# \$C^{3}\$-spline Methods for Solving Fractional Integro-differential Equations

Document Type : Original Article

Authors

1 Department of Mathematics‎, ‎Karaj Branch‎, ‎Islamic Azad University‎, ‎Karaj 31499-68111

2 Iran University of Science and Technology

3 3Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj

Abstract

Fractional integro-differential equations (FIDEs) constitute an important mathematical tool in modeling many dynamical processes‎. ‎To solve FIDEs‎, ‎several analytical and numerical methods have been proposed‎, ‎namely those based on symmetry and spline approaches‎. ‎This paper proposes quartic and sextic C3-spline methods for the numerical solution of FIDEs‎. ‎The convergence analysis of the proposed strategy is examined in detail‎. ‎Finally‎, ‎three numerical examples are given to illustrate the numerical accuracy and efficiency of the proposed strategy‎.

Keywords

Main Subjects

#### References

[1] A.M. Ahmad, K.M. Furati, N.E. Tatar, Asymptotic behavior of solutions for a class of fractional integro-differential equations, Mediterranean Journal of Mathematics, 15(5) 2018, 1-19.
[2] I. Ahmad, H. Ahmad, P. Thounthong, Y.M. Chu, C. Cesarano, Solution of multi-term timefractional PDE models arising in mathematical biology and physics by local meshless method, Symmetry, 12(7) 2020, 1195
[3] K. Atkinson, The numerical solution of integral equations of the second kind, cambridge, Cambridge university, 1997.
[4] Z. Avazzadeh, O. Nikan, A.T. Nguyen, et al, A localized hybrid kernel meshless technique for solving the fractional R-ayleigh-S-tokes problem for an edge in a viscoelastic fluid. Engineering Analysis with Boundary Elements,146 2023, 695 705.
[5] R.L. Bagley, P.Torvik, A theoretical basis for the application of fractional calculus to viscoelasticity, Journal of Rheology, 27(3) 1983, 201-210.
[6] R.T. Baillie, Long memory processes and fractional integration in econometrics. Journal of econometrics, 73(1) 1996, 5-59.
[7] H. Bateman, Higher transcendental functions, McGraw Hill, New York, 1 1953.
[8] N.H. Can, O. Nikan, M.N. Rasoulizadeh, H. Jafari, Y.S. Gasimov, Numerical computation of the time non-linear fractional generalized equal width model arising in shallow water channel, Thermal Science, 24(Suppl. 1) 2020, 49-58.
[9] M. Dehghan, J. Manafian, A.Saadatmandi, Solving nonlinear fractional partial differential equations using the homotopy analysis method. Numerical Methods for Partial Differential Equations: An International Journal, 26(2) 2010, 448-479.
[10] A.A. Elbeleze, A. Kılıçman, B.M. Taib, Approximate solution of integro-differential equation of fractional (arbitrary) order. Journal of King Saud University-Science, 28(1) 2016, 61-68
[11] M. Gülsu, Y. Öztürk, A. Anapalı, Numerical approach for solving fractional Fredholm integrodifferential equation. International Journal of Computer Mathematics, 90(7) 2013, 1413-1434.
[12] Q. Huang, O.Nikan, Z. Avazzadeh, Numerical analysis of alternating direction implicit orthogonal Spline collocation scheme for the hyperbolic integrodifferential equation with a weakly singular kernel. Mathematics, 10(18) 2022, 3390.
[13] N. Kosmatov, Integral equations and initial value problems for nonlinear differential equations of fractional order. Nonlinear Analysis: Theory, Methods and Applications, 70(7) 2009, 2521-2529.
[14] S. Larsson, M. Racheva, F. Saedpanah, Discontinuous Galerkin method for an integrodifferential equation modeling dynamic fractional order viscoelasticity. Computer Methods in Applied Mechanics and Engineering, 283 2015, 196-209.
[15] Y. Li, W. Zhao, Haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations. Applied mathematics and computation, 216(8) 2010, 2276-2285
[16] L.D. Long, B. Moradi, O. Nikan, Z. Avazzadeh, A.M. Lopes, Numerical approximation of the fractional Rayleigh–Stokes problem arising in a generalised maxwell fluid. Fractal and Fractional, 6(7) 2022, 377.
[17] M. Luo, W. Qiu, O. Nikan, Z. Avazzadeh, Second-order accurate, robust and efficient ADI Galerkin technique for the three-dimensional nonlocal heat model arising in viscoelasticity, Applied Mathematics and Computation, 440 2023, 127655.
[18] J.T. Machado, V. Kiryakova,: The chronicles of fractional calculus. Fractional Calculus and Applied Analysis, 20(2) 2017, 307-336.
[19] F. Mainardi, Fractional calculus. In: Fractals and fractional calculus in continuum mechanics, 1997 291-348.
[20] K.S. Miller, B.Ross, An introduction to the fractional calculus and fractional differential equations. Wiley 1993.
[21] R. Mittal, R.Nigam, Solution of fractional integro-differential equations by Adomian decomposition method. Int. J. Appl. Math. Mech, 4(2) 2008, 87-94.
[22] S. Mohammadizadeh, J. Rashidinia, R. Ezzati, M. Khumalo, C3-spline for solution of second order fractional integro-differential equations. Alexandria Engineering Journal, 59(5) 2020, 3635-3641
[23] N. Nie, Y. Zhao, M. Li, X. Liu, S. Jiménez, Y. Tang, L. Vázquez, Solving two-point boundary value problems of fractional differential equations via Spline collocation methods. International Journal of Modeling, Simulation, and Scientific Computing, 1(01) 2010, 117-132.
[24] O. Nikan, Z. Avazzadeh, An improved localized radial basis-pseudospectral method for solving fractional reaction-subdiffusion problem. Results in Physics, 23 2021, 104048.
[25] O. Nikan, Z. Avazzadeh, J.A Tenreiro Machado, Localized kernel-based meshless method for pricing financial options underlying fractal transmission system. Mathematical Methods in the Applied Sciences, 47(5) 2024, 3247-3260.
[26] O. Nikan, A. Golbabai, J.T. Machado, T. Nikazad, Numerical approximation of the time fractional cable model arising in neuronal dynamics. Engineering with Computers, 38(1) 2022,155-173.
[27] O. Nikan, S.M. Molavi-Arabshai, H. Jafari, Numerical simulation of the nonlinear fractional regularized long-wave model arising in ion acoustic plasma waves. Discrete and Continuous Dynamical Systems-S, 14(10) 2021, 3685.
[28] O. Nikan, J. Tenreiro Machado, A. Golbabai, T. Nikazad, Numerical investigation of the nonlinear modified anomalous diffusion process. Nonlinear Dynamics, 97(4) 2019, 2757-2775.
[29] A. Pedas, E. Tamme, Numerical solution of nonlinear fractional differential equations by spline collocation methods. Journal of Computational and Applied Mathematics, 255 2014, 216-230.
[30] I. Podlubny, Fractional differential equations, mathematics in science and engineering. Academic press New York, 1999.
[31] P.M. Prenter, et al. Splines and variational methods, John Wiley and Sons, New York, NY, USA. Courier Corporation, 2008.
[32] Y.A. Rossikhin, M.Shitikova, Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids. Applied Mechanics Reviews, 50(1) 2017, 15-67.
[33] R. Russell, L.F. Shampine, A collocation method for boundary value problems. Numerische Mathematik, 19(1) 1972, 1-28.
[34] A. Saadatmandi, M.Dehghan, A legendre collocation method for fractional integro-differential equations. Journal of Vibration and Control, 17(13) 2011, 2050-2058
[35] J. Sabatier, O.P. Agrawal, J.T. Machado, Advances in fractional calculus, 4 2007.
[36] H. Saeedi, M.M. Moghadam, Numerical solution of nonlinear Volterra integro-differential equations of arbitrary order by CAS wavelets. Communications in Nonlinear Science and Numerical Simulation, 16(3) 2011, 1216-1226.
[37] S. Sallam, A. Karaballi, A quartic c3-spline collocation method for solving second-order initial value problems. Journal of computational and applied mathematics, 75(2) 1996, 295-304.
[38] R. Ye, C. Wang, A. Shu, H. Zhang, Quasi-synchronization and quasi-uniform synchronization of Caputo fractional variable-parameter neural networks with probabilistic time-varying delays. Symmetry, 14(5) 20222, 1035.
[39] L. Yuanlu, Solving a nonlinear fractional differential equation using Chebyshev wavelets. Communications in Nonlinear Science and Numerical Simulation, 15(9) 2010, 2284-2292

### History

• Receive Date: 13 February 2024
• Revise Date: 15 March 2024
• Accept Date: 21 March 2024