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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 840573, 9 pages
http://dx.doi.org/10.1155/2014/840573
Research Article

Stability of Exact and Discrete Energy for Non-Fickian Reaction-Diffusion Equations with a Variable Delay

1School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China
2School of Computer Science and Engineering, Beihang University, Beijing 100191, China
3School of Computer Science, McGill University, Montreal, QC, Canada H3A 2K6
4Department of Mathematics and Statistics, McGill University, Montreal, QC, Canada H3A 2K6

Received 4 December 2013; Revised 28 December 2013; Accepted 11 January 2014; Published 5 March 2014

Academic Editor: Adem Kilicman

Copyright © 2014 Dongfang Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. R. P. Agarwal, L. Berezansky, E. Braverman, and A. Domoshnitsky, Nonoscillation Theory of Functional Differential Equations with Applications, Springer, New York, NY, USA, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  2. H. Brunner, Collocation Methods for Volterra Integral and Related Functional Differential Equations, Cambridge University Press, Cambridge, UK, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  3. J. H. Wu, Theory and Application of Partial Functional Differential Equation, vol. 119 of Applied Mathematical Sciences, Springer-Verlag, 1996. View at Publisher · View at Google Scholar · View at MathSciNet
  4. L. Berezansky, E. Braverman, and L. Idels, “Nicholson's blowflies differential equations revisited: main results and open problems,” Applied Mathematical Modelling, vol. 34, no. 6, pp. 1405–1417, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  5. D. Li, C. Zhang, and H. Qin, “LDG method for reaction-diffusion dynamical systems with time delay,” Applied Mathematics and Computation, vol. 217, no. 22, pp. 9173–9181, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  6. M. Mei, C.-K. Lin, C.-T. Lin, and J. W.-H. So, “Traveling wavefronts for time-delayed reaction-diffusion equation. I. Local nonlinearity,” Journal of Differential Equations, vol. 247, no. 2, pp. 495–510, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  7. M. Mei, C. Ou, and X.-Q. Zhao, “Global stability of monostable traveling waves for nonlocal time-delayed reaction-diffusion equations,” SIAM Journal on Mathematical Analysis, vol. 42, no. 6, pp. 2762–2790, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  8. J. Wu, D. Wei, and M. Mei, “Analysis on the critical speed of traveling waves,” Applied Mathematics Letters, vol. 20, no. 6, pp. 712–718, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  9. Z.-C. Wang, W.-T. Li, and S. Ruan, “Travelling wave fronts in reaction-diffusion systems with spatio-temporal delays,” Journal of Differential Equations, vol. 222, no. 1, pp. 185–232, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  10. D. D. Joseph and L. Preziosi, “Heat waves,” Reviews of Modern Physics, vol. 61, no. 1, pp. 41–73, 1989. View at Publisher · View at Google Scholar · View at MathSciNet
  11. S. Fedotov, “Traveling waves in a reaction-diffusion system: diffusion with finite velocity and Kolmogorov-Petrovskii-Piskunov kinetics,” Physical Review E, vol. 58, no. 4, pp. 5143–5145, 1998. View at Publisher · View at Google Scholar · View at MathSciNet
  12. F. Andreu, V. Caselles, and J. M. Mazón, “A Fisher-Kolmogorov equation with finite speed of propagation,” Journal of Differential Equations, vol. 248, no. 10, pp. 2528–2561, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  13. J. R. Branco, J. A. Ferreira, and P. da Silva, “Non-Fickian delay reaction-diffusion equations: theoretical and numerical study,” Applied Numerical Mathematics, vol. 60, no. 5, pp. 531–549, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  14. A. Araújo, J. R. Branco, and J. A. Ferreira, “On the stability of a class of splitting methods for integro-differential equations,” Applied Numerical Mathematics, vol. 59, no. 3-4, pp. 436–453, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  15. J. R. Branco, J. A. Ferreira, and P. de Oliveira, “Numerical methods for the generalized Fisher-Kolmogorov-Petrovskii-Piskunov equation,” Applied Numerical Mathematics, vol. 57, no. 1, pp. 89–102, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  16. J.-C. Chang, “Local existence for retarded Volterra integrodifferential equations with Hille-Yosida operators,” Nonlinear Analysis. Theory, Methods & Applications, vol. 66, no. 12, pp. 2814–2832, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  17. A. Araújo, J. A. Ferreira, and P. Oliveira, “Qualitative behavior of numerical traveling solutions for reaction-diffusion equations with memory,” Applicable Analysis, vol. 84, no. 12, pp. 1231–1246, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  18. A. Araújo, J. A. Ferreira, and P. Oliveira, “The effect of memory terms in diffusion phenomena,” Journal of Computational Mathematics, vol. 24, no. 1, pp. 91–102, 2006. View at MathSciNet
