A new optimization operational matrix algorithm for solving nonlinear variable-order time fractional convection-diffusion equation

Document Type : Original Paper

Authors

1 Department of َApplied Mathematics, Shahrekord University, Shahrekord , Iran

2 Department of Mathematics, Yasouj University, Yasouj, Iran.

Abstract

In this paper, a new and effective optimization algorithm is proposed for solving the nonlinear time fractional convection-diffusion equation with the concept of variable-order fractional derivative in the Caputo sense. For finding the solution, we first introduce the generalized polynomials (GPs) and construct the variable-order operational matrices. In the proposed optimization technique, the solution of the problem under consideration is expanded in terms of GPs with unknown free coefficients and control parameters. The main advantage of the presented method is to convert the variable-order fractional partial differential equation to a system of nonlinear algebraic equations. Also, we obtain the free coefficients and control parameters optimally by minimizing the error of the approximate solution. Finally, the numerical examples confirm the high accuracy and efficiency of the proposed method in solving the problem under study.

Keywords

Main Subjects


[1] Ciombra, C. F. M. (2013). Mechanics with variable-order differential operators, Ann. Phys., 12 (11-12), 692-703.
[2] Pedro, H. T. C., Kobayashi, M. H., Pereira, J. M. C. and Coimbra, C. F. M. (2008). Variable order modeling of diffusive-convective effects on the oscillatory flow past a sphere, J. Vib. Control, 14, 1569-1672.
[3] Ramirez, L. E. S. and Coimbra, C. F. M. (2011). On the variable order dynamics of the nonlinear wake caused by a sedimenting particle, Physica D, 240, 1111-1118.
[4] Shyu, J. J., Pei, S. C. and Chan, C. H. (2009). An iterative method for the design of variable fractional-order FIR different egrators, Signal Process., 89, 320-327.
[5] Sun, H. G., Chen, W. and Chen, Y. Q. (2009). Variable-order dractional differential operators in anomalous diffusion modeling, Phys. A, 388, 4586-4592.
[6] Zahra, W. K. and Hikal, M. M. (2017). Nonstandard finite difference method for solving variable order fractional control problems, J. Vib. Control, 23 (6), 948-958.
[7] Ramirez, L. E. S. and Coimbra, C. F. M. (2007). Variable order constitutive relation for viscoelasticity, Ann. Phys., 16, 543-552.
[8] Chen, C. M. (2013). Numerical methods for solving a two-dimensional variable-order modified diffusion equation. Appl. Math. Comput., 225, 62-78.
[9] Bhrawy, A. H. and Zaky, M. A. (2016). Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation, Nonlinear Dyn., 80 (1), 101-116.
[10] Shen, S., Liu, F., Chen, J., Turner, I. and Anh, V. (2012). Numerical techniques for the variable order time fractional diffusion equation, Appl. Math. Comput., 218, 10861-10870.
[11] Yang, X. J. and Tenreiro Machado, J. A. (2017). A new fractional operator of variable order: application in the description of anomalous diffusion equation, Physica A: Statistical Mechanics and its Applications, 481, 276-283.
[12] Dahaghin, M. Sh. and Hassani, H. (2017). An optimization method based on the generalized polynomials for nonlinear variable-order time fractional diffusion-wave equation, Nonlinear Dyn., 88 (3), 1587-1598.
[13] Li, X. Y. and Wu, B. (2015). A numerical technique for variable fractional functional boundary value problems, Appl. Math. Lett., 43, 108-113.
[14] Chen, C. M., Liu, F., Anh, V. and Turner, I. (2010). Numerical schemes with high spatial accuracy for a variable-order anomalous subdiffusion equation, SIAM J. Sci. Comput., 32 (4), 1740-1760.
[15] Bhrawy, A. H. and Zaky, M. A. (2017). An improved collocation method for multi-dimensional space-time variable-order fractional Schrödinger equations, Appl. Numer. Math., 11, 197-218.
