Abstract
The paper gives a survey of the author’s results on the grid-based numerical algorithms for solving the evolutionary equations (parabolic and hyperbolic) with the effect of heredity on a time variable. From uniform positions we construct analogs of schemes with weights for the one-dimensional heat conduction equation with delay of general form, analog of a method of variable directions for the equation of parabolic type with time delay and two spatial variables, analog of the scheme with weights for the equation of hyperbolic type with delay. For the one-dimensional heat conduction equation and the wave equation we obtained conditions on the weight coefficients that ensure stability on the prehistory of the initial function. Numerical algorithms are implemented in the form of software package Partial Delay Differential Equations (PDDE) toolbox.
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
Wu, J.: Theory and Applications of Partial Functional Differential Equations. Springer, New York (1996)
Bellen, A., Zennaro, M.: Numerical Methods for Delay Differential Equations. Oxford Science Publications, Oxford (2003)
Kim, A.V., Pimenov, V.G.: i-Smooth Analysis and Numerical Methods for Solving Functional Differential Equations. Regulyarn, Khaotichesk, Dinamika, Moscow-Izhevsk (2004) (in Russian)
Samarskii, A.A.: The Theory of Difference Schemes. Nauka, Moscow (1977)
Pimenov, V.G.: General Linear Methods for Numerical Solving Functional Differential Equations. Differential Equations 37(1), 105–114 (2001) (in Russian)
Tavernini, L.: Finite Difference Approximations for a Class of Semilinear Volterra Evolution Problems. SIAM J. Numer. Anal. 14(5), 931–949 (1977)
Kropielnicka, K.: Convergence of Implicit Difference Methods for Parabolic Functional Differential Equations. Int. Journal of Mat. Analysis 1(6), 257–277 (2007)
Pimenov, V.G., Lozhnikov, A.B.: Difference Schemes for the Numerical Solution of the Heat Conduction Equation with Aftereffect. Proceedings of the Steklov Institute of Mathematics 275(S. 1), 137–148 (2011)
Lekomtsev, A.V., Pimenov, V.G.: Convergence of the Alternating Direction Methods for the Numerical Solution of a Heat Conduction Equation with Delay. Proceedings of the Steklov Institute of Mathematics 272(1), 101–118 (2011)
Pimenov, V.G., Tashirova, E.E.: Numerical Methods for Solution of Hyperbolic Equation with Delay. Trudy Instituta Matematiki i Mekhaniki UrO RAN 18(2), 222–231 (2012) (in Russian)
Faria, T., Huang, W.: Stability of periodic solutions arising from Hopf bifurcation for a reaction-diffusion equation with time delay. Differential Equations and Dynamical Systems 31, 125–141 (2002)
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
Pimenov, V., Lozhnikov, A. (2013). Numerical Methods for Evolutionary Equations with Delay and Software Package PDDE. In: Dimov, I., Faragó, I., Vulkov, L. (eds) Numerical Analysis and Its Applications. NAA 2012. Lecture Notes in Computer Science, vol 8236. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-41515-9_49
Download citation
DOI: https://doi.org/10.1007/978-3-642-41515-9_49
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-41514-2
Online ISBN: 978-3-642-41515-9
eBook Packages: Computer ScienceComputer Science (R0)