Skip to main content

An Interval Finite Difference Method of Crank-Nicolson Type for Solving the One-Dimensional Heat Conduction Equation with Mixed Boundary Conditions

  • Conference paper

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 7134)

Abstract

In the paper an interval method for solving the one-dimensio-nal heat conduction equation with mixed boundary conditions is considered. The idea of the interval method is based on the finite difference scheme of the conventional Crank-Nicolson method adapted to the mixed boundary conditions. The interval method given in the form presented in the paper includes the error term of the conventional method.

Keywords

  • interval methods
  • finite difference methods
  • Crank-Nicolson method
  • partial differential equations
  • heat conduction equation with mixed boundary conditions

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (Canada)
  • 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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Jankowska, M.A., Marciniak, A.: An Interval Finite Difference Method for Solving the One-Dimensional Heat Equation. LNCS (accepted)

    Google Scholar 

  2. Lapidus, L., Pinder, G.F.: Numerical Solution of Partial Differential Equations in Science and Engineering. J. Wiley & Sons (1982)

    Google Scholar 

  3. Manikonda, S., Berz, M., Makino, K.: High-order verified solutions of the 3D Laplace equation. WSEAS Transactions on Computers 4(11), 1604–1610 (2005)

    Google Scholar 

  4. Marciniak, A.: An Interval Difference Method for Solving the Poisson Equation - the First Approach. Pro Dialog 24, 49–61 (2008)

    Google Scholar 

  5. Marciniak, A.: An Interval Version of the Crank-Nicolson Method – The First Approach. In: Jónasson, K. (ed.) PARA 2010, Part II. LNCS, vol. 7134, pp. 120–126. Springer, Heidelberg (2012)

    Google Scholar 

  6. Moore, R.E., Kearfott, R.B., Cloud, M.J.: Introduction to Interval Analysis. SIAM, Philadelphia (2009)

    CrossRef  MATH  Google Scholar 

  7. Nagatou, K., Hashimoto, K., Nakao, M.T.: Numerical verification of stationary solutions for Navier-Stokes problems. Journal of Computational and Applied Mathematics 199(2), 445–451 (2007)

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. Nakao, M.T.: Numerical verification methods for solutions of ordinary and partial differential equations. Numerical Functional Analysis and Optimization 22(3-4), 321–356 (2001)

    CrossRef  MathSciNet  MATH  Google Scholar 

  9. Smith, G.D.: Numerical Solution of Partial Differential Equations: Finite Difference Methods. Oxford University Press (1995)

    Google Scholar 

  10. Watanabe, Y., Yamamoto, N., Nakao, M.T.: A Numerical Verification Method of Solutions for the Navier-Stokes Equations. Reliable Computing 5(3), 347–357 (1999)

    CrossRef  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Jankowska, M.A. (2012). An Interval Finite Difference Method of Crank-Nicolson Type for Solving the One-Dimensional Heat Conduction Equation with Mixed Boundary Conditions. In: Jónasson, K. (eds) Applied Parallel and Scientific Computing. PARA 2010. Lecture Notes in Computer Science, vol 7134. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-28145-7_16

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-28145-7_16

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-28144-0

  • Online ISBN: 978-3-642-28145-7

  • eBook Packages: Computer ScienceComputer Science (R0)