Legendre wavelet method combined with the Gauss quadrature rule for numerical solution of fractional integro-differential equations

Document Type : SNA2020- 8th Seminar on Numerical Analysis and its Applications

Author

Department of Mathematics, Higher Education Complex of Saravan, Saravan, Iran.

Abstract

In this paper, we use a novel technique to solve the nonlinear fractional Volterra-Fredholm integro-differential equations (FVFIDEs). To this end, the Legendre wavelets are used in conjunction with the quadrature rule for converting the problem into a linear or nonlinear system of algebraic equations, which can be easily solved by applying mathematical programming techniques. Only a small number of Legendre wavelets are needed to obtain a satisfactory result. Better accuracies are also achieved within the method by increasing the number of polynomials. Furthermore, the existence and uniqueness of the solution are proved by preparing some theorems and lemmas. Also, error estimation and convergence analyses are given for the considered problem and the method. Moreover, some examples are presented and their results are compared to the results of Chebyshev wavelet, Nystro¨m, and Newton–Kantorovitch methods to show the capability and validity of this scheme.
 

Keywords

Main Subjects


1. Abbasbandy, S., Hashemi, M., Hashim, I.On convergence of homotopy analysis method and its application to fractional integro-differential equations, Quaest. Math. 36(1), (2013) 93–105.
2. Alkan, S., Hatipoglu, V.F.
Approximate solutions of Volterra-fredholm integro-differential equations of fractional order, Tbilisi Math. Journal 10(2), (2017) 1–13.
3. Amin, R., Alshahrani, B., Mahmoud, M., Abdel-Aty, A.H., Shah, K., Deebani, W.
Haar wavelet method for solution of distributed order timefractional differential equations, Alex. Eng. J. 60(3), (2021) 3295–3303.
4. Amin, R., Shah, K., Asif, M., Khan, I.
A computational algorithm for the numerical solution of fractional order delay differential equations, Appl. Math. Comput. 402, (2021) 125863.
5. Amin, R., Shah, K., Asif, M., Khan, I., Ullah, F.
An effcient algorithm for numerical solution of fractional integro-differential equations via Haar wavelet, J. Comput. Appl. Math. 381, (2021) 113028.
6. Bhrawy, A., Zaky, M., Van Gorder, R.A.A space-time Legendre spectraltau method for the two-sided space-time Caputo fractional diffusion-wave equation, Numer. Algorithms 71(1) (2016), 151–180.
7. Erfanian, M., Gachpazan, M., Beiglo, H.
A new sequential approach for solving the integro-differential equation via Haar wavelet bases, Comput. Math. Math. Phys. 57(2), (2017) 297–305.
8. Guner, O., Bekir, A.
Exp-function method for nonlinear fractional differential equations, Nonlinear Sci. Lett. A 8, (2017) 41–49 .
9. Hamoud, A., Ghadle, K.
The reliable modified of Laplace Adomian decomposition method to solve nonlinear interval Volterra-Fredholm integral equations, Korean J. Math. 25(3) (2017), 323–334.
10. Hashemi, M., Ashpazzadeh, E., Moharrami, M., Lakestani, M.
Fractional order Alpert multiwavelets for discretizing delay fractional differential equation of pantograph type, Appl. Numer. Math. 170 (2021), 1–13.
11. Hashemi, M.S., Baleanu, D.
Lie symmetry analysis of fractional differential equations, CRC Press (2020)
12. Hashemi, M.S., Darvishi, E., Inc, M.
A geometric numerical integration method for solving the Volterra integro-differential equations, Int. J. Comput. Math. 95(8), (2018) 1654–1665.
13. He, S., Sun, K., Wang, H.
Dynamics of the fractional-order Lorenz system based on Adomian decomposition method and its DSP implementation, IEEE/CAA Journal of Automatica Sinica (2016) doi: 10.1109/JAS.2016.7510133.
14. Heris, J.M.
Solving the integro-differential equations using the modified Laplace Adomian decomposition method, J. Math. Ext. 6 (2012).
15. Hesameddini, E., Rahimi, A., Asadollahifard, E.
On the convergence of a new reliable algorithm for solving multi-order fractional differential equations, Commun. Nonlinear Sci. Numer. Simul. 34, 154–164 (2016)
16. Hesameddini, E., Riahi, M.
Bernoulli Galerkin matrix method and its convergence analysis for solving system of Volterra-Fredholm integrodifferential equations, Iran. J. Sci. Technol. Trans. A Sci. 43(3) (2019), 1203–1214.
17. Hesameddini, E., Riahi, M., Latifizadeh, H.
A coupling method of homotopy technique and Laplace transform for nonlinear fractional differential equations, International Journal of Advances in Applied Sciences 1(4), (2012) 159–170.
18. Hesameddini, E., Shahbazi, M.Hybrid Bernstein block-pulse functions for solving system of fractional integro-differential equations, Int. J. Comput. Math. 95(11) (2018), , 2287–2307.
19. Liu, Z., Cheng, A., Li, X.
A second-order finite difference scheme for quasilinear time fractional parabolic equation based on new fractional derivative, Int. J. Comput. Math. 95(2) (2018), 396–411.
20. Mahdy, A.M., Mohamed, E.M.
Numerical studies for solving system of linear fractional integro-differential equations by using least squares method and shifted Chebyshev polynomials, Fluid Mechanics: Open Access. (2016)03. 10.4172/2476-2296.1000142.
21. Miller, K.S., Ross, B.
An introduction to the fractional calculus and fractional differential equations, A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1993.
22. Modanli, M., Akgül, A.
On solutions of fractional order telegraph partial differential equation by Crank-Nicholson finite difference method, Appl. Math. Nonlinear Sci. 5(1), (2020) 163–170.
23. Mohyud-Din, S.T., Khan, H., Arif, M., Rafiq, M.
Chebyshev wavelet method to nonlinear fractional Volterra–Fredholm integro-differential equations with mixed boundary conditions, Adv. Mech. Eng. 9(3), (2017) 1687814017694802.
24. Nazari S.D., Jahanshahi, M.
Numerical solution of nonlinear fractional Volterra-Fredholm integro-differential equations with mixed boundary conditions, International Journal of Industrial Mathematics 7(1), (2015) 63–69.
25. Nazari, D., Shahmorad, S.
Application of the fractional differential transform method to fractional-order integro-differential equations with nonlocal boundary conditions, J. Comput. Appl. Math. J. 234(3), (2010) 883–891.
26. Oldham, K., Spanier, J.
The fractional calculus theory and applications of differentiation and integration to arbitrary order, vol. 111. Elsevier 1974.
27. Ordokhani, Y., Rahimi, N.
Numerical solution of fractional Volterra integro-differential equations via the rationalized Haar functions, modern research physics 14(3), (2014) 211–2.
28. Pashayi, S., Hashemi, M.S., Shahmorad, S.
Analytical Lie group approach for solving fractional integro-differential equations, Commun. Nonlinear Sci. Numer. Simul. 51, 66–77 (2017)
29. Pedas, A., Tamme, E., Vikerpuur, M.
Spline collocation for a class of nonlinear fractional boundary value problems, AIP Conference Proceedings, vol. 1863, p. 160010. AIP Publishing, 2017.
30. Pirim, N.A., Ayaz, F.A new technique for solving fractional order systems: Hermite collocation method, Applied Mathematics 7(18), (2016) 2307.
31. Podlubny, I.
Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, vol. 198. Academic press, 1998.
32. Sahu, P., Ray, S.S.
A novel Legendre wavelet Petrov-Galerkin method for fractional Volterra integro-differential equations, Comput. Math. with Appl. (2016).
33. Samko, S.G., Kilbas, A.A., Marichev, O.I.
Fractional integrals and derivatives, Theory and Applications, Gordon and Breach, Yverdon 1993.
34. Shah, K., Khan, Z.A., Ali, A., Amin, R., Khan, H., Khan, A.
Haar wavelet collocation approach for the solution of fractional order Covid-19 model using Caputo derivative, Alex. Eng. J. 59(5), (2020) 3221–3231.
35. Singh, B.K.
Homotopy perturbation new integral transform method for numeric study of space-and time-fractional (n + 1)-dimensional heat-and wave-like equations, Waves, Wavelets and Fractals 4(1) (2018)
36. Sun, H., Zhao, X., Sun, Z.z.
The temporal second order difference schemes based on the interpolation approximation for the time multiterm fractional wave equation, J. Sci. Comput. 78(1), (2019) 467–498.
37. Sweilam, N., Nagy, A., Youssef, I.K., Mokhtar, M.M.
New spectral second kind Chebyshev wavelets scheme for solving systems of integro-differential equations, Int. J. Appl. Comput. Math. 3(2), (2017) 333–345.
38. Wang, Y., Zhu, L.
Solving nonlinear Volterra integro-differential equations of fractional order by using euler wavelet method, Adv. Differ. Equ. 2017(1), (2017) 27 pp.
39. Yin, X.B., Kumar, S., Kumar, D.
A modified homotopy analysis method for solution of fractional wave equations, Adv. Mech. Eng. 7(12), (2015) 1687814015620330.
40. Zhu, L., Fan, Q.
Solving fractional nonlinear Fredholm integro-differential equations by the second kind Chebyshev wavelet, Commun. Nonlinear Sci. Numer. Simul. 17(6), (2012) 2333–2341.
CAPTCHA Image