A positive and bounded convergent scheme for general space-fractional diffusion-reaction systems with inertial times

References

• M. Abe, M. Chase, B. David, M. Kohlweiss, R. Nishimaki, and M. Ohkubo, Constant-size structure-preserving signatures: Generic constructions and simple assumptions, J. Cryptol. 29(4) (2016), pp. 833–878.
   Web of Science ®Google Scholar
• A. Arakawa and V.R. Lamb, A potential entrophy and energy conserving scheme for the shallow water equations, Monthly Weather Rev. 109(1) (1981), pp. 18–36.
   Web of Science ®Google Scholar
• S. Badia and J. Bonilla, Monotonicity-preserving finite element schemes based on differentiable nonlinear stabilization, Comput. Methods. Appl. Mech. Eng. 313 (2017), pp. 133–158.
   Web of Science ®Google Scholar
• E. Balsa-Canto, A. López-Núñez, and C. Vázquez, Numerical methods for a nonlinear reaction–diffusion system modelling a batch culture of biofilm, Appl. Math. Model. 41 (2017), pp. 164–179.
   Web of Science ®Google Scholar
• M Banerjee, Turing and non-turing patterns in two-dimensional prey-predator models, in Applications of Chaos and Nonlinear Dynamics in Science and Engineering, Vol. 4, Springer, 2015, pp. 257–280.
   Google Scholar
• G. Ben-Yu, P.J. Pascual, M.J. Rodriguez, and L. Vázquez, Numerical solution of the sine-Gordon equation, Appl. Math. Comput. 18(1) (1986), pp. 1–14.
   Web of Science ®Google Scholar
• R. Bürger, G. Chowell, E. Gavilán, P. Mulet, and L.M. Villada, Numerical solution of a spatio-temporal predator-prey model with infected prey, Math. Biosci. Eng. 16 (2018), pp. 438–473.
   PubMed Web of Science ®Google Scholar
• A. Cascone, R. Manzo, B. Piccoli, and L. Rarità, Optimization versus randomness for car traffic regulation, Phys. Rev. E. 78(2) (2008), p. 026113.
   Web of Science ®Google Scholar
• A. Cutolo, C. De Nicola, R. Manzo, and L. Rarità, Optimal paths on urban networks using travelling times prevision, Model. Simulat. Engin. 2012 (2012), p. 9.
   Google Scholar
• C. D'Apice, R. Manzo, and L. Rarità, Splitting of traffic flows to control congestion in special events, Int. J. Math. Math. Sci. 2011 (2011), p. 18.
   Google Scholar
• M. de Falco, M. Gaeta, V. Loia, L. Rarità, and S. Tomasiello, Differential quadrature-based numerical solutions of a fluid dynamic model for supply chains, Commun. Math. Sci. 14(5) (2016), pp. 1467–1476.
   Web of Science ®Google Scholar
• A. De Wit, Spatial patterns and spatiotemporal dynamics in chemical systems, Adv. Chem. Phys. 109 (2007), pp. 435–513.
   Google Scholar
• V. Dufiet and J. Boissonade, Dynamics of Turing pattern monolayers close to onset, Phys. Rev. E. 53(5) (1996), p. 4883.
   Web of Science ®Google Scholar
• V. Ervin, J. Macías-Díaz, and J. Ruiz-Ramírez, A positive and bounded finite element approximation of the generalized burgers–huxley equation, J. Math. Anal. Appl. 424(2) (2015), pp. 1143–1160.
   Web of Science ®Google Scholar
• H. Fazli, H. Sun, and S. Aghchi, Existence of extremal solutions of fractional Langevin equation involving nonlinear boundary conditions, Int. J. Comput. Math. (2020), pp. 1–10. https://doi.org/10.1080/00207160.2020.1720662.
   Web of Science ®Google Scholar
• Z. Fei and L. Vázquez, Some conservative numerical schemes for an ordinary differential equation, Comput. Appl. Math. 10 (1991), pp. 59–69.
   Google Scholar
• A. Friedman, Foundations of Modern Analysis, Courier Corporation, New York, 1970.
   Google Scholar
• K.M. Furati, M. Yousuf, and A.Q.M. Khaliq, Fourth-order methods for space fractional reaction–diffusion equations with non-smooth data, Int. J. Comput. Math. 95(6–7) (2018), pp. 1240–1256.
   Web of Science ®Google Scholar
• D. Furihata, A stable and conservative finite difference scheme for the Cahn-Hilliard equation, Numerische Mathematik 87(4) (2001), pp. 675–699.
