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doi: 10.3934/dcdss.2020035

Dynamics of traveling waves of variable order hyperbolic Liouville equation: Regulation and control

1. 

Department of Mathematical Sciences, University of South Africa, Florida, 0003, South Africa

2. 

Institute for Groundwater Studies, University of the Free State, Bloemfontein, 9300, South Africa

* Corresponding author: franckemile2006@yahoo.ca

Received  June 2018 Revised  June 2018 Published  March 2019

Fund Project: This work was partially supported by the grant No: 105932 from the National Research Foundation (NRF) of South Africa

Traveling waves remain significant in Applied Sciences mostly because they involve the movement of energy carrier particles. In this paper, traveling waves described by a generalized system, the fractional variable order hyperbolic Liouville model is solved numerically by means of Crank-Nicholson scheme. Detailed analysis are performed and prove that the numerical method is stable and converges. Simulations reveal that the model's variable order derivative (a function of time and position variables) has a considerable impact on the dynamics of the whole system. It influences the movement and the shape of the resulting waves including their amplitude, their wavelength as well as their compression and rarefaction processes. Such a variable order derivative becomes, due to these results, a substantial parameter and non-constant tool for the regulation and control of models describing wave motion.

Citation: Emile Franc Doungmo Goufo, Abdon Atangana. Dynamics of traveling waves of variable order hyperbolic Liouville equation: Regulation and control. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020035
References:
[1]

J. F. Aguilar, T. Cordova-Fraga, J. Trres-Jimnez, R. F. Escobar-Jimnez, V. H. Olivares-Peregrino and G. V. Guerrero-Ramrez, Nonlocal transport processes and the fractional cattaneo-vernotte equation, Mathematical Problems in Engineering, 2016 (2016), Art. ID 7845874, 15 pp. doi: 10.1155/2016/7845874. Google Scholar

[2]

A. Atangana, On the stability and convergence of the time-fractional variable order telegraph equation, Journal of Computational Physics, 293 (2015), 104-114. doi: 10.1016/j.jcp.2014.12.043. Google Scholar

[3]

D. Baleanu, J. A. Tenreiro Machado and A. C. J. Luo, Fractional Dynamics and Control, Springer New York Dordrecht Heidelberg London, 2012. doi: 10.1007/978-1-4614-0457-6. Google Scholar

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E. Bonyah, M. A. Khan, K. O. Okosun and S. Islam, A theoretical model for Zika virus transmission, PloS One, 12 (2017), e0185540. doi: 10.1371/journal.pone.0185540. Google Scholar

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R. Bürger, Mathematical principles of mutationselection models, Genetica, 102/103 (1998), 279-298. Google Scholar

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C. M. ChenF. LiuI. Turner and V. Anh, A Fourier method for the fractional diffusion equation describing sub-diffusion, J. Comput. Phys., 227 (2007), 886-897. doi: 10.1016/j.jcp.2007.05.012. Google Scholar

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A. Coronel-EscamillaJ. F. Gmez-AguilarL. Torres and R. F. Escobar-Jimnez, A numerical solution for a variable-order reaction-diffusion model by using fractional derivatives with non-local and non-singular kernel, Physica A: Statistical Mechanics and its Applications, 491 (2018), 406-424. doi: 10.1016/j.physa.2017.09.014. Google Scholar

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A. Coronel-Escamilla, J. F. Gmez-Aguilar, D. Baleanu, T. Crdova-Fraga, R. F. Escobar-Jimnez, V. H. Olivares-Peregrino and M. M. A. Qurashi, BatemanF eshbach Tikochinsky and CaldirolaKanai Oscillators with New Fractional Differentiation, Entropy, 19 (2017), p55.Google Scholar

[9]

A. Coronel-EscamillaF. TorresJ. F. Gomez-AguilarR. F. Escobar-Jimenez and G. V. Guerrero-Ramrez, On the trajectory tracking control for an SCARA robot manipulator in a fractional model driven by induction motors with PSO tuning, Multibody System Dynamics, 43 (2018), 257-277. doi: 10.1007/s11044-017-9586-3. Google Scholar

[10]

A. Coronel-Escamilla, J.F. Gmez-Aguilar, D. Baleanu, T. Crdova-Fraga, R.F. Escobar-Jimnez, V.H. Olivares-Peregrino, and A. Abundez-Pliego, Formulation of Euler-Lagrange and Hamilton equations involving fractional operators with regular kernel, Advances in Difference Equations, 2016 (2016), Paper No. 283, 17 pp. doi: 10.1186/s13662-016-1001-5. Google Scholar

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J. Crank and P. Nicolson, A practical method for numerical evaluation of solutions of partial differential equations of the heat conduction type, Proc. Camb. Philol. Soc., 43 (1947), 50-67. doi: 10.1017/S0305004100023197. Google Scholar

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K. DiethelmN. J. Ford and A. D. Freed, Detailed error analysis for a fractional Adams method, Numer. Algorithms, 36 (2004), 31-52. doi: 10.1023/B:NUMA.0000027736.85078.be. Google Scholar

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E. F. Doungmo Goufo, Speeding up chaos and limit cycles in evolutionary language and learning processes, Mathematical Methods in the Applied Sciences, 40 (2017), 3055-3065. doi: 10.1002/mma.4220. Google Scholar

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E. F. Doungmo Goufo and S. Kumar, Shallow water wave models with and without singular kernel: Existence, uniqueness and similarities, Mathematical Problems in Engineering, 2017 (2017), Article ID 4609834, 9 pages. doi: 10.1155/2017/4609834. Google Scholar

[15]

E. F. Doungmo Goufo and J. J. Nieto, Attractors for fractional differential problems of transition to turbulent flows, Journal of Computational and Applied Mathematics, 339 (2018), 329-342. doi: 10.1016/j.cam.2017.08.026. Google Scholar

[16]

E. F. Doungmo Goufo, Solvability of chaotic fractional systems with 3D four-scroll attractors, Chaos, Solitons & Fractals, 104 (2017), 443-451. doi: 10.1016/j.chaos.2017.08.038. Google Scholar

[17]

E. F. Doungmo Goufo, Chaotic processes using the two-parameter derivative with non-singular and non-local kernel: Basic theory and applications, Chaos: An Interdisciplinary Journal of Nonlinear Science, 26 (2016), 084305, 10 pp. doi: 10.1063/1.4958921. Google Scholar

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R. Gorenflo and F. Mainardi, Fractional diffusion processes: Probability distribution and continuous time random walk, Lecture Notes Phys, 621 (2003), 148-166. doi: 10.1007/3-540-44832-2_8. Google Scholar

[19]

E. Hanert, On the numerical solution of space time fractional diffusion models, Comput. Fluids, 46 (2011), 33-39. doi: 10.1016/j.compfluid.2010.08.010. Google Scholar

[20]

R. Hilfer, Application of Fractional Calculus in Physics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/9789812817747. Google Scholar

[21] J. Hofbauer and K. Sigmund, Evolutionary Games and Replicator Dynamics, Cambridge University Press, Cambridge, 1998. doi: 10.1017/CBO9781139173179.
[22]

Y. KhanK. SayevandM. Fardi and M. Ghasemi, A novel computing multi-parametric homotopy approach for system of linear and nonlinear Fredholm integral equations, Applied Mathematics and Computation, 249 (2014), 229-236. doi: 10.1016/j.amc.2014.10.070. Google Scholar

[23]

N. L. Komarova, Replicator-mutator equation, universality property and population dynamics of learning, Journal of Theoretical Biology, 230 (2004), 227-239. doi: 10.1016/j.jtbi.2004.05.004. Google Scholar

[24]

N. L. KomarovaP. Niyogi and M. A. Nowak, Evolutionary dynamics of grammar acquisition, J. Theor. Biol., 209 (2001), 43-59. doi: 10.1006/jtbi.2000.2240. Google Scholar

[25]

C. P. Li and C. X. Tao, On the fractional Adams method, Computers and Mathematics with Applications, 58 (2009), 1573-1588. doi: 10.1016/j.camwa.2009.07.050. Google Scholar

[26]

R. LinF. LiuV. Anh and I. Turner, Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation, Appl. Math. Comput., 212 (2009), 435-445. doi: 10.1016/j.amc.2009.02.047. Google Scholar

[27]

C. F. Lorenzo and T. T. Hartley, Variable order and distributed order fractional operators, Nonlinear Dyn., 29 (2002), 57-98. doi: 10.1023/A:1016586905654. Google Scholar

