Optimal solution of the nonlinear time-fractional diffusion-wave equation using generalized Laguerre polynomials

Goudarzian, Narges; Hassani, Hossein; Dahaghin, Mohammad Shafi

doi:10.22034/cmde.2025.66412.3097

Optimal solution of the nonlinear time-fractional diffusion-wave equation using generalized Laguerre polynomials

Document Type : Research Paper

Authors

¹ Department of Applied Mathematics, Faculty of Mathematical Sciences, Shahrekord University, Shahrekord, Iran.

² Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam.

10.22034/cmde.2025.66412.3097

Abstract

Determining the numerical solutions of a nonlinear fractional differential equation has been of interest for a long time. This study presents an optimization method with Lagrange coefficients based on operational matrices of derivatives to approximate the solution by selecting appropriate basis functions as a linear combination of generalized Laguerre polynomials (GLPs). Achieving an exact solution using fewer basis functions is one of the outstanding features of this method. This feature and high accuracy make the use of this method inevitable. Finally, the application of the mentioned method is investigated in determining the approximate solution of the nonlinear fractional time diffusion-wave equation for different values of $\alpha$.

Keywords

Main Subjects

Fractional partial differential equations

References

[1] A. Afarideh, F. Saei, M. Lakestani, and B. Nemati, Pseudospectral method for solving fractional Sturm-Liouville problem using Chebyshev cardinal functions, Physica Scripta, 96(12) (2021), 125267.
[2] M. A. Attar, M. Roshani, K. Hosseinzadeh, and D. D. Ganji, Analytical solution of fractional differential equations by Akbari–Ganji’s method, Appl. Math, 6 (2022), 100450.
[3] Z. Avazzadeh, H. Hassani, M. J. Ebadi, P. Agarwal, M. Poursadeghfard, and E. Naraghirad, Optimal approximation of fractional order brain tumor model using generalized Laguerre polynomials, J. sci, 47(2) (2023), 501–513.
[4] S. Bonyadi, Y. Mahmoudi, M. Lakestani, and M. Jahangiri Rad, Numerical solution of space-time fractional PDEs with variable coefficients using shifted Jacobi collocation method, Comput. Methods. Diff. Equ, 11(1) (2023), 581–94.
[5] L. Changpin and Z. Wang, Numerical methods for the time fractional convection-diffusion-reaction equation, Numer. Func. Analys. Optimiz, 42(10) (2021), 1115–1153.
[6] M. S. Dahaghin and H. Hassani, An optimization method based on the generalized polynomials for nonlinear variable-order time fractional diffusion-wave equation, Nonlinear Dyn, 88 (2017), 1587–1598.
[7] K. D. Dwivedi, S. Das, and D. Baleanu, Numerical solution of nonlinear space–time fractional-order advection–reaction–diffusion equation, J. Comput. Nonlinear Dyn, 15(6) (2020), 061005.
[8] H. Hassani, J. A. Tenreiro Machado, Z. Avazzadeh, and E. Naraghirad, Generalized shifted Chebyshev polynomials: Solving a general class of nonlinear variable order fractional PDE, Commun. Nonlinear Sci. Numer. Simul, 85 (2020), 105229.
[9] H. Hassani, M. S. Dahaghin, and H. Heydari, A new optimized method for solving variable-order fractional differential equations, Commun. J. Math. Exten, 11 (2017), 85–98.
[10] H. Hassani, S. Mehrabi, E. Naraghirad, M. Naghmachi, and S. Yu¨zba¸si, An optimization method based on the generalized polynomials for a model of HIV infection of CD4+ T cells, Commun. J. Scie. Techno, 44(2) (2020), 407–416.
[11] M. H. Heydari and S. Zhagharian, A hybrid approach for piecewise fractional reaction–diffusion equations, Results. Phys, 51(2) (2023), 106651.
[12] A. Jannelli, M. Ruggier, and M. P. Speciale, Exact and numerical solutions of time-fractional advection–diffusion equation with a nonlinear source term by means of the Lie symmetries, Nonlinear Dyn, 92 (2018), 543–555.
[13] M. Lakestani and J. Manafian, Analytical treatments of the space–time fractional coupled nonlinear Schrödinger equations, Optic. Quantum. Elec, 50 (2018), 1–33.
[14] R. Lin, F. Liu, V. Anh, and I. Turner, Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation, Appl. Math. comput, 212(2) (2009), 435–445.
[15] Y. Ordokhani, S. Sabermahani, and P. Rahimkhani, Application of Chelyshkov wavelets and least squares support vector regression to solve fractional differential equations arising in optics and engineering, Math. Methods Appl. Sci, 48(2) (2025), 1996–2010.
[16] S. Sabermahani, Y. Ordokhani, and P. Agarwal, Application of general Lagrange scaling functions for evaluating the approximate solution time-fractional diffusion-wave equations, Comput. Methods. Diff. Equ, 13(2) (2025), 450–465.
[17] S. Sabermahani, Y. Ordokhani, and H. Hassani, General Lagrange scaling functions: application in general model of variable order fractional partial differential equations, Comput. Appl. Math, 40(8) (2021), 269.
[18] A. Safaei, A. H. Salehi Shayegan, and M. Shahriari, Two-dimensional temporal fractional advection-diffusion problem resolved through the Sinc-Galerkin method, Comput. Methods. Diff. Equ, 13(3) (2025), 1047–1058.
[19] H. Safdari, M. Rajabzadeh, and M. Khalighi, LDG approximation of a nonlinear fractional convection-diffusion equation using B-spline basis functions, Appl. Numer. Math, 171 (2022), 45–57.
[20] K. K. Saha, N. Sukavanam, and S. Pan, Existence and uniqueness of solutions to fractional differential equations with fractional boundary conditions, A. Engrg. J, 72 (2023), 147–155.
[21] N. Van Hoa, T. Allahviranloo, and W. Pedrycz, A new approach to the fractional Abel k-integral equations and linear fractional differential equations in a fuzzy environment, Fuzzy Sets. Systems, 481 (2024), 108895.
[22] Z. Yang, A finite difference method for fractional partial differential equation, Appl. Math. comput, 215(2) (2009), 24–529.
[23] B. Zheng and W. Chuanbao, Exact solutions for fractional partial differential equations by a new fractional sub-equation method, Advances in Difference Equations, 2013(1) (2013), 199.
[24] Y. H. Zhou, X. M. Wang, J. Z. Wang, and X. J. Liu, A wavelet numerical method for solving nonlinear fractional vibration, diffusion and wave equations, Computer. Model.Engrg. Sci, 77(2) (2011), 137.