Boundary value problem of fractional fuzzy Volterra-Fredholm systems

Volume 36, Issue 4, pp 432--443 https://dx.doi.org/10.22436/jmcs.036.04.06

Publication Date: August 24, 2024 Submission Date: June 02, 2024 Revision Date: July 08, 2024 Accteptance Date: July 20, 2024

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Authors

A‎. ‎J‎. ‎ Abdulqader - Department of Mathematics‎, ‎College of Education, ‎Al-Mustansiriyah University, Iraq. ‎R‎. ‎B‎. ‎ Abdulmaged - Department of Mathematics‎, ‎College of Education, ‎Al-Mustansiriyah University, Iraq.

Abstract

‎This paper investigates the suitable conditions for the uniqueness and existence results for a class of fuzzy fractional Caputo Volterra-Fredholm integro differential equations (FFCV-FIDEs) with boundary conditions‎. ‎The findings are based on Banach contraction principle and Schaefer's fixed point theorem‎. ‎Additionally‎, ‎the solution to the given problem is found using the Adomian decomposition technique (ADT)‎. ‎We support the concept with various instances‎. ‎The relationship between the lower and upper reduce approximations of the fuzzy solutions has been demonstrated numerically and graphically via MATLAB‎.

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Keywords

• Volterra-Fredholm equation
• Caputo fractional derivative
• fixed point technique
• ADT‎

MSC

•  34A12
•  26A33
•  49M27

References

• [1] S. Abbasbandy, M. Hashemi, Fuzzy integro-differential equations: formulation and solution using the variational iteration method, Nonlinear Sci. Lett. A, 1 (2010), 413–418
  • View Article
  • Google Scholar


• [2] G. Adomian, Solution of physical problems by decomposition, Comput. Math. Appl., 27 (1994), 145–154
  • View Article
  • MathSciNet
  • Google Scholar


• [3] G. Adomian, R. Rach, Equality of partial solutions in the decomposition method for linear or nonlinear partial differential equations, Comput. Math. Appl., 19 (1990), 9–12
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [4] N. Ahmad, A. Ullah, A. Ullah, S. Ahmad, K. Shah, I. Ahmad, On analysis of the fuzzy fractional order Volterrafredholm integro-differential equation, Alex. Eng. J., 60 (2021), 1827–1838
  • View Article
  • Google Scholar


• [5] M. Alaroud, M. Al-Smadi, R. R. Ahmad, U. K. S. Din, An analytical numerical method for solving fuzzy fractional Volterra integro-differential equations, Symmetry, 11 (2019), 19 pages
  • View Article
  • Google Scholar
  • MATH


• [6] M. Al-Smadi, O. A. Arqub, Computational algorithm for solving Fredholm time-fractional partial integrodifferential equations of Dirichlet functions type with error estimates, Appl. Math. Comput., 342 (2019), 280–294
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [7] A. Arikoglu, I. Ozkol, Solution of fractional differential equations by using differential transform method, Chaos Solitons Fractals, 34 (2007), 1473–1481
  • View Article
  • MathSciNet
  • Google Scholar


• [8] O. Arqub, Adaptation of reproducing kernel algorithm for solving fuzzy Fredholm-Volterra integrodifferential equations, Neural Comput. Appl., 28 (2017), 1591–1610
  • View Article
  • Google Scholar


• [9] O. A. Arqub, J. Singh, M. Alhodaly, Adaptation of kernel functions-based approach with AtanganaBaleanu-Caputo distributed order derivative for solutions of fuzzy fractional Volterra and Fredholm integrodifferential equations, Math. Methods Appl. Sci., 46 (2023), 7807–7834
  • View Article
  • Google Scholar


• [10] O. Arqub, J. Singh, B. Maayah, M. Alhodaly, Reproducing kernel approach for numerical solutions of fuzzy fractional initial value problems under the Mittag-Leffler kernel differential operator, Math. Methods Appl. Sci., 46 (2023), 7965– 7986
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [11] B. Bede, L. Stefanini, Generalized differentiability of fuzzy-valued functions, Fuzzy Sets Syst., 230 (2013), 119–141
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [12] F. E. Browder, W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl., 20 (1967), 197–228
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [13] S. S. L. Chang, L. A. Zadeh, On fuzzy mapping and control, IEEE Trans. Systems Man Cybernet., SMC-2 (1972), 30–34
  • View Article
  • MathSciNet
  • Google Scholar


• [14] P. Das, S. Rana, H. Ramos, On the approximate solutions of a class of fractional order nonlinear Volterra integro-differential initial value problems and boundary value problems of first kind and their convergence analysis, J. Comput. Appl. Math., 404 (2022), 15 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [15] D. Dubois, H. Prade, Operations on fuzzy numbers, Int. J. Syst. Sci., 9 (1978), 613–626
  • View Article
  • MathSciNet
  • Google Scholar


