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October  2021, 14(10): 3763-3783. doi: 10.3934/dcdss.2021022

A robust computational framework for analyzing fractional dynamical systems

Khosro Sayevand , and  Valeyollah Moradi

Faculty of Mathematical Sciences, Malayer University, Malayer, Iran

* Corresponding author: Khosro Sayevand

Received  February 2020 Revised  April 2020 Published  October 2021 Early access  March 2021

This study outlines a modified implicit finite difference method for approximating the local stable manifold near a hyperbolic equilibrium point for a nonlinear systems of fractional differential equations. The fractional derivative is described in the Caputo sense of order $ \alpha\; (0<\alpha \le1) $ which is approximated based on the modified trapezoidal quadrature rule of order $ O(\triangle t ^{2-\alpha}) $. The solution existence, uniqueness and stability of the proposed method is discussed. Three numerical examples are presented and comparisons are made to confirm the reliability and effectiveness of the proposed method.

Keywords: Caputo fractional derivatives, finite difference method, fractional dynamical systems, Hartman-Grobman theorem, stable manifold.
Mathematics Subject Classification: Primary: 41A58; 39A10 Secondary: 34K28; 41A10.
Citation: Khosro Sayevand, Valeyollah Moradi. A robust computational framework for analyzing fractional dynamical systems. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3763-3783. doi: 10.3934/dcdss.2021022
References:
[1]

M. M. Alsuyuti, E. Z. Doha, S. S. Ezz-Eldien, B. I. Bayoumi and D. Baleanu, Modified Galerkin algorithm for solving multitype fractional differential equations, Math. Methods Appl. Sci., 42 (2019), 1389-1412.  doi: 10.1002/mma.5431.

[2]

D. Baleanu, R. Darzi and B. Agheli, Existence results for Langevin equation involving Atangana-Baleanu fractional operators, Mathematics, 8 (2020), 408. doi: 10.3390/math8030408.

[3]

D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Fractional Calculus Models and Numerical Models (Series on Complexity, Nonlinearity and Chaos), Word Scientific, 2012. doi: 10.1142/9789814355216.

[4]

S. Bhatter, A. Mathur, D. Kumar and J. Singh, A new analysis of fractional Drinfeld-Sokolov-Wilson model with exponential memory, Physica A., 537 (2020), 122578, 13 pp. doi: 10.1016/j.physa.2019.122578.

[5]

A. Bueno-Orovio, D. Kay and K. Burrage, Fourier spectral methods for fractional in space reaction-diffusion equations, BIT Numer. Math., 54 (2014), 937-954.  doi: 10.1007/s10543-014-0484-2.

[6]

Y. Chen, X. Han and L. Liu, Numerical solution for a class of linear system of fractional differential equations by the haar wavelet method and the convergence analysis, Comput. Model. Eng. Sci., 97 (2014), 391-405. 

[7]

M. Dehghan and M. Safarpoor, Application of the dual reciprocity boundary integral equation approach to solve fourth-order time-fractional partial differential equations, Int. J. Comput. Math., 95 (2018), 2066-2081.  doi: 10.1080/00207160.2017.1365141.

[8]

A. Deshpande and V. Daftardar-Gejji, Local stable manifold theorem for fractional systems, Nonlinear Dynam., 83 (2016), 2435-2452.  doi: 10.1007/s11071-015-2492-4.

[9]

V. Daftardar-Gejji and A. Babakhani, Analysis of a system of fractional differential equations, J. Math. Anal. Appl., 293 (2004), 511-522.  doi: 10.1016/j.jmaa.2004.01.013.

[10]

H. Delavari, D. Baleanu and J. Sadati, Stability analysis of Caputo fractional order nonlinear systems revisited, Nonlinear Dyna., 67 (2012), 2433-2439.  doi: 10.1007/s11071-011-0157-5.

[11]

S. Esmaeili and R. Garrappa, A pseudo-spectral scheme for the approximate solution of a time-fractional diffusion equation, Int. J. Comput. Math., 92 (2015), 980-994.  doi: 10.1080/00207160.2014.915962.

[12]

R. M. Ganji, H. Jafari and D. Baleanu, A new approach for solving multi variable orders differential equations with Mittag-Leffler kernel, Chaos Soltion Fract., 130 (2020), 109405, 5 pp. doi: 10.1016/j.chaos.2019.109405.

[13]

M. M. Ghalib, A. A. Zafar, M. B. Riaz, Z. Hammouch and K. Shabbir, Analytical approach for the steady MHD conjugate viscous fluid flow in a porous medium with nonsingular fractional derivative, Physica A., 554 (2020), 123941, 15 pp. doi: 10.1016/j.physa.2019.123941.

[14]

M. M. Ghalib, A. A. Zafar, Z. Hammouch, M. B. Riaz and K. Shabbir, Analytical results on the unsteady rotational flow of fractional-order non-Newtonian fluids with shear stress on the boundary, Disret Contin. Dyn. S., 13 (2020), 683-693.  doi: 10.3934/dcdss.2020037.

[15]

R. M. Ganji, H. Jafari and D. Baleanu, A new approach for solving multi variable orders differential equations with Mittag-Leffler kernel, Chaos Soltion Fract., 130 (2020), 109405, 4 pp. doi: 10.1016/j.chaos.2019.109405.

[16]

V. R. Hosseini, W. Chen and Z. Avazzadeh, Numerical solution of fractional telegraph equation by using radial basis functions, Eng. Anal. Boundary Elements, 38 (2014), 31-39.  doi: 10.1016/j.enganabound.2013.10.009.

[17]

M. H. Heydari, Z. Avazzadeh, Y. Yang and C. A. Cattani, A cardinal method to solve coupled nonlinear variable-order time fractional sine-Gordon equations, Comput. Appl.Math., 39 (2020), 2-23.  doi: 10.1007/s40314-019-0936-z.

[18]

P. Hartman, Ordinary Differential Equations, John Wiley and Sons, New York, 1964.

[19]

D. Ingman and J. Suzdalnitsky, Control of damping oscillations by fractional differential operator with time-dependent order, Comput. Methods Appl. Mech. Eng., 193 (2004), 5585-5595.  doi: 10.1016/j.cma.2004.06.029.

[20]

A. Jhinga and V. Daftardar-Gejji, A new numerical method for solving fractional delay differential equations, Comput. Appl. Math., 38 (2019), 166-184.  doi: 10.1007/s40314-019-0951-0.

[21]

M. M. Khader, A. Shloof and H. Ali, On the numerical simulation and convergence study for system of non-linear fractional dynamical model of marriage, New Trends Math. Scie., 5 (2017), 130-141.  doi: 10.20852/ntmsci.2017.223.

[22]

D. Kumar, J. Singh and D. Baleanu, On the analysis of vibration equation involving a fractional derivative with Mittag-Leffler law, Math. Methods Appl. Sci., 43 (2019), 443-457.  doi: 10.1002/mma.5903.

[23]

S. Kazem and M. Dehghan, Semi-analytical solution for time-fractional diffusion equation based on finite difference method of lines (MOL), Eng. Comput., 35 (2019), 229-241.  doi: 10.1007/s00366-018-0595-5.

[24]

M. H. Kim, G. C. Ri and O. Hyong-Chol, Operational method for solving multi-term fractional differential equations with the generalized fractional derivatives, Fract. Calculus Appl. Anal., 17 (2014), 79-95.  doi: 10.2478/s13540-014-0156-6.

[25]

D. Kumar, R. P. Agarwal and J. Singh, A modified numerical scheme and conver- gence analysis for fractional model of lienard's equation, J. Comput. Appl. Math., 339 (2018), 405-413.  doi: 10.1016/j.cam.2017.03.011.

[26]

D. Kumar, F. Tchier, J. Singh and D. Baleanu, An efficient computational technique for fractal vehicular traffic flow, Entropy, 20 (2018), 259. doi: 10.3390/e20040259.

[27]

A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.

[28]

C. Li and F. Zeng, Finite difference methods for fractional differential equations, Int. J. Bifurcat Chaos, 22 (2012), 1230014, 28 pp. doi: 10.1142/S0218127412300145.

[29]

Y. Li, Y. Q. Chen and I. Podlubny, Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability, Comput. Math. Appl., 59 (2010), 1810-1821.  doi: 10.1016/j.camwa.2009.08.019.

[30]

F. Liu, P. Zhuang, I. Turner, K. Burrage and V. Anh, A new fractional finite volume method for solving the fractional diffusion equation, Appl. Math. Model., 38 (2014), 3871-3878.  doi: 10.1016/j.apm.2013.10.007.

[31]

C. Li and Y. Ma, Fractional dynamical system and its linearization theorem, Nonlinear Dynam., 71 (2013), 621-633.  doi: 10.1007/s11071-012-0601-1.

[32]

K. S. Miller and B. Ross, An Itroduction to the Fractional Calculus and Fractional Differential Equations, , Johan Willey and Sons, Inc. New York, 1993.

[33]

M. Malik and V. Kumar, Existence, stability and controllability results of coupled fractional dynamical system on time scales, Bull. Malays. Math. Sci. Soc., 43 (2020), 3369-3394.  doi: 10.1007/s40840-019-00871-0.

[34]

K. M. Owolabi and Z. Hammouch, Spatiotemporal patterns in the Belousov-Zhabotinskii reaction systems with Atangana-Baleanu fractional order derivative, Physica A., 523 (2019) 1072–1090. doi: 10.1016/j.physa.2019.04.017.

[35] K. B. Oldham and J. Spanier, The Fractional Calculus, Theory and Applications of Differentiation and Integration to Arbitrary Order, Academic Press, New York, 1974. 
[36] I. Podlubny, Fractional Differential Equations Calculus, Academic Press, New York, 1999. 
[37]

E. Pindza and K. M. Owolabi, Fourier spectral method for higher order space fractional reaction-diffusion equations, Commun. Nonlinear Sci. Numer. Simulat., 40 (2016), 112-128.  doi: 10.1016/j.cnsns.2016.04.020.

