Search Results (372)

In this paper, we study some existence and stability results related to the boundary value problem involving the Atangana–Baleanu–Caputo hybrid fractional derivative. More precisely, we transform the studied problem to an equivalent integral equation, and after that, by applying appropriate fixed-point theorems and using suitable conditions, we prove the existence of solutions. Furthermore, we derive sufficient conditions that guarantee the stability in the sense of Hyers–Ulam. To support the theoretical findings, we present illustrative examples along with numerical simulations. Full article

This paper establishes a rigorous analysis of a coupled hybrid fractional differential system involving a generalized Hilfer operator under integral and antiperiodic boundary conditions. The existence and uniqueness of solutions are proved using Dhage’s fixed point theorem for existence and the Banach contraction principle for uniqueness. Furthermore, we establish Ulam–Hyers stability by deriving the following explicit and computable bound estimate: $\left(\begin{array}{c}\parallel \widehat{u}-u\parallel \\ \parallel \widehat{v}-v\parallel \end{array}\right)\le {(I-\mathit{\chi })}^{-1}\left(\begin{array}{c}{C}_1{ϵ}_1\\ C_2{ϵ}_2\end{array}\right)$, where $C_1$ and $C_2$ are positive constants depending on the system parameters, ${ϵ}_1,{ϵ}_2$ denote the perturbation bounds, and $\mathit{\chi }$ is the associated Lipschitz matrix. This formulation provides a more detailed stability description than scalar criteria, as it captures the interactions among the system components through the entries of $\mathit{\chi }$, leading to a more informative stability estimate. Two illustrative examples confirm the theoretical results and demonstrate their potential applicability for modeling real-world phenomena where memory effects are present. Full article

We study a perturbed coupled system of generalized hybrid pantograph equations involving the deformable derivative of Zulfeqarr–Ujlayan–Ahuja. A central point of the revision is made explicit: for classically differentiable functions this derivative is local and satisfies $D^{\tau }u=(1-\tau )u+\tau u^{\prime }$. Therefore, in the present differentiable setting the memory or aftereffect is produced by the proportional pantograph delays, while the deformable order $\tau$ supplies an order-dependent local relaxation/drift term. After rewriting the system as an equivalent integral equation on $\mathcal{X}=C(I,{\mathbb{R}}^2)$, we establish invariant-ball conditions, existence and uniqueness within invariant balls, generalized Ulam–Hyers stability, and Lipschitz continuous dependence on the perturbation amplitude $\epsilon$. The assumptions and constants are stated so that the restrictive roles of the Lipschitz bounds, the interval length, and $\left|\epsilon \right|$ are transparent. We then provide numerical parameter sensitivity diagrams for illustrative pantograph systems and include step-size refinement checks and performance indices. The numerical and plasma-inspired sections are deliberately framed as exploratory delay-feedback examples rather than as first-principles plasma models or rigorous bifurcation theory. Full article

This article employs a fractional differential equation model to probe the dynamic mechanism of animal avoidance learning and memory retention. This model encompasses both linear and nonlinear scenarios. We first obtain the series-type analytical solution for the linear scenario and its absolute uniform convergence by Laplace transform and Mittag–Leffler function. Secondly, we establish the existence, uniqueness and Ulam–Hyers stability for the nonlinear scenario via the fixed point theorem and analytical techniques. Eventually, some examples and numerical simulations are provided to examine the effectiveness and availability of the main findings. Full article

Diabetes mellitus is a chronic disease with complex progression dynamics. This study introduces a fractional order compartmental model based on the Caputo derivative, a singular-kernel derivative, to describe disease progression across four compartments: susceptible, insulin-resistant, diabetic without complications, and diabetic with complications. The model is novel for integrating memory effects into disease-stage transitions while maintaining dimensional consistency. Key mathematical properties, including existence, uniqueness, positivity, boundedness, equilibrium analysis, and both local and global stability, are established. Ulam–Hyers stability is also examined to evaluate the robustness of the model solutions. Numerical approximations are obtained using the Adomian Decomposition Method and its Laplace variant. Simulations indicate that lower fractional orders enhance memory effects, slow disease progression, and influence long-term dynamics. These results demonstrate that the proposed approach provides a flexible and robust framework for studying chronic disease progression and makes a meaningful contribution to the literature on fractional diabetes models. Full article

This study formulates closed-form solution expressions for linear discrete matrix equations that involve several time delays, without requiring the coefficient matrices or the non-homogeneous term to commute. Using a generalized multinomial series and exponential matrix functions adapted to multiple delays, we establish fundamental solutions in a setting where matrix multiplication is not assumed to be commutative. These explicit representations are subsequently utilized to analyze the stability properties of the system, specifically establishing Hyers–Ulam stability. The analysis elucidates the influence of both delay structure and noncommutativity on solution behavior and robustness. A representative example is provided to illustrate the practical applicability of the proposed method and to highlight the significant qualitative effects induced by delays and noncommutative matrix interactions. Notably, the results extend classical theories by addressing noncommutative settings and yield novel contributions that remain significant even in the absence of delays. Full article

