The paper reports a spectral method for generating an approximate solution for the space-time fractional PDEs with variable coefficients based on the spectral shifted Jacobi collocation method in conjunction with the shifted Jacobi operational matrix of fractional derivatives. The spectral collocation method investigates both temporal and spatial discretizations. By applying the shifted Jacobi collocation method, the problem reduces to a system of algebraic equations, which greatly simplifies the problem. Numerical results are given to establish the validity and accuracy of the presented procedure for space-time fractional PDE.
[1] M. Akbarzade and J. Langari, Application of Homotopy perturbation method and variational iteration method to three dimensional diffusion problem, Int. J. Math. Anal., 5 (2011), 871–80.
[2] M. A. Bayrak and A. Demir, A new approach for space-time fractional partial differential equations by residual power series method, Appl. Math. Comput., 336 (2018), 215-230.
[3] D. A. Benson, S. W. Wheatcraft, and M. M. Meerschaert, Application of a fractional advection-dispersion equation, Water Resour. Res., 36(6) (2000), 1403-1412.
[4] A. H. Bhrawy, E. A. Ahmed, and D. Baleanu, An efficient collocation technique for solving generalized Fokker- Planck type equations with variable coefficients, In Proc. Rom. Acad. Ser. A., 15 (2014), 322-330.
[5] A. H. Bhrawy, T. M. Taha, and J. A. T. Machado, A review of operational matrices and spectral techniques for fractional calculus, Nonlinear Dyn., 81(3) (2015), 1023-1052.
[6] A. H. Bhrawy, M. M. Tharwat, and M. A. Alghamdi, A new operational matrix of fractional integration for shifted Jacobi polynomials, Bull. Malays. Math. Sci. Soc., 37(4) (2014), 983-995.
[7] A. H. Bhrawy, M. A. Zaky, and R. A. Van Gorder, A space-time Legendre spectral tau method for the two-sided space-time Caputo fractional diffusion- wave equation, Numer. Algor., 71(1) (2016), 151-180.
[8] A. H. Bhrawy and M. A. Zaky, A method based on the Jacobi tau approximation for solving multi-term time-space fractional partial differential equations, J. Comput. Phys., 281 (2015), 876-895.
[9] A. H. Bhrawy and M. A. Zaky, Shifted fractional-order Jacobi orthogonal functions: application to a system of fractional differential equations, Appl. Math. Model., 40(2) (2016), 832-845.
[10] A. H. Bhrawy and M. A. Zaky, Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation, Nonlinear Dyn., 80(1) (2015), 101-116.
[11] D. Brockmann, V. David, and A. M. Gallardo, Human mobility and spatial disease dynamics, Rev. Nonlinear Dyn. Complex., 2 (2009), 1-24.
[12] A. Cartea and D. del Castillo-Negrete, Fractional diffusion models of option prices in markets with jumps, Physica A: Stat. Mech. Appl., 374(2) (2007), 749-763.
[13] C. M. Chen, F. Liu, I. Turner, and V. Anh, A Fourier method for the fractional diffusion equation describing sub-diffusion, J. Comput. Phys., 227(2) (2007), 886–897.
[14] Y. Chen, Y. Wu, Y. Cui, Z. Wang, and D. Jin, Wavelet method for a class of fractional convection-diffusion equation with variable coefficients, J. Comput. Sci., 1(3)(2010) 146–149.
[15] M. Dehghan, S. A. Yousefi, and A. Lotfi, The use of He’s variational iteration method for solving the telegraph and fractional telegraph equations, Comm. Numer. Method Eng., 27 (2011), 219-231.
[16] Z. Q. Deng, J. L. De Lima, M. I. P. de Lima, and V. P. Singh, A fractional dispersion model for overland solute transport, Water Resour. Res., 42(3) (2006).
[17] E. H. Doha, On the construction of recurrence relations for the expansion and connection coefficients in series of Jacobi polynomial, J. Phys. A: Math. Gen., 37(3) (2004), 657-675.
[18] E. H. Doha, A. H. Bhrawy, and D. Baleanu, Numerical treatment of coupled nonlinear hyperbolic Klein-Gordon equations, Rom. J. Phys., 59(3-4) (2014), 247-264.
[19] E. H. Doha, A. H. Bhrawy, and S. S. Ezz-Eldien, A new Jacobi operational matrix, an application for solving fractional differential equations, Appl. Math. Model., 36(10) (2012), 4931-4943.
[20] M. El-Shahed and A. Salem, An extension of Wright function and its properties, J. Math., 2015 (2015).
[21] E. Hanert, On the numerical solution of space–time fractional diffusion models, Comput. Fluids., 46(1) (2011), 33–39.
[22] E. Hanert and C. Piret, A Chebyshev pseudo spectral method to solve the space-time tempered fractional diffusion equation, SIAM J. Sci. Comput., 36(4) (2014), A1797–A1812.
[23] Y. Hu, Y. Luo, and Z. Lu, Analytical solution of the linear fractional differential equation by Adomian decompo- sition method, J. Comput. Appl. Math., 215 (2008), 220-229.
