Novel Legendre-Jaiswal functions for solving time-space fractional partial differential equations

Document Type : Research Article

Authors

1 Department of Mathematics National Institute of Technology Rourkela

2 Department of Mathematics, NIT Rourkela, Odisha, India

Abstract

This paper examines a new fractional function based on Legendre and Jaiswal polynomials to solve linear and nonlinear time-space fractional partial differential equations of linear and nonlinear class. The Caputo sense is applied while using the fractional derivative. These problems can be solved using the collocation method, operational, and pseudo-operational matrices of integer and fractional-order integration. Using operational matrices, pseudo-operational matrices, and the collocation method, the problem is transformed into a system of algebraic equations. An upper bound on the error of the fractional-order integral operational matrix is computed. Furthermore, a detailed stability and convergence analysis of the collocation scheme presented to validate the robustness of the numerical approach. The applicability and effectiveness of the approach are demonstrated through several benchmark examples, including linear and non-linear fractional convection-diffusion, convection-diffusion-reaction, and nonlinear Fisher's equation. The numerical results confirm that the proposed method is stable, rapidly convergent, and highly accurate, outperforming several existing techniques in both efficiency and precision.

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Main Subjects


[1] A.H. Bhrawy, E.H. Doha, D. Baleanu, S.S. Ezz-Eldien, A spectral tau algorithm based on Jacobi
operational matrix for numerical solution of time fractional diffusion-wave equations, J. Comput.
Phys. 293 (2015) 142–156.
[2] W. Bu, S. Shu, X. Yue, A. Xiao, W. Zeng, Space–time finite element method for the multi-term
time–space fractional diffusion equation on a two-dimensional domain, Comput. Math. Appl. 78(5)
(2019) 1367–1379.
[3] Y. Chen, Y. Wu, Y. Cui, Z. Wang, D. Jin, Wavelet method for a class of fractional convection
diffusion equation with variable coefficients, J. Comput. Sci. 1(3) (2010) 146–149
[4] H. Dehestani, Y. Ordokhani, M. Razzaghi, Fractional–order Legendre–Laguerre functions and
their applications in fractional partial differential equations, Appl. Math. Comput. 336 (2018)
433–453.
[5] B. Gasmi, K. Arezki, Z. Hammouch, Various optical solutions to the (1+ 1)-Telegraph equation
with space-time conformable derivatives, Int. J. Nonlinear Analysis Appl. 12 (2021) 767-780.
[6] A.A. Hamou, E. H. Azroul, Z. Hammouch, A. L. Alaoui, A monotone iterative technique com
bined to finite element method for solving reaction-diffusion problems pertaining to non-integer
derivative, Eng. Comput. 39(4) (2023) 2515–2541.
[7] A.A.Hamou,Z.Hammouch,E.Azroul, P.Agarwal, Monotoneiterative technique for solving finite
difference systems of time fractional parabolic equations with initial/periodic conditions, Appl.
Numer. Math. 181 (2022) 561-593.
[8] Z. Hammouch, T. Mekkaoui. Traveling-wave solutions of the generalized Zakharov equation with
time-space fractional derivatives, Math. Eng. Sci. Aerosp. MESA. 5(4) (2014) 1-11.
[9] K. Hosseini, M. Samavat, M. Mirzazadeh, A New-dimensional Hirota Bilinear Equation: Its
B¨acklund Transformation and Rational-type Solutions, Regul. Chaot. Dyn. 25, (2020) 383–391.
[10] M. Inc, The approximate and exact solutions of the space and time–fractional Burgers equations
with initial conditions by variational iteration method, J. Math. Anal. Appl. 345(1) (2008) 476–484.
