Applying the Least Squares Approach to Solve Fredholm Fuzzy Fractional Integro-Differential Equations

Saif Aldeen M.  Jameel; Sharmila  karim; Ali Fareed  Jameel

doi:10.6180/jase.202510_28(10).0013

Applying the Least Squares Approach to Solve Fredholm Fuzzy Fractional Integro-Differential Equations

Computer Science and Information Engineering

Saif Aldeen M. Jameel^1,2This email address is being protected from spambots. You need JavaScript enabled to view it., Sharmila karim¹, and Ali Fareed Jameel³

¹School of Quantitative Sciences (SQS), College of Arts and Science, University Utara Malaysia (UUM), Kedah, Sintok, 06010, Malaysia

²Department of Computer Systems, Institute of Administration Al-Rusaffa, Middle Technical University, Baghdad, Iraq

³Faculty of Education and Arts, Sohar University, Sohar, 3111, Oman

Received: August 31, 2024
Accepted: December 26, 2024
Publication Date: January 24, 2025

Copyright The Author(s). This is an open access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are cited.

Download Citation: ||https://doi.org/10.6180/jase.202510_28(10).0013

This study introduces a methodology that integrates solution and analysis for linear and nonlinear fuzzy integro differential equations (FIDEs). A novel form approximates analytical method is presented to obtain a converge solution for fuzzy fractional Fredholm integral differential equations (FFIDEs). A control points method based on the principles least squares method (LSM) is presented. These control points have important role to ensuring the convergence of the approximate series solution. A new form of LSM is changed to FLSM includes an extra feature known as the convergence control points parameters, regarded as one of its most powerful tools that constructed from the use of some concepts of fuzzy set theory and fractional calculus. The properties of Caputo’s fractional derivative in the fuzzy domain, along with convergence analysis, improve and refine the methodology. The results validation of FLSM is demonstrated by addressing two numerical problems as test examples that include FFIDE subject to initial conditions. The results that displayed in the form of tables and f igures illustrated the effectiveness and simplicity of the proposed computation method.

Keywords: Fractional calculus; Fuzzy set theory; Fuzzy Fractional Integro-differential Equations; Least Squares method

