Open Access Articles Version 1 https://doi.org/10.12345/wqtg4e81

Fuzzy fractional diffusion equation with time delay having power law and Mittag-Leffler kernel

Vol. 1 No. 1 (2026): Volume 1, Issue 1, (January - June) 2026 Madhubala Madhubala1 Sachin Kumar2
  1. Government Degree College, Budaun, Uttar Pradesh, India , IN
  2. Government Degree College, Budaun, Uttar Pradesh, India , IN
Published: June 30, 2026

Abstract

 In this article, we study a new type of mathematical model of the diffusion equation. This model has space and time-fractional derivative with both singular and non-singular derivative i.e., having power-law kernel and Mittag-Leffler kernel. This model is taken in a fuzzy environment. A delay term is also added to this model. First, we approximate the unknown fuzzy function as a truncated series expansion of shifted Chebyshev polynomials. We find an approximate formula for fractional derivatives and delay function. The main advantage of this method in applying this method is, the taken fuzzy fractional partial differential equation (FFPDE) reduces into a system of a non-linear fuzzy algebraic equation. We solve this system by any method available in the literature and find the value of unknowns. We depict the validity and feasibility of this derived method by deriving numerical solution of particular cases of taken fuzzy model. We observe that this method is valid for a fuzzy fractional differential equation with delay term and has good accuracy as seen from the error tables.

Keywords:
Fractional PDE Fuzzy valued function diffusion equation time delay Chebyshev polynomial

References

Ahmadian, A., Suleiman, M., & Salahshour, S. (2013). An operational matrix based on Legendre polynomials for solving fuzzy fractional-order differential equations. Abstract and Applied Analysis, 2013. Allahviranloo, T., & Kermani, M. A. (2006). Solution of a fuzzy system of linear equation. Applied Mathematics and Computation, 175(1), 519–531. Bede, B., Stefanini, L., et al. (2013). Generalized differentiability of fuzzy-valued functions. Fuzzy Sets and Systems, 230(1), 119–114. Bertoluzza, C., Corral Blanco, N., & Salas, A. (1995). On a new class of distances between fuzzy numbers. Mathware & Soft Computing, 2(2). Darania, P., & Ebadian, A. (2007). A method for the numerical solution of the integro-differential equations. [Journal title and volume information not provided], 188, 657–668. [Bibliographic details require verification.] Diethelm, K., Ford, N. J., & Freed, A. D. (2002). A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dynamics, 29(1–4), 3–22. Dubois, D., Kerre, E., Mesiar, R., & Prade, H. (2000). Fuzzy interval analysis. In Fundamentals of fuzzy sets (pp. 483–581). Springer. Goetschel, R., Jr., & Voxman, W. (1983). Topological properties of fuzzy numbers. Fuzzy Sets and Systems, 10(1–3), 87–99. Hashim, I., Abdulaziz, O., & Momani, S. (2009). Homotopy analysis method for fractional IVPs. Communications in Nonlinear Science and Numerical Simulation, 14(3), 674–684. Heilpern, S. (1997). Representation and application of fuzzy numbers. Fuzzy Sets and Systems, 91(2), 259–268. Kaleva, O. (1990). The calculus of fuzzy-valued functions. Applied Mathematics Letters, 3(2), 55–59. Krätschmer, V. (2002). Some complete metrics on spaces of fuzzy subsets. Fuzzy Sets and Systems, 130(3), 357–365. Li, Y., & Sun, N. (2011). Numerical solution of fractional differential equations using the generalized block pulse operational matrix. Computers & Mathematics with Applications, 62(3), 1046–1054. Long, H. V., Son, N. T. K., & Tam, H. T. T. (2017). The solvability of fuzzy fractional partial differential equations under Caputo GH-differentiability. Fuzzy Sets and Systems, 309, 35–63. Lupulescu, V. (2015). Fractional calculus for interval-valued functions. Fuzzy Sets and Systems, 265, 63–85. Mazandarani, M., & Kamyad, A. V. (2013). Modified fractional Euler method for solving fuzzy fractional initial value problem. Communications in Nonlinear Science and Numerical Simulation, 18(1), 12–21. Salahshour, S., Ahmadian, A., & Chan, C. S. (2016). A novel technique for solving fuzzy differential equations of fractional order using Laplace and integral transforms. In 2016 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE) (pp. 1473–1477). IEEE. Salahshour, S., Allahviranloo, T., & Abbasbandy, S. (2012). Solving fuzzy fractional differential equations by fuzzy Laplace transforms. Communications in Nonlinear Science and Numerical Simulation, 17(3), 1372–1381. Stefanini, L., & Bede, B. (2014). Generalized fuzzy differentiability with LU-parametric representation. Fuzzy Sets and Systems, 257, 184–203. Suarez, L., & Shokooh, A. (n.d.). An eigenvector expansion method for the solution of motion containing fractional derivatives. Van Hoa, N., & Vu, H. (n.d.). A survey on the initial value problems of fuzzy implicit fractional differential equations. Fuzzy Sets and Systems. Van Hoa, N., Vu, H., & Duc, T. M. (2019). Fuzzy fractional differential equations under Caputo–Katugampola fractional derivative approach. Fuzzy Sets and Systems, 375, 70–99. Van Ngo, H., Lupulescu, V., & O’Regan, D. (2018). A note on initial value problems for fractional fuzzy differential equations. Fuzzy Sets and Systems, 347, 54–69. Zureigat, H., Ismail, A. I., & Sathasivam, S. (2019). Numerical solutions of fuzzy fractional diffusion equations by an implicit finite difference scheme. Neural Computing and Applications, 31(8), 4085–4094.