Cubic hat-functions approximation for linear and nonlinear fractional integral-differential equations with weakly singular kernels

Document Type : Research Article

Authors

• H. Ebrahimi
• J. Biazar

Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran.

10.22067/ijnao.2023.80249.1209

Abstract

In the current study, a new numerical algorithm is presented to solve a class of nonlinear fractional integral-differential equations with weakly singular kernels. Cubic hat functions (CHFs) and their properties are introduced for the first time. A new fractional-order operational matrix of integration via CHFs is presented. Utilizing the operational matrices of CHFs, the main problem is transformed into a number of trivariate polynomial equations. Error analysis and the convergence of the proposed method are evaluated, and the convergence rate is addressed. Ultimately, three examples are provided to illustrate the precision and capabilities of this algorithm. The numerical results are presented in some tables and figures.

Keywords

• Fractional integral-differential equations
• Numerical algorithms
• Weakly singular kernels
• Cubic hat functions
• Fractional operational matrix

20.1001.1.24236977.2023.13.3.8.3

Main Subjects

• Numerical Analysis

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References

[1] Abdelaty, A.M., Roshdy, M., Said, L.A. and Radwan, A. G. Numeri-cal simulations and FPGA implementations of fractional-order systems based on product integration rules, IEEE Access. 8 (2020), 102093–102105.
[2] Ahmed, S.A., Elzaki, T.M. and Hassan, A.A. Solution of integral dif-ferential equations by new double integral transform (Laplace–Sumudu transform), Abstr. Appl. Anal. (2020), Art. ID 4725150, 7.
[3] Al-Ahmad, S., Sulaiman, I.M. and Mamat, M. An efficient modifica-tion of differential transform method for solving integral and integro-differential equations, Aust. J. Math. Anal. Appl. 17(2) (2020), Art. 5, 15.
[4] Amin, R., Shah, K., Asif, M., Khan, I. and Ullah, F. An efficient algo-rithm for numerical solution of fractional integro-differential equations via Haar wavelet, J. Comput. Appl. Math. 381(113028) (2021), 17.

[5] Arsalan Sajjadi, S., Saberi Najafi, H. and Aminikhah, H. A numerical study on the non‐smooth solutions of the nonlinear weakly singular frac-tional Volterra integro‐differential equations, Math. Methods Appl. Sci. 46(4) (2023), 4070–4084.
[6] Babolian, E. and Mordad, M. A numerical method for solving systems of linear and nonlinear integral equations of the second kind by hat basis functions, Comput. Math. Appl. 62(1) (2011), 187–198.
[7] Behera, S. and Ray, S. Saha. An efficient numerical method based on Euler wavelets for solving fractional order pantograph Volterra delay-integro-differential equations, J. Comput. Appl. Math. 406(113825) (2022), 23.
[8] Behera, S. and Ray, S. Saha. On a wavelet-based numerical method for linear and nonlinear fractional Volterra integro-differential equations with weakly singular kernels, Comput. Appl. Math. 41(5) (2022), 1–32.
[9] Biazar, J. Solution of systems of integral–differential equations by Ado-mian decomposition method, Appl. Math. Comput. 168(2) (2005), 1232–1238.
[10] Biazar, J. and Ebrahimi, H. Orthonormal Bernstein polynomials for Volterra integral equations of the second kind, Int. J. Appl. Math. Res. 9(1) (2019), 9–20.
[11] Biazar, J. and Ebrahimi, H. A numerical algorithm for a class of non-linear fractional Volterra integral equations via modified hat functions, J. Integral Equ. Appl. 34(3) (2022), 295–316.
[12] Biazar, J. and Montazeri, R. Optimal homotopy asymptotic and multi-stage optimal homotopy asymptotic methods for solving system of volterra integral equations of the second kind, J. Appl. Math. 2019 (2019), Art. ID 3037273, 17.
[13] Derakhshan, M. A numerical scheme based on the Chebyshev functions to find approximate solutions of the coupled nonlinear sine-Gordon equa-tions with fractional variable orders, Abstr. Appl. Anal. 2021 (2021), Art. ID 8830727, 20.
[14] Du, H., Chen, Z. and Yang, T. A stable least residue method in repro-ducing kernel space for solving a nonlinear fractional integro-differential equation with a weakly singular kernel, Appl. Numer. Math. 157 (2020), 210–222.
[15] Hashemi Mehne, S.H. Differential transform method: A comprehensive review and analysis, Iranian Journal of Numerical Analysis and Opti-mization 12(3) (Special Issue), (2022), 629–657.

