In this work, the multi-term variable-order fractional multi-dimensional differential equations are studied based on Gegenbauer wavelet functions. The main aim of this paper is to develop the spectral method with the help of modified operational matrices, which are directly effective in the numerical process. Therefore, we discuss the novel method of obtaining the modified operational matrices (MOMs) of integration and variable-order (VO) fractional derivative. Then, the overall algorithm for solving multi-term VO-fractional differential equations and partial differential equations is introduced. We also discuss error analysis in detail. At last, we implement the numerical scheme in several examples that involve the damped mechanical oscillator equation, the VO-fractional mobile-immobile advection-dispersion equation, and the VO-fractional nonlinear Galilei invariant advection-diffusion equation. Also, to confirm the theoretical results and demonstrate the accuracy and efficiency of the method, we compare our numerical results with analytical solutions and other existing methods.
[1] M. A. Abdelkawy, M. A. Zaky, A. H. Bhrawy, and D. Baleanu, Numerical simulation of time variable fractional order mobile-immobile advection-dispersion model, Rom. Rep. Phys., 67(3) (2015), 773–791.
[2] M. A. Abd-Elkawy and R. T. Alqahtani, Space-time spectral collocation algorithm for the variable-order Galilei invariant advection diffusion equations with a nonlinear source term, Math. Model. Anal., 22(1) (2017), 1–20.
[3] O. A. Arqub and B. Maayah, Adaptive the Dirichlet model of mobile/immobile advection/dispersion in a timefractional sense with the reproducing kernel computational approach: Formulations and approximations, Int. J. Mod. Phys. B., 37(18) (2023), 2350179.
[4] O. A. Arqub, Computational algorithm for solving singular Fredholm time-fractional partial integrodifferential equations with error estimates, J. Appl. Math. Comput., 59(1) (2019), 227–243.
[5] O. A. Arqub and N. Shawagfeh, Application of reproducing kernel algorithm for solving Dirichlet time-fractional diffusion-Gordon types equations in porous media, J. Porous. Media., 22(4) (2019), 411–434.
[6] A. Babaaghaie and K. Maleknejad, Numerical solutions of nonlinear two-dimensional partial Volterra integrodifferential equations by Haar wavelet, J. Comput. Appl. Math., 317 (2017), 643–651.
[7] R. Bagley and P. Torvik, A theoretical basis for the application of fractional calculus to viscoelasticity, J. Rheol., 27 (1983), 201–210.
[8] R. L. Bagley and P. J. Torvik, Fractional calculus in the transient analysis of viscoelastically damped structures, AIAA J., 23 (1985), 918–925.
[9] R. T. Baillie, Long memory processes and fractional integration in econometrics, J. Econometrics., 73 (1996), 5–59.
[10] C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral methods, New York, NY: Springer, 2006.
[11] W. Chen, J. Zhang, and J. Zhang, A variable-order time fractional derivative model for chloride ions sub-diffusion in concrete structures, Fract. Calc. Appl. Anal., 16(1) (2013), 76–92.
[12] Y. M. Chen, Y. Q. Wei, D. Y. Liu, and H. Yu, Numerical solution for a class of nonlinear variable-order fractional differential equations with Legendre wavelets, Appl. Math. Lett., 46 (2015), 83–88.
[13] M. Q. Chen, C. Hwang, and Y. P. Shih, The computation of wavelet-Galerkin approximation on a bounded interval, Int. J. Numer. Methods Eng., 39 (1996), 2921–2944.
[14] C. M. Chen, F. Liu, V. Anh, and I. Turner, Numerical simulation for the variable-order Galilei invariant advection diffusion equation with a nonlinear source term, Appl. Math. Comput., 217(12) (2011), 5729–5742.
[15] C. F. M. Coimbra, Mechanics with variable-order differential operators, Ann. Phys., 12(11–12) (2003), 692–703.
[16] G. R. J. Cooper and D.R. Cowan, Filtering using variable-order vertical derivatives, Comput. Geosci., 30(5) (2004), 455–459.
[17] C. Chui, Wavelets: a mathematical tool for signal analysis, SIAM, 1997.
[18] H. Dehestani, Y. Ordokhani, and M. Razzaghi, Fractional-order Bessel wavelet functions for solving variable-order fractional optimal control problems with estimation error, Int. J. Syst. Sci., 51(6) (2020), 1032–1052.
[19] H. Dehestani and Y. Ordokhani, Designing an efficient algorithm for fractional partial integro-differential viscoelastic equations with weakly singular kernel, CMDE, 13(1) (2025), 214–232.
[20] H. Dehestani, Y. Ordokhani, and M. Razzaghi, Application of the modified operational matrices in multiterm variable-order time-fractional partial differential equations, Math. Meth. Appl. Sci., 42 (2019), 7296–7313.
[21] H. Dehestani, Y. Ordokhani, and M. Razzaghi, Application of fractional Gegenbauer functions in variable-order fractional delay-type equations with non-singular kernel derivatives, Chaos Solitons Fractals., 140 (2020), 110111.
[22] K. Diethelm and A. D. Freed, On the solution of nonlinear fractional-order differential equations used in the modeling of viscoplasticity, in: Scientific Computing in Chemical Engineering II, Springer, (1999), 217–224.
[23] A. A. El-Sayed and P. Agarwal, Numerical solution of multiterm variable-order fractional differential equations via shifted Legendre polynomials, Math. Meth. Appl. Sci., 42(11) (2019), 3978–3991.
