Boundary Value Problems for Nonlinear Fractional Differential Equations: Theory, Methods, and Applications

1. Introduction

This special issue is devoted to the advancement of fractional-order differential equations and their wide-ranging applications. Such problems often involve boundary conditions that model nonlinear phenomena across diverse fields, including physics, economics, finance, and the social sciences. For example, fractional models have been applied to the design of heating and cooling systems [1,2], the study of incompressible and viscous fluid flows [3], and the epidemiology of childhood diseases [4].

While many classical topological and numerical methods for differential equations remain applicable, fractional-order equations may exhibit different behaviors from their integer-order counterparts. In addition, fractal–fractional calculus incorporates both fractional-order and fractal properties, allowing for a more accurate representations of memory-dependent systems. Thus, boundary value problems (BVPs) involving nonlocal conditions are particularly important in systems with long-range interactions or memory [5].

Recent studies have extended these concepts to hybrid fractional systems [1], impulsive integro-differential equations, and integral boundary conditions [6]. For instance, novel approaches have been developed for fractional integro-differential systems with delay and impulsive terms [7], as well as for nonlinear equations with four-point $\mathsf{\Psi }$-Caputo boundary conditions [8]. Other recent works address existence, uniqueness, and Ulam–Hyers stability for fractional integro-differential equations with nonlocal boundary conditions [9,10], demonstrating the continuing expansion of theoretical frameworks for fractional BVPs.

On the numerical side, solving fractional-order evolution differential equations provides accurate predictions of the behavior of complex systems. Advanced numerical techniques include adaptive step-size schemes [11] and neural network-based methods [12,13], which improve both computational efficiency and accuracy in high-dimensional fractional problems. Such methods complement analytical approaches by enabling the practical resolution of complex real-world systems.

This special issue received 36 submissions, of which 12 papers were accepted, resulting in an acceptance rate of 33%. All submissions underwent rigorous peer review by at least two experts in the field, and each accepted paper presented novel contributions to theory, methodology, or applications. Contributions in theory include a comprehensive study of new types of fractional derivatives, the introduction of a class of contractions in normed spaces, and general coincidence degree theory for operators with impulsive terms, along with their applications in solvability. The results generalize previous work in related areas.

In addition to widely applied fixed-point theory, new approaches were derived on the existence, uniqueness, and stability of solutions for various BVPs. For example, parameterized identities and symmetric inequalities have been established and applied to a class of fractional operators. Furthermore, the introduction of a novel neural network method for solving time-fractional diffusion equations represents an advancement in numerical computation combined with deep learning.

Topics in this issue also include the modeling of dynamical systems, such as global health challenges, quasi-static contact problems in nonlinear thermo-elasticity, and fractional-order thermostats. Studies on impulsive differential inclusions and impulsive differential equations contribute to recent advancements in modeling processes and phenomena with sudden state changes [7,14].

In Section 2, we present a detailed summary of each individual contribution.

2. An Overview of the Published Articles

This section provides a brief overview of the papers published in this special issue. Each contribution is summarized with emphasis on its motivation, methodological approaches, and main results.

• Contribution 1.

This paper makes a direct contribution to modeling and applications. The authors introduced a hybrid Caputo fractional inclusion subject to a nonlocal multivalued boundary condition with a multivalued control variable. The proposed boundary value problem involves Chandrasekhar’s kernel and provides a mathematical model for thermostats.

To establish solvability and uniqueness, the problem is first converted to an equivalent integral equation, and then Dhage’s hybrid fixed-point theorem [15] is applied. Furthermore, the continuous dependence of the unique solution on the control variable and on the set of selections is investigated. Several examples are provided to illustrate the theoretical results.

Integral equations involving Chandrasekhar’s kernel have been extensively studied due to their wide applicability in various fields, including traffic theory, neutron transport theory, the kinetic theory of gases, and radiative transfer theory. This contribution extends earlier investigations on fractional-order thermostat models and generalizes the recent results of [1,2].

• Contribution 2.

Motivated by the recent work of four-point inequalities for functions whose first derivatives are s-convex [16], the authors of this contribution derived new fractional integral inequalities involving four points using the Riemann–Liouville fractional integral operator for functions whose first derivatives are s-$tgs$-convex [17]. The resulting inequalities were further applied to problems in numerical quadrature. To prove the results, a novel parameterized integral identity for differentiable functions is introduced.

Convexity plays a fundamental role in the theory of inequalities and is closely related to the qualitative analysis of solutions to ordinary, partial, and integral differential equations, as well as to numerical analysis, particularly in estimating errors for quadrature rules. The results presented in this contribution may stimulate further research in this area and lead to extensions in other frameworks, including multiplicative calculus and quantum calculus.

• Contribution 3.

The authors investigate a coupled system of Langevin fractional problems involving $\mathsf{\Psi }$-Caputo fractional derivatives with generalized slit-strip-type integral boundary conditions and impulsive effects. Coupled systems of fractional differential equations have been applied in a wide range of physical and practical models, including those describing disease dynamics, environmental processes, and chaotic systems [18,19]. Slit-strip boundary conditions arise naturally in applications such as imaging and acoustics using strip detectors [20,21].