[38]

J. Singh, D. Kumar, D. Baleanu and S. Rathore, On the local fractional wave equation in fractal strings, Math. Methods Appl. Sci., 42 (2019), 1588-1595.  doi: 10.1002/mma.5458.

[39]

K. Sayevand and K. Pichaghchi, Successive approximation: A survey on stable manifold of fractional differential systems, Fract. Calc. Appl. Anal., 18 (2015), 621-641.  doi: 10.1515/fca-2015-0038.

[40]

K. Sayevand and M. Rostami, Fractional optimal control problems: optimality conditions and numerical solution, IMA J. Math. Control Info., 35 (2018), 123-148.  doi: 10.1093/imamci/dnw041.

[41]

K. Sayevand and M. Rostami, General fractional variational problem depending on indefinite integrals, Numer. Algor., 72 (2016), 959-987.  doi: 10.1007/s11075-015-0076-5.

[42]

J. Stoer, R. Bulirsch and R. Bartels, Introduction to Numerical Analysis, Springer, 2002. doi: 10.1007/978-0-387-21738-3.

[43]

K. Sayevand, J. Tenreiro Machado and V. Moradi, A new non-standard finite difference method for analysing the fractional Navier-Stokes equations, Comput. Math. Appl., 78 (2019), 1681-1694.  doi: 10.1016/j.camwa.2018.12.016.

[44]

J. J. E. Slotine and W. Li, Applied Nonlinear Control, Prentice Hall, Englewood Cliffs, New Jersey, 1991.

[45]

J. Singh, D. Kumar, D. Baleanu and S. Rathore, An efficient numerical algorithm for the fractional Drinfeld-Sokolov-Wilson equation, Appl. Math. Comput., 335 (2018), 12-24.  doi: 10.1016/j.amc.2018.04.025.

[46]

V. E. Tarasov, Fractional Dynamics: Aapplications of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer Science, Business Media, 2010. doi: 10.1007/978-3-642-14003-7.

[47]

S. Ucar, E. Ucar, N. Ozdemir and Z. Hammouch, Mathematical analysis and numerical simulation for a smoking model with Atangana-Baleanu derivative, Chaos Solition Fract., 118 (2019), 300-306.  doi: 10.1016/j.chaos.2018.12.003.

[48]

V. V. Uchaikin, Fractional Derivatives for Physicists and Engineers, Vol. 2, Springer, Berlin, 2003. doi: 10.1007/978-3-642-33911-0.

[49]

J. Yu, C. Hu and H. Jiang, $\alpha$-stability and $\alpha$-synchronization for fractional-order neural networks, Neural Netw., 35 (2012), 82-87. 

[50]

M. A. Zaky, A legendre spectral quadrature tau method for the multi-term time-fractional diffusion equations, Comput. Appl. Math., 37 (2018), 3525-3538.  doi: 10.1007/s40314-017-0530-1.

[51]

X. Zhang, C. Zhu and Z. Wu, Solvability for a coupled system of fractional differential equations with impulses at resonance, Bound. Value Probl., 2013 (2013), 80-103.  doi: 10.1186/1687-2770-2013-80.

show all references

References:
[1]

M. M. Alsuyuti, E. Z. Doha, S. S. Ezz-Eldien, B. I. Bayoumi and D. Baleanu, Modified Galerkin algorithm for solving multitype fractional differential equations, Math. Methods Appl. Sci., 42 (2019), 1389-1412.  doi: 10.1002/mma.5431.

[2]

D. Baleanu, R. Darzi and B. Agheli, Existence results for Langevin equation involving Atangana-Baleanu fractional operators, Mathematics, 8 (2020), 408. doi: 10.3390/math8030408.

[3]

D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Fractional Calculus Models and Numerical Models (Series on Complexity, Nonlinearity and Chaos), Word Scientific, 2012. doi: 10.1142/9789814355216.

[4]

S. Bhatter, A. Mathur, D. Kumar and J. Singh, A new analysis of fractional Drinfeld-Sokolov-Wilson model with exponential memory, Physica A., 537 (2020), 122578, 13 pp. doi: 10.1016/j.physa.2019.122578.

[5]

A. Bueno-Orovio, D. Kay and K. Burrage, Fourier spectral methods for fractional in space reaction-diffusion equations, BIT Numer. Math., 54 (2014), 937-954.  doi: 10.1007/s10543-014-0484-2.

[6]

Y. Chen, X. Han and L. Liu, Numerical solution for a class of linear system of fractional differential equations by the haar wavelet method and the convergence analysis, Comput. Model. Eng. Sci., 97 (2014), 391-405. 

[7]

M. Dehghan and M. Safarpoor, Application of the dual reciprocity boundary integral equation approach to solve fourth-order time-fractional partial differential equations, Int. J. Comput. Math., 95 (2018), 2066-2081.  doi: 10.1080/00207160.2017.1365141.