[16] Zhang, H., Liu, F., Phanikumar, M, S. and Meerschaert, M. M. (2013). A novel numerical method for the time variable fractional order mobile-immobile advection-dispersion model, Comput. Math. Appl., 66, 693-701.
[17] Chen, S., Liu, F. and Burrage, K. (2014). Numerical simulation of a new two-dimensional variable-order fractional percolation equation in non-homogeneous porous media, Comput. Math. Appl., 68 (12), 2133-2141.
[18] Chen, Y. M., Wei, Y. Q., Liu, D. Y., Boutat, D. and Chen, X. K. (2016). Variable-order fractional numerical differentiation for noisy signals by wavelet denoising, J. Comput. Phys., 311, 338-347.
[19] Zhao, X., Sun, Z. Z. and Karniadakis, G. E. (2015). Second-order approximations for variable order fractional derivatives: Algorithms and applications, J. Comput. Phys., 293, 184-200.
[20] Li, X. Y. and Wu, B. (2015). A numerical technique for variable fractional functional boundary value problems, Appl. Math. Lett., 43, 108-113.
[21] Jia, Y. T., Xu, M. Q. and Lin, Y. Z. (2017). A numerical solution for variable order fractional differential equation, Appl. Math. Lett., 64, 125-130.
[22] Atangana, A. (2015). On the stability and convergence of the time-fractional variable order telegraph equation, J. Comput. Phys., 293, 104-114.
[23] Chen, C. M. (2013). Numerical methods for solving a two-dimensional variable-order modified diffusion equation, Appl. Math. Comput., 225, 62-78.
[24] Zhou, F. and Xu, X. (2016). The third kind Chebyshev wavelet collocation method for solving the time-fractional convection diffusion equations with variable coefficients, Appl. Math. Comput., 280, 11-29.
[25] Behroozifar, M. and Sazmand, A. (2017). An approximate solution based on Jacobi polynomials for time-fractional convection-diffusion equation, Appl. Math. Comput. 296, 1-17.
[26] Chen, M. H. and Deng, W. H. (2014). A second-order numerical method for two-dimensional two-sided space fractional convection diffusion equation, Appl. Math. Model. 38 (13), 3244-3259.
[27] Li, H. F., Cao, J. X. and Li, C. P. (2016). High-order approximation to Caputo derivatives and Caputo-type advection-diffusion equations (III), J. Comput. Appl. Math., 299, 159-175.
[28] Wang, Y. M. (2015). A compact finite difference method for a class of time fractional convection-diffusion-wave equations with variable coefficients, Numer. Algor., 70, 635-651.
[29] Dahaghin, M. Sh. and Hassani, H. (2017). A new optimization method for a class of time fractional convection-diffusion-wave equations with variable coefficients, Eur. Phys. J. Plus, 132: 130, DOI 10.1140/epjp/i2017-11407-y.
[30] Shen, S., Liu, F., Anh, V., Turner, I. and Chen, J. (2013). A characteristic difference method for the variable-order fractional advection-diffusion equation, J. Appl. Math. Comput., 42, 371-386.
[31] Wang, J., Liu, T., Li, H., Liu, Y. and He, S. (2017). Second-order approximation scheme combined with H1-Galerkin MFE method for nonlinear time fractional convection-diffusion convection-diffusion equation, Comput. Math. Appl., 73 (6), 1182-1196.
[32] Liu, T. (2018). A wavelet multiscale method for the inverse problem of a nonlinear convection-diffusion equation, J. Comput. Appl. Math., 330, 165-176.
[33] Srinivasan, S., Poggie, J. and Zhang, X. (2018). A positivity-preserving high order discontinuous Galerkin scheme for convection-diffusion equations, J. Comput. Phys., 366, 120-143.
[34] Ezz-Eldien, S. S., Doha, Bhrawy, A. H., El-Kalaawy, A. A. and Tenreiro Machado, J. A. (2018). A new operational approach for solving fractional variational problems depending on indefinite integrals, Commun. Nonlinear. Sci. Numer. Simulat., 57, 246-263.
[35] Kreyszig, E. (1978). Introductory Functional Analysis with Applications, John Wiley and Sons. Inc.