Atangana–Baleanu–Caputo (ABC), Caputo-Fabrizio (CF), and Caputo fractional derivative approaches in fuzzy time fractional cancer tumor growth models

Document Type : Research Article

Authors

1 Department of Mathematics, Panjab University, Chandigarh, India & Department of Mathematics, Government College Gurdaspur, Punjab, India

2 Department of Mathematics, Panjab University, Chandigarh, India

3 Centre for Nuclear Medicine, Panjab University Chandigarh, India

Abstract

This article introduces a new approach for solving a time-fractional cancer tumor model using Caputo, Caputo-Fabrizio (CF), and Atangana-Baleanu-Caputo (ABC) fractional derivatives, accounting for varying net-killing rates of cancer cells in an uncertain environment. The model with the Caputo derivative is initially tackled using an explicit finite difference method (EFDM) with a fully time-dependent net killing rate. Approximate solutions for two different net death rates are obtained using the Sumudu transformation (ST) combined with the Adomian decomposition method (ADM), providing more accurate approximations than the EFDM. The model's behavior is analyzed with 2D and 3D visualizations. Convergence and error analysis of the method for the Caputo fractional derivative have been performed. The ADM provides reliable approximations for fractional models with fuzzy parameters, outperforming the EFDM by achieving lower absolute errors. The results exhibit symmetric lower and upper approximations around zero, effectively capturing the fuzzy nature of the solution.  All methods converge to zero at higher cuts in fuzzy triangular numbers, i.e. \( v = 1 \).

Keywords

Main Subjects

References

[1] R. Abdollahi, A. Khastan, J. Nieto, and R. Rodr´ıguez-L´opez, Linear fuzzy model with Ca-
puto–Fabrizio operator, Boundary Value Probl. 91 (2018) 1–18
[2] R.P. Agarwal, V. Lakshmikantham, J.J. Nieto, Fractional equations with uncertainty, Nonlinear
Anal. TMA. 72 (2010) 2859–2862.
[3] S. Ahmad, A. Ullah, A. Akg¨ul, T. Abdeljawad,Computational analysis of fuzzy fractional order
non-dimensional Fisher equation, Phys. Scr. 96 (2021) 084004.
[4] T. Allahviranloo, S. Salahshour, S. Abbasbandy, Explicit solutions of fractional differential equa-
tions with uncertainty, Soft Comput. 16 (2012) 297–302.
[5] Arafa, Anas A. M., Newly Proposed Solutions Using Caputo, Caputo-Fabrizio and Atangana-
Baleanu Fractional Derivatives: A Comparison, Progr. Fract. Differ. Appl. 8 (2022) 231–241.
[6] A. Arafa, Solutions with Caputo and Atangana-Baleanu derivatives, Prog. Fract. Differ. Appl 8
(2022) 231–241.
[7] A. Atangana, D. Baleanu, Fractional derivatives in heat transfer, Thermal Sci. 20 (2016) 763–769.
[8] P.K. Burgess, P.M. Kulesa, J.D. Murray, E.C. Jr. Alvord,3D model of gliomas: Growth-diffusion
interaction, J. Neuropathol. Exp. Neurol. 56 (1997) 704–713.
[9] J.A. Conejero, J.Franceschi, E. Pic´o-Marco, Fractional vs. ordinary control systems, Mathematics
10 (2022) 2719.
[10] N. Debbouche, A. Ouannas, G. Grassi, A. A. Al-Hussein, F. R. Tahir, K. M. Saad, and A. A. Aly,
Chaos in cancer growth models, Comp. Math. Methods Med. (2022) 1–13.
[11] D. Dubois, H. Prade, Fuzzy differential calculus, Fuzzy Sets Syst. 8 (1982) 225–233.
[12] M.M. El-Borai, W. G. El-Sayed, and A. M. Jawad, Adomian decomposition method for fractional
equations, IRJET. 2 (2015) 296–305.
[13] O.S. Fard, An iterative scheme for the solution of the generalized system of linear fuzzy differential
equations, World Appl. Sci. J. 7 (2009) 1597–1604.
[14] A.N Gani, S.N.M Assarudeen, A new operation on triangular fuzzy numbers for solving fuzzy linear
programming problems, Appl. Math. Sci. 6(2012) 525–532.
[15] H. Gandhi, A. Tomar, D.Singh, Brain tumor growth model, Adv. Intell. Syst. Comput. 1169 (2021)
17–25.
[16] L.L. Huang, D. Baleanu, Z.W. Mo, G.C. Wu,Fractional diffusion equation with uncertainty, Phys.
A Stat. Mech. Appl. 508 (2018) 166–175.
[17] A. Iomin, Superdiffusion of cancer on a comb, J. Phys.: Conf. Ser. 7 (2005) 57–67.
[18] O.S. Iyiola, F.D. Zaman, Fractional diffusion equation model for tumors, AIP Adv. 4 (2014) 107–
121.
[19] M. Keshavarz, E. Qahremani, T. Allahviranloo, Solving a fuzzy fractional diffusion model for can-
cer tumor by using fuzzy transforms, Fuzzy Sets Syst. 443 ( 2022) 198–220.
[20] T.D. Laajala, J. Corander, N.M. Saarinen, K. M¨akel¨a, S. Savolainen, M.I. Suominen, E. Alhoniemi,
S. M¨akel¨a, M. Poutanen, T. Aittokallio, Statistical modeling of tumor growth, Clin. Cancer Res. 18
(2012) 4385–4396.
[21] Z. Korpinar, E. Hıncal, D. Baleanu, Residual power series algorithm for fractional cancer models,
Alex. Eng. J. 59 (2020) 1405–1412.
[22] Z. Odibat, D. Baleanu, Numerical simulation of initial value problems with generalized Caputo-
type fractional derivatives, Appl. Numer. Math. 156 (2020) 94–105.
[23] A. Padder, L. Almutairi, S. Qureshi, A. Soomro, A. Afroz, E. Hincal, A. Tassaddiq,Dynamical
analysis of tumor model with Caputo derivative, Fractal Fract. 7 (2023) 258.
[24] M. Qayyum, A. Tahir, S.T. Saeed, A. Akg¨ul, Fractional Fisher models with uncertainties, Chaos
Solitons Fractals. 172 (2023) 113502.
[25] F. A. Rihan, A. Hashish, D. Baleanu, A. H. El-Bary, Fractional order cancer model, J. Fract. Calc.
Appl. 3 (2012) 1–6.
[26] R. Saadeh, A. Qazza, K. Amawi, Solving cancer models using integral transform, Fractal Fract. 6
(2022) 490.
[27] T. Tang, Z. Shah, E. Bonyah, R. Jan, M. Shutaywi, N. Alreshidi, Breast cancer model with
chemotherapy effects, Comp. Math. Methods Med. (2022) 1–19.
[28] E. Uc¸ar, N. ¨Ozdemir, Cancer-immune model with Caputo derivatives, Eur. Phys. J. Plus 136 (2021)
43.
[29] L. Verma, R. Meher, Fuzzy computational study on the generalized fractional smoking model with
caputo gH-type derivatives, Int. J. Biomath. 17 (2024) 2350037.
[30] L. Verma, R. Meher, O. Nikan, Z. Avazzadeh, Numerical analysis on fuzzy fractional human liver
model using a novel double parametric approach, Physica Scripta. 99 (2024) 115202.
[31] L. Vieira, R. Costa, D. Val´erio,Fractional calculus in cancer research, Fractal Fract. 7 (2023) 595.
[32] R.R. Yager, D.P. Filev, Essentials of Fuzzy Modeling, Wiley 1994.
[33] H Zureigat, M Al-Smadi, A. Al-Khateeb, S. Al-Omari, S. Alhazmi, Numerical Solution for Fuzzy
Time-Fractional Cancer Tumor Model with a Time-Dependent Net Killing Rate of Cancer Cells,
Int. J. Environ. Res. Public. Health. 20 (2023) 3766.
Journal of Mathematical Modeling
Volume 13, Issue 3
July 2025
Pages 725-746

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