[8]

A. Deshpande and V. Daftardar-Gejji, Local stable manifold theorem for fractional systems, Nonlinear Dynam., 83 (2016), 2435-2452.  doi: 10.1007/s11071-015-2492-4.

[9]

V. Daftardar-Gejji and A. Babakhani, Analysis of a system of fractional differential equations, J. Math. Anal. Appl., 293 (2004), 511-522.  doi: 10.1016/j.jmaa.2004.01.013.

[10]

H. Delavari, D. Baleanu and J. Sadati, Stability analysis of Caputo fractional order nonlinear systems revisited, Nonlinear Dyna., 67 (2012), 2433-2439.  doi: 10.1007/s11071-011-0157-5.

[11]

S. Esmaeili and R. Garrappa, A pseudo-spectral scheme for the approximate solution of a time-fractional diffusion equation, Int. J. Comput. Math., 92 (2015), 980-994.  doi: 10.1080/00207160.2014.915962.

[12]

R. M. Ganji, H. Jafari and D. Baleanu, A new approach for solving multi variable orders differential equations with Mittag-Leffler kernel, Chaos Soltion Fract., 130 (2020), 109405, 5 pp. doi: 10.1016/j.chaos.2019.109405.

[13]

M. M. Ghalib, A. A. Zafar, M. B. Riaz, Z. Hammouch and K. Shabbir, Analytical approach for the steady MHD conjugate viscous fluid flow in a porous medium with nonsingular fractional derivative, Physica A., 554 (2020), 123941, 15 pp. doi: 10.1016/j.physa.2019.123941.

[14]

M. M. Ghalib, A. A. Zafar, Z. Hammouch, M. B. Riaz and K. Shabbir, Analytical results on the unsteady rotational flow of fractional-order non-Newtonian fluids with shear stress on the boundary, Disret Contin. Dyn. S., 13 (2020), 683-693.  doi: 10.3934/dcdss.2020037.

[15]

R. M. Ganji, H. Jafari and D. Baleanu, A new approach for solving multi variable orders differential equations with Mittag-Leffler kernel, Chaos Soltion Fract., 130 (2020), 109405, 4 pp. doi: 10.1016/j.chaos.2019.109405.

[16]

V. R. Hosseini, W. Chen and Z. Avazzadeh, Numerical solution of fractional telegraph equation by using radial basis functions, Eng. Anal. Boundary Elements, 38 (2014), 31-39.  doi: 10.1016/j.enganabound.2013.10.009.

[17]

M. H. Heydari, Z. Avazzadeh, Y. Yang and C. A. Cattani, A cardinal method to solve coupled nonlinear variable-order time fractional sine-Gordon equations, Comput. Appl.Math., 39 (2020), 2-23.  doi: 10.1007/s40314-019-0936-z.

[18]

P. Hartman, Ordinary Differential Equations, John Wiley and Sons, New York, 1964.

[19]

D. Ingman and J. Suzdalnitsky, Control of damping oscillations by fractional differential operator with time-dependent order, Comput. Methods Appl. Mech. Eng., 193 (2004), 5585-5595.  doi: 10.1016/j.cma.2004.06.029.

[20]

A. Jhinga and V. Daftardar-Gejji, A new numerical method for solving fractional delay differential equations, Comput. Appl. Math., 38 (2019), 166-184.  doi: 10.1007/s40314-019-0951-0.

[21]

M. M. Khader, A. Shloof and H. Ali, On the numerical simulation and convergence study for system of non-linear fractional dynamical model of marriage, New Trends Math. Scie., 5 (2017), 130-141.  doi: 10.20852/ntmsci.2017.223.

[22]

D. Kumar, J. Singh and D. Baleanu, On the analysis of vibration equation involving a fractional derivative with Mittag-Leffler law, Math. Methods Appl. Sci., 43 (2019), 443-457.  doi: 10.1002/mma.5903.

[23]

S. Kazem and M. Dehghan, Semi-analytical solution for time-fractional diffusion equation based on finite difference method of lines (MOL), Eng. Comput., 35 (2019), 229-241.  doi: 10.1007/s00366-018-0595-5.

[24]

M. H. Kim, G. C. Ri and O. Hyong-Chol, Operational method for solving multi-term fractional differential equations with the generalized fractional derivatives, Fract. Calculus Appl. Anal., 17 (2014), 79-95.  doi: 10.2478/s13540-014-0156-6.

[25]

D. Kumar, R. P. Agarwal and J. Singh, A modified numerical scheme and conver- gence analysis for fractional model of lienard's equation, J. Comput. Appl. Math., 339 (2018), 405-413.  doi: 10.1016/j.cam.2017.03.011.

[26]

D. Kumar, F. Tchier, J. Singh and D. Baleanu, An efficient computational technique for fractal vehicular traffic flow, Entropy, 20 (2018), 259. doi: 10.3390/e20040259.

[27]

A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.