• [16] D. Dubois, H. Prade, Towards fuzzy differential calculus part 1: integration of fuzzy mappings, Fuzzy Sets Syst., 8 (1982), 1–17
  • View Article
  • Google Scholar


• [17] A. A. Hamoud, A. Azeez, K. Ghadle, A study of some iterative methods for solving fuzzy Volterra-Fredholm integral equations, Indones. J. Electr. Eng. Comput. Sci., 11 (2018), 1228–1235
  • View Article
  • Google Scholar


• [18] A. Hamoud, M. SH. Bani Issa, K. Ghadle, Existence and uniqueness results for nonlinear Volterra-Fredholm integrodifferential equations, Nonlinear Funct. Anal. Appl., 23 (2018), 797–805
  • View Article
  • Google Scholar


• [19] A. A. Hamoud, K. P. Ghadle, Modified Adomian decomposition method for solving fuzzy Volterra-Fredholm integral equation, J. Indian Math. Soc., 85 (2018), 53–69
  • View Article
  • Google Scholar


• [20] A. A. Hamoud, K. P. Ghadle, The approximate solutions of fractional Volterra-Fredholm integro-differential equations by using analytical techniques, Probl. Anal. Issues Anal., 7(25) (2018), 41–58
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [21] A. A. Hamoud, K. P. Ghadle, Modified Laplace decomposition method for fractional Volterra-Fredholm integro-differential equations, J. Math. Model., 6 (2018), 91–104
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [22] A. A. Hamoud, K. Ghadle, S. M. Atshan, The approximate solutions of fractional integro-differential equations by using modified Adomian decomposition method, Khayyam J. Math., 5 (2019), 21–39
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [23] A. A. Hamoud, K. P. Ghadle, Homotopy analysis method for the first order fuzzy Volterra-Fredholm integro-differential equations, Indones. J. Electr. Eng. Comput. Sci., 11 (2018), 857–867
  • View Article
  • Google Scholar


• [24] A. A. Hamoud, A. D. Khandagale, R. Shah, K. P. Ghadle, Some new results on Hadamard neutral fractional nonlinear Volterra-Fredholm integro-differential equations, Discontinuity Nonlinearity Complex., 12 (2023), 893–903
  • View Article
  • Google Scholar


• [25] K. Ivaz, I. Alasadi, A. Hamoud, On the Hilfer fractional Volterra-Fredholm integro differential equations, IAENG Int. J. Appl. Math., 52 (2022), 426–431
  • View Article
  • Google Scholar


• [26] W. Jiang, T. Tian, Numerical solution of nonlinear Volterra integro-differential equations of fractional order by the reproducing kernel method, Appl. Math. Model., 39 (2015), 4871–4876
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [27] O. Kaleva, Fuzzy differential equations, Fuzzy Sets Syst., 24 (1987), 301–317
  • View Article
  • MathSciNet
  • Google Scholar


• [28] P. Linz, Analytical and numerical methods for Volterra equations, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1985)
  • View Article
  • MathSciNet
  • Google Scholar


• [29] X. Ma, C. Huang, Numerical solution of fractional integro-differential equations by a hybrid collocation method, Appl. Math. Comput., 219 (2013), 6750–6760
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [30] K. S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, John Wiley & Sons, New York (1993)
  • View Article
  • Google Scholar


• [31] S. Momani, M. A. Noor, Numerical methods for fourth-order fractional integro-differential equations, Appl. Math. Comput., 182 (2006), 754–760
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [32] M. Osman, Y. Xia, O. A. Omer, A. Hamoud, On the fuzzy solution of linear-nonlinear partial differential equations, Mathematics, 10 (2022), 49 pages
  • View Article
  • Google Scholar


• [33] I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Academic Press, San Diego (1999)
  • View Article
  • Google Scholar
  • MATH


• [34] K. Sayevand, Analytical treatment of Volterra integro-differential equations of fractional order, Appl. Math. Model., 39 (2015), 4330–4336
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [35] M. R. M. Shabestari, R. Ezzati, T. Allahviranloo, Numerical solution of fuzzy fractional integrodifferential equation via two-dimensional Legendre wavelet method, J. Intell. Fuzzy Syst., 34 (2018), 2453–2465
  • View Article
  • Google Scholar


• [36] A. A. Sharif, A. A. Hamoud, K. P. Ghadle, On existence and uniqueness of solutions to a class of fractional Volterra- Fredholm initial value problems, Discontinuity Nonlinearity Complex., 12 (2023), 905–916
  • View Article
  • Google Scholar


• [37] L. Zhu, Q. Fan, Numerical solution of nonlinear fractional-order-Volterra integro-differential equations by SCW, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 1203–1213
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH