An Efficient Numerical Approach for Approximating Nonlocal Variable-Order Weakly Singular Integro-Differential Equations

Subject Areas : Numerical Analysis

Nayereh Tanha ¹ , Behrouz Parsa Moghaddam ^{2 *} , Mousa Ilie ³

1 - Department of Mathematics, Lahijan Branch, Islamic Azad University, Lahijan, Iran
2 - Department of Mathematics, Lahijan Branch, Islamic Azad University, Lahijan, Iran
3 - Department of Mathematics, Rasht Branch, Islamic Azad University, Rasht, Iran

Received: 2024-10-08 Accepted : 2024-12-05 Published : 2024-02-11

Keywords: Fractional calculus \sep Variable-order fractional derivative \sep Fractional differential equations, Spline interpolation, Numerical optimization, Weakly singular integro-differential equations.,

Abstract :

This paper presents an efficient numerical method for approximating variable-order fractional derivatives using an Integro spline quasi-interpolation approach. The proposed technique is extended to address nonlocal variable-order weakly singular integro-differential equations. Several illustrative examples are provided to validate the effectiveness and performance of the numerical scheme. Additionally, the optimal error orders are determined by minimizing the mean absolute error, demonstrating the method’s accuracy and computational efficiency. 

 

References:


1. Alawneh, A., Al-Khaled, K., & Al-Towaiq, M. (2010). Reliable algorithms for solving integro-differential equations with applications. *International Journal of Computer Mathematics*, 87(7), 1538-1554. https://doi.org/10.1080/00207160802385818

2. Ahmad, J., Iqbal, A., & Ul Mahmood, Q. H. (2021). Study of nonlinear fuzzy integro-differential equations using mathematical methods and applications. *International Journal of Fuzzy Logic and Intelligent Systems*, 21(1), 76-85. https://doi.org/10.5391/IJFIS.2021.21.1.76

3. Bakirova, E. A., Assanova, A. T., & Kadirbayeva, Z. M. (2021). A problem with parameter for the integro-differential equations. *Mathematical Modelling and Analysis*, 26(1), 34-54. https://doi.org/10.3846/mma.2021.11977

4. Zeb, H., Sohail, M., Alrabaiah, H., & Naseem, T. (2021). Utilization of Chebyshev collocation approach for differential, differential-difference and integro-differential equations. *Arab Journal of Basic and Applied Sciences*, 28(1), 413-426. https://doi.org/10.1080/25765299.2021.1997442

5. Sunthrayuth, P., Ullah, R., Khan, A., Shah, R., Kafle, J., Mahariq, I., & Jarad, F. (2021). Numerical analysis of the fractional-order nonlinear system of Volterra integro-differential equations. *Journal of Function Spaces*, 2021, Article ID 1537958. https://doi.org/10.1155/2021/1537958

6. Mohammed, J. K., & Khudair, A. R. (2023). Integro-differential equations: Numerical solution by a new operational matrix based on fourth-order hat functions. *Partial Differential Equations in Applied Mathematics*, 8, 100529. https://doi.org/10.1016/j.padiff.2023.100529

7. Amin, R., Shah, K., Asif, M., Khan, I., & Ullah, F. (2021). An efficient algorithm for numerical solution of fractional integro-differential equations via Haar wavelet. *Journal of Computational and Applied Mathematics*, 381, 113028. https://doi.org/10.1016/j.cam.2020.113028

8. Gürbüz, B. (2022). A numerical scheme for the solution of neutral integro-differential equations including variable delay. *Mathematical Sciences*, 16(1), 13-21. https://doi.org/10.1007/s40096-021-00388-3

9. Samko, S. G., & Ross, B. (1993). Integration and differentiation to a variable fractional order. *Integral Transforms and Special Functions*, 1(4), 277-300. https://doi.org/10.1080/10652469308819027

10. Parsa Moghaddam, B., & Tenreiro Machado, J. A. (2017). A computational approach for the solution of a class of variable-order fractional integro-differential equations with weakly singular kernels. *Fractional Calculus and Applied Analysis*, 20(4), 1023-1042. https://doi.org/10.1515/fca-2017-0053

11. Amin, R., Sitthiwirattham, T., Hafeez, M. B., & Sumelka, W. (2023). Haar collocations method for nonlinear variable order fractional integro-differential equations. *Progress in Fractional Differentiation and Applications*, 9(2), 223-229. https://doi.org/10.18576/pfda/090203

12. Tuan, N. H., Nemati, S., Ganji, R. M., & Jafari, H. (2020). Numerical solution of multi-variable order fractional integro-differential equations using the Bernstein polynomials. *Engineering with Computers*, 38(Suppl 1), 139-147. https://doi.org/10.1007/s00366-020-01142-4

13. Mahdy, A. M. S. (2018). Numerical studies for solving fractional integro-differential equations. *Journal of Ocean Engineering and Science*, 3(2), 127-132. https://doi.org/10.1016/j.joes.2018.05.004

14. Parsa Moghaddam, B., Pishbin, M., Salamat Mostaghim, Z., Iyiola, O. S., Galhano, A., & Lopes, A. M. (2023). A numerical algorithm for solving nonlocal nonlinear stochastic delayed systems with variable-order fractional Brownian noise. *Fractal and Fractional*, 7(4), 293. https://doi.org/10.3390/fractalfract7040293

15. Mostaghim, Z. S., Parsa Moghaddam, B., & Haghgozar, H. S. (2018). Computational technique for simulating variable-order fractional Heston model with application in US stock market. *Mathematical Sciences*, 12, 277-283. https://doi.org/10.1007/s40096-018-0267-z

16. Indiaminov, R., Butaev, R., Isayev, N., Ismayilov, K., Yuldoshev, B., & Numonov, A. (2020). Nonlinear integro-differential equations of bending of physically nonlinear viscoelastic plates. *Materials Science and Engineering*, 869(5), 052048. https://doi.org/10.1088/1757-899X/869/5/052048

17. Durdiev, D. K., & Rakhmonov, A. A. (2020). The problem of determining the 2D kernel in a system of integro-differential equations of a viscoelastic porous medium. *Journal of Applied and Industrial Mathematics*, 14(2), 281-295. https://doi.org/10.1134/S1990478920020076

18. Abro, K. A., Atangana, A., & Gómez-Aguilar, J. F. (2023). A comparative analysis of plasma dilution based on fractional integro-differential equation: An application to biological science. *International Journal of Modelling and Simulation*, 43(1), 1-10. https://doi.org/10.1080/02286203.2021.2015818

19. MacCamy, R. C. (1977). An integro-differential equation with application in heat flow. *Quarterly of Applied Mathematics*, 35(1), 1-19. https://doi.org/10.1090/qam/452184

20. Caputo, M. (1967). Linear models of dissipation whose Q is almost frequency independent - II. *Geophysical Journal International*, 13(5), 529-539. https://doi.org/10.1111/j.1365-246X.1967.tb02162.x

21. Caputo, M. (1969). *Elasticità e dissipazione*. Bologna, Italy: Zanichelli.

22. Wu, J., Ge, W., & Zhang, X. (2020). Integro spline quasi-interpolants and their super convergence. *Computational and Applied Mathematics*, 39(3). https://doi.org/10.1007/s40314-020-01286-5

23. Wu, J., Shan, T., & Zhu, C. (2018). Integro quadratic spline quasi-interpolants. *Journal of Systems Science and Mathematical Sciences*, 38(12), 1407.

24. Mandal, B. N., & Chakrabarti, A. (2016). *Applied singular integral equations*. Boca Raton, FL: CRC Press.

25. Parsa Moghaddam, B., & Tenreiro Machado, J. A. (2017). SM-algorithms for approximating the variable-order fractional derivative of high order. *Fundamenta Informaticae*, 151(1-4), 293-311. https://doi.org/10.3233/FI-2017-1493