10.1007/s40096-020-00369-y

Laguerre polynomial solutions of linear fractional integro-differential equations

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  • Ayşegül Daşcıoğlu1,
  • Dilek Varol*1,
  1. Department of Mathematics, Pamukkale University, Denizli, TR

Published in Issue 2021-01-11

How to Cite

Daşcıoğlu, A., & Varol, D. (2021). Laguerre polynomial solutions of linear fractional integro-differential equations. Mathematical Sciences, 15(1 (March 2021). https://doi.org/10.1007/s40096-020-00369-y
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Abstract

Abstract In this paper, the numerical solutions of the linear fractional Fredholm–Volterra integro-differential equations have been investigated. For this purpose, Laguerre polynomials have been used to develop an approximation method. Precisely, using the suitable collocation points, a system of linear algebraic equations arises which is resulted by the transformation of the integro-differential equation. The fractional derivative is considered in the conformable sense, and the conformable fractional derivative of the Laguerre polynomials is obtained in terms of Laguerre polynomials. Additionally, for the first time in the literature, the exact matrix formula of the conformable derivatives of the Laguerre polynomials is established. Furthermore, the results of the proposed method have been given applying the method to some various examples.

Keywords

  • Laguerre polynomials,
  • Fractional Fredholm–Volterra integro-differential equations,
  • Conformable fractional derivative

