Abstract:
In this study, we introduce an efficient numerical approach for solving time-fractional differential equations that use the Atangana-Baleanu fractional derivative with the Mittag-Leffler kernel. We showcase the applicability of the proposed method via two representative time-fractional models: the HIV infection model that represents the dynamics of CD4+ T-cells and the Keller-Segel chemotaxis model. The first step of the procedure involves creating a numerical approximation for the Atangana-Baleanu fractional derivative and employing it to develop the Legendre operational matrix of fractional differential approximation. Once the proposed operational matrix is created, along with Legendre spectral collocation method, the governing fractional differential equation along with the associated initial and boundary conditions are converted into a system of nonlinear algebraic equations that we later solve to obtain solutions. Several numerical examples were presented in order to evaluate the effectiveness of the method developed and the obtained numerical results closely coincided with the correlated exact solutions that proved the correctness, efficiency, and proficiency of the proposed method of solving nonlinear time-fractional differential equations using the Mittag-Leffler kernel.

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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.

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Abstract:
In the present scientific article, an efficient operational matrix based on the famous Bernstein polynomials is applied for the numerical solution of two-dimensional non-linear variable order reaction-diffusion equation in porous media with given initial and boundary conditions. An operational matrix is constructed for fractional variable order differentiation w.r.to space variable $x, y$ and time $t$, so that our proposed model is converted into a system of non-linear algebraic equations with the help of collocation method, which can be solved employing the Newton-Iteration method. The salient features of the article are finding the stability analysis and error bounds of the proposed method and also the validation and the effectiveness of the method through the RMS, $ L_{\infty} $ and $L_{2}$ errors. The physical presentation of the these errors for considered two-dimensional non-linear variable order reaction-diffusion with their exact solutions shows the method is too good for finding the solution of these kind of problems.

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Abstract:
This article presents a Laguerre-Galerkin approximation for the numerical solution of time
fractional sub-diffusion equations involving the Caputo fractional derivative. The proposed
methodology employs the separation of variables technique to decompose the governing equa
tion into a spatial ordinary differential equation and a fractional temporal equation into spatial
and temporal components. The resulting spatial boundary value problem is approximated using
Laguerre polynomial basis functions, while the temporal component is represented through the
Mittag-Leffler function. Both Neumann and Dirichlet boundary conditions are investigated.
For problems with Dirichlet boundary conditions, a shooting technique is utilized to obtain an
equivalent Neumann formulation suitable for the proposed approach. Numerical experiments
are carried out for different values of the fractional order and approximation parameters. The
obtained numerical solutions are compared with available exact solutions through graphical and
error analysis. The results demonstrate that the proposed method provides accurate approxi
mations with good convergence characteristics and serves as an efficient Computationally tool
for the study of time-fractional sub-diffusion models.

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Abstract:
This study examines the HU-stability of a class of fractional functional differential systems including the “Caputo–Fabrizio derivative”, which is especially efficient in modelling systems with non-immediate impulses due of its non-singular, exponentially decaying kernel. Here the system is described across a piecewise interval with both differential and integral requirements, where the generalized fractional derivative is applied on subintervals and non-local integral restrictions are placed elsewhere. Functionals satisfying Lipschitz continuity control nonlinearities in the system, hence guaranteeing limitedness and regularity. The contraction criterion ρ < 1 is satisfied by the system over all admissible values of the fractional order δ ∈ (0,1) by use of analytical estimates and numerical validation. Tables and graphs show the changes of stability-related constants concerning δ, hence stressing the stabilizing function of the Caputo–Fabrizio derivative. The results verify that any approximate solution z(x) stays near the precise solution φ(x), hence confirming the HU-stability of the system and stressing its strength under perturbations.

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Abstract:
Fractional differential equations (FDEs) are a more advanced version of classical equations that consist of non-integer order derivatives. This modification of differential equations allows modeling complex systems with memory effects, hereditary properties, and anomalous behavior, which cannot be described using classical integer-order models. Over the last few years, fractional differential equations have found broad applications in various areas of science and engineering including physics, biological research, and finance. FDEs are employed in modeling unusual diffusion behavior in physics such as subdiffusion and superdiffusion in nonhomogeneous materials. In biology, FDEs are used to describe memory-based processes including the dynamics of brain networks and viscoelastic materials. In finance, fractional mathematical models help to explain certain aspects of financial market volatility and temporal dependence in financial time series. The field of fractional calculus has undergone extensive evolution, resulting in numerous definitions of fractional derivatives, including the classical Riemann-Liouville derivative, the modern Caputo derivative, and the more advanced Grünwald-Letnikov derivative. The aim of this paper is to succinctly demonstrate basic properties of finite difference equations (FDEs), introduce representative numerical and analytical methods used to resolve these equations, and comment on the wide applicability of these equations in various areas of science and technology. The advantages of the application of finite difference equations in modeling the processes in the real world, the problems of their computational realization, and the opportunities for their implementation in the course of solving scientific problems will be discussed.

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