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Original Articles

Positive and elementary stable explicit nonstandard Runge-Kutta methods for a class of autonomous dynamical systems

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Pages 2036-2054 | Received 07 Nov 2018, Accepted 01 Oct 2019, Published online: 16 Oct 2019

References

  • O. Adekanye and T. Washington, Nonstandard finite difference scheme for a Tacoma Narrows Bridge model, Appl. Math. Model. 62 (2018), pp. 223–236. doi: 10.1016/j.apm.2018.05.027
  • L.J.S. Allen, An Introduction to Mathematical Biology, Prentice Hall, New Jersey, NJ, 2007.
  • P. Amarasekare and H. Possingham, Patch dynamics and metapopulation theory: The case of successional species, J. Theor. Biol. 209 (2001), pp. 333–344. doi: 10.1006/jtbi.2001.2269
  • R. Anguelov and J.M.S. Lubuma, Nonstandard finite difference method by nonlocal approximation, Math. Comput. Simul. 61 (2003), pp. 465–475. doi: 10.1016/S0378-4754(02)00106-4
  • R. Anguelov, Y. Dumont, J.M.-S. Lubuma, and M. Shillor, Dynamically consistent nonstandard finite difference schemes for epidemiological models, J. Comput. Appl. Math. 255 (2014), pp. 161–182. doi: 10.1016/j.cam.2013.04.042
  • R. Appadu, M. Chapwanya, O.A. Jejeniwa, and J.M.-S. Lubuma, An explicit nonstandard finite difference scheme for the FitzHugh-Nagumo equations, Int. J. Comput. Math. Available at https://doi.org/10.1080/00207160.2018.1546849.
  • A.J. Arenas, G. González-Parra, and B. M.Chen-Charpentier, Construction of nonstandard finite difference schemes for the SI and SIR epidemic models of fractional order, J. Comput. Appl. Math. 121 (2016), pp. 48–63.
  • U. Ascher, L. Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, Society for Industrial and Applied Mathematics, Philadelphia, 1998.
  • M. Chapwanya, J.M.-S. Lubuma, and R.E. Mickens, Positivity-preserving nonstandard finite difference schemes for cross-diffusion equations in biosciences, Comput. Math. Appl. 68 (2014), pp. 1071–1082. doi: 10.1016/j.camwa.2014.04.021
  • B.M. Chen-Charpentier, D.T. Dimitrov, and H.V. Kojouharov, Combined nonstandard numerical methods for ODEs with polynomial right-hand sides, Math. Comput. Simul. 73 (2006), pp. 105–113. doi: 10.1016/j.matcom.2006.06.008
  • G.J. Cooper and J.H. Verner, Some explicit Runge-Kutta methods of high order, SIAM J. Numer. Anal. 9 (1972), pp. 389–405. doi: 10.1137/0709037
  • J. Cresson and F. Pierret, Non standard finite difference scheme preserving dynamical properties, J. Comput. Appl. Math. 303 (2016), pp. 15–30. doi: 10.1016/j.cam.2016.02.007
  • Q.A. Dang and M.T. Hoang, Dynamically consistent discrete metapopulation model, J. Difference Equ. Appl. 22 (2016), pp. 1325–1349. doi: 10.1080/10236198.2016.1197213
  • Q.A. Dang and M.T. Hoang, Exact finite difference schemes for three-dimensional linear systems with constant coefficients, Vietnam J. Math. 46 (2018), pp. 471–492. doi: 10.1007/s10013-017-0249-8
  • Q.A. Dang and M.T. Hoang, Lyapunov direct method for investigating stability of nonstandard finite difference schemes for metapopulation models, J. Difference Equ. Appl. 24 (2018), pp. 15–47. doi: 10.1080/10236198.2017.1391235
  • Q.A. Dang and M.T. Hoang, Complete global stability of a metapopulation model and its dynamically consistent discrete models, Qual. Theory Dyn. Syst. 18 (2019), pp. 461–475. doi: 10.1007/s12346-018-0295-y
  • Q.A. Dang and M.T. Hoang, Nonstandard finite difference schemes for a general predator-prey system, J. Comput. Sci. Available at https://doi.org/10.1016/j.jocs.2019.07.002.
  • Q.A. Dang, M.T. Hoang, and Q.L. Dang, Nonstandard finite difference schemes for solving a modified epidemiological model for computer viruses, J. Comput. Sci. Cybernet. 32(2) (2018), pp. 171–185. doi:10.15625/1813-9663/32/2/13078
  • D.T. Dimitrov and H.V. Kojouharov, Complete mathematical analysis of predator-prey models with linear prey growth and beddington-DeAngelis functional response, Appl. Math. Comput. 162 (2005), pp. 523–538.
  • D.T. Dimitrov and H.V. Kojouharov, Nonstandard finite-difference schemes for general two-dimensional autonomous dynamical systems, Appl. Math. Lett. 18 (2005), pp. 769–774. doi: 10.1016/j.aml.2004.08.011
  • D.T. Dimitrov and H.V. Kojouharov, Positive and elementary stable nonstandard numerical methods with applications to predator-prey models, J. Comput. Appl. Math. 189 (2006), pp. 98–108. doi: 10.1016/j.cam.2005.04.003
  • D.T. Dimitrov and H.V. Kojouharov, Stability-preserving finite-difference methods for general multi-dimensional autonomous dynamical systems, Int. J. Numer. Anal. Model. 4 (2007), pp. 280–290.
