Solutions to the mathematical model of human liver with fuzzy data under generalized Hukuhara differentiability

Document Type : Original Article

Author

Department of Mathematics, Savadkooh Branch, Islamic Azad University, Savadkooh, Iran.

Abstract

A simple mathematical model of human liver action is a linear system consisting of two first-order differential equations. Since laboratory data and clinical values are recorded through measurement and estimation, they have the characteristic of uncertainty and inaccuracy. Taking into account the uncertainty of the data leads to the production of higher-quality results. Fuzzy logic is a powerful tool for dealing with uncertainty, which provides the possibility of observing the effects of data uncertainty in the problem-solving process. In this paper, from a theoretical point of view, we study and obtain the solutions of the mentioned model along with fuzzy data. To this end, we use the generalized Hukuhara differentiability concept related to fuzzy-valued functions. First, we analyze the process of solving the problem in the fully fuzzy case and obtain the sufficient conditions for the existence of a unique solution. Next, we study the problem in two separate special cases: (1) the case where the coefficients are real numbers and the initial value is fuzzy, and (2) the case where the coefficients are symmetric triangular fuzzy numbers with the same width and the initial value is a real number. In both cases, we obtain the solution formulas and, at the end, by presenting two examples, we practically explain and apply the results. 

Keywords

Main Subjects

References

[1] J. Casasnovas and F. Rossell, “Averaging fuzzy biopolymers,” Fuzzy Sets Syst., vol. 152, pp. 139-158, 2005.
[2] Z. Ding, H. Shen, and A. Kandel, “Performance analysis of service composition based on fuzzy differential equations,” IEEE Trans. Fuzzy Syst., vol. 19, pp. 164-178, 2010, doi: 10.1109/TFUZZ.2010.2089633.
[3] M. Hanss, Applied Fuzzy Arithmetic: An Introduction with Engineering Applications. Berlin, Germany: Springer-Verlag, 2005.
[4] H. Moradi Farahani, J. Asgari, and M. Zakeri, “A Surveying on Type-2 Fuzzy Logic: Its Genesis and Its Application,” Soft Comput. J., vol. 2, no. 1, pp. 22-43, 2013, dor: 20.1001.1.23223707.1392.2.1.58.2 [In Persian].
[5] A. Akrami and M. Parsamanesh, “Investigation of a mathematical fuzzy epidemic model for the spread of coronavirus in a population,” Soft Comput. J., vol. 11, no. 1, pp. 2-9, 2022, doi: 10.22052/scj.2022.246053.1045 [In Persian].
[6] R. Akhoondi and R. Hosseini, “A Novel Fuzzy-Genetic Differential Evolutionary Algorithm for Optimization of A Fuzzy Expert Systems Applied to Heart Disease Prediction,” Soft Comput. J., vol. 6, no. 2, pp. 32-47, 2018, dor: 20.1001.1.23223707.1396.6.2.3.7 [In Persian].
[7] C. Wu and Z. Gong, “On Henstock integral of fuzzy-number-valued functions I,” Fuzzy Sets Syst., vol. 120, pp. 523-532, 2001, doi: 10.1016/S0165-0114(99)00057-3.
[8] L. Celechovska, “A simple mathematical model of the human liver,” Appl. Math., vol. 49, pp. 227-246, 2004, doi: 10.1023/B:APOM.0000042364.85016.7d.
[9] L.P. Lebedev and M.J. Cloud, The Calculus of Variations and Functional Analysis with Optimal Control and Applications in Mechanics. Singapore: World Scientific, 2003, pp. 1-98, doi: 10.1142/5374.
[10] M. Shabibi, Z. Zeinalabedini Charandabi, H. Mohammadi, and S. Rezapour, “Investigation of mathematical model of human liver by Caputo fractional derivative approach,” J. Adv. Math. Model., vol. 11, no. 4, pp. 750-760, 2021, doi: 10.22055/jamm.2022.37102.1918 [In Persian].
[11] L.C. Barros and F.S. Pedro, “Fuzzy differential equations with interactive derivative,” Fuzzy Sets Syst., vol. 309, pp. 64-80, 2017, doi: 10.1016/j.fss.2016.04.002.
[12] J.J. Buckley and T. Feuring, “Fuzzy differential equations,” Fuzzy Sets Syst., vol. 110, pp. 43-54, 2000, doi: 10.1016/S0165-0114(98)00141-9.
[13] Y. Chalco-Cano, H. Román-Flores, and M.D. Jimenez-Gamero, “Generalized derivative and π-derivative for set-valued functions,” Inf. Sci., vol. 181, pp. 2177-2188, 2011, doi: 10.1016/j.ins.2011.01.023.
