# Laguerre polynomial solutions of linear fractional integro-differential equations

## 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.

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## References

1. 1.

Yüzbaşı, Ş.: A numerical approximation for Volterra’s population growth model with fractional order. Appl. Math. Model. 37, 3216–3227 (2013)

2. 2.

Sweilam, N.H., Khader, M.M.: A Chebyshev pseudo-spectral method for solving fractional-order integro-differential equations. ANZIAM J. 51, 464–475 (2010)

3. 3.

Saaedi, H., Mohseni Moghadam, M.: Numerical solution of nonlinear Volterra integro-differential equations of arbitrary order by CAS wavelets. Commun. Nonlinear Sci. Numer. Simulat. 16, 1216–1226 (2011)

4. 4.

Wang, Y., Zhu, L.: Solving nonlinear Volterra integro-differential equations of fractional order by using Euler wavelet method. Adv. Differ. Equ. 1, 27 (2017)

5. 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. 6.

Awawdeh, F., Rawashdeh, E.A., Jaradat, H.M.: Analytic solution of fractional integro-differential equations. Ann. Univ. Craiova Math. Comput. Sci. Ser. 38(1), 1–10 (2011)

7. 7.

Arikoglu, A., Ozkol, I.: Solution of fractional integro-differential equations by using fractional differential transform method. Chaos, Solitons Fractals 40, 521–529 (2009)

8. 8.

Sayevand, K., Fardi, M., Moradi, E., Hemati Boroujeni, F.: Convergence analysis of homotopy perturbation method for Volterra integro-differential equations of fractional order. Alexandria Eng. J. 52, 807–812 (2013)

9. 9.

Nawaz, Y.: Variational iteration method and homotopy perturbation method for fourth-order fractional integro-differential equations. Comput. Math Appl. 61, 2330–2341 (2011)

10. 10.

Saadatmandi, A., Dehghan, M.: A Legendre collocation method for fractional integro-differential equations. J. Vib. Control 17(13), 2050–2058 (2011)

11. 11.

Yang, Y., Chen, Y., Huang, Y.: Convergence analysis of the Jacobi spectral-collocation method for fractional integro-differential equations. Acta Math. Sci. Ser. B 34(3), 673–690 (2014)

12. 12.

Ma, X., Huang, C.: Spectral collocation method for linear fractional integro-differential equations. Appl. Math. Model. 38, 1434–1448 (2014)

13. 13.

Saleh, M.H., Amer, S.M., Mohamed, M.A., Abdelrhman, N.S.: Approximate solution of fractional integro-differential equation by Taylor expansion and Legendre wavelets methods. Cubo A Math. J. 15(3), 89–103 (2013)

14. 14.

Nemati, S., Sedaghat, S., Mohammadi, I.: A fast numerical algorithm based on the second kind Chebyshev polynomials for fractional integro-differential equations with weakly singular kernels. J. Comput. Appl. Math. 308, 231–242 (2016)

15. 15.

Kumar, K., Pandey, R.K., Sharma, S.: Comparative study of three numerical schemes for fractional integro-differential equations. J. Comput. Appl. Math. 315, 287–302 (2017)

16. 16.

Turmetov, B., Abdullaev, J.: Analytic solutions of fractional integro-differential equations of Volterra type. IOP conference series. J. Phys: Conf. Ser. 890, 012113 (2017)

17. 17.

Maleki, M., Kajani, M.T.: Numerical approximations for Volterra’s population growth model with fractional order via a multi-domain pseudospectral method. Appl. Math. Model. 39, 4300–4308 (2015)

18. 18.

Karimi Vanani, S., Aminataei, A.: Operational Tau approximation for a general class of fractional integro-differential equations. Comput. Appl. Math. 30(3), 655–674 (2011)

19. 19.

Nazari Susahab, D., Shahmorad, S., Jahanshahi, M.: Efficient quadrature rules for solving nonlinear fractional integro-differential equations of the Hammerstein type. Appl. Math. Model. 39, 5452–5458 (2015)

20. 20.

Zhao, J., Xiao, J., Ford, N.J.: Collocation methods for fractional integro-differential equations with weakly singular kernels. Numer. Algor. 65, 723–743 (2014)

21. 21.

Zhu, L., Fan, Q.: Numerical solution of nonlinear fractional-order Volterra integro-differential equations by SCW. Commun. Nonlinear Sci. Numer. Simulat. 18, 1203–1213 (2013)

22. 22.

Jiang, W., Tian, T.: Numerical solution of nonlinear Volterra integro-differential equations of fractional order by the reproducing kernel method. Appl. Math. Model. 39, 4871–4876 (2015)

23. 23.

Fahim, A., Fariborzi Araghi, M.A., Rashidinia, J., Jalalvand, M.: Numerical solution of Volterra partial integro-differential equations based on sinc-collocation method. Adv. Differ. Equ. 1, 362 (2017)

24. 24.

Alkan, S.: A numerical method for solution of integro-differential equations of fractional order. Sakarya Üniversitesi Fen Bilimleri Enstitüsü Dergisi 21(2), 82–89 (2017)

25. 25.

Nemati, S., Lima, P.M.: Numerical solution of nonlinear fractional integro-differential equations with weakly singular kernels via a modification of hat functions. Appl. Math. Comput. 327, 79–92 (2018)

26. 26.

