Skip to main content
SpringerLink
Log in

Boundary Value Problems

  • Chapter
  • First Online:
Book cover Solving Differential Equations in R

Part of the book series: Use R! ((USE R))

  • 9208 Accesses

Abstract

When solving initial value problems for ordinary differential equations, differential algebraic equations or partial differential equations, as discussed in previous chapters, a unique solution to the equations, if it exists, is obtained by specifying the values of all the components at the starting point of the range of integration. With boundary value problems (BVPs), the conditions are specified at more than one point, usually (but not necessarily) at the boundaries of the independent variable. Because of this it is not guaranteed that BVPs have a unique solution; they may have no solution at all or many solutions. The theory of BVPs, such as the proof of existence and uniqueness of solutions, is considerably more difficult than it is in the initial value case. Also, software for BVPs is much less well developed than for IVPs. In this chapter we will deal mainly with two-point boundary value problems which have the boundary conditions specified at both ends of a finite range of integration. We discuss two distinct methods to solve BVPs, namely shooting and finite difference methods.

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 49.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 64.99
Price excludes VAT (USA)
  • Compact, lightweight 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

References

  1. Ascher, U. M., & Petzold, L. R. (1998). Computer methods for ordinary differential equations and differential-algebraic equations. Philadelphia: SIAM.

    Google Scholar 

  2. Ascher, U. M., & Spiteri, R. J. (1994). Collocation software for boundary value differential-algebraic equations. SIAM Journal on Scientific Computing, 15(4), 938–952.

    Google Scholar 

  3. Ascher, U. M., Christiansen, J., & Russell, R. D. (1979). COLSYS–a collocation code for boundary value problems. In B. Childs et al. (Ed.), Lecture notes in computer science 76 (pp. 164–185). New York: Springer.

    Google Scholar 

  4. Ascher, U. M., Christiansen, J., & Russell, R. D. (1981). Collocation software for boundary-value ODEs. ACM Transactions on Mathematical Software, 7, 209–222.

    Google Scholar 

  5. Ascher, U. M., Mattheij, R. M. M., & Russell, R. D. (1995). Numerical solution of boundary value problems for ordinary differential equations. Philadelphia: SIAM.

    Google Scholar 

  6. Bader, G., & Ascher, U. M. (1987). A new basis implementation for a mixed order boundary value ODE solver. SIAM Journal on Scientific and Statistical Computing, 8, 483–500.

    Google Scholar 

  7. Brugnano, L., Mazzia, F., & Trigiante, D. (2011). Fifty years of stiffness. In T. E. Simos (Ed.), Recent advances in computational and applied mathematics. Dordrecht/New York: Springer.

    Google Scholar 

  8. Brugnano, L., & Trigiante, D. (1996). On the characterization of stiffness for ODEs. Dynamics of Continuous, Discrete and Impulsive Systems, 2(3), 317–335.

    Google Scholar 

  9. Brugnano, L., & Trigiante, D. (1998). Solving differential problems by multistep initial and boundary value methods: Vol. 6. Stability and control: Theory, methods and applications. Amsterdam: Gordon and Breach.

    Google Scholar 

  10. Cash, J. R. (1975). A class of implicit Runge–Kutta methods for the numerical integration of stiff ordinary differential equations. Journal of Alternative and Complementary Medicine, 22, 504.

    Google Scholar 

  11. Cash, J. R. (2004). A survey of some global methods for solving two-point boundary value problems. Applied Numerical Analysis and Computational Mathematics, 1, 1–17.

    Google Scholar 

  12. Cash, J. R., & Mazzia, F. (2005). A new mesh selection algorithm, based on conditioning, for two-point boundary value codes. Journal of Computational and Applied Mathematics, 184, 362–381.

    Google Scholar 

  13. Cash, J. R., & Mazzia, F. (2006). Hybrid mesh selection algorithms based on conditioning for two-point boundary value problems. Journal of Numerical Analysis, Industrial and Applied Mathematics, 1(1), 81–90.

    Google Scholar 

  14. Cash, J. R., & Mazzia, F. (2009). Conditioning and hybrid mesh selection algorithms for two-point boundary value problems. Scalable Computing: Practice and Experience, 10(4), 347–361.

    Google Scholar 

  15. Cash, J. R., & Singhal, A. (1982). High order methods for the numerical solution of two-point boundary value problems. BIT, 22, 184.

    Google Scholar 

  16. Cash, J. R., & Wright, M. H. (1991). A deferred correction method for nonlinear two-point boundary value problems: Implementation and numerical evaluation. SIAM journal on scientific and statistical computing, 12, 971–989.

    Google Scholar 

  17. Cash, J. R., Moore, G., & Wright, R. W. (1995). An automatic continuation strategy for the solution of singularly perturbed linear two-point boundary value problems. Journal of Computational Physics, 122, 266–279.