[16] Kukreja, V. K. An improvised collocation algorithm with specific end con-ditions for solving modified Burgers equation, Numer. Methods Partial Differ. Equ. 37(1) (2021), 874–896.
[17] Kythe, P.K. and Puri, P. Computational methods for linear integral equations, Birkhäuser Boston, Inc., Boston, MA, 2002.
[18] Leitman, M.J. An integro-differential equation for plane waves propa-gating into a random fluid: asymptotic behavior, SIAM J. Math. Anal. 12(4) (1981), 560–571.
[19] Mirzaee, F. and Hadadiyan, E. Numerical solution of Volterra–Fredholm integral equations via modification of hat functions, Appl. Math. Com-put. 280 (2016), 110–123.
[20] Moosavi Noori, S.R. and Taghizadeh, N. Modified differential transform method for solving linear and nonlinear pantograph type of differential and Volterra integro-differential equations with proportional delays, Adv. Differ. Equ. 2020(1) (2020), 1–25.
[21] Ndiaye, A. and Mansal, F. Existence and uniqueness results of Volterra-Fredholm integro-differential equations via Caputo fractional derivative, J. Math. (2021), Art. ID 5623388, 8.
[22] Nemati, S. and Lima, P M. Numerical solution of nonlinear fractional integro-differential equations with weakly singular kernels via a modifi-cation of hat functions, Appl. Math. Comput. 327 (2018), 79–92.
[23] Özaltun, G., Konuralp, A. and Gümgüm, S. Gegenbauer wavelet solu-tions of fractional integro-differential equations, J. Comput. Appl. Math. 420(114830) (2023), 11.
[24] Podlubny, I. Fractional differential equations, Math. Sci. Eng. 198 (1999), Academic Press.
[25] Qiao L. and Xu, D. A fast ADI orthogonal spline collocation method with graded meshes for the two-dimensional fractional integro-differential equation, Adv. Comput. Math. 47(5) (2021), 1–22.
[26] Quentin, R., King, J.R., Sallard, E., Fishman, N., Thompson, R., Buch E.R. and Cohen, L.G. Differential brain mechanisms of selection and maintenance of information during working memory J. Neurosci. 39(19) (2019), 3728–3740.
[27] Rabbath, C.A. and Corriveau, D. A comparison of piecewise cubic Her-mite interpolating polynomials, cubic splines and piecewise linear func-tions for the approximation of projectile aerodynamics, Defence Tech-nology 15(5) (2019), 741–757.

[28] Rakshit G. and Rane, A.S. Asymptotic expansion of iterated Galerkin solution of Fredholm integral equations of the second kind with Green’s kernel, J. Integral Equ. Appl. 32(4) (2020), 495–507.
[29] Riahi Beni, M. Legendre wavelet method combined with the Gauss quadra-ture rule for numerical solution of fractional integro-differential equa-tions, Iranian Journal of Numerical Analysis and Optimization 12(1) (2022), 229–249.
[30] Sabatier, J., Aoun, M., Oustaloup, A., Gregoire, G., Ragot, F. and Roy, P. Fractional system identification for lead acid battery state of charge estimation, Signal Process 86(10) (2006), 2645–2657.
[31] Vinagre, B.M. ,Monje, C.A. ,Calderón A.J. and Suárez, J.I. Fractional PID controllers for industry application. A brief introduction, J. Vib. Control, 13(9-10) (2007), 1419–1429.
[32] Wang, Y. and Zhu, Li. SCW method for solving the fractional integro-differential equations with a weakly singular kernel, Appl. Math. Comput. 275 (2022), 72–80.
[33] Xie, J., Wang, T., Ren, Z., Zhang J. and Quan, L. Haar wavelet method for approximating the solution of a coupled system of fractional-order integral–differential equations, Math. Comput. Simul 163 (2019), 80–89.
[34] Yang, Z. Gröbner Bases for Solving Multivariate Polynomial Equations, Computing Equilibria and Fixed Points: The Solution of Nonlinear In-equalities (1999) 265–288.