[28]

C. Li and F. Zeng, Finite difference methods for fractional differential equations, Int. J. Bifurcat Chaos, 22 (2012), 1230014, 28 pp. doi: 10.1142/S0218127412300145.

[29]

Y. Li, Y. Q. Chen and I. Podlubny, Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability, Comput. Math. Appl., 59 (2010), 1810-1821.  doi: 10.1016/j.camwa.2009.08.019.

[30]

F. Liu, P. Zhuang, I. Turner, K. Burrage and V. Anh, A new fractional finite volume method for solving the fractional diffusion equation, Appl. Math. Model., 38 (2014), 3871-3878.  doi: 10.1016/j.apm.2013.10.007.

[31]

C. Li and Y. Ma, Fractional dynamical system and its linearization theorem, Nonlinear Dynam., 71 (2013), 621-633.  doi: 10.1007/s11071-012-0601-1.

[32]

K. S. Miller and B. Ross, An Itroduction to the Fractional Calculus and Fractional Differential Equations, , Johan Willey and Sons, Inc. New York, 1993.

[33]

M. Malik and V. Kumar, Existence, stability and controllability results of coupled fractional dynamical system on time scales, Bull. Malays. Math. Sci. Soc., 43 (2020), 3369-3394.  doi: 10.1007/s40840-019-00871-0.

[34]

K. M. Owolabi and Z. Hammouch, Spatiotemporal patterns in the Belousov-Zhabotinskii reaction systems with Atangana-Baleanu fractional order derivative, Physica A., 523 (2019) 1072–1090. doi: 10.1016/j.physa.2019.04.017.

[35] K. B. Oldham and J. Spanier, The Fractional Calculus, Theory and Applications of Differentiation and Integration to Arbitrary Order, Academic Press, New York, 1974. 
[36] I. Podlubny, Fractional Differential Equations Calculus, Academic Press, New York, 1999. 
[37]

E. Pindza and K. M. Owolabi, Fourier spectral method for higher order space fractional reaction-diffusion equations, Commun. Nonlinear Sci. Numer. Simulat., 40 (2016), 112-128.  doi: 10.1016/j.cnsns.2016.04.020.

[38]

J. Singh, D. Kumar, D. Baleanu and S. Rathore, On the local fractional wave equation in fractal strings, Math. Methods Appl. Sci., 42 (2019), 1588-1595.  doi: 10.1002/mma.5458.

[39]

K. Sayevand and K. Pichaghchi, Successive approximation: A survey on stable manifold of fractional differential systems, Fract. Calc. Appl. Anal., 18 (2015), 621-641.  doi: 10.1515/fca-2015-0038.

[40]

K. Sayevand and M. Rostami, Fractional optimal control problems: optimality conditions and numerical solution, IMA J. Math. Control Info., 35 (2018), 123-148.  doi: 10.1093/imamci/dnw041.

[41]

K. Sayevand and M. Rostami, General fractional variational problem depending on indefinite integrals, Numer. Algor., 72 (2016), 959-987.  doi: 10.1007/s11075-015-0076-5.

[42]

J. Stoer, R. Bulirsch and R. Bartels, Introduction to Numerical Analysis, Springer, 2002. doi: 10.1007/978-0-387-21738-3.

[43]

K. Sayevand, J. Tenreiro Machado and V. Moradi, A new non-standard finite difference method for analysing the fractional Navier-Stokes equations, Comput. Math. Appl., 78 (2019), 1681-1694.  doi: 10.1016/j.camwa.2018.12.016.

[44]

J. J. E. Slotine and W. Li, Applied Nonlinear Control, Prentice Hall, Englewood Cliffs, New Jersey, 1991.

[45]

J. Singh, D. Kumar, D. Baleanu and S. Rathore, An efficient numerical algorithm for the fractional Drinfeld-Sokolov-Wilson equation, Appl. Math. Comput., 335 (2018), 12-24.  doi: 10.1016/j.amc.2018.04.025.

[46]

V. E. Tarasov, Fractional Dynamics: Aapplications of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer Science, Business Media, 2010. doi: 10.1007/978-3-642-14003-7.

[47]

S. Ucar, E. Ucar, N. Ozdemir and Z. Hammouch, Mathematical analysis and numerical simulation for a smoking model with Atangana-Baleanu derivative, Chaos Solition Fract., 118 (2019), 300-306.  doi: 10.1016/j.chaos.2018.12.003.

[48]

V. V. Uchaikin, Fractional Derivatives for Physicists and Engineers, Vol. 2, Springer, Berlin, 2003. doi: 10.1007/978-3-642-33911-0.

[49]

J. Yu, C. Hu and H. Jiang, $\alpha$-stability and $\alpha$-synchronization for fractional-order neural networks, Neural Netw., 35 (2012), 82-87. 

[50]

M. A. Zaky, A legendre spectral quadrature tau method for the multi-term time-fractional diffusion equations, Comput. Appl. Math., 37 (2018), 3525-3538.  doi: 10.1007/s40314-017-0530-1.

