniform mesh for the solution of non-linear two point boundary value problems with singularity. Int J Comput Math 82:341–353

    Article  MathSciNet  Google Scholar 

  12. Mohanty RK, Evans DJ (2005) Highly accurate two parameter CAGE parallel algorithms for non-linear singular two point boundary value problems. Int J Comput Math 82:433–444

    Article  MathSciNet  Google Scholar 

  13. Mohanty RK, Khosla N (2005) A third order accurate variable mesh TAGE iterative method for the numerical solution of two point non-linear singular boundary value problems. Int J Comput Math 82:1261–1273

    Article  MathSciNet  Google Scholar 

  14. Mohanty RK, Talwar J (2015) A new coupled reduced alternating group explicit method for non-linear singular two point boundary value problems on a variable mesh. Numer Anal Appl 8:55–67

    Article  MathSciNet  Google Scholar 

  15. Mohanty RK, Evans DJ, Jha N (2006) A sixth order accurate AGE iterative method for non-linear singular two point boundary value problems. J Comput Methods Sci Eng 6:57–69

    MATH  Google Scholar 

  16. Mohanty RK, Arora U (2006) A TAGE iterative method for the solution of non-linear singular two point boundary value problems using a sixth order discretization. Appl Math Comput 180:538–548

    MathSciNet  MATH  Google Scholar 

  17. Mohanty RK, Talwar J (2012) A combined approach using coupled reduced alternating group explicit (CRAGE) algorithm and sixth order off-step discretization for the solution of two point nonlinear boundary value problems. Appl Math Comput 219:248–259

    MathSciNet  MATH  Google Scholar 

  18. Dahalan AA, Sulaiman J (2016) Half-sweep two parameter AGE iterative method applied to fuzzy Poisson equation. Appl Math Sci 10:45–57

    Google Scholar 

  19. Dahalan AA, Aziz NSA, Sulaiman J (2017) QSAGE iterative method for the numerical solution of two point fuzzy boundary value problem. J Eng Appl Sci 12:4068–4074

    Google Scholar 

  20. Rahman R, Ali NAM, Sulaiman J, Muhiddin FA (2018) Caputo’s finite difference solution of fractional two point boundary value problems using AGE iteration. IOP Conf Ser J Phys Conf Ser 1123:012044

    Article  Google Scholar 

  21. Shi Y, Zhao P (2018) A 6th–order alternating group iterative scheme for two point BVP. Adv Intell Syst Res 159:303–306

    Google Scholar 

  22. Chawla MM (1979) A sixth order tridiagonal finite difference method for general nonlinear two point boundary value problems. J Inst Math Appl 24:35–42

    Article  MathSciNet  Google Scholar 

  23. Mohanty RK, Dhall D (2009) Third order accurate variable mesh discretization and application of TAGE iterative method for the nonlinear two-point boundary value problems with homogeneous functions in integral form. Appl Math Comput 215:2024–2034

    MathSciNet  MATH  Google Scholar 

  24. Mohanty RK, Jain MK, Dhall D (2011) A cubic spline approximation and application of TAGE iterative method for the solution of two-point boundary value problems with forcing function in integral form. Appl Math Model 35:3036–3047

    Article  MathSciNet  Google Scholar 

  25. Hageman LA, Young DM (2004) Applied iterative methods. Dover Publications, New York

    MATH  Google Scholar 

  26. Evans DJ (1999) Iterative methods for solving non-linear two point boundary value problems. Int J Comput Math 72:395–401

    Article  MathSciNet  Google Scholar 

  27. Polyanin AD, Zaitsev VF (2003) Handbook of exact solutions for ordinary differential equations, 2nd edn. Chapman & Hall, Boca Raton

    MATH  Google Scholar 

  28. Kubicek M, Hlavacek V (1983) Numerical solutions of nonlinear boundary value problems with applications. Prentice Hall, Englewood Cliffs

    MATH  Google Scholar 

Download references

Acknowledgements

The author thanks the reviewers for their valuable suggestions, which substantially improved the quality of the paper.

Author information

Authors and Affiliations

  1. Department of Computer Science, College of Science, Purdue University, West Lafayette, IN, 47907-2107, USA

    Pinaki Ranjan Mohanty

Authors
  1. Pinaki Ranjan Mohanty
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pinaki Ranjan Mohanty.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mohanty, P.R. A new sixth-order approximation for nonlinear two-point boundary value problems: application of single-step alternating group explicit iteration method to engineering problems. Engineering with Computers 37, 3541–3550 (2021). https://doi.org/10.1007/s00366-020-01016-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00366-020-01016-9

Keywords

Mathematics Subject Classification

Search

Navigation