    Google Scholar 

  18. Davis, H. T. (1962). Introduction to nonlinear differential and integral equations. New York: Dover.

    Google Scholar 

  19. Enright, W. H., & Muir, P. (1980). Efficient classes of Runge–Kutta methods for two point boundary value problems. Computing, 37, 315.

    Google Scholar 

  20. Fox, L. (1957). The numerical solution of two point boundary value problems in ordinary differential equations. London: Clarendon Press.

    Google Scholar 

  21. Hairer, E., Norsett, S. P., & Wanner, G. (2009). Solving ordinary differential equations I: Nonstiff problems. Second revised edition. Heidelberg: Springer.

    Google Scholar 

  22. Hairer, E., & Wanner, G. (1996). Solving ordinary differential equations II: Stiff and differential-algebraic problems. Heidelberg: Springer.

    Google Scholar 

  23. Iavernaro, F., Mazzia, F., & Trigiante, D. (2006). Stability and conditioning in numerical analysis. Journal of Numerical Analysis, Industrial and Applied Mathematics, 1(1), 91–112.

    Google Scholar 

  24. Lentini, M., & Pereyra, V. (1977). An adaptive finite difference solver for nonlinear two point boundary value problems with mild boundary layers. SIAM Journal of Numerical Mathematics, 14, 91–111.

    Google Scholar 

  25. Lindberg, B. (1980). Error estimation and iterative improvement for discetization algorithms. BIT, 20, 486.

    Google Scholar 

  26. Mazzia, F., & Nagy, A. M. (2010). Stiffness detection strategy for explicit Runge–Kutta methods. AIP Conference Proceedings, 1281(1), 239–242.

    Google Scholar 

  27. Mazzia, F., & Sgura, I. (2002). Numerical approximation of nonlinear BVPs by means of BVMs. Applied Numerical Mathematics, 42(1–3), 337–352. Ninth Seminar on Numerical Solution of Differential and Differential-Algebraic Equations (Halle, 2000).

    Google Scholar 

  28. Mazzia, F., & Trigiante, D. (1992). Numerical methods for second order singular perturbation problems. Computers and Mathematics with Applications, 23(11), 81–89.

    Google Scholar 

  29. Mazzia, F., & Trigiante, D. (1993). Numerical solution of singular perturbation problems. Calcolo, 30(4), 355–369 (1995).

    Google Scholar 

  30. Mazzia, F., & Trigiante, D. (2004). A hybrid mesh selection strategy based on conditioning for boundary value ODE problems. Numerical Algorithms, 36(2), 169–187.

    Google Scholar 

  31. Mazzia, F., & Trigiante, D. (2010). Efficient strategies for solving nonlinear problems in BVPs codes. Nonlinear Studies, 17(4), 309–326.

    Google Scholar 

  32. Mazzia, F., Sestini, A., & Trigiante, D. (2009). The continuous extension of the B-spline linear multistep methods for BVPs on non-uniform meshes. Applied Numerical Mathematics, 59(3–4), 723–738.

    Google Scholar 

  33. Muir, P. H., & Owren, B. (1993). Order barriers and charaterisations of continuous mono-implicit Runge–Kutta schemes. Mathematics of Computation, 61, 675.

    Google Scholar 

  34. Shampine, L. F., Kierzenka, J., & Reichelt, M. W. (2000). Solving boundary value problems for ordinary differential equations in MATLAB with bvp4c. In Matlab Guide, D.J. Higham and N.J. Higham, pp 163–169, Philadelphia: SIAM.

    Google Scholar 

  35. Shampine, L. F., Gladwell, I., & Thompson, S. (2003). Solving ODEs with MATLAB. Cambridge: Cambridge University Press.

    Google Scholar 

  36. Shampine, L. F., Muir, P. H., & Xu, H. (2006). A user-friendly Fortran BVP solver. Journal of Numerical Analysis, Industrial and Applied Mathematics, 1(2), 201–217.

    Google Scholar 

  37. Skeel, R. D. (1982). A theoretical framework for providing accurracy results for deferred corrections. SINUM Journal on Numerical Analysis, 19, 171–196.

    Google Scholar 

  38. Strang, G., & Fix, G. (1973). Analysis of the finite element method. Englewood Cliffs, NJ: Prentice Hall.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department Ecosystem Studies, Royal Netherlands Institute for Sea Research, Yerseke, The Netherlands

    Karline Soetaert

  2. Mathematics, Imperial College, South Kensington Campus, London, UK

    Jeff Cash

  3. Dipartimento di Matematica, University of Bari, Bari, Italy

    Francesca Mazzia

Authors
  1. Karline Soetaert
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jeff Cash
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Francesca Mazzia
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Soetaert, K., Cash, J., Mazzia, F. (2012). Boundary Value Problems. In: Solving Differential Equations in R. Use R!. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-28070-2_10

Download citation

Publish with us

Policies and ethics

Search

Navigation