By applying the Banach contraction principle and Schaefer’s fixed point theorem, the authors establish results on the existence and uniqueness of solutions. In addition, Ulam–Hyers stability is analyzed, ensuring that approximate solutions remain close to exact solutions within a prescribed region. This property is particularly important in approximation theory and the numerical analysis of related fractional problems.

The Langevin equation plays a fundamental role in modeling phenomena associated with Brownian motion and has been widely used to describe processes in economics, engineering, medicine, and other applied sciences. This contribution further enriches research on fractional Langevin-type models and their applications.

• Contribution 4.

This paper addresses nonlinear impulsive Caputo–Hadamard fractional Langevin and Sturm–Liouville equations subject to nonlocal boundary conditions within a unified framework. The authors formulate a general nonlinear impulsive coupled implicit system and investigate its solvability and stability using the coincidence degree theory [22] and inequality techniques [23,24].

In applications, the Langevin equation is widely used to model Brownian motion. Compared with their integer-order counterparts, fractional-order Langevin equations offer a more accurate description of random motion in complex viscoelastic fluids. Moreover, Sturm–Liouville equations, including the Helmholtz, Bessel, and Legendre equations, play a fundamental role in mathematical physics and engineering. Most existing studies treat these two classes of equations separately. This work fills a gap in the literature by providing a unified approach to nonlocal boundary value problems for implicit and impulsive fractional coupled systems.

• Contribution 5.

This paper introduces a new class of fractional differential operators, namely the p-proportional $\omega$-weighted $\kappa$-Hilfer fractional differential operator of order $\sigma \in (1,2)$ and type v. Fundamental properties of this newly defined operator are established. This operator generalizes many well-known fractional derivatives, including the Caputo proportional, Hilfer–Hadamard, Hilfer–Katugampola, $\kappa$-Riemann–Liouville, $\kappa$-Caputo, $\kappa$-Hilfer, and proportional $\kappa$-Riemann–Liouville operators [25,26].

The authors further investigates the existence and stability of solutions for a coupled system of impulsive fractional differential inclusions involving the proportional weighted $\kappa$-Hilfer fractional derivative. By employing fixed-point theory for multivalued mappings, existence and stability results were obtained. It is shown that many existing results in the literature can be recovered as special cases, demonstrating the generality of the proposed framework.

It is noted that this type of fractional differential inclusion has not been studied previously, highlighting the originality of this contribution. The paper also outlines possible directions for future research and provides illustrative examples to demonstrate the applicability of the theoretical results.

• Contribution 6.

This paper makes direct contributions to both modeling and applications by studying a fractional nonlinear singular second-order integro-differential equation with an initial condition and a one-point boundary condition, involving the Caputo derivative of order $0<\sigma \le 1$.

The model captures a general one-dimensional quasi-static contact problem in fractional thermo-elasticity, formulated using the fractional heat equation with a Bessel operator [27]. By defining appropriate function spaces, the authors prove the uniqueness of the solution within a fractional Sobolev space and establish existence using the Schauder fixed-point theorem.

Singular integro-differential equations—featuring singularities in coefficients, the integral term, or the solution—pose significant analytical challenges compared to regular integro-differential equations. The results advance our understanding of nonlocal initial boundary value problems for such equations, providing a rigorous framework for their well-posedness and solution.

• Contribution 7.

This paper investigates fixed-point theorems for contractive mappings in b-metric spaces, cone b-metric spaces, and the newly introduced extended b-metric spaces [28]. Applications of b-metric spaces span similarity and pattern recognition, string matching, trademark shape analysis, ice floe tracking, and optimal transport in probability measures, with pattern recognition being particularly relevant to data analytics and machine learning [29].

Following an abstract-to-concrete approach, the authors first examine general equations involving the Urysohn integral operator, establish positive solutions in cone b-metric spaces, and extend these results to the newly introduced extended b-metric spaces by generalizing the Hölder continuity condition. The theoretical results are then applied to a previously studied fractional boundary value problem, with a numerical simulation illustrating their effectiveness and providing intuitive insight.

• Contribution 8.

In the literature, various approaches have extended the classical Banach fixed-point theorem, including enlarging the domain of the mapping, introducing generalized contraction criteria, and exploring multi-point analogs of classical operators. Notably, recent studies have proposed innovative mapping techniques capable of contracting the perimeters of triangles [30] and constructing generalized Kannan-type mappings [31]. Later, Berinde and Păcurar [32] investigated enriched Kannan operators and demonstrated their applications to split feasibility and variational inequality problems.

Building on this body of work, the authors introduce the concept of generalized enriched Kannan mappings, a three-point analog that unifies and extends the previous enriched Kannan operators. This new framework allows for a broader class of fixed-point problems to be analyzed, providing more flexible and powerful tools for both theoretical investigation and applied problem solving.

To demonstrate the utility of their results, the authors apply these mappings to establish the existence of solutions for a boundary value problem involving a fractional differential equation. By extending fixed-point theorems to more general settings, this study not only deepens the theoretical understanding of multi-point and enriched operators but also enhances the practical applicability of fixed-point theory in solving fractional differential and boundary value problems. The inclusion of illustrative examples and rigorous proofs underscores both the robustness and the potential impact of these results in applied mathematics and computational modeling.

• Contribution 9.

This paper provides a comprehensive study on the maximal possible regularity of solutions for Hybrid proportional Caputo derivatives. The authors consider the problem not only in spaces of absolutely continuous functions but also in little Hölder spaces, which offer a natural and flexible framework for analyzing classical Riemann–Liouville integral operators as inverses of fractional derivatives [33,34]. In particular, the study examines the Hölder regularity of the operator values, proving that Hölder spaces are natural domains for both differential and integral operators, ensuring their invertibility while maintaining maximal regularity.

Moreover, the investigation extends these results to the more general setting of Hilfer hybrid derivatives, representing a first study of this type in the context of proportional calculus. The theoretical findings are complemented by applications to Langevin-type equations, where the results are used to construct right and left inverses for operators involving proportional Caputo derivatives. This work not only establishes a rigorous functional analytic framework for these operators but also provides practical tools for applying fractional calculus to boundary value problems, highlighting both the theoretical depth and potential applicability of hybrid proportional derivatives.

• Contribution 10.

Building on recent developments with the ABC fractional operator [35], which has proven effective in capturing complex dynamics in real-world systems, subsequent studies introduced the modified ABC (mABC) operator [36] and further extended it to a piecewise mABC (pmABC) fractional operator [37], combining classical and modified fractional derivatives to capture hybrid behaviors.

Leveraging these advances, the authors develop a novel mathematical model for diarrhea dynamics using a system of fractional differential equations that incorporates the pmABC operator. This framework captures the disease’s transmission dynamics, including crossover effects between classical and fractional behaviors. The study rigorously analyzes the existence and uniqueness of solutions, their positivity and boundedness, and evaluates key epidemiological metrics such as the basic reproduction number, along with the sensitivity analysis and local and global stability of the disease-free equilibrium.

The results highlight that increases in the effective contact rate lead to higher reproduction numbers, emphasizing the critical role of preventive measures in controlling disease spread. By integrating advanced fractional operators into epidemiological modeling, this work not only deepens the theoretical understanding of disease dynamics but also provides practical insights for public health interventions.

• Contribution 11.

This paper investigates a three-dimensional system of fractal–fractional-order evolution differential equations subject to terminal boundary conditions. The authors derive sufficient criteria for the existence and stability of solutions and apply their results to a general problem, providing graphical representations of the solutions for various fractal and fractional orders to demonstrate the effectiveness of the proposed approach.

To illustrate the practical relevance, the study includes a numerical example with plots showing the system’s behavior under different fractal–fractional orders. The analysis reveals that the fractional order primarily affects the smoothness and stability of the system, while the fractal dimension governs the rate and pattern of change, highlighting the distinct roles of these parameters in shaping the dynamics.

Evolution equations [38,39] are fundamental in modeling real-world phenomena, as they allow solutions to be constructed from given initial or boundary conditions. By combining fractal and fractional derivatives, this work provides a versatile framework for analyzing complex dynamical systems, offering both theoretical insights and practical tools for applications in mathematical modeling and applied sciences.

• Contribution 12.

This paper presents a neural network-based approach for solving time-fractional partial differential equations (TFPDEs) with Dirichlet boundary conditions by combining machine learning techniques with the Method of Lines (MOL). Building on previous methods for TFPDEs—including finite difference, finite element, spectral, homotopy, spline, and MOL techniques [40]—the authors extend these approaches by leveraging neural networks to approximate solutions efficiently and accurately.

Typically, MOL is a semi-analytical method applied to integer-order diffusion equations, discretizing the spatial domain while keeping the time domain continuous. The main challenges in machine learning approaches lie in designing an effective neural network architecture and defining an appropriate loss function. To overcome these challenges, the authors employ Gauss–Jacobi quadrature to construct a suitable loss function for the neural network, ensuring an accurate approximation of the TFPDE solution.

Numerical experiments demonstrate that the proposed method is straightforward to implement and effective in handling 1D, 2D, and 3D time–fractional diffusion problems, providing both computational efficiency and flexibility. This study illustrates the growing synergy between fractional calculus and machine learning, offering a powerful tool for solving complex fractional differential equations in applied science and engineering.

3. Conclusions

Fractional calculus has the advantage of modeling complex, memory-dependent, and nonlocal phenomena in science and engineering [41]. Common formulations, such as Caputo and Riemann–Liouville, balance physical interpretability with mathematical rigor, while Hadamard-type Caputo derivatives extend this framework using a logarithmic kernel to generalize classical RL operators.

By extending differentiation to non-integer orders, fractional calculus enables the analysis of a wide range of applications, including viscoelasticity, anomalous diffusion, control systems, and bioengineering. The contributions in this Special Issue advance both the theoretical foundations and practical applications of fractional-order differential equations, offering new insights and tools for researchers.

The success of this issue is evident in its strong readership and citations, and a second edition has already been launched, promising to further enrich this rapidly evolving field.