Solving a system of fractional Volterra integro-differential equations using cubic Hermit spline functions

Document Type : Research Paper

Authors

1 Faculty of Mathematics, Statistics and Computer Science, University of Tabriz, Tabriz, Iran.

2 Department of Mathematics, Faculty of Science, University of Jiroft, Jiroft, Iran.

3 Research Center of Performance and Productivity Analysis, Istinye University, Istanbul, Türkiye.

Abstract

In this article, we solve systems of fractional Volterra integro-differential equations in the sense of the Caputo fractional derivative, using cubic Hermite spline functions. We first construct the operational matrix for the fractional derivative of the cubic Hermite spline functions. Then, using this matrix and key properties of these functions, we transform systems of fractional Volterra integro-differential equations into a system of algebraic equations, which can be solved numerically to obtain approximate solutions. Numerous examples show that the results obtained by this method align closely with the results presented by some previous works.

Keywords

Main Subjects

References

  • [1] M. Abbaszadeh, A. Bagheri Salec, A. Al-Khafaji, and S. Kamel, The Effect of Fractional-Order Derivative for Pattern Formation of Brusselator Reaction–Diffusion Model Occurring in Chemical Reactions, Iranian Journal of Mathematical Chemistry, 14(4) (2023), 243-269.
  • [2] M. A. Abdelkawy, A. Z. M. Amin, and A. M. Lopes, Fractional-order shifted Legendre collocation method for solving non-linear variable-order fractional Fredholm integro-differential equations, Computational and Applied Mathematics, 41(1) (2022), 2.
  • [3] B. N. Abood, Approximate solutions and existence of solution for a Caputo nonlocal fractional volterra fredholm integro-differential equation, International Journal of Applied Mathematics, 33(6) (2020), 1049.
  • [4] R. Akbari and L. Navaei, Fractional Dynamics of Infectious Disease Transmission with Optimal Control, Mathematics Interdisciplinary Research, 9(2) (2024), 199-213.
  • [5] M. Akbar, R. Nawaz, S. Ahsan, K. S. Nisar, A. H. AbdelAty, and H. Eleuch, New approach to approximate the solution for the system of fractional order Volterra integro-differential equations, Results in Physics, 19 (2020), 103453.
  • [6] M. Asgari, Numerical Solution for solving a system of Fractional Integro-differential Equations, IAENG International Journal of Applied Mathematics, 45(2) (2015), 85-91.
  • [7] D. Baleanu, A. Saadatmandi, A. Kadem, and M. Dehghan, The fractional linear systems of equations within an operational approach, Journal of Computational and Nonlinear Dynamics, 8(2) (2013), 021011.
  • [8] M. Bergounioux, A. Leaci, G. Nardi, and F. Tomarelli, Fractional Sobolev Spaces and Functions of Bounded Variation of One Variable, Fractional Calculus and Applied Analysis, 20(4) (2017), 936-962.
  • [9] J. Biazar, H. Ghazvini, and M. Eslami, He’s homotopy perturbation method for systems of integro-differential equations, Chaos, Solitons and Fractals, 39(3) (2007), 1253-1258.
  • [10] A. M. Bijura, Systems of Singularly Perturbed Fractional Integral Equations II, IAENG International Journal of Applied Mathematics, 42(4) (2012), 198-203.
  • [11] C. M. Chen, F. Liu, I. Turner, and V. Anh, A Fourier method for the fractional diffusion equation describing sub-diffusion, Journal of Computational Physics, 227(2) (2007), 886-897.
  • [12] W. Dahmen, B. Han, R.Q, Jia, and A. Kunoth, Biorthogonal multiwavelets on the interval: cubic Hermite splines, Constructive Approximation, 16 (2000), 221-259.
  • [13] W. Dahmen, A. Kunoth, and K. Urban, Biorthogonal Spline Wavelets on the Interval-Stability and Moment Conditions, Applied and Computational Harmonic Analysis, 6 (1999), 132-196.
  • [14] S. Das, Functional Fractional Calculus for System Identification and Controls, Springer, New York, 2008.
  • [15] A. Hamoud, Existence and uniqueness of solutions for fractional neutral volterra-fredholm integro differential equations, Advances in the Theory of Nonlinear Analysis and its Application, 4(4) (2020), 321-331.
  • [16] M. H. Heydari, M. R. Hooshmandasl, and F. Mohammadi, Wavelets method for solving systems of nonlinear singular fractional Volterra integro-differential equations, Communications in Nonlinear Science and Numerical Simulation, 19(1) (2014), 37-48.
  • [17] I. Hashim, O. Abdulaziz, and S. Momani, Homotopy analysis method for fractional IVPs, Commun. Communications in Nonlinear Science and Numerical Simulation, 14(3) (2009), 674-684.
  • [18] E. Hesameddini and A. Rahimi, A new numerical scheme for solving systems of integro-differential equations, Computational Methods for Differential Equations, 1(2) (2013), 108-119.
  • [19] K. Diethelm, The analysis of fractional differential equations, Berlin, Springer-Verlag, 2010.
  • [20] M. M. Khader and N. H. Swilam, On the approximate solutions for system of fractional integro-differential equations using Chebyshev pseudo-spectral method, Applied Mathematical Modelling, 37(24) (2013), 9819-9828.
  • [21] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, San Diego, 2006.
  • [22] A. M. Mahdi and E. M. Mohamed, Numerical studies for solving system of linear fractional integro-differential equations by using least squares method and shifted Chebyshev polynomials, Journal of Abstract and Computational Mathematics, 1(1) (2016), 24-32.
  • [23] V. Mazja, Sobolev Spaces, New York, Springer-Verlag, 1985.
  • [24] K. S. Miller and B. Ross, An Introduction to The Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.
  • [25] R. Mohammadzadeh, M. Lakestani, and M. Dehghan, Collocation method for the numerical solutions of LaneEmden type equations using cubic Hermite spline functions, Mathematical Methods in the Applied Sciences, 37(9) (2014), 1303-1717.
  • [26] S. Momani and R. Qaralleh, An efficient method for solving systems of fractional integro-differential equations, Computersand Mathematics with Applications, 52(3-4) (2006), 459-470.
  • [27] S. Momani and Z. Odibat, Analytical approach to linear fractional partial differential equations arising in fluid mechanics, Phys Lett A, 355(4-5) (2006), 271-279.
  • [28] K. B. Oldham and J. Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, Academic Press, New York, 1974.
  • [29] I. Podlubny, Fractional Differential Equations, Academic Press, New York, NY, USA 1999.
  • [30] E. A. Rawashdeh, Numerical solution of fractional integro-differential equations by collocation method, Applied mathematics and computation, 176(1) (2006), 1-6.
  • [31] S. Sabermahani, Y. Ordokhani, and P. Rahimkhani, Spectral methods for solving integro-differential equations and bibiliometric analysis, Topics in Integral and Integro-Differential Equations. Theory and Applications, (2021), 169-214.
  • [32] S. Sabermahani and Y. Ordokhani, A new operational matrix of Muntz-Legendre polynomials and Petrov-Galerkin method for solving fractional Volterra-Fredholm integro-differential equations, Computational Methods for Differential Equations, 8(3) (2020), 408-423.
  • [33] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, Yverdon, 1993.
  • [34] S. Shahmorad, S. Pashaei, and M. S. Hashemi, Numerical solution of a nonlinear fractional integro-differential equation by a geometric approach, Differential Equations and Dynamical Systems, 29 (2021), 585-596.
  • [35] H. Singh, H. Dutta, and M. M. Cavalcanti, Topics in Integral and Integro-Differential Equations, Springer International Publishing., 2021.
  • [36] Y. Yan, K. Pal, and N. J. Ford, Higher order numerical methods for solving fractional differential equations, BIT Numer Math, 54 (2014), 555-584.
  • [37] M. Zurigat, S. Momani, and A. Alawneh, Homotopy analysis method for systems of fractional integro-differential equations, Neural, Parallel and Scientific Computations, 17(2) (2009), 169.
Computational Methods for Differential Equations
Volume 13, Issue 3
July 2025
Pages 980-994

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History

  • Receive Date: 12 July 2024
  • Revise Date: 10 April 2025
  • Accept Date: 19 April 2025

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