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

[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