[1] A.H.BhrawyandM.A.Alghamdi,(2012)“Ashifted Jacobi-Gauss-Lobattocollocationmethodforsolvingnon linearfractionalLangevinequationinvolvingtwofrac tional orders indifferent intervals"BoundaryValue Problems2012:1–13.DOI:10.1186/1687-2770-2012 62.
[2] Y.Yang,Y.Chen, andY.Huang, (2014)“Spectral collocationmethod for fractional Fredholm integro differentialequations"Journal of the Korean Mathematical Society 51(1): 203–224. DOI: 10.4134/JKMS.2014.51.1.203.
[3] O.AbuArqub,(2017)“Adaptationofreproducing kernel algorithm for solving fuzzy Fredholm–Volterra integrodif ferential equations" Neural Computing and Applica tions 28: 1591–1610. DOI: 10.1007/s00521-015-2110-x.
[4] O.AbuArqub, R. Mezghiche, and B. Maayah, (2023) “Fuzzy M-fractional integrodifferential models: theoretical existence and uniqueness results, and approximate solu tions utilizing the Hilbert reproducing kernel algorithm" Frontiers in Physics 11: 1252919. DOI: 10.3389/fphy.2023.1252919.
[5] O. Abu Arqub, J. Singh, and M. Alhodaly, (2023) “Adaptation of kernel functions-based approach with Atangana–Baleanu–Caputo distributed order derivative for solutions of fuzzy fractional Volterra and Fredholm integrodifferential equations" Mathematical Methods in the Applied Sciences 46(7): 7807–7834. DOI: 10.1002/mma.7228.
[6] O.AbuArqub,J.Singh, B. Maayah, and M.Alhodaly, (2023) “Reproducing kernel approach for numerical solu tions of fuzzy fractional initial value problems under the Mittag–Leffler kernel differential operator" Mathemat ical Methods in the Applied Sciences 46(7): 7965 7986. DOI: 10.1002/mma.7305.
[7] D.Baleanu, B. Ghanbari, J. H. Asad, A. Jajarmi, and H. M.Pirouz, (2020) “Planar system-masses in an equi lateral triangle: numerical study within fractional calcu lus" ComputerModelinginEngineering&Sciences 124(3): 953–968. DOI: 10.32604/cmes.2020.010236.
[8] A. Jajarmi, D. Baleanu, K. Zarghami Vahid, and S. Mobayen, (2022) “A general fractional formulation and tracking control for immunogenic tumor dynamics" Mathematical Methods in the Applied Sciences 45(2): 667–680. DOI: 10.1002/mma.7804.
[9] D.Baleanu, A. Jajarmi, H. Mohammadi, and S. Reza pour, (2020) “Anewstudyonthemathematicalmodelling of human liver with Caputo–Fabrizio fractional deriva tive" Chaos, Solitons & Fractals 134: 109705. DOI: 10.1016/j.chaos.2020.109705.
[10] L.Moradi, F. Mohammadi,andD.Baleanu,(2019) “A direct numerical solution of time-delay fractional optimal control problems by using Chelyshkov wavelets" Journal of Vibration and Control 25(2): 310–324. DOI: 10.1177/1077546318777338.
[11] F. Mohammadi, L. Moradi, D. Baleanu, and A. Ja jarmi, (2018) “A hybrid functions numerical scheme for fractional optimal control problems: application to nonanalytic dynamic systems" Journal of Vibration and Control 24(21): 5030–5043. DOI: 10.1177/1077546317741769.
[12] I. Hashim, O. Abdulaziz, and S. Momani, (2009) “Ho motopy analysis method for fractional IVPs" Communi cations in Nonlinear Science and Numerical Simu lation 14(3): 674–684. DOI: 10.1016/j.cnsns.2007.09.014.
[13] A.Rivaz,O.S.Fard,andT.A.Bidgoli, (2016) “Solving fuzzy fractional differential equations by a generalized differential transform method" SeMA Journal 73: 149 170. DOI: 10.1007/s40324-015-0061-x.
[14] D. Kumar, A. R. Seadawy, and A. K. Joardar, (2018) “Modified Kudryashov method via new exact solutions for some conformable fractional differential equations arising in mathematical biology" Chinese journal of physics 56(1): 75–85. DOI: 10.1016/j.cjph.2017.11.020.
[15] Z. Odibat and S. Momani, (2009) “The variational iter ation method: an efficient scheme for handling fractional partial differential equations in fluid mechanics" Com puters & Mathematics with Applications 58(11-12): 2199–2208. DOI: 10.1016/j.camwa.2009.03.009.
[16] J.-C. Pedjeu and G. S. Ladde, (2012) “Stochastic frac tional differential equations: Modeling, method and analy sis" Chaos, Solitons & Fractals 45(3): 279–293. DOI: 10.1016/j.chaos.2011.12.009.
[17] J. Tenreiro Machado, M. F. Silva, R. S. Barbosa, I. S. Jesus, C. M. Reis, M. G. Marcos, and A. F. Galhano, (2010) “Some applications of fractional calculus in en gineering" Mathematical problems in engineering 2010(1): 639801. DOI: 10.1155/2010/639801.
[18] R.P.Agarwal,D.Baleanu, J. J. Nieto, D. F. Torres, and Y. Zhou, (2018) “A survey on fuzzy fractional differential and optimal control nonlocal evolution equations" Jour nal of Computational and Applied Mathematics 339: 3–29. DOI: 10.1016/j.cam.2017.09.039.
[19] S. A. M. Jameel, S. A. Rahman, and A. A. Hamoud, (2024) “Analysis of Hilfer fractional Volterra-Fredholm system" Nonlinear Functional Analysis and Appli cations: 259–273. DOI: 10.22771/nfaa.2024.29.01.16.
[20] M. A. Matlob and Y. Jamali, (2019) “The concepts and applications of fractional order differential calculus in modeling of viscoelastic systems: A primer" Critical Reviews™ in Biomedical Engineering 47(4): DOI: 10.48550/arXiv.1706.06446.
[21] B. Maayah and O. A. Arqub, (2024) “Uncertain M fractional differential problems: existence, uniqueness, and approximations using Hilbert reproducing technique pro visioner with the case application: series resistor-inductor circuit" Physica Scripta 99(2): 025220. DOI: 10.1088/1402-4896/ad1738.
