Iterative solution for nonlinear impulsive advection- reaction-diffusion equations

Volume 9, Issue 6, pp 4070--4077 http://dx.doi.org/10.22436/jnsa.009.06.50

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 1533 Downloads
• 2700 Views

Authors

Xinan Hao - School of Mathematical Sciences, Qufu Normal University, Qufu 273165, Shandong, P. R. China. Lishan Liu - School of Mathematical Sciences, Qufu Normal University, Qufu 273165, Shandong, P. R. China. - Department of Mathematics and Statistics, Curtin University, Perth, WA6845, Australia. Yonghong Wu - Department of Mathematics and Statistics, Curtin University, Perth, WA6845, Australia.

Abstract

Through solving equations step by step and by using the generalized Banach fixed point theorem, under simple conditions, the authors present the existence and uniqueness theorem of the iterative solution for nonlinear advection-reaction-diffusion equations with impulsive effects. An explicit iterative scheme for the solution is also derived. The results obtained generalize and improve some known results.

Share and Cite

Keywords

• Iterative solution
• nonlinear advection-reaction-diffusion equations
• impulse.

MSC

•  35K57
•  35R12

References

• [1] R. P. Agarwal, M. Meehan, D. O'Regan, Fixed point theory and applications, Cambridge University Press, Cambridge (2001)


• [2] D. G. Aronson, H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation. Partial differential equations and related topics (Program, Tulane Univ., New Orleans, La., 1974), Lecture Notes in Math., Springer, Berlin, 446 (1975), 5-49


• [3] Z. Bu, Z.Wang, N. Liu, Asymptotic behavior of pulsating fronts and entire solutions of reaction-advection-diffusion equations in periodic media, Nonlinear Anal. Real World Appl., 28 (2016), 48-71.


• [4] K. Deimling, Nonlinear functional analysis, Springer, Berlin (1985)


• [5] P. C. Fife, Mathematical aspects of reacting and diffusing systems, Springer, Berlin (1979)


• [6] J. Ge, K. I. Kim, Z. Lin, H. Zhu, A SIS reaction-diffusion-advection model in a low-risk and high-risk domain, J. Differential Equations, 259 (2015), 5486-5509.


• [7] X. Hao, L. Liu, Y. Wu, Positive solutions for second order impulsive differential equations with integral boundary conditions, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 101-111.


• [8] S. Hsu, L. Mei, F. Wang, On a nonlocal reaction-diffusion-advection system modelling the growth of phytoplankton with cell quota structure, J. Differential Equations, 259 (2015), 5353-5378.


• [9] J. Keener, J. Sneyd, Mathematical physiology, Springer, New York (1998)


• [10] E. Kreyszig , Introductory functional analysis with applications, John Wiley and Sons, Inc., New York (1989)


• [11] V. Lakshmikantham, D. D. Bainov, P. S. Simeonov, Theory of impulsive differential equations, World Scientific, Singapore (1989)


• [12] L. Liu, L. Hu, Y. Wu , Positive solutions of two-point boundary value problems for systems of nonlinear second- order singular and impulsive differential equations, Nonlinear Anal., 69 (2008), 3774-3789.


• [13] N. Liu, W. Li, Z. Wang, Pulsating type entire solutions of monostable reaction-advection-diffusion equations in periodic excitable media, Nonlinear Anal., 75 (2012), 1869-1880.


• [14] L. Liu, Y. Wu, X. Zhang , On well-posedness of an initial value problem for nonlinear second-order impulsive integro-differential equations of Volterra type in Banach spaces, J. Math. Anal. Appl., 317 (2006), 634-649.


• [15] N. N. Nefedov, L. Recke, K. R. Schneider, Existence and asymptotic stability of periodic solutions with an interior layer of reaction-advection-diffusion equations, J. Math. Anal. Appl., 405 (2013), 90-103.


• [16] J. I. Ramos, Picard's iterative method for nonlinear advection-reaction-diffusion equations, Appl. Math. Comput., 215 (2009), 1526-1536.


• [17] J. Ruiz-Ramirez, J. E. Macias-Diaz, A skew symmetry-preserving computational technique for obtaining the positive and the bounded solutions of a time-delayed advection-diffusion-reaction equation, J. Comput. Appl. Math., 250 (2013), 256-269.


• [18] A. M. Samoilenko, N. A. Perestyuk, Impulsive differential equations, World Scientific, Singapore (1995)


• [19] D. H. Sattinger, On the stability of waves of nonlinear parabolic systems, Advances in Math., 22 (1976), 312-355.


• [20] A. I. Volpert, V. A. Volpert, Traveling-wave solutions of parabolic systems with discontinuous nonlinear terms, Nonlinear Anal., 49 (2002), 113-139.


• [21] F. A. Williams , Combustion theory, The Benjamin/Cummings Publishing Company Inc., Menlo Park, CA (1985)


• [22] X. Xin, Existence and uniqueness of travelling waves in a reaction-diffusion equation with combustion nonlinearity, Indiana Univ. Math. J., 40 (1991), 985-1008.


• [23] G. Zhao, S. Ruan, Time periodic traveling wave solutions for periodic advection-reaction-diffusion systems, J. Differential Equations, 257 (2014), 1078-1147.