[51]

X. Zhang, C. Zhu and Z. Wu, Solvability for a coupled system of fractional differential equations with impulses at resonance, Bound. Value Probl., 2013 (2013), 80-103.  doi: 10.1186/1687-2770-2013-80.

Figure 1.  Comparison of $ x_{1} $ for different values of $ t $ in Example 1 and corresponding to results obtained by Ref. [39] for $ \alpha = 0.3 $
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Figure 2.  Comparison of $ x_{2} $ for different values of $ t $ in Example 1 and corresponding to results obtained by Ref. [39] for $ \alpha=0.3 $
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Figure 3.  Comparison of $ x_{1} $ for different values of $ t $ in Example 1 and corresponding to results obtained by Ref. [39] for $ \alpha=0.7 $
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Figure 4.  Comparison of $ x_{2} $ for different values of $ t $ in Example 1 and corresponding to results obtained by Ref. [39] for $ \alpha=0.7 $
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Figure 5.  Comparison of $ x_{1} $ for different values of $ t $ in Example 2 and corresponding to results obtained by Ref. [39] for $ \alpha=0.2 $
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Figure 6.  Comparison of $ x_{2} $ for different values of $ t $ in Example 2 and corresponding to results obtained by Ref. [39] for $ \alpha=0.2 $
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Figure 7.  Comparison of $ x_{3} $ for different values of $ t $ in Example 2 and corresponding to results obtained by Ref. [39] for $ \alpha=0.2 $
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Figure 8.  Comparison of $ x_{1} $ for different values of $ t $ in Example 2 and corresponding to results obtained by Ref. [39] for $ \alpha=0.9 $
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Figure 9.  Comparison of $ x_{2} $ for different values of $ t $ in Example 2 and corresponding to results obtained by Ref. [39] for $ \alpha=0.9 $
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Figure 10.  Comparison of $ x_{3} $ for different values of $ t $ in Example 2 and corresponding to results obtained by Ref. [39] for $ \alpha = 0.9 $
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Figure 11.  Comparison of $ x_{1} $ for different values of $ t $ in Example 3 and corresponding to results obtained by Ref. [8] for $ \alpha = 0.7 $ and $ h = 0.1 $
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Figure 12.  Comparison of $ x_{2} $ for different values of $ t $ in Example 3 and corresponding to results obtained by Ref. [8] for $ \alpha=0.7 $ and $ h=0.1 $
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Figure 13.  Comparison of $ x_{3} $ for different values of $ t $ in Example 3 and corresponding to results obtained by Ref. [8] for $ \alpha = 0.7 $ and $ h = 0.1 $
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Figure 14.  Projection $ \sigma_3 $ versus $ \sigma_1 $ for different values of $ \alpha $ in Example 3
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Figure 15.  Projection $ \sigma_2 $ versus $ \sigma_1 $ for different values of $ \alpha $ in Example 3
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Figure 16.  Projection local stable manifold for different values of $ \alpha $ in Example 3
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Table 1.  Numerical results of Example 1 for different values of $ t $ and $ \alpha = 0.3 $ and $ h = 0.1 $
$ \alpha=0.3, a_{1}=0.015 $
$ x_{1} $ $ x_{2} $