  19. C. Zhang and S. Vandewalle, “General linear methods for Volterra integro-differential equations with memory,” SIAM Journal on Scientific Computing, vol. 27, no. 6, pp. 2010–2031, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  20. W. Wang, C. Zhang, and D. Li, “Asymptotic stability of exact and discrete solutions for neutral multidelay-integro-differential equations,” Applied Mathematical Modelling, vol. 35, no. 9, pp. 4490–4506, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  21. W. Wang and C. Zhang, “Preserving stability implicit Euler method for nonlinear Volterra and neutral functional differential equations in Banach space,” Numerische Mathematik, vol. 115, no. 3, pp. 451–474, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  22. W. Wang and C. Zhang, “Analytical and numerical dissipativity for nonlinear generalized pantograph equations,” Discrete and Continuous Dynamical Systems A, vol. 29, no. 3, pp. 1245–1260, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  23. L. Wen, Y. Yu, and W. Wang, “Generalized Halanay inequalities for dissipativity of Volterra functional differential equations,” Journal of Mathematical Analysis and Applications, vol. 347, no. 1, pp. 169–178, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  24. V. Méndez, S. Fedotov, and W. Horsthemke, Reaction-Transport Systems: Mesoscopic Foundations, Fronts, and Spatial Instabilities, Springer Series in Synergetics, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  25. A. Bellen and M. Zennaro, Numerical Methods for Delay Differential Equations, Oxford University Press, Oxford, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  26. H. Brunner, “Recent advances in the numerical analysis of Volterra functional differential equations with variable delays,” Journal of Computational and Applied Mathematics, vol. 228, no. 2, pp. 524–537, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  27. H. Brunner, “Current work and open problems in the numerical analysis of Volterra functional equations with vanishing delays,” Frontiers of Mathematics in China, vol. 4, no. 1, pp. 3–22, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  28. A. Yadav, S. Fedotov, V. Méndez, and W. Horsthemke, “Propagating fronts in reaction C- transport systems with memory,” Physics Letters A, vol. 371, pp. 374–378, 2007.
  29. S. Fedotov, “Non-Markovian random walks and nonlinear reactions: subdiffusion and propagating fronts,” Physical Review E, vol. 81, no. 1, Article ID 011117, 7 pages, 2010. View at Publisher · View at Google Scholar
  30. S. Fedotov, A. Iomin, and L. Ryashko, “Non-Markovian models for migration-proliferation dichotomy of cancer cells: anomalous switching and spreading rate,” Physical Review E, vol. 84, no. 6, Article ID 061131, 8 pages, 2011. View at Publisher · View at Google Scholar
  31. C. Zhang and S. Zhou, “The asymptotic stability of theoretical and numerical solutions for systems of neutral multidelay-differential equations,” Science in China A, vol. 41, no. 11, pp. 1151–1157, 1998. View at Publisher · View at Google Scholar · View at MathSciNet
  32. C.-J. Zhang and S.-F. Li, “Dissipativity and exponentially asymptotic stability of the solutions for nonlinear neutral functional-differential equations,” Applied Mathematics and Computation, vol. 119, no. 1, pp. 109–115, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
  33. D. Li, C. Zhang, and W. Wang, “Long time behavior of non-Fickian delay reaction-diffusion equations,” Nonlinear Analysis. Real World Applications, vol. 13, no. 3, pp. 1401–1415, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  34. J. R. Cannon and Y. P. Lin, “A priori L2 error estimates for finite-element methods for nonlinear diffusion equations with memory,” SIAM Journal on Numerical Analysis, vol. 27, no. 3, pp. 595–607, 1990. View at Publisher · View at Google Scholar · View at MathSciNet
  35. J. R. Cannon and Y. Lint, “Error estimates for semidiscrete finite element methods for parabolic integro-Differential equations,” Mathematics of Computation, vol. 187, no. 53, pp. 121–139, 1989.
  36. H. Tian, “Numerical and analytic dissipativity of the θ-method for delay differential equations with a bounded variable lag,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 14, no. 5, pp. 1839–1845, 2004. View at Publisher · View at Google Scholar · View at MathSciNet