   Web of Science ®Google Scholar
• D. Furihata, Finite-difference schemes for nonlinear wave equation that inherit energy conservation property, J. Comput. Appl. Math. 134(1) (2001), pp. 37–57.
   Web of Science ®Google Scholar
• D. Furihata and T. Matsuo, Discrete Variational Derivative Method: A Structure-preserving Numerical Method for Partial Differential Equations, CRC Press, New York, 2010.
   Google Scholar
• D. Furihata, S. Sato, and T. Matsuo, A novel discrete variational derivative method using“average-difference methods”, JSIAM Lett. 8 (2016), pp. 81–84.
   Google Scholar
• B. Ghanbari, A. Yusuf, and D. Baleanu, The new exact solitary wave solutions and stability analysis for the (2+ 1) (2+1)-dimensional Zakharov–Kuznetsov equation, Adv. Differ. Equ. 2019(1) (2019), p. 49.
   Web of Science ®Google Scholar
• W.G. Glöckle and T.F. Nonnenmacher, A fractional calculus approach to self-similar protein dynamics, Biophys. J. 68(1) (1995), pp. 46–53.
   PubMed Web of Science ®Google Scholar
• C. Huang, J. Cao, M. Xiao, A. Alsaedi, and F.E. Alsaadi, Controlling bifurcation in a delayed fractional predator–prey system with incommensurate orders, Appl. Math. Comput. 293 (2017), pp. 293–310.
   Web of Science ®Google Scholar
• A. Ibrahimbegovic and S. Mamouri, Energy conserving/decaying implicit time-stepping scheme for nonlinear dynamics of three-dimensional beams undergoing finite rotations, Comput. Methods. Appl. Mech. Eng. 191(37-38) (2002), pp. 4241–4258.
   Web of Science ®Google Scholar
• T. Ide, Some energy preserving finite element schemes based on the discrete variational derivative method, Appl. Math. Comput. 175(1) (2006), pp. 277–296.
   Web of Science ®Google Scholar
• T. Ide and M. Okada, Numerical simulation for a nonlinear partial differential equation with variable coefficients by means of the discrete variational derivative method, J. Comput. Appl. Math. 194(2) (2006), pp. 425–459.
   Web of Science ®Google Scholar
• O.S. Iyiola, E. Asante-Asamani, K.M. Furati, A.Q.M. Khaliq, and B.A. Wade, Efficient time discretization scheme for nonlinear space fractional reaction–diffusion equations, Int. J. Comput. Math. 95(6–7) (2018), pp. 1274–1291.
   Web of Science ®Google Scholar
• A.Q.M. Khaliq, X. Liang, and K.M. Furati, A fourth-order implicit-explicit scheme for the space fractional nonlinear Schrödinger equations, Numer. Algorithms. 75(1) (2017), pp. 147–172.
   Web of Science ®Google Scholar
• A.N. Kochubei and Y. Kondratiev, Growth equation of the general fractional calculus, Mathematics7(7) (2019), p. 615.
   Web of Science ®Google Scholar
• R. Koeller, Applications of fractional calculus to the theory of viscoelasticity, ASME Trans. J. Appl. Mech. 51 (1984), pp. 299–307. ISSN 0021-8936.
   Web of Science ®Google Scholar
• H. Kuramae and T. Matsuo, An alternating discrete variational derivative method for coupled partial differential equations, JSIAM Lett. 4 (2012), pp. 29–32.
   Google Scholar
• N. Laskin, Fractional Schrödinger equation, Phys. Rev. E. 66(5) (2002), p. 056108.
   Web of Science ®Google Scholar
• H.-L. Li, A. Muhammadhaji, L. Zhang, and Z. Teng, Stability analysis of a fractional-order predator–prey model incorporating a constant prey refuge and feedback control, Adv. Differ. Equ.2018(1) (2018), p. 325.
   Google Scholar
• J.E. Macías-Díaz, Numerical study of the transmission of energy in discrete arrays of sine-Gordon equations in two space dimensions, Phys. Rev. E. 77(1) (2008), p. 016602.
   Web of Science ®Google Scholar
• J.E. Macías-Díaz, A structure-preserving method for a class of nonlinear dissipative wave equations with Riesz space-fractional derivatives, J. Comput. Phys.351 (2017), pp. 40–58.