$ t $ Ref. [39] $ OM $ $ AE $ Ref. [39] $ OM $ $ AE $
$ 0.3 $ 0.00824 0.00835 0.00010 -0.00002 -0.00003 0.00000
$ 0.6 $ 0.00744 0.00753 0.00009 -0.00002 -0.00005 0.00003
$ 0.9 $ 0.00697 0.00705 0.00008 -0.00002 -0.00008 0.00006
$ 1.2 $ 0.00663 0.00676 0.00012 -0.00002 -0.00012 0.00009
$ 1.5 $ 0.00638 0.00655 0.00017 -0.00001 -0.00017 0.00015
$ 1.8 $ 0.00617 0.00637 0.00020 -0.00001 -0.0.00024 0.00022
$ 2 $ 0.00605 0.00627 0.00021 -0.00001 -0.00030 -0.00028
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$ \alpha=0.3, a_{1}=0.015 $
$ x_{1} $ $ x_{2} $
$ t $ Ref. [39] $ OM $ $ AE $ Ref. [39] $ OM $ $ AE $
$ 0.3 $ 0.00824 0.00835 0.00010 -0.00002 -0.00003 0.00000
$ 0.6 $ 0.00744 0.00753 0.00009 -0.00002 -0.00005 0.00003
$ 0.9 $ 0.00697 0.00705 0.00008 -0.00002 -0.00008 0.00006
$ 1.2 $ 0.00663 0.00676 0.00012 -0.00002 -0.00012 0.00009
$ 1.5 $ 0.00638 0.00655 0.00017 -0.00001 -0.00017 0.00015
$ 1.8 $ 0.00617 0.00637 0.00020 -0.00001 -0.0.00024 0.00022
$ 2 $ 0.00605 0.00627 0.00021 -0.00001 -0.00030 -0.00028
Table 2.  Numerical results of Example 1 for different values of $ t $ and $ \alpha=0.7 $ and $ h=0.1 $
$ \alpha=0.7 , a_{1}=0.05 $
$ x_{1} $ $ x_{2} $
$ t $ Ref. [39] $ OM $ $ AE $ Ref. [39] $ OM $ $ AE $
$ 0.3 $ 0.03226 0.03276 0.00049 -0.00037 -0.00029 0.00007
$ 0.6 $ 0.02539 0.02596 0.00056 -0.00025 -0.00014 0.00011
$ 0.9 $ 0.02109 0.02170 0.00060 -0.00019 -0.00007 0.00011
$ 1.2 $ 0.01808 0.01884 0.00076 -0.00015 -0.00004 0.00011
$ 1.5 $ 0.01584 0.01683 0.00098 -0.00013 -0.00003 0.00010
$ 1.8 $ 0.01411 0.01523 0.00111 -0.00011 -0.00002 0.00008
$ 2 $ 0.01315 0.01432 0.00116 -0.00010 -0.00002 0.00007
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$ \alpha=0.7 , a_{1}=0.05 $
$ x_{1} $ $ x_{2} $
$ t $ Ref. [39] $ OM $ $ AE $ Ref. [39] $ OM $ $ AE $
$ 0.3 $ 0.03226 0.03276 0.00049 -0.00037 -0.00029 0.00007
$ 0.6 $ 0.02539 0.02596 0.00056 -0.00025 -0.00014 0.00011
$ 0.9 $ 0.02109 0.02170 0.00060 -0.00019 -0.00007 0.00011
$ 1.2 $ 0.01808 0.01884 0.00076 -0.00015 -0.00004 0.00011
$ 1.5 $ 0.01584 0.01683 0.00098 -0.00013 -0.00003 0.00010
$ 1.8 $ 0.01411 0.01523 0.00111 -0.00011 -0.00002 0.00008
$ 2 $ 0.01315 0.01432 0.00116 -0.00010 -0.00002 0.00007
Table 3.  Numerical results of Example 2 for different values of $ t $ and $ \alpha=0.2 $ and $ h=0.1 $
$ \alpha=0.2 , a_{1}=0.02,a_{2}=0.03 $
$ x_{1} $ $ x_{2} $ $ x_{3} $
$ t $ Ref. [39] $ OM $ $ AE $ Ref. [39] $ OM $ $ AE $ Ref. [39] $ OM $ $ AE $
$ 0.3 $ 0.01064 0.01075 0.0011 0.01092 0.01101 0.0009 0.00000 0.00000 0.00000
$ 0.6 $ 0.00994 0.01003 0.00009 0.00997 0.00998 0.00001 0.00000 0.00000 0.00000
$ 0.9 $ 0.00952 0.00960 0.00007 0.00943 0.00941 0.00002 0.00000 0.00000 0.00000
$ 1.2 $ 0.00923 0.00935 0.00012 0.00907 0.00907 0.00000 0.00000 0.00000 0.00000
$ 1.5 $ 0.00901 0.00917 0.00016 0.00879 0.00880 0.00001 0.00000 0.00000 0.00000
$ 1.8 $ 0.00882 0.00901 0.00018 0.00856 0.00859 0.00002 0.00000 0.00001 0.00001
$ 2 $ 0.00872 0.00891 0.00019 0.00843 0.00846 0.00002 0.00000 -0.00001 0.00001
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$ \alpha=0.2 , a_{1}=0.02,a_{2}=0.03 $
$ x_{1} $ $ x_{2} $ $ x_{3} $
$ t $ Ref. [39] $ OM $ $ AE $ Ref. [39] $ OM $ $ AE $ Ref. [39] $ OM $ $ AE $
$ 0.3 $ 0.01064 0.01075 0.0011 0.01092 0.01101 0.0009 0.00000 0.00000 0.00000
$ 0.6 $ 0.00994 0.01003 0.00009 0.00997 0.00998 0.00001 0.00000 0.00000 0.00000
$ 0.9 $ 0.00952 0.00960 0.00007 0.00943 0.00941 0.00002 0.00000 0.00000 0.00000
