New collocation scheme for solving fractional partial differential equations

Year 2020, , 1107 - 1125, 02.06.2020

Chang Phang , Afshan Kanwal Jian Rong Loh

https://doi.org/10.15672/hujms.459621

Cited By: 16

Abstract

This article concerned about the numerical solution of time fractional partial differential equations (FPDEs). The proposed technique is using shifted Chebyshev-Gauss-Lobatto (CGL) collocation points in conjunction with an operational matrix of Caputo sense derivatives via Genocchi polynomials. The system of linear algebraic equations is obtained when the main equation along with the initial as well as boundary conditions is collocated by using shifted CGL collocation points. The main approach to this method is to transform the FPDEs to system of algebraic equations, hence, greatly simplify the numerical scheme. Comparison of the obtained results with the existing methods depicts that the suggested method is highly effect, more efficient and have less computational work. Some examples are given to illustrate the effectiveness and applicability of the proposed technique.

Keywords

Collocation method, shifted Chebyshev Gauss Lobatto, fractional partial differential equations, operational matrix

References

• [1] K. Al-Khaled, Numerical solution of time-fractional partial differential equations using Sumudu decomposition method, Rom. J. Phys. 60 (1-2), 99-110, 2015.
• [2] A.H. Bhrawy and M.A. Zaky, Fractional-order Jacobi Tau method for a class of timefractional PDEs with variable coefficients, Math. Methods Appl. Sci. 16 (4), 490-498, 2015.
• [3] 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, 876-895, 2015.
• [4] A.H. Bhrawy and M.A. Zaky, An improved collocation method for multi-dimensional space-time variable-order fractional Schrodinger equations, Appl. Numer. Math. 111, 197-218, 2017.
• [5] A.H. Bhrawy and M.A. Zaky, Numerical simulation of multi-dimensional distributedorder generalized Schrodinger equations, Nonlinear Dyn. 89 (2), 1415-1432, 2017.
• [6] M.A.M. Ghandehari and M. Ranjbar, A numerical method for solving a fractional partial differential equation through converting it into an NLP problem, Comput. Math. Appl. 65 (7), 975-982, 2013.
• [7] A. Isah, C. Phang and P. Phang, Collocation method based on Genocchi operational matrix for solving generalized fractional Pantograph equations, Int. J. Differ. Equ. 2017, 2017.
• [8] 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), 3377-3388, 2012.
• [9] F. Liu, M. Meerschaert, R. McGough, P. Zhuang and Q. Liu, Numerical methods for solving the multi-term time-fractional wave-diffusion equation, Fract. Calc. Appl. Anal. 16 (1), 9-25, 2013.
• [10] J.R. Loh, C. Phang and A. Isah, New operational matrix via Genocchi polynomials for solving Fredholm-Volterra fractional integro-differential equations (FIDEs), Adv. Math. Phys. 2017, 2017.
• [11] K.S. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations, Wiley-Interscience, 1993.
• [12] A. Mohebbi, M. Abbaszadeh and M. Dehghan, High-order difference scheme for the solution of linear time fractional Klein-Gordon equations, Numer. Methods Partial Diff. Equ. 30 (4), 1234-1253, 2014.
• [13] S. Nemati and Y. Ordokhani, Legendre expansion methods for the numerical solution of nonlinear 2D Fredholm integral equations of the second kind, J. Appl. Math. Informatics, 31 (5-6), 609-621. 2013.
• [14] K. Oldham and J. Spanier, The fractional calculus theory and applications of differentiation and integration to arbitrary order, Elsevier, 1974.
• [15] N. Ozdemir and M. Yavuz, Numerical solution of fractional Black-Scholes equation by using the multivariate Pade approximation, Acta Phys. Pol. A, 132 (3), 1050-1053, 2017.
• [16] V.K. Patel, S. Somveer and V.K. Singh, Two-dimensional shifted Legendre polynomial collocation method for electromagnetic waves in dielectric media via almost operational matrices, Math. Methods Appl. Sci., 2017.
• [17] C. Phang, N.F. Ismail, A. Isah and J.R. Loh, A new efficient numerical scheme for solving fractional optimal control problems via a Genocchi operational matrix of integration, J. Vib. Control 24 (14), 3036-3048, 2018.
• [18] S.Y. Reutskiy, A new semi-analytical collocation method for solving multi-term fractional partial differential equations with time variable coefficients, Appl. Math. Model. 45, 238-254, 2017.
• [19] A. Saadatmandi, M. Dehghan and M.R. Azizi, The Sinc-Legendre collocation method for a class of fractional convection-diffusion equations with variable coefficients, Commun. Nonlinear Sci. Numer. Simul. 17 (11), 4125-4136, 2012.
• [20] E. Tohidi, A.H. Bhrawy and K. Erfani, A collocation method based on Bernoulli operational matrix for numerical solution of generalized pantograph equation, Appl. Math. Model. 37 (6), 4283-4294, 2013.
• [21] V. Turut and N. Güzel, On solving partial differential equations of fractional order by using the variational iteration method and multivariate Padé approximations, Eur. J. Pure Appl. Math. 6 (2), 147-171, 2013.
• [22] V.A. Vyawahare and P.S.V. Nataraj, Fractional-order modeling of neutron transport in a nuclear reactor, Appl. Math. Model. 37 (23), 9747-9767, 2013.
• [23] M. Yavuz and N. Ozdemir, A different approach to the European option pricing model with new fractional operator, Math. Model. Nat. Pheno. 13 (1), 12, 2018.
• [24] M. Yavuz, N. Ozdemir, and H.M. Baskonus, Solutions of partial differential equations using the fractional operator involving Mittag-Leffler kernel, Eur. Phys. J. Plus, 133 (6), 215, 2018.
• [25] M. Yi, J. Huang and J. Wei, Block pulse operational matrix method for solving fractional partial differential equation, Appl. Math. Comput. 221, 121-131, 2013.
• [26] M.A. Zaky, An improved Tau method for the multi-dimensional fractional Rayleigh- Stokes problem for a heated generalized second grade fluid, Comput. Math. Appl. 75 (7), 2243-2258, 2018.
• [27] M.A. Zaky, A Legendre spectral quadrature tau method for the multi-term timefractional diffusion equations, Computat. Appl. Math. 37 (3), 3525-3538, 2018.
• [28] M.A. Zaky, Recovery of high order accuracy in Jacobi spectral collocation methods for fractional terminal value problems with non-smooth solutions, J. Comput. Appl. Math. 357, 103-122, 2019.
• [29] M.A. Zaky, E.H. Doha and J.T. Machado, A spectral framework for fractional variational problems based on fractional Jacobi functions, Appl. Numer. Math. 132, 51-72, 2018.
• [30] F. Zhou and X. Xu, The third kind Chebyshev wavelets collocation method for solving the time-fractional convection diffusion equations with variable coefficients, Appl. Math. Comput. 280, 11-29, 2016.
• [31] B. Zogheib, E. Tohidi and S. Shateyi, Bernoulli collocation method for solving linear multidimensional diffusion and wave equations with Dirichlet boundary conditions, Adv. Math. Phys. 2017, 2017.