   Web of Science ®Google Scholar
• J.E. Macías-Díaz, Persistence of nonlinear hysteresis in fractional models of Josephson transmission lines, Commun. Nonlinear Sci. Numer. Simulat. 53 (2017), pp. 31–43.
   Web of Science ®Google Scholar
• J.E. Macías-Díaz, Numerical simulation of the nonlinear dynamics of harmonically driven Riesz-fractional extensions of the Fermi–Pasta–Ulam chains, Commun. Nonlinear Sci. Numerical Simulat. 55 (2018), pp. 248–264.
   Web of Science ®Google Scholar
• J.E. Macías-Díaz, On the solution of a Riesz space-fractional nonlinear wave equation through an efficient and energy-invariant scheme, Int. J. Comput. Math. 96(2) (2019), pp. 337–361.
   Web of Science ®Google Scholar
• J.E. Macías-Díaz and J. Villa-Morales, A deterministic model for the distribution of the stopping time in a stochastic equation and its numerical solution, J. Comput. Appl. Math. 318 (2017), pp. 93–106.
   Web of Science ®Google Scholar
• T. Matsuo, Dissipative/conservative Galerkin method using discrete partial derivatives for nonlinear evolution equations, J. Comput. Appl. Math.218(2) (2008), pp. 506–521.
   Web of Science ®Google Scholar
• R.E. Mickens, Dynamic consistency: a fundamental principle for constructing nonstandard finite difference schemes for differential equations, J. Differ. Equ. Appl. 11(7) (2005), pp. 645–653.
   Web of Science ®Google Scholar
• M.D. Morales-Hernández, I.E. Medina-Ramírez, F.J. Avelar-González, and J.E. Macías-Díaz, An efficient recursive algorithm in the computational simulation of the bounded growth of biological films, Inter. J. Comput. Methods 9(04) (2012), p. 1250050.
   Google Scholar
• A. Mvogo, J.E. Macías-Díaz, and T.C. Kofané, Diffusive instabilities in a hyperbolic activator-inhibitor system with superdiffusion, Phys. Rev. E. 97(3) (2018), p. 032129.
   PubMed Web of Science ®Google Scholar
• V. Namias, The fractional order Fourier transform and its application to quantum mechanics, IMA J. Appl. Math. 25(3) (1980), pp. 241–265.
   Google Scholar
• M.D. Ortigueira, Fractional central differences and derivatives, IFAC Proc. Vol. 39(11) (2006), pp. 58–63.
   Google Scholar
• M.D. Ortigueira, Riesz potential operators and inverses via fractional centred derivatives, Int. J. Math. Math. Sci. 2006 (2006), p. 12.
   Google Scholar
• M.D. Ortigueira, Fractional Calculus for Scientists and Engineers, Vol. 84, New York: Springer, 2011.
   Google Scholar
• M.D. Ortigueira and A.G. Batista, On the relation between the fractional brownian motion and the fractional derivatives, Phys. Lett. A 372(7) (2008), pp. 958–968.
   Web of Science ®Google Scholar
• M.D. Ortigueira and F. Coito, From differences to derivatives, Fract. Calc. Appl. Anal. 7(4) (2004), p. 459.
   Google Scholar
• M.D. Ortigueira and J.J. Trujillo, Generalized gl fractional derivative and its laplace and fourier transform, in International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Vol. 49019, 2009, pp. 1227–1231.
   Google Scholar
• K. Pen-Yu, Numerical methods for incompressible viscous flow, Sci. Sin. 20 (1977), pp. 287–304.
   Google Scholar
• Y. Peng and T. Zhang, Turing instability and pattern induced by cross-diffusion in a predator-prey system with Allee effect, Appl. Math. Comput. 275 (2016), pp. 1–12.
   Web of Science ®Google Scholar
• I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Vol. 198, Amsterdam: Elsevier, 1998.
   Google Scholar
• Y. Povstenko, Theory of thermoelasticity based on the space-time-fractional heat conduction equation, Physica Scripta 2009(T136) (2009), p. 014017.
   Google Scholar
• F. Rao and Y. Kang, The complex dynamics of a diffusive prey–predator model with an allee effect in prey, Ecol. Complex. 28 (2016), pp. 123–144.
   Web of Science ®Google Scholar
• L. Rarità, C. D'Apice, B. Piccoli, and D. Helbing, Sensitivity analysis of permeability parameters for flows on barcelona networks, J. Differ. Equ. 249(12) (2010), pp. 3110–3131.
