Abstract
The one-dimensional heat conduction equation with the term concerning some heat sources, together with the mixed boundary conditions is considered. Such problems occur in the area of the bioheat transfer and their well-known example is given by the Pennes equation. The paper deals with some interval finite difference method based on the Crank-Nicolson finite difference scheme. In the approach presented, the local truncation error of the conventional method is bounded by some interval values. A method of approximation of such error term intervals is also presented.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Anderson, D.A., Tannehill, J.C., Pletcher, R.H.: Computational fluid mechanics and heat transfer. Hemisphere Publishing, New York (1984)
Jankowska, M.A.: Remarks on Algorithms Implemented in Some C++ Libraries for Floating-Point Conversions and Interval Arithmetic. In: Wyrzykowski, R., Dongarra, J., Karczewski, K., Wasniewski, J. (eds.) PPAM 2009, Part II. LNCS, vol. 6068, pp. 436–445. Springer, Heidelberg (2010)
Jankowska, M.A.: An Interval Finite Difference Method of Crank-Nicolson Type for Solving the One-Dimensional Heat Conduction Equation with Mixed Boundary Conditions. In: Jónasson, K. (ed.) PARA 2010, Part II. LNCS, vol. 7134, pp. 157–167. Springer, Heidelberg (2012)
Jankowska, M.A.: The Error Term Approximation in Interval Method of Crank-Nicolson Type. Differential Equations and Dynamical Systems (2012), doi:10.1007/s12591-012-0144-4
Jankowska, M.A., Sypniewska-Kaminska, G., Kaminski, H.: Evaluation of the Accuracy of the Solution to the Heat Conduction Problem with the Interval Method of Crank-Nicolson Type. Acta Mechanica et Automatica 6(1), 36–43 (2012)
Lapidus, L., Pinder, G.F.: Numerical Solution of Partial Differential Equations in Science and Engineering. J. Wiley & Sons (1982)
Manikonda, S., Berz, M., Makino, K.: High-order verified solutions of the 3D Laplace equation. WSEAS Transactions on Computers 4(11), 1604–1610 (2005)
Minamoto, T.: Numerical verification of solutions for nonlinear hyperbolic equations. Applied Mathematics Letters 10(6), 91–96 (1997)
Minamoto, T., Nakao, M.T.: Numerical Verifications of Solutions for Nonlinear Parabolic Equations in One-Space Dimensional Case. Reliable Computing 3(2), 137–147 (1997)
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)
Nakao, M.T.: A numerical verification method for the existence of weak solutions for nonlinear boundary value problems. Journal of Mathematical Analysis and Applications 164(2), 489–507 (1992)
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)
Nakao, M.T., Watanabe, Y., Yamamoto, N., Nishida, T.: Some computer assisted proofs for solutions of the heat convection problems. Reliable Computing 9(5), 359–372 (2003)
Szyszka, B.: The Central Difference Interval Method for Solving the Wave Equation. In: Wyrzykowski, R., Dongarra, J., Karczewski, K., Waśniewski, J. (eds.) PPAM 2011, Part II. LNCS, vol. 7204, pp. 523–532. Springer, Heidelberg (2012)
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)
Watanabe, Y., Yamamoto, N., Nakao, M.T., Nishida, T.: A numerical verification of nontrivial solutions for the heat convection problem. Journal of Mathematical Fluid Mechanics 6(1), 1–20 (2004)
Xu, F., Seffen, K.A., Lu, T.J.: Non-Fourier analysis of skin biothermomechanics. International Journal of Heat and Mass Transfer 51, 2237–2259 (2008)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Jankowska, M.A., Sypniewska-Kaminska, G. (2013). Interval Finite Difference Method for Solving the One-Dimensional Heat Conduction Problem with Heat Sources. In: Manninen, P., Öster, P. (eds) Applied Parallel and Scientific Computing. PARA 2012. Lecture Notes in Computer Science, vol 7782. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36803-5_36
Download citation
DOI: https://doi.org/10.1007/978-3-642-36803-5_36
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-36802-8
Online ISBN: 978-3-642-36803-5
eBook Packages: Computer ScienceComputer Science (R0)