[22] A. Ali, M. Sarwar, M. B. Zada, and K. Shah, (2021) “Existence of solution to fractional differential equation with fractional integral type boundary conditions" Math ematical Methods in the Applied Sciences 44(2): 1615–1627. DOI: 10.1002/mma.6864.
[23] K. Shah and R. A. Khan, (2017) “Study of solution to a toppled system of fractional differential equations with integral boundary conditions" International Journal of Applied and Computational Mathematics 3(3): 2369–2388. DOI: 10.1007/s40819-016-0243-y.
[24] Z. Alijani, D. Baleanu, B. Shiri, and G.-C. Wu, (2020) “Spline collocation methods for systems of fuzzy fractional differential equations" Chaos, Solitons & Fractals 131: 109510. DOI: 10.1016/j.chaos.2019.109510.
[25] M.Arfan, K. Shah, A. Ullah, S. Salahshour, A. Ahma dian, and M. Ferrara, (2022) “A novel semi-analytical method for solutions of two dimensional fuzzy fractional wave equation using natural transform" Discrete and Continuous Dynamical Systems-S 15(2): 315–338. DOI: 10.3934/dcdss.2021011.
[26] M. Zahid, F. Ud Din, K. Shah, and T. Abdeljawad, (2024) “Fuzzy fixed point approach to study the existence of solution for Volterra type integral equations using fuzzy Sehgal contraction" Plos one 19(6): e0303642. DOI: 10.1371/journal.pone.0303642.
[27] R. P. Agarwal, V. Lakshmikantham, and J. J. Nieto, (2010) “On the concept of solution for fractional differen tial equations with uncertainty" Nonlinear Analysis: Theory, Methods & Applications 72(6): 2859–2862. DOI: 10.1016/j.na.2009.11.029.
[28] S. Salahshour, T. Allahviranloo, S. Abbasbandy, and D. Baleanu, (2012) “Existence and uniqueness results for fractional differential equations with uncertainty" Ad vances in Difference Equations 2012: 1–12. DOI: 10.1186/1687-1847-2012-112.
[29] A. Da¸scıo˘glu and D. V. Bayram, (2019) “Solving frac tional Fredholm integro-differential equations by Laguerre polynomials": DOI: 10.17576/jsm-2019-4801-29.
[30] M.F. KazemandA.Al-Fayadh. “Solving Fredholm integro-differential equation of fractional order by us ing Sawihomotopyperturbationmethod”.In:Journal of Physics: Conference Series. 2322. 1. IOP Publishing. 2022, 012056. DOI: 10.1088/1742-6596/2322/1/012056.
[31] T. Schürmann, (2016) “The uncertainty principle in terms of isoperimetric inequalities" arXiv preprint arXiv:1606.07936: DOI: 10.4236/am.2017.83025.
[32] H.Vu,T.V.An,andN.VanHoa,(2019)“Ontheinitial value problem for random fuzzy differential equations with Riemann-Liouville fractional derivative: Existence theory andanalytical solution" Journal of Intelligent & Fuzzy Systems 36(6): 6503–6520. DOI: 10.3233/JIFS-182876.
[33] S. Salahshour, A. Ahmadian, F. Ismail, D. Baleanu, and N. Senu, (2015) “A new fractional derivative for dif ferential equation of fractional order under interval uncer tainty" Advances in Mechanical Engineering 7(12): 1687814015619138. DOI: 10.1177/1687814015619138.
[34] S. R. Raj and M. Saradha, “Solving Hybrid Fuzzy Frac tional Differential Equations by Runge Kutta Fehlberg 6th Order Method": DOI: 10.22457/apam.v17n2a6.
[35] M. Alaroud, M. Al-Smadi, R. Rozita Ahmad, and U. K. Salma Din, (2019) “An analytical numerical method for solving fuzzy fractional Volterra integro differential equations" Symmetry 11(2): 205. DOI: 10.3390/sym11020205.
[36] S. Hasan, B. Maayah, S. Bushnaq, and S. Momani, (2023) “A modified reproducing kernel Hilbert space method for solving fuzzy fractional integro-differential equations" Boletim da Sociedade Paranaense de Matemática 41: 1–16. DOI: 10.5269/bspm.52289.
[37] O. Taiwo and O. Odetunde, (2014) “Numerical ap proximation of fractional Integro-differential equations by an iterative decomposition method" Ilorin Journal of Science 1(1): 195–203.
[38] M. Mohammed*, (2014) “Numerical Solution of Frac tional Integro-Differential Equations by Least Squares Method and Shifted Chebyshev Polynomial" Sociologie 5(1): 1–11.
[39] T.Oyedepo,C.Ishola, O. Uwaheren, M. Olaosebikan, M. Ajisope, and A. Victor, (2022) “Least-squares Bern stein Method for Solving Fractional Integro-differential Equations" ATBU Journal of Science, Technology and Education 10(1): 105–114.
[40] B. C˘aruntu and M. S. Pa¸sca, (2021) “The Polynomial Least Squares Method for Nonlinear Fractional Volterra and Fredholm Integro-Differential Equations" Mathe matics 9(18): 2324. DOI: 10.3390/math9182324.
[41] S. Bodjanova, (2006) “Median alpha-levels of a fuzzy number" Fuzzy sets and Systems 157(7): 879–891. DOI: 10.1016/j.fss.2005.10.015.
[42] C. Duraisamy and B. Usha. “Another approach to so lution of fuzzy differential equations by Modified Euler’s method”. In: 2010 International Conference on Communication and Computational Intelligence (IN COCCI). IEEE. 2010, 52–56.
[43] M. T. Mizukoshi, L. d. Barros, Y. Chalco-Cano, H. Román-Flores, and R. C. Bassanezi, (2007) “Fuzzy differential equations and the extension principle" Infor mation Sciences 177(17): 3627–3635. DOI: 10.1016/j.ins.2007.02.039.
[44] K. Shah, T. Abdeljawad, A. Ali, and M. A. Alqudah, (2022) “Investigation of integral boundary value prob lem with impulsive behavior involving non-singular derivative" Fractals 30(08): 2240204. DOI: 10.1142/S0218348X22402046.
[45] A. F. Jameel, N. R. Anakira, A. Alomari, D. Alsharo, and A. Saaban, (2019) “New semi-analytical method for solving two point nth order fuzzy boundary value prob lem" International Journal of Mathematical Mod elling and Numerical Optimisation 9(1): 12–31.