Skip to main content
SpringerLink
Log in

Approximating Solutions of BVPs Using Cubic B-Splines

  • Chapter
  • First Online:
Boundary Value Problems for Engineers

  • 1588 Accesses

Abstract

Cubic B-spline function is a piecewise third degree polynomial function, constructed from a linear combination of some recursive functions , called cubic B-spline basis. The derivation of B-spline basis and the construction of B-spline function are discussed elsewhere (Salomon in Curves and surfaces for computer graphics. Springer Science + Business Media, Inc, New York, 2006 [1].

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 54.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 69.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 99.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    In these problems, the method is more involved with intense algebraic computations of relatively large sized matrices. Therefore, they are not included in this chapter.

  2. 2.

    An analogous study is described in [25].

  3. 3.

    Readers may refer to [24] for more detailed derivation of the method.

References

  1. Salomon D (2006) Curves and surfaces for computer graphics. Springer Science + Business Media Inc, New York

    MATH  Google Scholar 

  2. Bickley WG (1968) Piecewise cubic interpolation and two-point boundary problems. Comput J 11(2):206–208

    Article  MathSciNet  MATH  Google Scholar 

  3. Albasiny EL, Hoskins WD (1969) Cubic spline solutions to two-point boundary value problems. Comput J 12(2):151–153

    Article  MathSciNet  MATH  Google Scholar 

  4. Al-Said EA (1998) Cubic spline method for solving two-point boundary-value problems. J Appl Math Comput 5(3):669–680

    MathSciNet  MATH  Google Scholar 

  5. Khan A (2004) Parametric cubic spline solution of two point boundary value problems. Appl Math Comput 154(1):175–182

    MathSciNet  MATH  Google Scholar 

  6. Caglar H, Caglar N, Elfaituri K (2006) B-spline interpolation compared with finite difference, finite element and finite volume methods which applied to two-point boundary value problems. Appl Math Comput 175(1):72–79

    MathSciNet  MATH  Google Scholar 

  7. Xu G, Wang G-Z (2008) Extended cubic uniform B-spline and [alpha]-B-spline. Acta Autom Sinica 34(8):980–984

    Article  Google Scholar 

  8. Hamid NNA, Majid AA, Ismail AIM (2011) Extended cubic B-spline method for linear two-point boundary value problems. Sains Malaysiana 40(11):1285–1290

    MATH  Google Scholar 

  9. Kumar M, Gupta Y (2010) Methods for solving singular boundary value problems using splines: a review. J Appl Math Comput 32:265–278

    Article  MathSciNet  MATH  Google Scholar 

  10. Kanth ASVR, Bhattacharya V (2006) Cubic spline for a class of nonlinear singularboundary value problems arising in physiology. Appl Math Comput 174:768–774

    MathSciNet  MATH  Google Scholar 

  11. Kadalbajoo MK, Aggarwal VK (2005) Numerical solution of singular boundary value problems via Chebyshev polynomial and B-spline. Appl Math Comput 160:851–863

    MathSciNet  MATH  Google Scholar 

  12. Chawla MM, Subramanian R, Sathi HL (1988) A fourth-order spline method for singular two-point boundary-value problems. J Comput Appl Math 21:189–202

    Article  MathSciNet  MATH  Google Scholar 

  13. Fyfe DJ (1989) The use of cubic splines in the solution of two-point boundary value problems. Comput J 12:188–192

    Article  MathSciNet  MATH  Google Scholar 

  14. Guoqiang H (1993) Spline finite difference methods and their extrapolation for singular two-point boundary value problems. J Comput Math 11:289–296

    MathSciNet  MATH  Google Scholar 

  15. Iyengar SRK, Jain P (1987) Spline finite difference methods for singular two point boundary value problems. Numer Math 50:363–376

    Article  MathSciNet  MATH  Google Scholar 

  16. Kadalbajoo MK, Kumar V (2007) B-spline method for a class of singular two-point boundary value problems using optimal grid. Appl Math Comput 188:1856–1869

    MathSciNet  MATH  Google Scholar 

  17. Kumar M (2007) Higher order method for singular boundary value problems by using spline function. Appl Math Comput 192(1):175–179

