Search Results (1,545)

Time-fractional diffusion equations describe subdiffusive transport in heterogeneous media, but their numerical treatment is complicated by the nonlocal Caputo derivative and by the weak singularity that the solution develops at the initial time. We study a global-in-time discretization that combines spectral collocation in time—on the fractional power basis ${\left\{t^{\mathsf{\ell }\alpha }\right\}}_{\mathsf{\ell }=0}^N$, evaluated at Chebyshev–Gauss–Lobatto nodes, which reproduces the leading terms of the singular expansion of the solution—with a second-order conservative finite-difference stencil in space that uses harmonic averaging of the diffusivity at the cell faces and therefore remains accurate across discontinuous media. The resulting fully discrete problem is a large, nonsymmetric, dense-in-time linear system whose two-norm condition number grows like the inverse square of the spatial mesh size, so that Krylov subspace iteration without preconditioning stalls under refinement. Exploiting the Kronecker sum structure of the discrete operator, we build a preconditioner by fast diagonalization of the spatial factor through the discrete sine transform. For constant diffusivity the preconditioner reproduces the operator exactly and yields a direct solver; for variable diffusivity it is spectrally equivalent to the operator, and we prove that the eigenvalues of the preconditioned system cluster in a disk centered at one whose radius depends only on the coefficient contrast, and not on the mesh, the number of temporal degrees of freedom, or the fractional order. Numerical experiments in one and two space dimensions confirm second-order spatial accuracy and a preconditioned iteration count that stays flat—twelve iterations from $M=32$ up to $M=1024$ in one dimension and eleven up to $M=256$ per direction in two—while the unpreconditioned count grows by more than two orders of magnitude. In time, the accuracy is spectral until round-off in the ill-conditioned Vandermonde matrix of the power basis takes over: the barrier is reached at $N=9,10,13$ for $\alpha =0.3,0.5,0.7$, where the attainable error is about ${10}^{-6}$. A benchmark against the L1 scheme on uniform and graded meshes, the Alikhanov L2-$1_{\sigma }$ scheme and Grünwald–Letnikov convolution quadrature quantifies when the global approach pays: on forced problems and on modes with $\kappa \lambda T^{\alpha }\lesssim 2$ it reaches a prescribed accuracy one to two orders of magnitude faster and with several times less memory, while for strongly damped modes the fractional power basis converges only algebraically and graded time marching is preferable below a relative error of ${10}^{-2}$. Full article

This paper addresses the finite-time global Mittag-Leffler projective synchronization problem for a class of uncertain fractional-order delayed neural networks with heterogeneous fractional orders, parameter uncertainties, and multiple time delays. A novel integral sliding mode control approach is developed by designing a delayed integral sliding manifold and an appropriate robust reaching law, such that the synchronization error trajectories are driven to the sliding surface within finite time. Furthermore, sufficient algebraic conditions are established to guarantee the global Mittag-Leffler stability of the reduced-order error dynamics on the sliding manifold. Under the proposed control scheme, finite-time projective synchronization is achieved in the presence of heterogeneous fractional orders, uncertain parameters, time delays, and nonlinear coupling effects. Finally, a numerical example is provided to demonstrate the validity and effectiveness of the proposed theoretical results. Full article

Four-dimensional chaotic systems have garnered significant attention due to their complex nonlinear dynamics, high-dimensional complexity, and wide range of applications in science and engineering. This paper proposes and investigates a novel four-dimensional chaotic system in both constant- and variable-order frameworks to reveal the influence of memory effects on its dynamical behavior. The variable-order formulation is established using the Liouville–Caputo fractional derivative, while an efficient numerical scheme based on Lagrange interpolation is developed to approximate the variable-order derivative accurately. A rigorous local stability analysis is first conducted to characterize the equilibrium points and establish their instability and non-hyperbolic nature under the different derivative formulations. The nonlinear dynamics of the proposed system are then comprehensively examined through phase portraits, time series, bifurcation diagrams, and Lyapunov exponent analysis. The results demonstrate that the variable-order model preserves the fundamental topological characteristics of the chaotic attractors while introducing adaptive transient responses and significantly richer dynamical behaviors than the corresponding constant fractional-order model. Furthermore, the proposed system generates previously unreported chaotic attractors and phase-space patterns, enriching the class of known four-dimensional chaotic systems and demonstrating the variable-order framework’s enhanced capability to produce diverse nonlinear phenomena through adaptive memory effects. Full article

This paper investigates how opinion dynamics with memory effects and random communication uncertainties achieve consensus in an inhomogeneous Hegselmann–Krause (H-K) model. The Caputo fractional derivative is introduced to describe the influence of historical opinions, while multiplicative noise captures random disturbances in communication. We establish sufficient conditions under which all followers almost surely converge to the leader’s opinion. Moreover, we characterize the algebraic (power-law) convergence behavior determined by the fractional order. The analysis is based on fractional Lyapunov techniques and fractional Grönwall estimates, which enable the treatment of the nonlinear stochastic system with leadership. The results reveal how memory effects, leadership strength and stochastic perturbations jointly influence consensus formation. Numerical simulations are provided to verify the theoretical results and illustrate the effects of the model parameters on the convergence dynamics. Full article