[14] V.F. Wasques, E. Esmi, L. C. Barros, and P. Sussner, “The generalized fuzzy derivative is interactive,” Inf. Sci., vol. 519, pp. 93-109, 2020, doi: 10.1016/j.ins.2020.01.042.
[15] B. Bede, I.J. Rudas, and A.L. Bencsik, “First order linear fuzzy differential equations under generalized differentiability,” Inf. Sci., vol. 177, pp. 1648-1662, 2007, doi: 10.1016/j.ins.2006.08.021.
[16] B. Bede and S.G. Gal, “Solutions of fuzzy differential equations based on generalized differentiability,” Commun. Math. Anal., vol. 9, no. 2, pp. 22-41, 2010.
[17] M. Chehlabi and T. Allahviranloo, “Positive or negative solutions to first-order fully fuzzy linear differential equations under generalized differentiability,” Appl. Soft Comput., vol. 70, pp. 359-370, 2018, doi: 10.1016/j.asoc.2018.05.040.
[18] M. Chehlabi and T. Allahviranloo, “Existence of generalized Hukuhara differentiable solutions to a class of first-order fuzzy differential equations in dual form,” Fuzzy Sets Syst., vol. 478, p. 108839, 2024, doi: 10.1016/j.fss.2023.108839.
[19] A. Khastan, J.J. Nieto, and R. Rodriguez-Lopez, “Variation of constant formula for first order fuzzy differential equations,” Fuzzy Sets Syst., vol. 177, pp. 20-33, 2011, doi: 10.1016/j.fss.2011.02.020.
[20] A. Khastan and R. Rodriguez-Lopez, “On the solutions to first order linear fuzzy differential equations,” Fuzzy Sets Syst., vol. 295, pp. 114-135, 2016, doi: 10.1016/j.fss.2015.06.005.
[21] E. ElJaoui, S. Melliani, and L. S. Chadli, “Solving second-order fuzzy differential equations by the fuzzy Laplace transform method,” Adv. Differ. Equ., vol. 2015, p. 66, 2015, doi: 10.1186/s13662-015-0414-x.
[22] L. Hooshangian, “Analytic solution of fuzzy second order differential equations under H-derivation,” Theory Approx. Appl., vol. 11, no. 1, pp. 99-115, 2017.
[23] B. Bede and S.G. Gal, “Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations,” Fuzzy Sets Syst., vol. 151, pp. 581-599, 2005, doi: 10.1016/j.fss.2004.08.001.
[24] B. Bede and L. Stefanini, “Solution of fuzzy differential equations with generalized differentiability using LU-parametric representation,” in Proc. 7th Conf. Eur. Soc. Fuzzy Logic Technol. (EUSFLAT), 2011, pp. 785-790, doi: 10.2991/eusflat.2011.106.
[25] T. Allahviranloo and M. Chehlabi, “Solving fuzzy differential equations based on the length function properties,” Soft Comput., vol. 19, pp. 307-320, 2015, doi: 10.1007/s00500-014-1254-4.
[26] A. Khastan and R. Rodriguez-Lopez, “On linear fuzzy differential equations by differential inclusions’ approach,” Fuzzy Sets Syst., vol. 387, pp. 49-67, 2020, doi: 10.1016/j.fss.2019.05.014.
[27] M.Z. Ahmad, M.K. Hasan, and B. De Baets, “Analytical and numerical solutions of fuzzy differential equations,” Inf. Sci., vol. 236, pp. 156-167, 2013, doi: 10.1016/j.ins.2013.02.026.
[28] B. Bede, T.C. Bhaskar, and V. Lakshmikantham, “Perspectives of fuzzy initial value problems,” Commun. Appl. Anal., vol. 11, no. 3, pp. 339-358, 2007.
[29] M. Chehlabi, “Trapezoidal approximation operators preserving the most indicators of fuzzy numbers-relationships and applications,” Soft Comput., vol. 26, pp. 7081-7105, 2022, doi: 10.1007/s00500-022-07172-y.
[30] P. Diamond and P. Kloeden, Metric Spaces of Fuzzy Sets. Singapore: World Scientific, 1994, doi: 10.1142/9789814447010_0001.
[31] Y. Chalco-Cano and H. Roman-Flores, “Comparation between some approaches to solve fuzzy differential equations,” Fuzzy Sets Syst., vol. 160, pp. 1517-1527, 2009, doi: 10.1016/j.fss.2008.10.002.
[32] Y. Chalco-Cano and H. Roman-Flores, “On new solutions of fuzzy differential equations,” Chaos Solitons Fractals, vol. 38, pp. 112-119, 2008, doi: 10.1016/j.chaos.2006.10.043.
[33] T.M. Apostol, Linear Algebra: A First Course with Applications to Differential Equations, 1st Ed., Wiley-Interscience, 1997.
[34] R. Ezzati, “Solving fuzzy linear systems,” Soft Comput., vol. 15, pp. 193-197, 2011, doi: 10.1007/s00500-009-0537-7.
Soft Computing Journal
Volume 14, Issue 1 - Serial Number 27
September 2025
Pages 36-53

Files

History

  • Receive Date: 19 December 2023
  • Revise Date: 24 January 2024
  • Accept Date: 23 February 2024

Share

How to cite

Statistics