Advances in Fractional Modeling and Computation

1. Introduction

Fractional calculus, the study of derivatives and integrals of arbitrary order [1,2,3], has a history dating back to Leibniz in 1695, who first considered a half-order derivative. Over time, mathematicians such as Liouville, Riemann, and Grünwald formalized fractional operators. Today, commonly used definitions include the Riemann–Liouville, Caputo, and Grünwald–Letnikov derivatives [4], while other important definitions, such as Weyl, Hadamard, Marchaud, and Riesz fractional derivatives, are widely applied in mathematical physics and engineering.

In the third quarter of the 20th century, fractional calculus experienced renewed interest due to its ability to model systems with memory and nonlocal effects. Applications emerged in viscoelasticity, anomalous diffusion, and control theory, supported by advances in analytical and numerical methods. Theoretical foundations, including stability, existence, and uniqueness of solutions, were also developed.

The 21st century has seen rapid expansion in fractional calculus, driven by the introduction of novel operators such as the Caputo–Fabrizio (2015) and Atangana–Baleanu–Caputo (2016) derivatives [5,6]. These definitions utilize nonsingular kernels to overcome the limitations of classical formulations and offer improved modeling of memory effects. Among these, the Hilfer fractional derivative has garnered significant attention for providing greater flexibility in capturing intermediate and mixed memory behaviors [7]. Collectively, these advances have significantly expanded the applicability of fractional calculus across various disciplines including physics, biology, finance, and engineering [8,9,10,11,12].

This Special Issue on ‘Advances in Fractional Modeling and Computation’ reflects current trends in the rapid development of fractional calculus as a powerful framework for modeling complex, memory-dependent, and nonlocal phenomena. Out of 64 received submissions, 21 high-quality papers were accepted for publication, resulting in an acceptance rate of 37.5%. The collected papers address recent theoretical advances, efficient numerical and computational methods, and diverse applications of fractional differential equations in science and engineering. Emphasis is placed on modern fractional operators and high-accuracy algorithms, highlighting the role of fractional models in anomalous diffusion, viscoelasticity, control theory, and applied physics. Together, these contributions demonstrate how fractional modeling and computation continue to evolve in response to contemporary scientific and technological challenges.

2. An Overview of Published Articles

This volume presents recent advances in both analytical and numerical approaches to fractional differential equations, emphasizing their role in modeling real-world phenomena and demonstrating the practical value of fractional calculus for addressing challenges in science and engineering. The contributions in this reprint are organized into three main themes, reflecting the scope of the Special Issue.

2.1. Theoretical Analysis of Fractional Operators and Systems

The theoretical properties of fractional operators and systems are explored in the following papers. Contribution 1 study a class of boundary value problems for impulsive fractional integro-differential equations involving the Caputo–Hadamard fractional derivative. Sufficient conditions for the existence and uniqueness of solutions are established using the Banach contraction principle and Krasnoselskii’s fixed-point theorem. Contribution 2 introduce a fractional order chaotic system with nonlinear terms and study its dynamic behavior. Various characteristics such as stability of equilibrum points, phase portraits, bifurcation diagrams and Lyapunov exponents are examined to illistrate the behavior of the system and design a feedback control scheme to stabilize the system. Contribution 3 propose a fully intuitionistic fuzzy multi-objective, multi-item solid fractional transportation model with fractional objective functions to handle imprecision in logistical parameters. The proposed model is linearized and reduced to a crisp multi-objective problem, which is then solved using three types of membership functions. Contribution 4 analyze neutral fractional systems with Riemann–Liouville derivatives and distributed delays, proving the existence and uniqueness of solutions when the initial functions are discontinuous. In addition, they introduce a new concept of weighted stability in the Lyapunov, Ulam–Hyers, and Ulam–Hyers–Rassias senses, extending classical stability notions to systems with Riemann–Liouville derivatives and establishing key stability results. Contribution 5 introduce a generalized degenerate version of two-dimensional Appell polynomials using tools from fractional calculus. Several fundamental properties of these new polynomials are established, including their generating function, explicit series representation, recurrence relations, and determinant form, showing that well-known two-dimensional and classical Appell polynomials arise as special cases. Contribution 6 investigate a tempered fractional sub-diffusion model with a changing-sign perturbation and weak singularity, which is relevant to anomalous diffusion phenomena. A novel power factor is introduced into the Green function, ensuring that the associated linear operator has a positive lower bound. By applying a priori estimates and Schauder’s fixed-point theorem, the authors establish the existence of a positive solution.