    MathSciNet  MATH  Google Scholar 

  18. Caglar N, Caglar H (2009) B-spline method for solving linear system of second order boundary value problems. Comput Math Appl 27:757–762

    Article  MathSciNet  MATH  Google Scholar 

  19. Munguia M, Bhatta D (2015) Use of cubic B-Spline in approximating solutions of boundary value problems. Int J Appl Appl Math 10(2):750–771

    MathSciNet  MATH  Google Scholar 

  20. Faires JD, Burden RL (2003) Numerical Methods, Thomson/Brooks/Cole, p 531

    Google Scholar 

  21. Benabidallah M, Cherruault Y (2004) Application of the adomian method for solving a class of boundary problems. Kybernetes 33(1):118–132

    Article  MathSciNet  MATH  Google Scholar 

  22. Doolan EP, Miller JJH, Schilders WHA (1980) Uniform numerical methods for problems with initial and boundary layers. Boole Press, Dublin, Ireland

    MATH  Google Scholar 

  23. Aziz T, Arshad Khan A (2002) A spline method for second-order singularly perturbed boundary-value problems. J Comput Appl Math 147:445–452

    Article  MathSciNet  MATH  Google Scholar 

  24. Abukhaled M, Khuri SAA, Sayfy A (2011) A numerical approach for solving a class of singular boundary value problems arising in physiology. Int J Num Anal Model 8(2):353–363

    MathSciNet  MATH  Google Scholar 

  25. Caglar SH, Caglar HN, Ozer M (2009) B-spline solution of non-linear singular boundary value problems arising in physiology. Chaos, Solitons Fractals 39(3):1232–1237

    Article  MATH  Google Scholar 

  26. Caglar HN, Caglar SH (2006) B-spline solution of singular boundary value problems. Appl Math Comput 182:1509–1513

    MathSciNet  MATH  Google Scholar 

  27. Sayfy A, Khuri S (2008) A generalized algorithm for the order verification of numerical methods. Far East J Appl Math 33(2):295–306

    MathSciNet  MATH  Google Scholar 

  28. Thula K, Roul P (2018) A high-order B-Spline collocation method for solving nonlinear singular boundary value problems arising in engineering and applied science. Mediterr J Math 15:176

    Article  MathSciNet  MATH  Google Scholar 

  29. Caglar HN, Caglar SH, Twizell EH (1999) The numerical solution of fifth-order boundary-value problems with sixth-degree B-spline functions. Appl Math Lett 12:25–30

    Article  MathSciNet  MATH  Google Scholar 

  30. Rández L (1992) Improving the efficiency of the multiple shooting technique. Comput Math Appl 24(7):127–132

    Article  MathSciNet  MATH  Google Scholar 

  31. Cash JR, Wright MH (1991) A deferred correction method for nonlinear two point boundary value problems: implementation and numerical evaluation. SIAM J Numer Anal 12:971–989

    MathSciNet  MATH  Google Scholar 

  32. Raul P (2019) A fast and accurate computational technique for efficient numerical solution of nonlinear singular boundary value problems. Int J Comput Math 96(1):51–72

    Article  MathSciNet  Google Scholar 

  33. Ascher UM, Mattheij RMM, Russell RD, (1995) Numerical solution of boundary value problems for ordinary differential equations, SIAM, p 192

    Google Scholar 

  34. Goh J, Majid AA, Ismail AIM (2011) Extended cubic uniform B-spline for a class of singular boundary value problems. Science Asia 37:79–82

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Biomedical Engineering, Yeditepe University, Istanbul, Turkey

    Ali Ümit Keskin

Authors
  1. Ali Ümit Keskin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ali Ümit Keskin .

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Keskin, A.Ü. (2019). Approximating Solutions of BVPs Using Cubic B-Splines. In: Boundary Value Problems for Engineers. Springer, Cham. https://doi.org/10.1007/978-3-030-21080-9_9

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-21080-9_9

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-21079-3

  • Online ISBN: 978-3-030-21080-9

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation