جواب‌های مدل ریاضی عملکرد کبد انسان با داده‌های فازی تحت مفهوم مشتق‌پذیری هاکوهارای توسعه‌یافته

نوع مقاله : مقاله پژوهشی

نویسنده

گروه علوم پایه، واحد سوادکوه، دانشگاه آزاد اسلامی، سوادکوه، ایران.

چکیده

یک مدل ریاضی ساده از عملکرد کبد انسان، دستگاهی خطی متشکل از دو معادله دیفرانسیل مرتبه اول است. از آنجا که داده‌های آزمایشگاهی و مقادیر بالینی از روی اندازه‌گیری و سنجش ثبت می‌شوند، از ویژگی عدم قطعیت و نادقیقی برخوردارند. لحاظ نمودن عدم قطعیت داده‌ها، منجر به تولید نتایج با کیفیت بیشتر می‌شود. منطق فازی یک ابزار توانا جهت برخورد با عدم قطعیت است که امکان مشاهده اثرات عدم قطعیت داده‌ها، در فرآیند حل مساله را فراهم می‌کند. در این مقاله، به مطالعه نظری و یافتن جواب‌های مدل مذکور همراه با داده‌های فازی، می‌پردازیم. برای این منظور، از مفهوم مشتق هاکوهارای توسعه‌یافته مربوط به تابع‌های فازی مقدار، استفاده می‌کنیم. ابتدا به تحلیل فرآیند حل مساله در حالت کاملا فازی، می‌پردازیم و شرایط کافی وجود جواب یکتا را به دست می‌آوریم. سپس، مساله را در دو حالت خاص مجزا، مورد مطالعه قرار می‌دهیم: (1) حالتی که ضرایب اعداد حقیقی باشند و مقدار اولیه فازی باشد و (2) حالتی که ضرایب اعداد فازی مثلثی متقارن و هم پهنا باشند و مقدار اولیه، عددی حقیقی باشد. در هر دو حالت، فرمول‌های جواب را به دست می‌آوریم و در انتها، با ارائه دو مثال، به تشریح و بکارگیری عملی نتایج، می‌پردازیم.

کلیدواژه‌ها

موضوعات


عنوان مقاله [English]

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

نویسنده [English]

  • Mehran Chehlabi
Department of Mathematics, Savadkooh Branch, Islamic Azad University, Savadkooh, Iran.
چکیده [English]

A simple mathematical model of human liver function is a linear system of differential equations
consisting of two unknown functions and their first order derivative.Since laboratory data and clinical
values are recorded by measurement and evaluation, they have the characteristic of uncertainty and
inaccuracy. This paper deals with the analysis and solving of the model with fuzzy data. At first, the
problem is investigated in fully fuzzy case from the point of view of generalized Hukuhara differentiability,
and sufficient conditions for the existence of a unique solution are found. Next, the simpler
forms of the problem are studied in two separate cases of the fuzziness of parameters, and in each
case, the solution formulas are obtained, in details. In the end, two numerical examples are solved to
describe and apply the results in practice.

کلیدواژه‌ها [English]

  • Fuzzy numbers
  • Fuzzy valued functions
  • Fuzzy differential equations
  • Hukuhara derivative
  • Generalized Hukuhara differentiability
[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.