[11] H. Jafari, R.M. Ganji, S.M. Narsale, M. Kgarose, V.T. Nguyen, Application of Hosoya Polynomial
to solve a class of time-fractional diffusion equations, Fractals 31(04) (2023) 2340059.
[12] H. Jafari, S. Salati, M. Matinfar, V.T. Nguyen, A numerical scheme for a class of nonlinear multi
order fractional differential equations, Fractals 33(06) (2025) 2540132.
[13] D.V. Jaiswal, On polynomials related to Tchebichef polynomials of the second kind, The Fibonacci
Quart. 12(3) (1974) 263–265.
[14] S. Kazem, S. Abbasbandy, S. Kumar, Fractional-order Legendre functions for solving fractional
order differential equations, Appl. Math. Model. 37(7) (2013) 5498–5510.
[15] A.A. Kilbas, Theory and Applications of Fractional Differential Equations North–Holland Mathe
matics Studies 204 (2006).
[16] E. Kreyszig, Introductory Functional Analysis with Applications, John Wiley & Sons, 17 1991.
[17] V.S. Krishnasamy, S. Mashayekhi, M. Razzaghi, Numerical solutions of fractional differential
equations by using fractional Taylor basis, IEEE/CAA J. Autom. Sin. 4(1) (2017) 98–106.
[18] Y. Luchko, R. Gorenflo, An operational method for solving fractional differential equations with
the Caputo derivatives, Acta Math. Vietnam 24(2) (1999) 207–233.
[19] T. Mekkaoui, Z. Hammouch, Approximate analytical solutions to the Bagley-Torvik equation by
the fractional iteration method, Ann. Univ. Craiova Math. 39(2) (2012) 251–256.
[20] I. Podlubny, Fractional Differential Equations: an Introduction to Fractional Derivatives, Frac
tional Differential Equations, to Methods of Their Solution and Some of Their Applications, Aca
demic Press, California, 1998.
[21] P. Rahimkhani, Y. Ordokhani, E. Babolian, A new operational matrix based on Bernoulli wavelets
for solving fractional delay differential equations, Numer. Algorithms 74 (2017) 223–245.
[22] S. Rashid, Z. Hammouch, D. Baleanu, Y.-M. Chu, New generalizations in the sense of the weighted
non-singular fractional integral operator, Fractals 28(08) (2020) 2040003.
[23] P. Roul, V. Rohil, An accurate numerical method and its analysis for time-fractional Fisher’s equa
tion, Soft Comput. 28(19)(2024) 11495–11514.
[24] A. Saadatmandi, M. Dehghan, 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) (2012) 4125–4136.
[25] F.S. Sarvestani, M.H. Heydari, A. Niknam, Z. Avazzadeh, A wavelet approach for the multi-term
time fractional diffusion–wave equation, Int. J. Comput. Math. 96(3) (2019) 640–661.
[26] L. Su, W. Wang, Q. Xu, Finite difference methods for fractional dispersion equations, Appl. Math.
Comput. 216(11) (2010) 3329–3334.
[27] S. Tarei, A. Kanaujiya, J. Mohapatra Efficient Numerical Method for Pricing Option with Underly
ing Asset Follows a Fractal Stochastic Process Comput. Methods Differ. Equ. (2025).
[28] M.F. Uddin, M.G. Hafez, Z. Hammouch, H. Rezazadeh, D. Baleanu, Traveling wave with beta
derivative spatial-temporal evolution for describing the nonlinear directional couplers with meta
materials via two distinct methods, Alex. Eng. J. 60(1) (2021) 1055-1065.
[29] M. Uddin, S. Haq, Rbfs approximation method for time fractional partial differential equations,
Commun. Nonlinear Sci. Numer. Simul. 16(11) (2011) 4208–4214.
[30] A. Yıldırım, H. Koc¸ak, Homotopy perturbation method for solving the space–time fractional
advection–dispersion equation. Adv. Water Resour. 32(12) (2009) 1711–1716.
[31] F. Zhou, X. Xu, The third kind chebyshev wavelets collocation method for solving the time
fractional convection diffusion equations with variable coefficients, Appl. Math. Comput. 280
(2016) 11–29