   Web of Science ®Google Scholar
• F. Rihan, S. Lakshmanan, A. Hashish, R. Rakkiyappan, and E. Ahmed, Fractional-order delayed predator–prey systems with holling type-II functional response, Nonlinear. Dyn.80(1-2) (2015), pp. 777–789.
   Web of Science ®Google Scholar
• I. Romero and F. Armero, An objective finite element approximation of the kinematics of geometrically exact rods and its use in the formulation of an energy–momentum conserving scheme in dynamics, Int. J. Numer. Methods. Eng. 54(12) (2002), pp. 1683–1716.
   Web of Science ®Google Scholar
• E. Scalas, R. Gorenflo, and F. Mainardi, Fractional calculus and continuous-time finance, Phys. A Stat. Mech. Appl. 284(1) (2000), pp. 376–384.
   Google Scholar
• J. Singh, D. Kumar, and D. Baleanu, On the analysis of fractional diabetes model with exponential law, Adv. Differ. Equ. 2018(1) (2018), p. 231.
   Google Scholar
• F. Soltani Sarvestani, M.H. Heydari, A. Niknam, and Z. Avazzadeh, A wavelet approach for the multi-term time fractional diffusion-wave equation, Int. J. Comput. Math. 96(3) (2019), pp. 640–661.
   Web of Science ®Google Scholar
• W. Strauss and L. Vazquez, Numerical solution of a nonlinear Klein-Gordon equation, J. Comput. Phys. 28(2) (1978), pp. 271–278.
   Web of Science ®Google Scholar
• Y. Suzuki and M. Ohnawa, Generic formalism and discrete variational derivative method for the two-dimensional vorticity equation, J. Comput. Appl. Math. 296 (2016), pp. 690–708.
   Web of Science ®Google Scholar
• Y.-F. Tang, L. Vázquez, F. Zhang, and V. Pérez-García, Symplectic methods for the nonlinear Schrödinger equation, Comput. Math. Appl. 32(5) (1996), pp. 73–83.
   Web of Science ®Google Scholar
• S. Tomasiello, A note on three numerical procedures to solve volterra integrodifferential equations in structural analysis, Comput. Math. Appl. 62 (2011), pp. 3183–3193.
   Web of Science ®Google Scholar
• S. Tomasiello, Some remarks on a new dq-based method for solving a class of volterra integro-differential equations, Appl. Math. Comput. 219 (2012), pp. 399–407.
   Google Scholar
• R. Upadhyay, V. Volpert, and N. Thakur, Propagation of turing patterns in a plankton model, J. Biol. Dyn. 6(2) (2012), pp. 524–538.
   PubMed Web of Science ®Google Scholar
• A. Vahadane, T. Peng, A. Sethi, S. Albarqouni, L. Wang, M. Baust, K. Steiger, A.M. Schlitter, I. Esposito, and N. Navab, Structure-preserving color normalization and sparse stain separation for histological images, IEEE. Trans. Med. Imaging. 35(8) (2016), pp. 1962–1971.
   PubMed Web of Science ®Google Scholar
• Y. Wang and Z. Li, A family of convexity-preserving subdivision schemes, J. Math. Res. Appl 37(4) (2017), pp. 489–495.
   Google Scholar
• Y.-M. Wang and L. Ren, Efficient compact finite difference methods for a class of time-fractional convection–reaction–diffusion equations with variable coefficients, Int. J. Comput. Math. 96(2) (2019), pp. 264–297.
   Web of Science ®Google Scholar
• X. Wang, F. Liu, and X. Chen, Novel second-order accurate implicit numerical methods for the Riesz space distributed-order advection-dispersion equations, Adv. Math. Phys. 2015 (2015), p. 14.
   Web of Science ®Google Scholar
• L. Wang, Y. Li, and S Lazebnik, Learning deep structure-preserving image-text embeddings, in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2016, pp. 5005–5013.
   Google Scholar
• J. Wang, H. Cheng, H. Liu, and Y. Wang, Periodic solution and control optimization of a prey-predator model with two types of harvesting, Adv. Differ. Equ. 2018(1) (2018), p. 41.
   Google Scholar
• Y. Xiao and L. Chen, Modeling and analysis of a predator–prey model with disease in the prey, Math. Biosci. 171(1) (2001), pp. 59–82.
   PubMed Web of Science ®Google Scholar
• A. Yusuf, A.I. Aliyu, Baleanu, D., et al., Conservation laws, soliton-like and stability analysis for the time fractional dispersive long-wave equation, Adv. Differ. Equ. 2018(1) (2018), p. 319.
   Google Scholar