Solutions of Linear and Nonlinear Fractional Fredholm Integro-Differential Equations

Main Article Content

Abdelhalim Ebaid
Hind K. Al-Jeaid

Abstract

The present paper analyzes a class of first-order fractional Fredholm integro differential equations in terms of Caputo fractional derivative. In the literature, such kind of fractional integro-differential equations have been solved using several numerical methods, while the exact solutions were not obtained. However, the exact solutions are obtained in this paper for various linear and nonlinear examples. It is shown that the exact solution of the linear problems is unique, while multiple exact solutions exist for the nonlinear ones. Moreover, the obtained results reduce to the classical ones in the relevant literature as the fractional order becomes unity. The obtained exact solutions can be further invested by other researchers to validate their numerical/approximation methods.

Article Details

References

  1. R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publishing Company, Singapore, 2000.
  2. J.H. He, Approximate analytical solution for seepage flow with fractional derivatives in porous media, Comput. Meth. Appl. Mech. Eng. 167 (1998), 57-68. https://doi.org/10.1016/S0045-7825(98)00108-X.
  3. O.P. Agrawal, A new Lagrangian and a new Lagrange equation of motion for fractionally damped systems, J. Appl. Mech. 68 (2001), 339–341. https://doi.org/10.1115/1.1352017.
  4. V. Kiryakova, Generalized Fractional Calculus and Applications, Pitman Research Notes in Mathematics, Longman, Harlow, 1994.
  5. Yu. F. Luchko, H.M Srivastava, The exact solution of certain differential equations of fractional order by using operational calculus, Computers Math. Appl. 29(8) (1995), 73-85. https://doi.org/10.1016/0898-1221(95)00031-S.
  6. N. Sebaa, Z.E.A. Fellah, W. Lauriks, C. Depollier, Application of fractional calculus to ultrasonic wave propagation in human cancellous bone, Signal Process. 86 (2006), 2668-2677. https://doi.org/10.1016/j.sigpro.2006.02.015.
  7. Yongsheng Ding, Haiping Yea, A fractional-order differential equation model of HIV infection of CD4+T-cells, Math. Comput. Model. 50 (2009), 386-392. https://doi.org/10.1016/j.mcm.2009.04.019.
  8. A. Ebaid, D.M.M. El-Sayed, M.D. Aljoufi, Fractional calculus model for damped mathieu equation: approximate analytical solution, Appl. Math. Sci. 6 (2012), 4075–4080.
  9. Lei Song, Shiyun Xu, Jianying Yang, Dynamical models of happiness with fractional order, Commun. Nonlinear Sci. Numer. Simulat. 15 (2010), 616-628. https://doi.org/10.1016/j.cnsns.2009.04.029.
  10. A. Ebaid, Analysis of projectile motion in view of the fractional calculus, Appl. Math. Model. 35 (2011), 1231-1239. https://doi.org/10.1016/j.apm.2010.08.010.
  11. A. Ebaid, E.R. El-Zahar, A.F. Aljohani, Bashir Salah, Mohammed Krid, J. Tenreiro Machado, Analysis of the twodimensional fractional projectile motion in view of the experimental data, Nonlinear Dyn. 97 (2019), 1711-1720. https://doi.org/10.1007/s11071-019-05099-y.
  12. E.R. Elzahar, A.A. Gaber, A.F. Aljohani, J.T. Machado, A. Ebaid, Generalized Newtonian fractional model for the vertical motion of a particle, Appl. Math. Model. 88 (2020), 652-660. https://doi.org/10.1016/j.apm.2020.06.054.
  13. E.R. El-Zahar, A.M. Alotaibi, A. Ebaid, A.F. Aljohani, J.F. Gómez Aguilar, The Riemann-Liouville fractional derivative for Ambartsumian equation, Results Phys. 19 (2020), 103551. https://doi.org/10.1016/j.rinp.2020.103551.