[24] M. M. Khader, On the numerical solutions for the fractional diffusion equation, Commun. Nonlinear. Sci. Numer. Simulat., 16 (2011), 2535-2542.
[25] S. Kumar, R. S. Damor, and A. K. Shukla, Numerical study on thermal therapy of triple layer skin tissue using fractional bio heat model, Int. J. Bio math., 11(04) (2018), 1850052.
[26] S. Kumar and C. Piret, Numerical solution of space-time fractional PDEs using RBF-QR and Chebyshev polyno- mials, Appl. Numer. Math., 143 (2019), 300-315.
[27] X. Li and C. Xu, A space-time spectral method for the time fractional diffusion equation, SIAM J. Numer. Anal., 47(3) (2009), 2108–2131.
[28] F. Liu, P. Zhuang, V. Anh, I. Turner, and K. Burrage, Stability and convergence of the difference methods for the space–time fractional advection–diffusion equation, Appl. Math. Comput., 191(1) (2007), 12–20.
[29] Y. Luke, The Special Functions and Their Approximations, Academic Press, New York, 1969.
[30] F. Mainardi and G. Paqnini, The role of the Fox–Wright functions in fractional sub-diffusion of distributed order, J. Comput. Appl. Math., 207(2) (2007), 245–257.
[31] K. S. Miller and B. Ross, An introduction to the Fractional Calculus and Fractional Differential Equations, Wiley- Interscience, 1993.
[32] K. B. Oldham and J. Spanier, The fractional Calculus, Academic Press, New York and London, 1974.
[33] I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.
[34] I. Podlubny, Laplace transform method for linear differential equations of the fractional order, 1997.
[35] M. U. Rehman and R. A. Khan, Numerical solutions to initial and boundary value problems for linear fractional partial differential equations, Appl. Math. Model., 37 (2013) 5233–5244.
[36] A. Saadatmandi and M. Dehgan, A new operational matrix for solving fractional-order differential equations, Comput. Math. Appl., 59 (2010), 1326–1336.
[37] L. F. Wang, Y. P. Ma, and Z. J. Meng, wavelet method for solving fractional partial differential equations numer- ically, Appl. Math. Comput., 227 (2014), 66–76.
[38] H. Wang, K. Wang, and T. Sircar, A direct o (n log 2 n) finite difference method for fractional diffusion equations, J. Comput. Phys., 229(21) (2010), 8095-8104.
[39] L. Wei, H. Dai, D. Zhang, and Z. Si, Fully discrete local discontinuous Galerkin method for solving the fractional telegraph equation, Calcolo., 51(1) (2014), 175-192.
[40] M. A. Zaky and I. G. Ameen, A priori error estimates of a Jacobi spectral method for nonlinear systems of fractional boundary value problems and related Volterra-Fredholm integral equations with smooth solutions, Numer. Algor., 84(1) (2020), 63-89.
[41] M. A. Zaky, An accurate spectral collocation method for nonlinear systems of fractional differential equations and related integral equations with nonsmooth solutions, Appl. Numer. Math., 154 (2020), 205-222.
[42] M. A. Zaky and A. S. Hendy, An efficient dissipation-preserving Legendre-Galerkin spectral method for the Higgs boson equation in the de Sitter spacetime universe, Appl. Numer. Math., 160 (2021), 281-295.
[43] Z. Zhao, Y. Zheng, and P. Guo, A Galerkin finite element scheme for time–space fractional diffusion equation, Int. J. Comput. Math., 93(7) (2016), 1212-1225.
Bonyadi, S., Mahmoudi, Y., Lakestani, M., & Jahangiri rad, M. (2023). Numerical solution of space-time fractional PDEs with variable coefficients using shifted Jacobi collocation method. Computational Methods for Differential Equations, 11(1), 81-94. doi: 10.22034/cmde.2022.49901.2077
MLA
Samira Bonyadi; Yaghoub Mahmoudi; Mehrdad Lakestani; Mohammad Jahangiri rad. "Numerical solution of space-time fractional PDEs with variable coefficients using shifted Jacobi collocation method". Computational Methods for Differential Equations, 11, 1, 2023, 81-94. doi: 10.22034/cmde.2022.49901.2077
HARVARD
Bonyadi, S., Mahmoudi, Y., Lakestani, M., Jahangiri rad, M. (2023). 'Numerical solution of space-time fractional PDEs with variable coefficients using shifted Jacobi collocation method', Computational Methods for Differential Equations, 11(1), pp. 81-94. doi: 10.22034/cmde.2022.49901.2077
VANCOUVER
Bonyadi, S., Mahmoudi, Y., Lakestani, M., Jahangiri rad, M. Numerical solution of space-time fractional PDEs with variable coefficients using shifted Jacobi collocation method. Computational Methods for Differential Equations, 2023; 11(1): 81-94. doi: 10.22034/cmde.2022.49901.2077
)
