### A numerical solution of variable order diffusion and wave equations

Document Type : Research Paper

Authors

1 Department of Mathematics, Faculty of Basic Sciences, Bozorgmehr University of Qaenat, Qaenat, Iran.

2 Department of Mathematics, University of Mazandaran, Babolsar, Iran. Department of Mathematical Sciences, University of South Africa, UNISA0003, South Africa.

3 Department of Mathematics, University of Mazandaran, Babolsar, Iran.

Abstract

In this work, we consider variable order difusion and wave equations. The derivative is described in the Caputo sence of variable order. We use the Genocchi polynomials as basic functions and obtain operational matrices via these polynomials. These matrices and collocation method help us to convert variable order diffusion and wave equations to an algebraic system. Some examples are given to show the validity of the presented method.

Keywords

[1] T. M. Atanackovic, S. Pilipovic, B. Stankovic, and D. Zorica, Fractional calculus with applications in mechanics,Wiley, London, 2014.
[2] A. H. Bhrawya, and M. A. Zakyc, A method based on the Jacobi tau approximation for solving multi-term time-space fractional partial differential equations, Journal of Computational Physics, 281 (2015), 876–895.
[3] M. A. Firoozjaee, and S. A. Yousefi, A numerical approach for fractional partial differential equations by using Ritz approximation, Applied Mathematics and Computation, 338 (2018), 711–721.
[4] R. M. Ganji, and H. Jafari, A numerical approach for multi-variable orders differential equations using Jacobi polynomials, International Journal of Applied and Computational Mathematics, 5 (2019).
[5] R. M. Ganji, and H. Jafari, Numerical solution of variable order integro-differential equations, Advanced Mathematical Models & Applications, 4(1) (2019), 64–69.
[6] R. M. Ganji, and H. Jafari, A new approach for solving nonlinear Volterra integro-differential equations with Mittag–Leffler kernel, Proceedings of the Institute of Mathematics and Mechanics, 46(1) (2020), 144–158.
[7] R. M. Ganji, H. Jafari, and A. R. Adem, A numerical scheme to solve variable order diffusion-wave equations,Thermal Science, 23 (2019), 371–371.
[8] R. M. Ganji, H. Jafari and D. Baleanu, A new approach for solving multi variable orders differential equations with Mittag-Leffler kernel, Chaos, Solitons & Fractals, 130 (2020), 109405.
[9] R. M. Ganji, H. Jafari, and S. Nemati, A new approach for solving integro-differential equations of variable order, Journal of Computational and Applied Mathematics, 379 (2020), 112946.
[10] E. Doha, M. Abdelkawy, A. Z. M. Amin and D. Baleanu, Spectral technique for solving variable-order fractional Volterra integro-differential equations, Numerical Methods for Partial Differential Equations, 34(5) (2018), 1659–1677.
[11] M. Gasca, and T. Sauer, On the history of multivariate polynomial interpolation, Journal of Computational and Applied Mathematics, 122(1-2) (2000), 23–35.
[12] H. Hassani, and E. Naraghirad, A new computational method based on optimization scheme for solving variableorder time fractional Burgers equation, Mathematics and Computers in Simulation, 162 (2019), 1–17.
[13] M. H. Heydari, Z. Avazzadeh, and M. F. Haromi, A wavelet approach for solving multi-term variable-order time fractional diffusion-wave equation, Applied Mathematics and Computation, 341 (2019), 215–228.
[14] V. R. Hosseini, E. Shivanian, and W. Chen, Local integration of 2-D fractional telegraph equation via local radial point interpolant approximation, The European Physical Journal Plus, 130(33) (2015).
[15] H. Jafari, M. Saeidy, and J. Vahidi, The homotopy analysis method for solving fuzzy system of linear equation,
International Journal of Fuzzy Systems, 11(4) (2009), 308–313.
[16] H. Jafari, H. Tajadodi, and D. Baleanu, A modified variational Iteration method for solving fractional Riccati differential equation by Adomian polynomials, Fractional Calculus and Applied Analysis, 16 (2013), 109–122.
[17] H. Jafari, H. Tajadodi, and R. M. Ganji, A numerical approach for solving variable order differential equations based on Bernstein polynomials, Computational and Mathematical Methods, 1(5) (2019).
[18] W. Jiang, and B. Guo, A new numerical method for solving two-dimensional variable-order anomalous subdiffusion equation, Thermal Science, 20(3) (2016), 701–710.
[19] X. Li, and B. Wu, A numerical technique for variable fractional functional boundary value problems, Applied Mathematics Letters, 43 (2014), 108–113.
[20] K. Maleknejad, K. Nouri, and L. Torkzadeh, Operational matrix of fractional integration based on the shifted second kind chebyshev polynomials for solving fractional differential equations, Mediterranean Journal of Mathematics, 13(2016),1377–1390.
[21] P. Rahimkhani, Y. Ordokhani, and E. Babolian, Fractional-order Bernoulli functions and their applications in solving fractional Fredholem-Volterra integro-differential equations, Applied Numerical Mathematics, 122 (2017), 66–81.
[22] S. Sadeghi Roshan, H. Jafari, and D. Baleanu, Solving FDEs with Caputo-Fabrizio derivative by operational matrix based on Genocchi polynomials, Mathematical Methods in the Applied Sciences, 41(18) (2018), 9134–9141.
[23] S. Sadeghi Roshan, H. Jafari, and S. Nematia, Operational matrix for Atangana–Baleanu derivative based on Genocchi polynomials for solving FDEs, Chaos, Solitons & Fractals, 135 (2020), 109736.
[24] S. G. Samko, and B. Ross, Integration and differentiation to a variable fractional order, Integral Transforms and Special Functions, 1 (1993), 277–300.
[25] C. M. Soon, C. F. M. Coimbra, and M. H. Kobayashi, The variable viscoelasticity oscillator, Annalen der Physik, 14(6) (2005), 378–389.
[26] N. H. Tuan, S. Nemati, R. M. Ganji, and H. Jafari, Numerical solution of multi-variable order fractional integro differential equations using the Bernstein polynomials, Engineering with Computers, (2020).
[27] J. de. Villiers, Mathematics of Approximation, Atlantis Press, 2012.
[28] Y. Xu, and V. Suat Ert¨urk, A Finite Difference Technique For Solving Variable-Order Fractional IntegroDifferential Equations, Bulletin of the Iranian Mathematical Society, 40(3) (2014), 699–712.
[29] J. Yanga, An efficient numerical method for variable order fractional functional differential equation, Applied Mathematics Letters, 76 (2018), 221–226.
[30] X. J. Yang, Fractional derivatives of constant and variable orders applied to anomalous relaxation models in heat-transfer problems, Thermal Science, 21(3) (2017), 1161–1171.
###### Volume 12, Issue 1May 2021Pages 27-36
• Receive Date: 20 July 2020
• Revise Date: 12 August 2020
• Accept Date: 20 August 2020