$ 1.2 $ 0.00923 0.00935 0.00012 0.00907 0.00907 0.00000 0.00000 0.00000 0.00000
$ 1.5 $ 0.00901 0.00917 0.00016 0.00879 0.00880 0.00001 0.00000 0.00000 0.00000
$ 1.8 $ 0.00882 0.00901 0.00018 0.00856 0.00859 0.00002 0.00000 0.00001 0.00001
$ 2 $ 0.00872 0.00891 0.00019 0.00843 0.00846 0.00002 0.00000 -0.00001 0.00001
Table 4.  Numerical results of Example 2 for different values of $ t $ and $ \alpha = 0.9 $ and $ h = 0.1 $
$ \alpha=0.9 , a_{1}=0.02,a_{2}=0.01 $
$ x_{1} $ $ x_{2} $ $ x_{3} $
$ t $ Ref. [39] $ OM $ $ AE $ Ref. [39] $ OM $ $ AE $ Ref. [39] $ OM $ $ AE $
$ 0.3 $ 0.01416 0.01434 0.00018 0.00515 0.00536 0.00020 0.00000 0.00000 0.00000
$ 0.6 $ 0.01062 0.01090 0.00028 0.00305 0.00327 0.00021 0.00000 0.00000 0.00000
$ 0.9 $ 0.00817 0.00849 0.00031 0.00194 0.00211 0.00016 0.00000 0.00000 0.00000
$ 1.2 $ 0.00640 0.00675 0.00034 0.00132 0.00144 0.00012 0.00000 0.00000 0.00000
$ 1.5 $ 0.00511 0.00547 0.00036 0.00095 0.00104 0.00008 0.00000 0.00000 0.00000
$ 1.8 $ 0.00413 0.00450 0.00037 0.00072 0.00078 0.00006 0.00000 0.00000 0.00000
$ 2 $ 0.00362 0.00398 0.00036 0.00061 0.00066 0.00004 0.00000 0.00000 0.00000
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$ \alpha=0.9 , a_{1}=0.02,a_{2}=0.01 $
$ x_{1} $ $ x_{2} $ $ x_{3} $
$ t $ Ref. [39] $ OM $ $ AE $ Ref. [39] $ OM $ $ AE $ Ref. [39] $ OM $ $ AE $
$ 0.3 $ 0.01416 0.01434 0.00018 0.00515 0.00536 0.00020 0.00000 0.00000 0.00000
$ 0.6 $ 0.01062 0.01090 0.00028 0.00305 0.00327 0.00021 0.00000 0.00000 0.00000
$ 0.9 $ 0.00817 0.00849 0.00031 0.00194 0.00211 0.00016 0.00000 0.00000 0.00000
$ 1.2 $ 0.00640 0.00675 0.00034 0.00132 0.00144 0.00012 0.00000 0.00000 0.00000
$ 1.5 $ 0.00511 0.00547 0.00036 0.00095 0.00104 0.00008 0.00000 0.00000 0.00000
$ 1.8 $ 0.00413 0.00450 0.00037 0.00072 0.00078 0.00006 0.00000 0.00000 0.00000
$ 2 $ 0.00362 0.00398 0.00036 0.00061 0.00066 0.00004 0.00000 0.00000 0.00000
Table 5.  Numerical results of Example 3 for different values of $ t $ and $ \alpha = 0.7 $ and $ h = 0.1 $
$ \alpha=0.7, \sigma_{1}=0.001 $
$ x_{1} $ $ x_{2} $ $ x_{3} $
$ t $ Ref. [8] $ OM $ $ AE $ Ref. [8] $ OM $ $ AE $ Ref. [8] $ OM $ $ AE $
$ 0.3 $ 0.00064 0.00065 0.00000 0.00000 0.00000 0.00000 -0.00020 0.00000 0.00000
$ 0.6 $ 0.00051 0.00052 0.00001 0.00000 0.00000 0.00000 -0.00020 0.00000 0.00000
$ 0.9 $ 0.00042 0.00043 0.00001 0.00000 0.00000 0.00000 -0.00020 0.00000 0.00000
$ 1.2 $ 0.00036 0.00037 0.00001 0.00001 0.00001 0.00000 -0.00020 0.00000 -0.00001
$ 1.5 $ 0.00031 0.00033 0.00002 0.00002 0.00003 0.00001 -0.00021 0.00000 -0.00003
$ 1.8 $ 0.00028 0.00030 0.00002 0.00005 0.00009 0.00003 -0.00022 0.00002 -0.00007
$ 2 $ 0.00026 0.00028 0.00002 0.00009 0.00016 0.00006 -0.00024 -0.00003 -0.00012
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$ \alpha=0.7, \sigma_{1}=0.001 $
$ x_{1} $ $ x_{2} $ $ x_{3} $
$ t $ Ref. [8] $ OM $ $ AE $ Ref. [8] $ OM $ $ AE $ Ref. [8] $ OM $ $ AE $
$ 0.3 $ 0.00064 0.00065 0.00000 0.00000 0.00000 0.00000 -0.00020 0.00000 0.00000
$ 0.6 $ 0.00051 0.00052 0.00001 0.00000 0.00000 0.00000 -0.00020 0.00000 0.00000
$ 0.9 $ 0.00042 0.00043 0.00001 0.00000 0.00000 0.00000 -0.00020 0.00000 0.00000
$ 1.2 $ 0.00036 0.00037 0.00001 0.00001 0.00001 0.00000 -0.00020 0.00000 -0.00001
$ 1.5 $ 0.00031 0.00033 0.00002 0.00002 0.00003 0.00001 -0.00021 0.00000 -0.00003
$ 1.8 $ 0.00028 0.00030 0.00002 0.00005 0.00009 0.00003 -0.00022 0.00002 -0.00007
$ 2 $ 0.00026 0.00028 0.00002 0.00009 0.00016 0.00006 -0.00024 -0.00003 -0.00012
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