Abstract
The aim of this paper is to introduce a new wavelet method for presenting approximate solutions of multitype variable-order (VO) fractional partial differential equations arising from the modeling of phenomena. In specific, this paper focuses on the numerical solution of the VO-fractional mobile-immobile advection-dispersion equation, Klein Gordon equation and Burgers equation. These equations are converted into a system of algebraic equations with the assistance of the bivariate Genocchi wavelet functions, their operational matrices, and the variable-order fractional Caputo derivative operator. Also, we present a new technique to get the operational matrix of integration and VO-fractional derivative. The modified operational matrices for solving the proposed equations are powerful and effective. So that, the accuracy of these matrices directly affects the implementation process. Finally, we consider numerical examples to confirm the superiority of the scheme, and for each example, exhibit the results through graphs and tables.
Similar content being viewed by others
References
Sun, H.G., Chen, W., Wei, H., Chen, Y.Q.: A comparative study of constant-order and variable-order fractional models in characterizing memory property of systems. Eur. Phys. J. Spec. Top. 193(1), 185 (2011)
Shyu, J.J., Pei, S.C., Chan, C.H.: An iterative method for the design of variable fractional-order FIR differintegrators. Signal Process. 89(3), 320–327 (2009)
Samko, S.G., Ross, B.: Integration and differentiation to a variable fractional order. Integr. Transform. Spec. Funct. 1(4), 277–300 (1993)
Liu, F., Zhuang, P., Anh, V., Turner, I.: A fractional-order implicit difference approximation for the space-time fractional diffusion equation. ANZIAM. J. 47, 48–68 (2005)
Lorenzo, C.F., Hartley, T.T.: Variable order and distributed order fractional operators. Nonlinear Dyn. 29(1–4), 57–98 (2002)
Lorenzo, C.F., Hartley, T.T.: Initialized fractional calculus. NASA Glenn Research Center (2000)
Ramirez, L.E., Coimbra, C.F.: On the selection and meaning of variable order operators for dynamic modeling. Int. J. Differ. Equat. (2010). https://doi.org/10.1155/2010/846107
Ramirez, L.E., Coimbra, C.F.: A variable order constitutive relation for viscoelasticity. Annalen der, Physik. 16(7–8), 543–552 (2007)
Zhuang, P., Liu, F., Anh, V., Turner, I.: Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term. SIAM. J. Numer. Anal. 47(3), 1760–1781 (2009)
Hajipour, M., Jajarmi, A., Baleanu, D., Sun, H.: On an accurate discretization of a variable-order fractional reaction-diffusion equation. Commun. Nonlinear Sci. Numer. Simulat. 69, 119–133 (2019)
Dehestani, H., Ordokhani, Y., Razzaghi, M.: Application of the modified operational matrices in multiterm variable-order time-fractional partial differential equations. Math. Meth. Appl. Sci. 42(18), 7296–7313 (2019)
Lin, R., Liu, F., Anh, V., Turner, I.: Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation. Appl. Math. Comput. 212(2), 435–445 (2009)
Doha, E.H., Abdelkawy, M.A., Amin, A.Z.M., Lopes, A.M.: A space-time spectral approximation for solving nonlinear variable-order fractional sine and Klein-Gordon differential equations. Comput. Appl. Math. 37(5), 6212–6229 (2018)
Abd-Elkawy, M.A., Alqahtani, R.T.: Space-time spectral collocation algorithm for the variable-order Galilei invariant advection diffusion equations with a nonlinear source term. Math. Model. Anal. 22(1), 1–20 (2017)
Nagy, A.M., Sweilam, N.H.: Numerical simulations for a variable order fractional cable equation. Acta. Math. Sci. 38(2), 580–590 (2018)
Hassani, H., Naraghirad, E.: A new computational method based on optimization scheme for solving variable-order time fractional Burgers’ equation. Math. Comput. Simul. 162, 1–17 (2019)
Jiang, W., Liu, N.: A numerical method for solving the time variable fractional order mobile-immobile advection-dispersion model. Appl. Numer. Math. 119, 18–32 (2017)
Doha, E.H., Abdelkawy, M.A., Amin, A.Z.M., Baleanu, D.: Spectral technique for solving variable-order fractional Volterra integro-differential equations. Numer. Methods Partial Differ. Eq. 34(5), 1659–1677 (2018)
Dehestani, H., Ordokhani, Y., Razzaghi, M.: On the applicability of Genocchi wavelet method for different kinds of fractional-order differential equations with delay. Numer. Linear. Algebra Appl. 26(5), e2259 (2019)
Isah, A., Phang, C.: Genocchi Wavelet-like operational matrix and its application for solving non-linear fractional differential equations. Open Phys. 14(1), 463–472 (2016)
Heydari, M.H., Hooshmandasl, M.R., Ghaini, F.M., Cattani, C.: Wavelets method for solving fractional optimal control problems. Appl. Math. Comput. 286, 139–154 (2016)
Sahu, P.K., Ray, S.S.: Legendre wavelets operational method for the numerical solutions of nonlinear Volterra integro-differential equations system. Appl. Math. Comput. 256, 715–723 (2015)
Yuttanan, B., Razzaghi, M.: Legendre wavelets approach for numerical solutions of distributed order fractional differential equations. Appl. Math. Model. 70, 350–364 (2019)
Kajani, M.T., Ghasemi, M., Babolian, E.: Comparison between the homotopy perturbation method and the sine-cosine wavelet method for solving linear integro-differential equations. Comput. Math. Appl. 54(7–8), 1162–1168 (2007)
Yuanlu, L.I.: Solving a nonlinear fractional differential equation using Chebyshev wavelets. Commun. Nonlinear. Sci. Numer. Simul. 15(9), 2284–2292 (2010)
Heydari, M.H., Hooshmandasl, M.R., Mohammadi, F., Cattani, C.: Wavelets method for solving systems of nonlinear singular fractional Volterra integro-differential equations. Commun. Nonlinear Sci. Numer. Simul. 19(1), 37–48 (2014)
Singh, R., Guleria, V., Singh, M.: Haar wavelet quasilinearization method for numerical solution of Emden-Fowler type equations. Math. Comput. Simul. 174, 123–133 (2020)
Ma, H., Yang, Y.: Jacobi spectral collocation method for the time variable-order fractional mobile-immobile advection-dispersion solute transport model. E. Asian J. Appl. Math. 6(3), 337–352 (2016)
Zhang, Y., Benson, D.A., Reeves, D.M.: Time and space nonlocalities underlying fractional-derivative models: Distinction and literature review of field applications. Adv. Water. Resour. 32, 561–581 (2009)
Schumer, R., Benson, D.A., Meerschaert, M.M., Baeumer, B.: Fractal mobile/immobile solute transport. Water Resour. Res. 39, 1–12 (2003)
Momani, S., Odibat, Z.M.: Fractional green function for linear time-fractional inhomogeneous partial differential equations in fluid mechanics. J. Appl. Math. Comput. 24(1–2), 167–178 (2007)
Esen, A., Tasbozan, O.: Numerical solutions of time fractional Burgers equation. Acta Univ. Sapientiae Math. 7(2), 167–185 (2015)
Akram, T., Abbas, M., Riaz, M.B., Ismail, A.I., Ali, N.M.: An efficient numerical technique for solving time fractional Burgers equation. Alex. Eng. J. 59(4), 2201–2220 (2020)
Avazzadeh, Z., Hassani, H.: Transcendental Bernstein series for solving reaction-diffusion equations with nonlocal boundary conditions through the optimization technique. Numer. Methods. Partial. Differ. Equ. 35(6), 2258–2274 (2019)
Dehestani, H., Ordokhani, Y., Razzaghi, M.: Fractional-order Legendre-Laguerre functions and their applications in fractional partial differential equations. Appl. Math. Comput. 336, 433–453 (2018)
Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral methods: fundamentals in single domains. Springer Science & Business Media (2007)
Zhang, H., Liu, F., Phanikumar, M.S., Meerschaert, M.M.: A novel numerical method for the time variable fractional order mobile-immobile advection-dispersion model. Comput. Math. Appl. 66(5), 693–701 (2013)
Sweilam, N.H., Al-Mekhlafi, S.M., Albalawi, A.O.: A novel variable-order fractional nonlinear Klein Gordon model: a numerical approach. Numer. Methods Partial Differ. Equ. 35(5), 1617–1629 (2019)
Acknowledgements
This work is supported by the National Elite Foundation and Alzahra University. We express our sincere thanks to the anonymous referees for valuable suggestions that improved the final manuscript.
Author information
Authors and Affiliations
Contributions
All authors contributed equally and significantly in writing this paper. Also, all authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Dehestani, H., Ordokhani, Y. & Razzaghi, M. Modified wavelet method for solving multitype variable-order fractional partial differential equations generated from the modeling of phenomena. Math Sci 16, 343–359 (2022). https://doi.org/10.1007/s40096-021-00425-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40096-021-00425-1
Keywords
- Genocchi wavelet functions
- Variable-order Caputo fractional derivative
- Modified operational matrix
- Error estimation