[28]

D. Matignon, Stability results for fractional differential equations with applications to control processing, in: Computational Eng. in Sys. Appl., 2 (1996), 963.Google Scholar

[29]

M. M. Meerschaert and C. Tadjeran, Finite difference approximations for fractional advection dispersion equations, J. Comput. Appl. Math., 172 (2004), 65-77. doi: 10.1016/j.cam.2004.01.033. Google Scholar

[30]

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993. Google Scholar

[31]

W. G. Mitchener and M. A. Nowak, Chaos and language, Proceedings of the Royal Society B: Biological Sciences, 271 (2004), 701. doi: 10.1098/rspb.2003.2643. Google Scholar

[32]

M. Nowak and K. Sigmund, Chaos and the evolution of cooperation, Proceedings of the National Academy of Sciences, 90 (1993), 5091-5094. doi: 10.1073/pnas.90.11.5091. Google Scholar

[33]

D. Pais and N. E. Leonard, Limit cycles in replicator-mutator network dynamics, in 50th IEEE Conference on Decision and Control, 2011, 3922–3927. doi: 10.1109/CDC.2011.6160995. Google Scholar

[34]

I. PodlubnyA. ChechkinT. SkovranekY. Q. Chen and B. M. Vinagre Jara, Matrix approach to discrete fractional calculus II: Partial fractional differential equations, J. Comput. Phys., 228 (2009), 3137-3153. doi: 10.1016/j.jcp.2009.01.014. Google Scholar

[35] I. Podlubny, Fractional Differential Equations, Academic Press, California, USA, 1999.
[36]

A. D. Polyanin and V. F. Zaitsev, Handbook of Nonlinear Partial Differential Equations, Chapman & Hall/CRC, Boca Raton, FL, 2004. Google Scholar

[37]

L. E. S. Ramirez and C. F. M. Coimbra, On the selection and meaning of variable order operators for dynamic modelling, Int. J. Differ. Equ., 2010 (2010), 846107, 16 pp. doi: 10.1155/2010/846107. Google Scholar

[38]

B. Ross and S. G. Samko, Fractional integration operator of a variable order in the Holder spaces $H_(x),$, Int. J. Math. Math. Sci., 18 (1995), 777-788. doi: 10.1155/S0161171295001001. Google Scholar

[39]

Z. ShahT. GulS. IslamM. A. KhanE. BonyahF. Hussain and M. Ullah, Three dimensional third grade nanofluid flow in a rotating system between parallel plates with Brownian motion and thermophoresis effects, Results in Physics, 10 (2018), 36-45. doi: 10.1016/j.rinp.2018.05.020. Google Scholar

[40]

P. F. Stadler and P. Schuster, Mutation in autocatalytic reaction networks-an analysis based on perturbation theory, J. Math. Biol., 30 (1992), 597-631. doi: 10.1007/BF00948894. Google Scholar

[41]

C. TadjeranM. M. Meerschaert and H. P. Scheffler, A second order accurate numerical approximation for the fractional diffusion equation, J. Comput. Phys., 213 (2006), 205-213. doi: 10.1016/j.jcp.2005.08.008. Google Scholar

[42]

S. Umarov and S. Steinberg, Variable order differential equations and diffusion with changing modes, Z. Anal. Anwend., 28 (2009), 431-450. doi: 10.4171/ZAA/1392. Google Scholar

[43]

L. YangF. ZhichaoL. Hong and H. Siriguleng, A mixed finite element method for a time-fractional fourth-order partial differential equation, Appl. Math. Comput., 243 (2014), 703-717. doi: 10.1016/j.amc.2014.06.023. Google Scholar

[44]

K. Yosida, Fonctional Analysis, Sixth Edition, Springer- Verlag, 1980. Google Scholar

[45]

S. B. Yuste and L. Acedo, An explicit finite difference method and a new Von Neumann-type stability analysis for fractional diffusion equations, SIAM J. Numer. Anal., 42 (2005), 1862-1874. doi: 10.1137/030602666. Google Scholar

[46]

Y. Zhang, A finite difference method for fractional partial differential equation, Appl. Math. Comput., 215 (2009), 524-529. doi: 10.1016/j.amc.2009.05.018. Google Scholar

[47]

P. ZhuangF. LiuV. Anh and I. Turner, Numerical methods for the variable-order fractional advection–diffusion equation with a nonlinear source term, SIAM J. Numer. Anal., 47 (2009), 1760-1781. doi: 10.1137/080730597. Google Scholar

show all references

References:
[1]

J. F. Aguilar, T. Cordova-Fraga, J. Trres-Jimnez, R. F. Escobar-Jimnez, V. H. Olivares-Peregrino and G. V. Guerrero-Ramrez, Nonlocal transport processes and the fractional cattaneo-vernotte equation, Mathematical Problems in Engineering, 2016 (2016), Art. ID 7845874, 15 pp. doi: 10.1155/2016/7845874. Google Scholar

[2]

A. Atangana, On the stability and convergence of the time-fractional variable order telegraph equation, Journal of Computational Physics, 293 (2015), 104-114. doi: 10.1016/j.jcp.2014.12.043. Google Scholar

[3]

D. Baleanu, J. A. Tenreiro Machado and A. C. J. Luo, Fractional Dynamics and Control, Springer New York Dordrecht Heidelberg London, 2012. doi: 10.1007/978-1-4614-0457-6. Google Scholar

[4]

E. Bonyah, M. A. Khan, K. O. Okosun and S. Islam, A theoretical model for Zika virus transmission, PloS One, 12 (2017), e0185540. doi: 10.1371/journal.pone.0185540. Google Scholar

[5]

R. Bürger, Mathematical principles of mutationselection models, Genetica, 102/103 (1998), 279-298. Google Scholar

[6]

C. M. ChenF. LiuI. Turner and V. Anh, A Fourier method for the fractional diffusion equation describing sub-diffusion, J. Comput. Phys., 227 (2007), 886-897. doi: 10.1016/j.jcp.2007.05.012. Google Scholar

[7]

A. Coronel-EscamillaJ. F. Gmez-AguilarL. Torres and R. F. Escobar-Jimnez, A numerical solution for a variable-order reaction-diffusion model by using fractional derivatives with non-local and non-singular kernel, Physica A: Statistical Mechanics and its Applications, 491 (2018), 406-424. doi: 10.1016/j.physa.2017.09.014. Google Scholar

[8]

A. Coronel-Escamilla, J. F. Gmez-Aguilar, D. Baleanu, T. Crdova-Fraga, R. F. Escobar-Jimnez, V. H. Olivares-Peregrino and M. M. A. Qurashi, BatemanF eshbach Tikochinsky and CaldirolaKanai Oscillators with New Fractional Differentiation, Entropy, 19 (2017), p55.Google Scholar

[9]

A. Coronel-EscamillaF. TorresJ. F. Gomez-AguilarR. F. Escobar-Jimenez and G. V. Guerrero-Ramrez, On the trajectory tracking control for an SCARA robot manipulator in a fractional model driven by induction motors with PSO tuning, Multibody System Dynamics, 43 (2018), 257-277. doi: 10.1007/s11044-017-9586-3. Google Scholar

[10]

A. Coronel-Escamilla, J.F. Gmez-Aguilar, D. Baleanu, T. Crdova-Fraga, R.F. Escobar-Jimnez, V.H. Olivares-Peregrino, and A. Abundez-Pliego, Formulation of Euler-Lagrange and Hamilton equations involving fractional operators with regular kernel, Advances in Difference Equations, 2016 (2016), Paper No. 283, 17 pp. doi: 10.1186/s13662-016-1001-5. Google Scholar

[11]

J. Crank and P. Nicolson, A practical method for numerical evaluation of solutions of partial differential equations of the heat conduction type, Proc. Camb. Philol. Soc., 43 (1947), 50-67. doi: 10.1017/S0305004100023197. Google Scholar

[12]

K. DiethelmN. J. Ford and A. D. Freed, Detailed error analysis for a fractional Adams method, Numer. Algorithms, 36 (2004), 31-52. doi: 10.1023/B:NUMA.0000027736.85078.be. Google Scholar

[13]

E. F. Doungmo Goufo, Speeding up chaos and limit cycles in evolutionary language and learning processes, Mathematical Methods in the Applied Sciences, 40 (2017), 3055-3065. doi: 10.1002/mma.4220. Google Scholar

[14]

E. F. Doungmo Goufo and S. Kumar, Shallow water wave models with and without singular kernel: Existence, uniqueness and similarities, Mathematical Problems in Engineering, 2017 (2017), Article ID 4609834, 9 pages. doi: 10.1155/2017/4609834. Google Scholar

[15]

E. F. Doungmo Goufo and J. J. Nieto, Attractors for fractional differential problems of transition to turbulent flows, Journal of Computational and Applied Mathematics, 339 (2018), 329-342. doi: 10.1016/j.cam.2017.08.026. Google Scholar

[16]

E. F. Doungmo Goufo, Solvability of chaotic fractional systems with 3D four-scroll attractors, Chaos, Solitons & Fractals, 104 (2017), 443-451. doi: 10.1016/j.chaos.2017.08.038. Google Scholar

[17]

E. F. Doungmo Goufo, Chaotic processes using the two-parameter derivative with non-singular and non-local kernel: Basic theory and applications, Chaos: An Interdisciplinary Journal of Nonlinear Science, 26 (2016), 084305, 10 pp. doi: 10.1063/1.4958921. Google Scholar

[18]

R. Gorenflo and F. Mainardi, Fractional diffusion processes: Probability distribution and continuous time random walk, Lecture Notes Phys, 621 (2003), 148-166. doi: 10.1007/3-540-44832-2_8. Google Scholar

[19]

E. Hanert, On the numerical solution of space time fractional diffusion models, Comput. Fluids, 46 (2011), 33-39. doi: 10.1016/j.compfluid.2010.08.010. Google Scholar

[20]

R. Hilfer, Application of Fractional Calculus in Physics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/9789812817747. Google Scholar

[21] J. Hofbauer and K. Sigmund, Evolutionary Games and Replicator Dynamics, Cambridge University Press, Cambridge, 1998. doi: 10.1017/CBO9781139173179.
[22]

Y. KhanK. SayevandM. Fardi and M. Ghasemi, A novel computing multi-parametric homotopy approach for system of linear and nonlinear Fredholm integral equations, Applied Mathematics and Computation, 249 (2014), 229-236. doi: 10.1016/j.amc.2014.10.070. Google Scholar

[23]

N. L. Komarova, Replicator-mutator equation, universality property and population dynamics of learning, Journal of Theoretical Biology, 230 (2004), 227-239. doi: 10.1016/j.jtbi.2004.05.004. Google Scholar

[24]

N. L. KomarovaP. Niyogi and M. A. Nowak, Evolutionary dynamics of grammar acquisition, J. Theor. Biol., 209 (2001), 43-59. doi: 10.1006/jtbi.2000.2240. Google Scholar

[25]

C. P. Li and C. X. Tao, On the fractional Adams method, Computers and Mathematics with Applications, 58 (2009), 1573-1588. doi: 10.1016/j.camwa.2009.07.050. Google Scholar

[26]

R. LinF. LiuV. Anh and I. Turner, Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation, Appl. Math. Comput., 212 (2009), 435-445. doi: 10.1016/j.amc.2009.02.047. Google Scholar

[27]

C. F. Lorenzo and T. T. Hartley, Variable order and distributed order fractional operators, Nonlinear Dyn., 29 (2002), 57-98. doi: 10.1023/A:1016586905654. Google Scholar

[28]

D. Matignon, Stability results for fractional differential equations with applications to control processing, in: Computational Eng. in Sys. Appl., 2 (1996), 963.Google Scholar

[29]

M. M. Meerschaert and C. Tadjeran, Finite difference approximations for fractional advection dispersion equations, J. Comput. Appl. Math., 172 (2004), 65-77. doi: 10.1016/j.cam.2004.01.033. Google Scholar

[30]

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993. Google Scholar

[31]

W. G. Mitchener and M. A. Nowak, Chaos and language, Proceedings of the Royal Society B: Biological Sciences, 271 (2004), 701. doi: 10.1098/rspb.2003.2643. Google Scholar

[32]

M. Nowak and K. Sigmund, Chaos and the evolution of cooperation, Proceedings of the National Academy of Sciences, 90 (1993), 5091-5094. doi: 10.1073/pnas.90.11.5091. Google Scholar

[33]

D. Pais and N. E. Leonard, Limit cycles in replicator-mutator network dynamics, in 50th IEEE Conference on Decision and Control, 2011, 3922–3927. doi: 10.1109/CDC.2011.6160995. Google Scholar

[34]

I. PodlubnyA. ChechkinT. SkovranekY. Q. Chen and B. M. Vinagre Jara, Matrix approach to discrete fractional calculus II: Partial fractional differential equations, J. Comput. Phys., 228 (2009), 3137-3153. doi: 10.1016/j.jcp.2009.01.014. Google Scholar

[35] I. Podlubny, Fractional Differential Equations, Academic Press, California, USA, 1999.
[36]

A. D. Polyanin and V. F. Zaitsev, Handbook of Nonlinear Partial Differential Equations, Chapman & Hall/CRC, Boca Raton, FL, 2004. Google Scholar

[37]

L. E. S. Ramirez and C. F. M. Coimbra, On the selection and meaning of variable order operators for dynamic modelling, Int. J. Differ. Equ., 2010 (2010), 846107, 16 pp. doi: 10.1155/2010/846107. Google Scholar

[38]

B. Ross and S. G. Samko, Fractional integration operator of a variable order in the Holder spaces $H_(x),$, Int. J. Math. Math. Sci., 18 (1995), 777-788. doi: 10.1155/S0161171295001001. Google Scholar

[39]

Z. ShahT. GulS. IslamM. A. KhanE. BonyahF. Hussain and M. Ullah, Three dimensional third grade nanofluid flow in a rotating system between parallel plates with Brownian motion and thermophoresis effects, Results in Physics, 10 (2018), 36-45. doi: 10.1016/j.rinp.2018.05.020. Google Scholar

[40]

P. F. Stadler and P. Schuster, Mutation in autocatalytic reaction networks-an analysis based on perturbation theory, J. Math. Biol., 30 (1992), 597-631. doi: 10.1007/BF00948894. Google Scholar

[41]

C. TadjeranM. M. Meerschaert and H. P. Scheffler, A second order accurate numerical approximation for the fractional diffusion equation, J. Comput. Phys., 213 (2006), 205-213. doi: 10.1016/j.jcp.2005.08.008. Google Scholar

[42]

S. Umarov and S. Steinberg, Variable order differential equations and diffusion with changing modes, Z. Anal. Anwend., 28 (2009), 431-450. doi: 10.4171/ZAA/1392. Google Scholar

[43]

L. YangF. ZhichaoL. Hong and H. Siriguleng, A mixed finite element method for a time-fractional fourth-order partial differential equation, Appl. Math. Comput., 243 (2014), 703-717. doi: 10.1016/j.amc.2014.06.023. Google Scholar

[44]

K. Yosida, Fonctional Analysis, Sixth Edition, Springer- Verlag, 1980. Google Scholar

[45]

S. B. Yuste and L. Acedo, An explicit finite difference method and a new Von Neumann-type stability analysis for fractional diffusion equations, SIAM J. Numer. Anal., 42 (2005), 1862-1874. doi: 10.1137/030602666. Google Scholar

[46]

Y. Zhang, A finite difference method for fractional partial differential equation, Appl. Math. Comput., 215 (2009), 524-529. doi: 10.1016/j.amc.2009.05.018. Google Scholar

[47]

P. ZhuangF. LiuV. Anh and I. Turner, Numerical methods for the variable-order fractional advection–diffusion equation with a nonlinear source term, SIAM J. Numer. Anal., 47 (2009), 1760-1781. doi: 10.1137/080730597. Google Scholar

Figure 1.  An example of water wave showing particles (in red) being propagated, after each oscillation, from one location to another accros the water. Successive positions held by the same particle are marked by blue dots and appear to be rectilinear
Figure 2.  An example of two-dimensional plot comparison between the analytical solution and analytical solution for the model (8) with $ \sigma(x, t) = 2, $. The figures are performed in the variable position $ x $ for some fixed time $ t: $ $ t_1 = 0, \ t_2 = 0.55, \ t_3 = 0.9. $
Figure 3.  An example of plot illustrating the dependence of traveling waves on the VOD. It shows different shapes, movements and displacements for resulting traveling waves for the model (8)–(10). Traveling waves show amplitudes larger and wavelengths shorter in (a) and (b) compared to (c) and (d). The compression and rarefaction processes also change from one traveling wave to another as the VOD varies
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