  • D.T. Dimitrov and H.V. Kojouharov, Nonstandard finite-difference methods for predator-prey models with general functional response, Math. Comput. Simul. 78 (2008), pp. 1–11. doi: 10.1016/j.matcom.2007.05.001
  • M. Ehrhardt and R.E. Mickens, A nonstandard finite difference scheme for convection-diffusion equations having constant coefficients, Appl. Math. Comput.219 (2013), pp. 6591–6604.
  • S.M. Garba, A.B. Gumel, and J.M.-S. Lubuma, Dynamically-consistent non-standard finite difference method for an epidemic model, Math. Comput. Model. 53 (2011), pp. 131–150. doi: 10.1016/j.mcm.2010.07.026
  • G. González-Parra, A.J. Arenas, and B.M. Chen-Charpentier, Combination of nonstandard schemes and Richardson's extrapolation to improve the numerical solution of population models, Math. Comput. Model. 52 (2010), pp. 1030–1036. doi: 10.1016/j.mcm.2010.03.015
  • A. Gerisch and R. Weiner, The positivity of low-order explicit Runge-Kutta schemes applied in splitting methods, Comput. Math. Appl. 45 (2003), pp. 53–67. doi: 10.1016/S0898-1221(03)80007-X
  • E. Hairer, G. Wanner, Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, Springer, Berlin, 1996.
  • E. Hairer, S.P. Nørsett, G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems, Springer, Berlin, 1993.
  • M.T. Hoang and A.M. Nagy, Uniform asymptotic stability of a logistic model with feedback control of fractional order and nonstandard finite difference schemes, Chaos Solitons Fractals 123 (2019), pp. 24–34. doi: 10.1016/j.chaos.2019.03.031
  • Z. Horváth, Positivity of Runge-Kutta methods and diagonally split Runge-Kutta methods, Appl. Numer. Math. 28 (1998), pp. 306–326. doi: 10.1016/S0168-9274(98)00050-6
  • Z. Horváth, On the positivity step size threshold of Runge-Kutta methods, Appl. Numer. Math. 53 (2005), pp. 341–356. doi: 10.1016/j.apnum.2004.08.026
  • J.E. Keymer, P.A. Marquet, J.X. Velasco-Hernandez, and S.A. Levin, Extinction thresholds and metapopulation persistence in dynamic landscapes, Am. Nat. 156 (2000), pp. 478–494. doi: 10.1086/303407
  • A. Korpusik, A nonstandard finite difference scheme for a basic model of cellular immune response to viral infection, Commun. Nonlinear Sci. Numer. Simul. 43 (2017), pp. 369–384. doi: 10.1016/j.cnsns.2016.07.017
  • J.F.B.M. Kraaijevanger, Contractivity of Runge-Kutta methods, BIT 31 (1991), pp. 482–528. doi: 10.1007/BF01933264
  • C.M. Kribs-Zaleta and J.X. Velasco-Hernández, A simple vaccination model with multiple endemic states, Math. Biosci. 164 (2000), pp. 183–201. doi: 10.1016/S0025-5564(00)00003-1
  • J. Martín-Vaquero, A. Martín del Rey, A.H. Encinas, J.D. Hernández Guillén, A. Queiruga-Dios, and G. Rodríguez Sánchez, Higher-order nonstandard finite difference schemes for a MSEIR model for a malware propagation, J. Comput. Appl. Math. 317 (2017), pp. 146–156. doi: 10.1016/j.cam.2016.11.044
  • J. Martín-Vaquero, A. Martín del Rey, A.H. Encinas, J.D. Hernández Guillén, A. Queiruga-Dios, and G. Rodríguez Sánchez, Variable step length algorithms with high-order extrapolated non-standard finite difference schemes for a SEIR model, J. Comput. Appl. Math. 330 (2018), pp. 848–854. doi: 10.1016/j.cam.2017.03.031
  • R.E. Mickens, Applications of Nonstandard Finite Difference Schemes, World Scientific, Singapore, 1994.
  • R.E. Mickens, Nonstandard Finite Difference Models of Differential Equations, World Scientific, Singapore, 2000.
  • R.E. Mickens, A nonstandard finite-difference scheme for the Lotka-Volterra system, Appl. Numer. Math. 45 (2003), pp. 309–314. doi: 10.1016/S0168-9274(02)00223-4
  • R.E. Mickens, Advances in the Applications of Nonstandard Finite Difference Schemes, World Scientific, Singapore, 2005.
  • K.C. Patidar, Nonstandard finite difference methods: Recent trends and further developments, J. Difference Equ. Appl. 22 (2016), pp. 817–849. doi: 10.1080/10236198.2016.1144748
  • D.T. Wood and H.V. Kojouharov, A class of nonstandard numerical methods for autonomous dynamical systems, Appl. Math. Lett. 50 (2015), pp. 78–82. doi: 10.1016/j.aml.2015.06.008
  • D.T. Wood, H.V. Kojouharov, and D.T. Dimitrov, Universal approaches to approximate biological systems with nonstandard finite difference methods, Math. Comput. Simul. 133 (2017), pp. 337–350. doi: 10.1016/j.matcom.2016.04.007
  • L. Yang, X. Yang, Q. Zhua, and L. Wen, A computer virus model with graded cure rates, Nonlinear Anal. 14 (2013), pp. 414–422. doi: 10.1016/j.nonrwa.2012.07.005

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