[24] M. S. Hashemi, E. Ashpazzadeh, M. Moharrami, and M. Lakestani, Fractional order Alpert multiwavelets for discretizing delay fractional differential equation of pantograph type, Appl. Numer. Math., 170 (2021), 1–13.
[25] B. Hussain, A. Afroz, and A. Abdullah, Haar wavelet based numerical method for solving proportional delay variant of Dirichlet boundary value problems, IJNAA., 14(1) (2023), 287–298.
[26] M. H. Heydari, A. Atangana, Z. Avazzadeh, and M.R. Mahmoudi, An operational matrix method for nonlinear variable-order time fractional reaction–diffusion equation involving Mittag-Leffler kernel, Eur. Phys. J. Plus., 135(2) (2020), 1–19.
[27] M. H. Heydari, M. R. Hooshmandasl, C. Cattani, and G. Hariharan, An optimization wavelet method for multi variable-order fractional differential equations, Fundam. Inform., 151(1–4) (2017), 255–273.
[28] J. Jia, X. Zheng, H. Fu, P. Dai, and H. Wang, A fast method for variable-order space-fractional diffusion equations, Numer. Algor., 85(4D) (2020), 1519–1540.
[29] X. Li and B. Wu, A new reproducing kernel method for variable-order fractional boundary value problems for functional differential equations, J. Comput. Appl. Math., 311 (2017), 387–393.
[30] F. Mainardi, Fractional calculus: some basic problems in continuum and statistical mechanics, in: Fractals and Fractional Calculus in Continuum Mechanics (Udine, 1996), in: CISM Courses and Lect., Springer, Vienna, 378 (1997), 291–348.
[31] H. T. C. Pedro, M. H. Kobayashi, J. M. C. Pereira, and C. F. M. Coimbra, Variable-order modeling of diffusiveconvective effects on the oscillatory flow past a sphere, J. Vib. Control., 14(9–10) (2008), 1659–1672.
[32] L. E. S. Ramirez and C. F. M. Coimbra, A variable-order constitutive relation for viscoelasticity, Ann. Phys., 16(7–8) (2007), 543–552.
[33] Y. A. Rossikhin and M. V. Shitikova, Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids, Appl. Mech. Rev., 50 (1997), 15–67.
[34] S. G. Samko and B. Ross, Integration and differentiation to a variable fractional order, Integral Transforms Spec. Funct., 1(4) (1993), 277–300.
[35] H. G. Sun, W. Chen, H. Sheng, and Y. Q. Chen, Onmean square displacement behaviors of anomalous diffusions with variable and random orders, Phys. Lett. A., 374 (2010), 906–910.
[36] M. Tavassoli Kajania, M. Ghasemi, and E. Babolian, Comparison between the homotopy perturbation method and the sine–cosine wavelet method for solving linear integro-differential equations, Comput. Math. Appl., 54 (2007), 1162–1168.
[37] C. Tseng, Design of variable and adaptive fractional order FIR differentiators, Signal Process., 86(10) (2006), 2554–2566.
[38] M. Usman, M. Hamid, R. U. Haq, and W. Wang, An efficient algorithm based on Gegenbauer wavelets for the solutions of variable-order fractional differential equations, Eur. Phys. J. Plus., 133 (2018), 327.
[39] M. Usman, M. Hamid, T. Zubair, R. U. Haq, W. Wang, and M. B. Liu, Novel operational matrices-based method for solving fractional-order delay differential equations via shifted Gegenbauer polynomials, Appl. Math. Comput., 372 (2020), 124985.
[40] K. Veselic, Damped oscillations of linear systems-A mathematical introduction, Springer, 2011.
[41] S. Yaghoobi, B. Parsa Moghaddam, and K. Ivaz, An efficient cubic spline approximation for variable-order fractional differential equations with time delay, Nonlinear. Dyn., 87(2) (2017), 815–826.
[42] H. Zhang, F. Liu, P. Zhuang, I. Turner, and V. Anh, Numerical analysis of a new space-time variable fractional order advection dispersion equation, Appl. Math. Comput., 242 (2014), 541–550.
[43] H. Zhang, F. Liu, M. S. Phanikumar, and M. M. Meerschaert, A novel numerical method for the time variable fractional order mobile–immobile advection–dispersion model, Comput. Math. Appl., 66(5) (2013), 693–701.
Dehestani, H. and Ordokhani, Y. (2025). A highly accurate wavelet approach for multi-term variable-order fractional multi-dimensional differential equations. Computational Methods for Differential Equations, 13(3), 850-869. doi: 10.22034/cmde.2024.62926.2793
MLA
Dehestani, H. , and Ordokhani, Y. . "A highly accurate wavelet approach for multi-term variable-order fractional multi-dimensional differential equations", Computational Methods for Differential Equations, 13, 3, 2025, 850-869. doi: 10.22034/cmde.2024.62926.2793
HARVARD
Dehestani, H., Ordokhani, Y. (2025). 'A highly accurate wavelet approach for multi-term variable-order fractional multi-dimensional differential equations', Computational Methods for Differential Equations, 13(3), pp. 850-869. doi: 10.22034/cmde.2024.62926.2793
CHICAGO
H. Dehestani and Y. Ordokhani, "A highly accurate wavelet approach for multi-term variable-order fractional multi-dimensional differential equations," Computational Methods for Differential Equations, 13 3 (2025): 850-869, doi: 10.22034/cmde.2024.62926.2793
VANCOUVER
Dehestani, H., Ordokhani, Y. A highly accurate wavelet approach for multi-term variable-order fractional multi-dimensional differential equations. Computational Methods for Differential Equations, 2025; 13(3): 850-869. doi: 10.22034/cmde.2024.62926.2793