References

  1. Yüzbaşı (2013) A numerical approximation for Volterra’s population growth model with fractional order (pp. 3216-3227) https://doi.org/10.1016/j.apm.2012.07.041
  2. Sweilam and Khader (2010) A Chebyshev pseudo-spectral method for solving fractional-order integro-differential equations (pp. 464-475) https://doi.org/10.1017/S1446181110000830
  3. Saaedi and Mohseni Moghadam (2011) Numerical solution of nonlinear Volterra integro-differential equations of arbitrary order by CAS wavelets (pp. 1216-1226) https://doi.org/10.1016/j.cnsns.2010.07.017
  4. Wang and Zhu (2017) Solving nonlinear Volterra integro-differential equations of fractional order by using Euler wavelet method https://doi.org/10.1186/s13662-017-1085-6
  5. Maleknejad, K., Sahlan, M. N., Ostadi, A.: Numerical solution of fractional integro-differential equation by using cubic B-spline wavelets. In Proceedings of the World Congress on Engineering 2013 Vol I, London, UK, 3–5 July 2013
  6. Awawdeh et al. (2011) Analytic solution of fractional integro-differential equations 38(1) (pp. 1-10)
  7. Arikoglu and Ozkol (2009) Solution of fractional integro-differential equations by using fractional differential transform method (pp. 521-529) https://doi.org/10.1016/j.chaos.2007.08.001
  8. Sayevand et al. (2013) Convergence analysis of homotopy perturbation method for Volterra integro-differential equations of fractional order (pp. 807-812) https://doi.org/10.1016/j.aej.2013.08.008
  9. Nawaz (2011) Variational iteration method and homotopy perturbation method for fourth-order fractional integro-differential equations (pp. 2330-2341) https://doi.org/10.1016/j.camwa.2010.10.004
  10. Saadatmandi and Dehghan (2011) A Legendre collocation method for fractional integro-differential equations 17(13) (pp. 2050-2058) https://doi.org/10.1177/1077546310395977
  11. Yang et al. (2014) Convergence analysis of the Jacobi spectral-collocation method for fractional integro-differential equations 34(3) (pp. 673-690) https://doi.org/10.1016/S0252-9602(14)60039-4
  12. Ma and Huang (2014) Spectral collocation method for linear fractional integro-differential equations (pp. 1434-1448) https://doi.org/10.1016/j.apm.2013.08.013
  13. Saleh et al. (2013) Approximate solution of fractional integro-differential equation by Taylor expansion and Legendre wavelets methods 15(3) (pp. 89-103) https://doi.org/10.4067/S0719-06462013000300009
  14. Nemati et al. (2016) A fast numerical algorithm based on the second kind Chebyshev polynomials for fractional integro-differential equations with weakly singular kernels (pp. 231-242) https://doi.org/10.1016/j.cam.2016.06.012
  15. Kumar et al. (2017) Comparative study of three numerical schemes for fractional integro-differential equations (pp. 287-302) https://doi.org/10.1016/j.cam.2016.11.013
  16. Turmetov and Abdullaev (2017) Analytic solutions of fractional integro-differential equations of Volterra type. IOP conference series
  17. Maleki and Kajani (2015) Numerical approximations for Volterra’s population growth model with fractional order via a multi-domain pseudospectral method (pp. 4300-4308) https://doi.org/10.1016/j.apm.2014.12.045
  18. Karimi Vanani and Aminataei (2011) Operational Tau approximation for a general class of fractional integro-differential equations 30(3) (pp. 655-674) https://doi.org/10.1590/S1807-03022011000300010
  19. Nazari Susahab et al. (2015) Efficient quadrature rules for solving nonlinear fractional integro-differential equations of the Hammerstein type (pp. 5452-5458) https://doi.org/10.1016/j.apm.2015.01.008
  20. Zhao et al. (2014) Collocation methods for fractional integro-differential equations with weakly singular kernels (pp. 723-743) https://doi.org/10.1007/s11075-013-9710-2
  21. Zhu and Fan (2013) Numerical solution of nonlinear fractional-order Volterra integro-differential equations by SCW (pp. 1203-1213) https://doi.org/10.1016/j.cnsns.2012.09.024
  22. Jiang and Tian (2015) Numerical solution of nonlinear Volterra integro-differential equations of fractional order by the reproducing kernel method (pp. 4871-4876) https://doi.org/10.1016/j.apm.2015.03.053
  23. Fahim et al. (2017) Numerical solution of Volterra partial integro-differential equations based on sinc-collocation method https://doi.org/10.1186/s13662-017-1416-7
  24. Alkan (2017) A numerical method for solution of integro-differential equations of fractional order 21(2) (pp. 82-89) https://doi.org/10.16984/saufenbilder.296796
  25. Nemati and Lima (2018) Numerical solution of nonlinear fractional integro-differential equations with weakly singular kernels via a modification of hat functions (pp. 79-92)
  26. Pedas et al. (2016) Spline collocation for fractional weakly singular integro-differential equations (pp. 204-214) https://doi.org/10.1016/j.apnum.2016.07.011
  27. Elbeleze et al. (2016) Approximate solution of integro-differential equation of fractional (arbitrary) order (pp. 61-68) https://doi.org/10.1016/j.jksus.2015.04.006
  28. Alkan and Hatipoglu (2017) Approximate solutions of Volterra-Fredholm integro-differential equations of fractional order 10(2) (pp. 1-13) https://doi.org/10.1515/tmj-2017-0021
  29. Dehestani et al. (2019) Hybrid functions for numerical solution of fractional Fredholm-Volterra functional integro-differential equations with proportional delays 32(5) https://doi.org/10.1002/jnm.2606
  30. Keshavarz et al. (2019) Numerical solution of nonlinear mixed Fredholm-Volterra integro-differential equations of fractional order by Bernoulli wavelets 7(2) (pp. 163-176)
  31. Loh, J. R., Phang, C., Isah, A.: New operational matrix via Genocchi polynomials for solving Fredholm-Volterra fractional integro-differential equations. Adv. Math. Phys. (2017)
  32. Ali et al. (2019) Solution of fractional Volterra-Fredholm integro-differential equations under mixed boundary conditions by using the HOBW method 2019(1) https://doi.org/10.1186/s13662-019-2044-1
  33. Dehestani et al. (2021) Combination of Lucas wavelets with Legendre-Gauss quadrature for fractional Fredholm-Volterra integro-differential equations https://doi.org/10.1016/j.cam.2020.113070
  34. Bayram et al. (2018) A solution method for integro-differential equations of conformable fractional derivative 22(1) (pp. 7-14) https://doi.org/10.2298/TSCI170624266B
  35. Schoeffel (1999) An elegant and fast method to solve QCD evolution equations Application to the determination of the gluon content of the Pomeron (pp. 439-445) https://doi.org/10.1016/S0168-9002(98)01316-3
  36. Kobayashi et al. (1995) FORTRAN program for a numerical solution of the nonsinglet Altarelli-Parisi equation (pp. 264-278) https://doi.org/10.1016/0010-4655(94)00159-Y
  37. Baykus Savasaneril and Sezer (2016) Laguerre polynomial solution of high-order linear Fredholm integro-differential equations 4(2) (pp. 273-284) https://doi.org/10.20852/ntmsci.2016218534
  38. Gürbüz et al. (2014) Laguerre collocation method for solving Fredholm integro-differential equations with functional arguments (pp. 1-12)
  39. Yüzbaşı (2014) Laguerre approach for solving pantograph-type Volterra integro-differential equations (pp. 1183-1199)
  40. Al-Zubaidy (2013) A Numerical solution of parabolic-type Volterra partial integro-differential equations by Laguerre collocation method 8(4) (pp. 51-55)
  41. Gürbüz and Sezer (2017) A new computational method based on Laguerre polynomials for solving certain nonlinear partial integro differential equations 132(3) (pp. 561-563) https://doi.org/10.12693/APhysPolA.132.561
  42. Gürbüz and Sezer (2017) A numerical solution of parabolic-type Volterra partial integro-differential equations by Laguerre collocation method 7(1) (pp. 49-58) https://doi.org/10.17706/ijapm.2017.7.1.49-58
  43. Gürbüz and Sezer (2017) Laguerre polynomial solutions of a class of delay partial functional differential equations 132(3) (pp. 558-560) https://doi.org/10.12693/APhysPolA.132.558
  44. Mahdy and Shwayyea (2016) Numerical solution of fractional integro-differential equations by least squares method and shifted Laguerre polynomials pseudo-spectral method 7(4) (pp. 1589-1596)
  45. Varol Bayram and Daşcıoğlu (2018) A method for fractional Volterra integro-differential equations by Laguerre polynomials https://doi.org/10.1186/s13662-018-1924-0
  46. Daşcıoğlu and Varol Bayram (2019) Solving fractional Fredholm integro-differential equations by Laguerre polynomials 48(1) (pp. 251-257) https://doi.org/10.17576/jsm-2019-4801-29
  47. Khalil et al. (2014) A new definition of fractional derivative (pp. 65-70) https://doi.org/10.1016/j.cam.2014.01.002
  48. Abdeljawad (2015) On conformable fractional calculus (pp. 57-66) https://doi.org/10.1016/j.cam.2014.10.016
  49. Bell (1968) D. Van Nostrand Company
  50. Lebedev (1972) Dover Publications