2.2. Numerical Methods for Fractional Differential Equations

Numerical methods for solving fractional differential equations are discussed in the following six papers. Contribution 7 develop a new compact difference–Galerkin spectral method for solving multi-dimensional fourth-order partial differential equations with a time-fractional derivative of order $\alpha \in (1,2)$ under complex boundary conditions. The stability, convergence, and solvability of the proposed numerical scheme are rigorously established. Contribution 8 develop a numerical approach for solving a class of fractional nonlinear mixed integro-differential equations that model various physical phenomena. This work introduces a quadrature Nyström method to numerically treat this challenging class of equations. The study provides a rigorous theoretical analysis, including proofs of existence, uniqueness, convergence, and error bounds for the solution. Contributions 9 and 10 construct expansion formulas of order $2-\alpha$ for approximations of the Caputo fractional derivative, parameter-dependent discretizations of the first derivative, and new approximations of the fractional derivative. The two papers examine applications of the approximations for the numerical solution of classical and fractional differential equations. Proofs of the convergence of the numerical methods and error estimates are presented. Contribution 11 develop a hybrid numerical method for solving fractional mixed diffusion and wave–diffusion equations involving Caputo derivatives of different orders. The approach combines the Laplace transform with a local radial basis function (RBF) scheme and the Stehfest algorithm, which circumvent traditional time-stepping procedures, thereby reducing computational complexity and improving efficiency. Contribution 12 introduce a Genocchi polynomial method to numerically solve a class of nonlinear fractional integral equations. The paper establishes the existence and uniqueness of solutions in a Hilbert space setting and analyzes the stability and error of the numerical approximations.

2.3. Fractional Models of Memory-Based Systems

Applications of fractional calculus to epidemiology, physics, and materials science are studied in the following nine papers. Contribution 13 propose a novel fractional discrete reaction–diffusion model to study the spread of infectious diseases. The linear stability of the equilibrium solutions and the global asymptotic stability of the model are analyzed using a specialized Lyapunov function. Contribution 14 develop a sophisticated mathematical model to describe HIV transmission dynamics by incorporating a fractal–fractional operator and three distinct levels of infection. The existence and uniqueness of the solutions are established, and the stability of both disease-free and endemic equilibria is analyzed using tools from fractional calculus. Contribution 15 formulate a fractional order model for measles that captures disease transmission dynamics and assesses the influence of vaccination on the population over time. The study formulates and solves an optimal control problem aimed at minimizing both the disease burden and the economic costs of intervention strategies. Contribution 16 develop a fractional order epidemic model to study the dynamics of a vector-borne disease, incorporating preventive control measures. A rigorous mathematical analysis is conducted to establish the existence, uniqueness, and boundedness of the model’s solutions and the stability of the equilibrium states. Contribution 17 investigate an SIR epidemic model that incorporates spatial diffusion and employs a generalized fractional derivative to describe disease dynamics. Rigorous mathematical analysis is presented to establish the existence and uniqueness of solutions to the model, and a numerical scheme is developed for its solution. Simulation results demonstrate the effectiveness of the proposed method and highlight the significant roles that the fractional order and diffusion play in influencing and controlling epidemic dynamics. Contribution 18 propose a fractional order tabu learning neuron model by incorporating fractional calculus into a classical integer-order tabu learning neuron framework. The authors analyze the model’s dynamics, including stability, bifurcation behavior, and chaotic phenomena, using both theoretical analysis and numerical simulations. Stability analysis based on the eigenvalues of the Jacobian matrix shows that a Hopf bifurcation occurs in the single-neuron model at a lower memory decay rate than in its integer-order counterpart. Contribution 19 introduce a fractional order viscoelastic creep model that captures the temperature-dependent creep behavior of concrete by employing fractional derivatives to represent its viscoelastic response. Experimental creep data obtained at various temperatures are used to validate the model, indicating that the fractional order approach provides a more accurate and parameter-efficient description of creep behavior than traditional empirical models. Contribution 20 integrate fractional calculus into cosmology to study scalar field models with conformal and nonminimal couplings to gravity. The study analyzes both exact and numerical solutions for different coupling forms, demonstrating that fractional order dynamics can reproduce accelerated cosmic expansion and provide a generalized framework for investigating the interplay between gravity and scalar fields in the evolution of the universe. Contribution 21 derive a new general fractional calculus model for time-dependent nuclear reactor neutron diffusion that incorporates the effects of delayed neutrons. Numerical studies for slab, cylindrical, and spherical reactor geometries are carried out using two analytical approximation techniques, the Temimi–Ansari method (TAM) and the Sumudu residual power series method (SRPSM), confirming that both methods yield consistent and physically realistic neutron diffusion results.

3. Concluding Remarks

In conclusion, the works presented here affirm that fractional calculus has evolved from a well-established mathematical field into an essential cross-disciplinary framework for modeling complex systems. Its growing impact across physics, engineering, biology, and the applied sciences demonstrates its unique effectiveness in capturing memory-dependent, nonlocal, and multiscale phenomena/features often inadequately described by classical integer-order approaches. We extend our deepest gratitude to the authors for their outstanding contributions, to the reviewers for their thorough and diligent work, and to the editorial staff of Fractal and Fractional for their support in making this Special Issue a valuable resource for the scientific community and for future research.