Pedas, A., Tamme, E., Vikerpuur, M.: Spline collocation for fractional weakly singular integro-differential equations. Appl. Numer. Math. 110, 204–214 (2016)

27. 27.

Elbeleze, A.A., Kılıçman, A., Taib, M.T.: Approximate solution of integro-differential equation of fractional (arbitrary) order. J. King Saud Univ. Sci. 28, 61–68 (2016)

28. 28.

Alkan, S., Hatipoglu, V.F.: Approximate solutions of Volterra-Fredholm integro-differential equations of fractional order. Tbilisi Math. J. 10(2), 1–13 (2017)

29. 29.

Dehestani, H., Ordokhani, Y., Razzaghi, M.: Hybrid functions for numerical solution of fractional Fredholm-Volterra functional integro-differential equations with proportional delays. Int. J. Numer. Model. Electron. Netw. Dev. Fields 32(5), e2606 (2019)

30. 30.

Keshavarz, E., Ordokhani, Y., Razzaghi, M.: Numerical solution of nonlinear mixed Fredholm-Volterra integro-differential equations of fractional order by Bernoulli wavelets. Comput. Methods Differ. Equ. 7(2), 163–176 (2019)

31. 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. 32.

Ali, M.R., Hadhoud, A.R., Srivastava, H.M.: Solution of fractional Volterra-Fredholm integro-differential equations under mixed boundary conditions by using the HOBW method. Adv. Differ. Equ. 2019(1), 115 (2019)

33. 33.

Dehestani, H., Ordokhani, Y., Razzaghi, M.: Combination of Lucas wavelets with Legendre-Gauss quadrature for fractional Fredholm-Volterra integro-differential equations. J. Comput. Appl. Math. 382, 113070 (2021)

34. 34.

Bayram, M., Hatipoglu, V.F., Alkan, S., Das, S.E.: A solution method for integro-differential equations of conformable fractional derivative. Thermal Sci. 22(1), 7–14 (2018)

35. 35.

Schoeffel, L.: An elegant and fast method to solve QCD evolution equations Application to the determination of the gluon content of the Pomeron. Nuclear Instrum. Methods Phys. Res. A 423, 439–445 (1999)

36. 36.

Kobayashi, R., Konuma, M., Kumano, S.: FORTRAN program for a numerical solution of the nonsinglet Altarelli-Parisi equation. Comput. Phys. Commun. 86, 264–278 (1995)

37. 37.

Baykus Savasaneril, N., Sezer, M.: Laguerre polynomial solution of high-order linear Fredholm integro-differential equations. New Trends Math. Sci. 4(2), 273–284 (2016)

38. 38.

Gürbüz, B., Sezer, M., Güler, C.: Laguerre collocation method for solving Fredholm integro-differential equations with functional arguments. J. Appl. Math. Article ID 682398, 1–12 (2014)

39. 39.

Yüzbaşı, Ş.: Laguerre approach for solving pantograph-type Volterra integro-differential equations. Appl. Math. Comput. 232, 1183–1199 (2014)

40. 40.

Al-Zubaidy, K.A.: A Numerical solution of parabolic-type Volterra partial integro-differential equations by Laguerre collocation method. Int. J. Sci. Technol. 8(4), 51–55 (2013)

41. 41.

Gürbüz, B., Sezer, M.: A new computational method based on Laguerre polynomials for solving certain nonlinear partial integro differential equations. Acta Phys. Polon. A 132(3), 561–563 (2017)

42. 42.

Gürbüz, B., Sezer, M.: A numerical solution of parabolic-type Volterra partial integro-differential equations by Laguerre collocation method. Int. J. Appl. Phys. Math. 7(1), 49–58 (2017)

43. 43.

Gürbüz, B., Sezer, M.: Laguerre polynomial solutions of a class of delay partial functional differential equations. Acta Phys. Polon. A 132(3), 558–560 (2017)

44. 44.

Mahdy, A.M.S., Shwayyea, R.T.: Numerical solution of fractional integro-differential equations by least squares method and shifted Laguerre polynomials pseudo-spectral method. Int. J. Sci. Eng. Res. 7(4), 1589–1596 (2016)

45. 45.

Varol Bayram, D., Daşcıoğlu, A.: A method for fractional Volterra integro-differential equations by Laguerre polynomials. Adv. Differ. Equ. 2018, 466 (2018)

46. 46.

Daşcıoğlu, A., Varol Bayram, D.: Solving fractional Fredholm integro-differential equations by Laguerre polynomials. Sains Malaysiana 48(1), 251–257 (2019)

47. 47.

Khalil, R., Horani, M.A., Yousef, A., Sababheh, M.: A new definition of fractional derivative. J. Comput. Appl. Math. 264, 65–70 (2014)

48. 48.

Abdeljawad, T.: On conformable fractional calculus. J. Comput. Appl. Math. 279, 57–66 (2015)

49. 49.

Bell, W.W.: Special Functions for Scientists and Engineers. D. Van Nostrand Company, London (1968)

50. 50.

Lebedev, N.N.: Special Functions and Their Applications. Dover Publications, New York (1972)

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Correspondence to Dilek Varol.

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Daşcıoğlu, A., Varol, D. Laguerre polynomial solutions of linear fractional integro-differential equations. Math Sci (2021). https://doi.org/10.1007/s40096-020-00369-y

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### Keywords

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

• 26A33
• 33C45
• 34A08
• 45J05