  14. A. Ebaid, C. Cattani, A.S. Al Juhani1, E.R. El-Zahar, A novel exact solution for the fractional Ambartsumian equation, Adv. Differ. Equ. 2021 (2021), 88. https://doi.org/10.1186/s13662-021-03235-w.
  15. A. Peedas, E. Tamme, Spline collocation method for integro-differential equations with weakly singular kernels, J. Comput. Appl. Math. 197 (2006), 253-269. https://doi.org/10.1016/j.cam.2005.07.035.
  16. H. Jafari, V. Daftardar-Gejji, Solving a system of nonlinear fractional differential equations using Adomian decomposition, J. Comput. Appl. Math. 196(2) (2006), 644-651. https://doi.org/10.1016/j.cam.2005.10.017.
  17. D. Nazari, S. Shahmorad, Application of the fractional differential transform method to fractional-order integrodifferential equations with nonlocal boundary conditions, J. Comput. Appl. Math. 234 (2010), 883-891. https://doi.org/10.1016/j.cam.2010.01.053.
  18. B. Ghazanfari, A.G. Ghazanfari, F. Veisi, Homotopy perturbation method for nonlinear fractional integro-differential equations, Aust. J. Basic Appl. Sci. 12(4) (2010), 5823-5829.
  19. S. Karimi Vanani, A. Aminataei, Operational Tau approximation for a general class of fractional integro-differential equations, Comput. Appl. Math. 3(30) (2011), 655-674. https://doi.org/10.1590/S1807-03022011000300010.
  20. Y. Ordokhani, N. Rahimi, Numerical solution of fractional Volterra integro-differential equations via the rationalized Haar functions, J. Sci. Kharazmi Univ. 14(3) (2014), 211-224.
  21. S. Bushnaq, B. Maayah, S. Momani, A. Alsaedi, A reproducing kernel Hilbert space method for solving systems of fractional integrodifferential equations, Abstr. Appl. Anal. 2014 (2014), Article ID 103016. https://doi.org/10.1155/2014/103016.
  22. M.R. Eslahchi, M. Dehghanb, M. Parvizi, Application of the collocation method for solving nonlinear fractional integro-differential equations, J. Comput. Appl. Math. 257 (2014), 105-128. https://doi.org/10.1016/j.cam.2013.07.044.
  23. M.H. Heydari, M.R. Hooshmandasl, F. Mohammadi, C. Cattani, Wavelets method for solving systems of nonlinear singular fractional Volterra integro-differential equations, Commun. Nonlinear Sci. 19 (2014), 37-48. https://doi.org/10.1016/j.cnsns.2013.04.026.
  24. W. Jiang, T. Tian, Numerical solution of nonlinear Volterra integro-differential equations of fractional order by the reproducing kernel method, Appl. Math. Model. 39 (2015), 4871-4876. https://doi.org/10.1016/j.apm.2015.03.053.
  25. D. Nazari Susahab, M. Jahanshahi, Numerical solution of nonlinear fractional Volterra Fredholm integro-differential equations with mixed boundary conditions, Int. J. Ind. Math. 7 (2017), 63-69.
  26. A.A. Al-Marashi, Approximate solution of the system of linear fractional integro-differential equations of Volterra using B-spline method, Amer. Res. Math. Stat. 3(2) (2015), 39-47. https://doi.org/10.15640/arms.v3n2a6.
  27. R.L. Jian, P. Chang, A. Isah, New operational matrix via Genocchi polynomials for solving Fredholm Volterra fractional integro-differential equations, Adv. Math. Phys. 2017 (2017), 3821870. https://doi.org/10.1155/2017/3821870.
  28. H. Khan, M. Arif, S.T. Mohyud-Din, S. Bushnaq, Numerical solutions to systems of fractional Voltera integro differential equations, using Chebyshev wavelet method, J. Taibah Univ. Sci. 12 (2018), 584–591. https://doi.org/10.1080/16583655.2018.1510149.
  29. A.M. Wazwaz, Linear and Nonlinear Integral Equations, Methods and Applications, Higher Education Press, Beijing and Springer-Verlag, Berlin Heidelberg (2011).