This paper investigates the existence, uniqueness, and stability of solutions for a new class of coupled systems of sequential fractional differential equations involving the Hilfer fractional derivative. Generalizing previous works based on Caputo derivatives, we employ the Hilfer operator, which interpolates between Riemann–Liouville and Caputo derivatives. The nonlinear terms are fully coupled, and the boundary conditions are nonlocal and coupled. The main results are established using the Banach Contraction Principle and Schaefer’s Fixed Point Theorem, with rigorous, detailed proofs for each step, addressing specific methodological requirements regarding operator invariance and space completeness. Furthermore, we provide a comprehensive analysis of the Ulam–Hyers stability of the proposed system, with explicitly tracked stability constants. An illustrative example with numerical verification is provided to validate the theoretical findings. Full article

This research presents a fractional-order formulation and mathematical analysis of the Rössler chaotic attractor. By utilizing the Nabla discrete Atangana–Baleanu fractional difference derivative in the Caputo sense, the classical integer-order attractor is extended into the fractional domain. The existence and uniqueness of solutions for the resulting fractional system are established via the fixed-point theorem, thereby ensuring that the recommended attractor is well-posed. Furthermore, the Ulam–Hyers stability is investigated within the Nabla discrete Atangana–Baleanu fractional difference derivative in the Caputo sense framework. For numerical investigations, an Euler numerical scheme adapted to the fractional difference derivative is developed and implemented, yielding high-quality phase portraits of a chaotic attractor. The results highlight the effectiveness of fractional-order modeling and numerical methods in capturing the dynamics and stability of the Rössler chaotic system. Full article

This paper examines an m-cyclic coupled system of sequential $(k,\psi )$-Hilfer and $(k,\psi )$-Caputo fractional differential equations with boundary conditions. The nonlinearities follow a cyclic pattern: for $j=1,\dots ,m-1$, $f_j$ depends on $x_j$ and $x_{j+1}$ and $f_m$ depends on $x_m$ and $x_1$, forming a closed loop of interactions. We convert the system into an equivalent integral equation and establish existence and uniqueness results using four fixed-point theorems: the Banach contraction principle, Schaefer’s theorem, Krasnosel’skiĭ’s theorem, and the Leray–Schauder alternative. A thorough Ulam–Hyers stability analysis is presented with explicit stability constants. Numerical examples illustrate the applicability of the theoretical findings. Full article

In this paper, we investigate the stability and superstability of a specific class of functional inequalities associated with centrally extended ∗-derivations on Banach ∗-algebras. A CE ∗-derivation $\delta :\mathcal{R}\to \mathcal{R}$ is defined as an additive mapping satisfying $\delta (x+y)-\delta \left(x\right)-\delta \left(y\right)\in Z\left(\mathcal{R}\right)$ and $\delta \left(xy\right)-\delta \left(x\right)y^{\ast }-x\delta \left(y\right)\in Z\left(\mathcal{R}\right)$ for all $x,y\in \mathcal{R}$, where $Z\left(\mathcal{R}\right)$ denotes the center of the ring. We consider the functional inequality $\parallel [a_1\delta \left(x_1\right)+a_2\delta \left(x_2\right)+a_3\delta \left(x_3\right),w]\parallel \le \parallel [\delta (a_1{x}_1+a_2{x}_2+a_3{x}_3),w]\parallel + \mathrm{\Phi }(x_1,x_2,x_3,w),$ where $\mathrm{\Phi }$ is a perturbing term. By employing the direct method, we establish several theorems concerning the Hyers–Ulam stability of this inequality in the context of unital Banach ∗-algebras. Furthermore, we provide sufficient conditions under which these functional inequalities exhibit superstability. We also explore the implications of our results for linear ∗-derivations in semiprime Banach ∗-algebras with no nonzero central ideals. Full article

This paper presents a unified analytical and computational framework for the study of typhoid fever transmission dynamics governed by a Caputo fractional-order compartmental model of order $\kappa \in (0,1]$. The population is stratified into five epidemiological classes, namely susceptible $\left(\mathcal{S}\right)$, asymptomatic $\left(\mathcal{A}\right)$, symptomatic $\left(\mathcal{I}\right)$, hospitalised $\left(\mathcal{H}\right)$, and recovered $\left(\mathcal{R}\right)$, and the governing system explicitly incorporates asymptomatic transmission, treatment dynamics, and temporary immunity with waning. The use of the Caputo fractional derivative is motivated by the well-documented existence of chronic asymptomatic Salmonella Typhi carriers, whose heavy-tailed sojourn times in the carrier state are naturally encoded by the Mittag–Leffler waiting-time distribution arising from the fractional operator. A complete qualitative analysis of the fractional system is carried out: the basic reproduction number ${\mathcal{R}}_0$ is derived via the next-generation matrix method; local and global asymptotic stability of both the disease-free equilibrium ${\mathcal{E}}_0$ (when ${\mathcal{R}}_0\le 1$) and the endemic equilibrium ${\mathcal{E}}^{*}$ (when ${\mathcal{R}}_0>1$) are established using fractional Lyapunov theory and the LaSalle invariance principle; and the normalised sensitivity indices of ${\mathcal{R}}_0$ are computed to identify transmission-amplifying and transmission-suppressing parameters. Existence, uniqueness, and Ulam–Hyers stability of solutions are established via Banach and Leray–Schauder fixed-point arguments. To complement the analytical results, a fractional physics-informed neural network ($\mathcal{PINN}$) framework is developed to simultaneously reconstruct compartmental trajectories and identify unknown biological parameters from sparse synthetic observations. $\mathcal{PINN}$ embeds the L1-Caputo discretisation directly into the training residuals and employs a four-stage Adam–L-BFGS optimisation strategy to recover five trainable parameters $\mathbf{\Theta }$ = $\{\varphi ,\mu ,\sigma ,\psi ,\beta \}$ across three fractional orders $\kappa \in \{1.0,0.95,0.9\}$. The estimated parameters show strong agreement with the true values at the classical limit $\kappa =1.0$ ($\mathrm{MAPE}=2.27\%$), with the natural mortality rate $\mu$ recovered with $\mathrm{APE}\le 0.51\%$ and the transmission rate $\beta$ with $\mathrm{APE}\le 3.63\%$ across all fractional orders, confirming the structural identifiability of the model. Pairwise correlation analysis of the learned parameters establishes the absence of equifinality, validating that $\beta$ can be reliably included in the trainable set. Noise robustness experiments under Gaussian perturbations of $1\%$, $3\%$, and $5\%$ demonstrate graceful degradation ($\mathrm{MAPE}$: $0.82\%\to 3.10\%\to 7.31\%$), confirming the reliability of the proposed framework under realistic observational conditions. Full article

This paper presents a unified theoretical and numerical investigation of Hilfer-type fuzzy fractional control systems with infinite continuous delay. By employing contraction mapping principles and compact semigroup theory, we establish rigorous solvability conditions together with Ulam–Hyers–Rassias stability results expressed in terms of Mittag–Leffler functions. To complement the analytical framework, we design and implement numerical schemes based on Euler and IMEX approaches, which confirm the theoretical predictions through simulations. The computational experiments demonstrate the robustness of the proposed framework under delayed feedback and fractional memory effects, highlighting its relevance to practical domains such as biological regulation, porous media transport, and intelligent traffic systems. The contribution of this study lies in the bridge between mathematical rigor and computational implementation, thus advancing the theory of fractional differential inclusions and providing a versatile tool for the stability analysis and control of complex systems with uncertainty and hereditary dynamics. Full article

Generalized incommensurate fractional differential systems (GIFDSs) unify classical fractional frameworks via weight functions, capturing non-uniform multicomponent system dynamics. This paper fills a critical research gap by analyzing GIFDSs for both commensurate and incommensurate weight functions. For commensurate weights ($w_i\left(t\right)=w\left(t\right)$), classical IFDS equivalence is established via state transformation. Linear homogeneous mild solutions are derived using the incommensurate Mittag–Leffler function. Existence and uniqueness of nonlinear solutions are proved under continuity and Lipschitz assumptions. Hyers–Ulam stability is verified for linear non-homogeneous systems. For incommensurate weights (distinct $w_i\left(t\right))$, a novel framework is developed: by the integral bound lemma and Picard iteration, local existence (existence on $[a,t_1]$) is established, then it is extended to the full interval. The global uniqueness is obtained by Gronwall-type inequality via combined substitution. These results are applied to Hopfield Neural Networks, showing that one-layer HNNs with tanh or sigmoid activations admit unique mild solutions under GIFDS dynamics. Full article

Fractional Langevin models are found to be useful in the study of physical phenomena such as diffusion processes, gait variability, etc. Langevin equations involving different fractional–order operators and boundary conditions have been addressed by many researchers. In this article, we formulate a new Langevin model consisting of a coupled system of Riemann–Liouville and Hadamard–type nonlinear fractional differential equations and coupled multipoint–integral boundary conditions. We present the existence and Ulam–Hyers stability criteria for solutions of the given model problem. Our study is based on the tools of the fixed–point theory. Numerical examples with graphical representations of solutions are offered to demonstrate the application of the obtained results. Our work is novel and useful in the given configuration, and specializes to some new results. Full article

This work investigates stability under Hyers–Ulam criteria for a class of Hadamard Neutral fractional stochastic differential equations (HNFSDE). The analysis applies a fixed-point theorem (FPT) combined with principles of stochastic integration. To illustrate the applicability of the derived theoretical results, two demonstrative cases are examined. Full article