A new efficient algorithm based on fifth-kind Chebyshev polynomials for solving multi-term variable-order time-fractional diffusion-wave equation

References

• W.M. Abd-Elhameed and Y.H. Youssri, Spectral solutions for fifth-order boundary value problems using generalized Jacobi operational matrix of derivatives, Int. J. Appl. Comput. Math. 3 (2017), pp. 883–901. doi:https://doi.org/10.1007/s40819-017-0388-3.
   Google Scholar
• W.M. Abd-Elhameed and Y.H. Youssri, Fifth-kind orthonormal Chebyshev polynomial solutions for fractional differential equations, Comput. Appl. Math. 37 (2018), pp. 2897–2921.
   Web of Science ®Google Scholar
• W.M. Abd-Elhameed and Y.H. Youssri, Sixth-kind Chebyshev spectral approach for solving fractional differential equations, Int. J. Nonlin. Sci. Num. 20(2) (2019), pp. 191–203. doi:https://doi.org/10.1515/ijnsns-2018-0118.
   Web of Science ®Google Scholar
• A.H. Bhrawy and M.A. Zaky, Numerical algorithm for the variable-order Caputo fractional functional differential equation, Nonlinear Dyn. 85 (2016), pp. 1815–1823.
   Web of Science ®Google Scholar
• A. Borhanifar and K. Sadri, A generalized operational method for solving integro-partial differential equations based on Jacobi polynomials, Hacet. J. Math. Stat. 45(2) (2016), pp. 311–335.
   Web of Science ®Google Scholar
• C. Chen, H. Liu, X. Zheng, and H. Wang, A two-grid MMOC finite element method for nonlinear variable-order time-fractional mobile/immobile advection-diffusion equations, Comput. Math. Appl.79(9) (2020), pp. 2771–2783.
   Web of Science ®Google Scholar
• Y.M. Chen, Y.Q. Wei, D.Y. Liu, and H. Yu, Numerical solution for a class of nonlinear variable-order fractional differential equations with Legendre wavelets, Appl. Math. Lett. 46 (2015), pp. 83–88.
   Web of Science ®Google Scholar
• C.F.M. Coimbra, Mechanics with variable-order differential operators, Ann. Phys. 12 (2003), pp. 692–703.
   Web of Science ®Google Scholar
• E.H. Doha and Y.H. Youssri, On the connection coefficients and recurrence relations arising from expansions in series of modified generalized Laguerre polynomials: applications on a semi-infinite domain, Nonlinear Eng. 8(1) (2019), pp. 318–327. doi:https://doi.org/https://doi.org/10.1515/nleng-2018-0073.
   Google Scholar
• L. Galue, S.L. Kalla, and B.N. Al-Saqabi, Fractional extensions of the temperature field problems in oil strata, Appl. Math. Comput. 186(1) (2007), pp. 35–44.
   Web of Science ®Google Scholar
• B.Y. Guo and L.L. Wang, Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces, J. Approx. Theory 128 (2004), pp. 1–41.
   Web of Science ®Google Scholar
• M. Hammad, R.M. Hafez, Y.H. Youssri, and E.H. Doha, Exponential Jacobi–Galerkin method and its applications to multidimensional problems in unbounded domains, Appl. Num. Math 157 (2020), pp. 88–109.
   Web of Science ®Google Scholar
• M.H. Heydari, A new direct method based on the Chebyshev cardinal functions for variable-order fractional optimal control problems, J. Frankl. I 355(12) (2018), pp. 4970–4995.
   Web of Science ®Google Scholar
• M.H. Heydari and A. Atangana, An optimization method based on the generalized Lucas polynomials for variable-order space-time fractional mobile-immobile advection-dispersion equation involving derivatives with non-singular kernels, Chaos. Soliton. Fract. 132 (2020), p. 109588. doi:https://doi.org/10.1016/j.chaos.2019.109588.
   Web of Science ®Google Scholar
• M.H. Heydari and Z. Avazzadeh, Legendre wavelets optimization method for variable-order fractional poisson equation, Chaos Solitons Fract. 112 (2018), pp. 180–190.
   Web of Science ®Google Scholar
• M.H. Heydari and Z. Avazzadeh, An operational matrix method for solving variable-order fractional biharmonic equation, Comput. Appl. Math 37 (2018), pp. 4397–4411. 0580-z.
   Web of Science ®Google Scholar
• M.H. Heydari and Z. Avazzadeh, A new wavelet method for variable-order fractional optimal control problems, Asian J. Control 20(5) (2018), pp. 1804–1817.
   Web of Science ®Google Scholar
• M.H. Heydari and Z. Avazzadeh, Chebyshev–Gauss–Lobatto collocation method for variable-order time fractional generalized Hirota–Satsuma coupled KdV system, Eng. Comput-Germany (2020). doi: https://doi.org/10.1007/s00366-020-01125-5
   Web of Science ®Google Scholar
• M.H. Heydari, Z. Avazzadeh, and C. Cattani, Numerical solution of variable-order space–time fractional KdV–Burgers–Kuramoto equation by using discrete Legendre polynomials, Eng. Comput-Germany (2020). doi:https://doi.org/10.1007/s00366-020-01181-x.
   Web of Science ®Google Scholar
• M.H. Heydari, Z. Avazzadeh, and C. Cattani, Discrete Chebyshev polynomials for nonsingular variable-order fractional KdV Burgers' equation, Math. Method. Appl. Sci. 44(6) (2021), pp. 2158–2170.
   Web of Science ®Google Scholar
• M.H. Heydari, Z. Avazzadeh, and M. Farzi Haromi, A wavelet approach for solving multi-term variable-order time fractional diffusion-wave equation, App. Math. Comput. 341 (2019), pp. 215–228.
   Web of Science ®Google Scholar
• R. Hilfer, Applications of Fractional Calculus in Physics, World Sci. Publishing, River Edge, 2000.
   Google Scholar
• M. Hosseininia, M.H. Heydari, M.R. Hooshmandasl, F.M. Maalek Ghaini, and Z. Avazzadeh, A numerical method based on the Chebyshev cardinal functions for variable-order fractional version of the fourth-order 2D Kuramoto–Sivashinsky equation, Math. Method. Appl. Sci. 44(2) (2021), pp. 1831–1842.
   Web of Science ®Google Scholar
• D. Ingman and J. Suzdalnitsky, Control of damping oscillations by fractional differential operator with time-dependent order, Comput. Methods. Appl. Mech. Eng. 193(52) (2004), pp. 5585–5595.
   Web of Science ®Google Scholar
• H. Jiang, F. Liu, I. Turner, and K. Burrage, Analytical solutions for the multi-term time-fractional diffusion-wave/diffusion equations in a finite domain, Comput. Math. Appl. 64(10) (2012), pp. 3377–3388.
   Web of Science ®Google Scholar
• Z. Li, H. Wang, R. Xiao, and S. Yang, A variable-order fractional differential equation model of shape memory polymers, Chaos Solitons Fract. 102 (2017), pp. 473–485.
   Web of Science ®Google Scholar
• R.R. Nigmatullin, To the theoretical explanation of the universal response, Phys. Status (B): Basic Res. 123 (1984), pp. 739–746.
   Web of Science ®Google Scholar
• R.R. Nigmatullin, Realization of the generalized transfer equation in a medium with fractal geometry, Phys. Status (B): Basic Res. 133 (1986), pp. 425–430.
   Web of Science ®Google Scholar
• P.W. Ostalczyk, P. Duch, D.W. Brzezinski, and D. Sankowski, Order functions selection in the variable fractional-order PID controller, in Advances in Modelling and Control of Non-integer Order Systems, Lecture Notes. Electr. Eng. Vol. 320, 2015, pp. 159–170.
   Google Scholar
• L.E.S. Ramirez and C.F.M. Coimbra, Variable-order constitutive relation for viscoelasticity, Ann. Phys. 16 (2007), pp. 543–552.
   Web of Science ®Google Scholar
• S.G. Samko and B. Ross, Integration and differentiation to a variable fractional-order, Integr. Transf. Spec. Funct. 1(4) (1993), pp. 277–300.
   Google Scholar
• H. Sheng, H.G. Sun, C. Coopmans, Y.Q. Chen, and G.W. Bohannan, A physical experimental study of variable-order fractional integrator and differentiator, Eur. Phys. J. Spec. Top 193 (2011), pp. 93–104.
   Web of Science ®Google Scholar
• C.M. Soon, C.F.M. Coimbra, and M.H. Kobayashi, The variable viscoelasticity operator, Ann. Phys.14 (2005), pp. 378–389.
   Web of Science ®Google Scholar
• C.M. Soon, C.F.M. Coimbra, and M.H. Kobayashi, The variable viscoelasticity oscillator, Ann. Phys.14(6) (2005), pp. 378–389.
   Web of Science ®Google Scholar
• Y.H. Youssri and R.M. Hafez, Chebyshev collocation treatment of Volterra–Fredholm integral equation with error analysis, Arab. J. Math. 9 (2019), pp. 471–480. doi:https://doi.org/10.1007/s40065-019-0243-y.
   Web of Science ®Google Scholar
• W.K. Zahra and M.M. Hikal, Non standard finite difference method for solving variable-order fractional optimal control problems, J. Vib. Control 23(6) (2017), pp. 948–958. doi:https://doi.org/10.1177/1077546315586646.
   Web of Science ®Google Scholar