A novel and efficient operational matrix method for solving multi-term variable-order fractional differential equations

Document Type : Research Article

Authors

1 Yau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, China

2 School of Mathematics and Computer Science, Iran University of Science and Technology (IUST), Tehran, 16846 13114, Iran

Abstract

The main aim of this research is to present a novel and efficient method based on the Müntz-Legendre polynomials for solving differential equations involving variable-order fractional Caputo derivatives. For the first time, based on the M$\rm{\ddot{u}}$ntz-Legendre polynomials, a formula for the operational matrix of the variable-order fractional Caputo differential operator is derived. By using this operational matrix via the collocation method, we convert the proposed problem into a system of equations. Then, we solve the obtained system by the Newton method to provide an approximate solution for the problem. Furthermore, we obtain an error bound for the approximation. Finally, we solve four test problems to confirm the reliability and effectiveness of the proposed method.

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Main Subjects

References

[1] N. Aghazadeh, A.A. Khajehnasiri, Solving nonlinear two-dimensional volterra integro-differential
equations by block-pulse functions, Math. Sci. 7 (2013) 1–7.
[2] A.H. Bhrawyi, M.A. Zaky, M. Abdel-Aty, A fast and precise numerical algorithm for a class of
variable-order fractional differential equations, Proc. Romanian. Acad. Ser. Math. Phys. Tech. Sci.
Inf. Sci. 18 (2017) 17–24.
[3] H. Brunner, Collocation Methods for Volterra Integral and Related Functional Equations, Cam-
bridge University Press, Cambridge, 2004.
[4] Y.M. Chen, Y.Q. Wei, D.Y. Liu, H. Yu, Numerical solution for a class of nonlinear variable order
fractional differential equations with Legendre wavelets, Appl. Math. Lett. 46 (2015) 83–88.
[5] C.F. M. Coimbra, C.M. Soon, M.H. Kobayashi, The variable viscoelasticity operator, Ann. Phys.
14 (2005) 378–389.
[6] A.A. El-Sayed, D. Baleanu, P. Agarwal, A novel Jacobi operational matrix for numerical solution of
multi-term variable-order fractional differential equations, J. Taibah Univ. Sci. 14 (2020) 963–974.
[7] S.H. Esmaeili, M. Shamsi, Y. Luchkob, Numerical solution of fractional differential equations with
a collocation method based on M ¨untz polynomials, Comput. Math. Appl. 62 (2011) 918–929.
[8] R. Gupta, S. Kumar, Chebyshev spectral method for the variable–order fractional mobile–immobile
advection–dispersion equation arising from solute transport in heterogeneous media, J. Eng. Math.
142, 1 (2023).
[9] M. Hosseinini, M.H. Heydari, Z. Avazzadeh, F.M. Maalek Ghaini, two-dimensional Legendre
wavelets for solving variableorder fractional nonlinear advection-diffusion equation with variable
coefficients, Int. J. Nonlinear. Sci. Num. 19 (2018) 793–802.
[10] D. Ingman, J. Suzdalnitsky, Control of damping oscillations by fractional differential operator with
time-dependent order, Comput. Methods. Appl. Mech. Eng. 193 (2004) 5585–5595.
[11] S. Irandoust-pakchin, H. Kheiri, S. Abdi-mazraeh, Efficient computational algorithms for solving
one class of fractional boundary value problems, Comp. Math. Math. Phys. 53 (2013) 920–932.
[12] A.J. Jerri, Introduction to Integral Equations with Applications, Wiley, NewYork, 1999.
[13] J. Liu, X. Li, L. Wu, An operational matrix of fractional differentiation of the second kind of
Chebyshev polynomial for solving multi-term variable order fractional differential equation, Math.
Probl. Eng. 2016 (2016) 7126080.
[14] Z. Li, H. Wang, R. Xiao, S. Yang, A variable-order fractional differential equation model of shape
memory polymers, Chaos. Solitons. Fract. 102 (2017) 473–485.
[15] A.M. Nagy, N.H. Sweilam, A.A. El-Sayed, New operational matrix for solving multi-term variable
order fractional differential equations, J. Comp. Nonlinear. Dyn. 13 (2018) 011001–011007.
[16] P.W. Ostalczyk, P. Duch, D.W. Brzezinski, D. Sankowski, Order Functions selection in the
variable-, fractional-order PID controller, advances in modelling and control of non-integerorder
systems, Lecture. Notes. Electr. Eng. 320 (2015) 159–170.
[17] S. Sabermahani, Y. Ordokhani, P.M. Lima, A Novel Lagrange Operational Matrix and Tau-
Collocation Method for Solving Variable-Order Fractional Differential Equations, Iran J. Sci. Tech-
nol. Trans. Sci. 44 (2020) 127–135.
[18] S.G. Samko, B. Ross, Integration and differentiation to a variable fractional order, Integr. Transf.
Spec. Funct. 1 (1993) 277–300.
[19] K. Veseli´c, Damped Oscillations of Linear systems – a Mathematical Introduction, Heidelberg,
Springer, 2011.
[20] Y. Xu, Z. He, Existence and uniqueness results for Cauchy problem of variable–order fractional
differential equations, J. Appl. Math. Comput. 43 (2013) 295–306.
Journal of Mathematical Modeling
Volume 13, Issue 3
July 2025
Pages 629-644

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