Integrated Modular Avionics (IMA) integrates safety-critical functions on shared computing and network resources, creating coupling channels through which a local fault may escalate into a catastrophic system-level scenario. This study develops a graph-based fractional-order model for cascade escalation in IMA architectures with communication delays and reconfiguration failures. The architecture is represented as a weighted directed graph of core processing modules, network switches, and remote data concentrators, where each node carries functional degradation and queue-backlog states. The proposed delayed Caputo fractional-order dynamics incorporate degradation propagation, backlog spillover, mixed-criticality priority conflict, and a state-dependent reconfiguration-failure mechanism. We establish well-posedness and positive invariance of the feasible state domain, derive a sufficient cascade threshold that separates a delay-independent, globally Mittag–Leffler stable nominal regime from a supercritical regime in which bistability and catastrophic attractors may occur, and characterize delay-induced oscillatory instability together with a memory-stabilization effect. Numerical experiments on a synthetic 22-node IMA configuration show fault absorption below the threshold, reconfiguration-contained cascades under sufficient supervisory capacity, and global escalation when reconfiguration collapses under load. The results indicate that backlog growth is an early warning signal and that maintaining the cascade threshold below unity while provisioning reconfiguration capacity above the tipping point can support safer reconfiguration-policy design in certifiable avionics. Full article

We investigate the asymptotic behavior of a Timoshenko beam system in which the free end carries a tip mass subject to a restoring spring force and a tempered Caputo fractional damping force with parameter $\eta >0$ (the case $\eta =0$ is left as an open problem). Using a diffusive state space reformulation of the fractional term, the original problem is embedded into an augmented first-order evolution system on a carefully constructed Hilbert space. Well-posedness is established via the Lumer–Phillips theorem. A spectral analysis of the governing operator, combined with the Arendt–Batty–Lyubich–Vũ theorem, shows that the associated $C_0$-semigroup is strongly asymptotically stable even when the classical equal-wave-speeds condition for the Timoshenko system is violated, provided $\eta >0$. Moreover, resorting to the Borichev–Tomilov resolvent method, we reduce the polynomial energy decay to a single resolvent exponent $\mathcal{\ell }>0$, so that the energy of every solution issued from a datum in the domain of the generator decays at least as fast as $t^{-1/\mathcal{\ell }}$ as $t\to +\infty$. We establish the estimates that control ℓ; we identify the mechanism that governs it—the inertia of the tip mass, which screens the damper at high frequency—and we measure ℓ numerically. In particular, the exponent is not dictated by the second-order character of the Timoshenko operator, contrary to what a comparison with the fourth–order beam might suggest. Full article

This paper investigates the qualitative and topological behavior of random solutions for a class of partial fractional random differential equations governed by the Darboux problem. Unlike classical configurations that rely on bounded or finite delays, our theoretical framework explicitly addresses systems involving unbounded infinite delay. The dynamics of the state transitions are formulated using left-sided mixed Riemann–Liouville fractional integrals and joint Caputo fractional derivatives of order $\epsilon =({\epsilon }_1,{\epsilon }_2)\in (0,1]\times (0,1]$. Because of the infinite historical horizon, the underlying model is constructed and analyzed within abstract, semi-normed axiomatic phase spaces defined over topological Fréchet spaces. By avoiding restrictive compactness assumptions on the nonlinear operational bounds, we establish novel random mild existence theorems. The structural proofs are achieved through a combination of a regular, sublinear family of axiomatic measures of noncompactness and an advanced generalization of the classical Darbo fixed-point theorem tailored for Fréchet domains. Finally, a concrete mathematical example is systematically analyzed to confirm the validity, consistency, and practical applicability of the established theoretical bounds. Full article

In this paper, we consider a nonlinear fractional superdiffusion equation involving a Caputo derivative of order $\alpha \in (1,2)$. Solutions to this class of problems typically exhibit weak singularities near the initial time. To handle this singularity, we introduce an auxiliary variable $p:=D_t^{\alpha /2}(u-t{u}_1)$ and reformulate the original problem as an equivalent coupled system. This system is then discretized using the nonuniform Alikhanov scheme in time, the conforming Galerkin finite element method in space, and Newton linearization for the nonlinear term. We prove that the numerical solutions remain uniformly bounded, independent of the temporal and spatial mesh parameters. Combining this result with a discrete fractional Grönwall inequality and a temporal-spatial splitting argument, we establish unconditional optimal error estimates of order $O(N^{-min\{2,r\alpha /2\}}+h^k)$. Finally, numerical experiments are presented to verify the sharpness of the theoretical convergence rates. Full article

We study a delayed semilinear evolution equation with a Caputo time derivative and the spectral fractional Dirichlet Laplacian on a bounded connected domain. The model separates two forms of memory: the Caputo operator retains the distributed Volterra history, whereas the nonlinear production samples the single past state $u(t-\tau )$. Working in the strongly continuous phase space $C_0(\Omega )$, we prove local well-posedness, positivity, a sup-norm continuation criterion, and a compatible weak formulation. In the delayed-source case with $\mu =0$, the solution exists globally and remains bounded on every finite time interval, while the first Dirichlet mode admits an explicit recursive sequence of positive lower bounds across successive delay windows. In the dissipative case $\mu >0$, $p>q>1$, histories satisfying the explicit smallness conditions remain in an invariant order interval and the $L^2$-energy decays at a Mittag–Leffler rate. The scalar computations are presented only as heuristic first-mode surrogate experiments. In addition, an independent spatially resolved sine spectral-Galerkin/L1 computation of the PDE, with temporal and spectral refinement studies, is included as a numerical illustration. Full article

In this study, random-order FDEs are studied with a focus on the variability of their solutions and random orders when the order of differentiation is governed by a probability distribution. Fractional differential equations (FDEs) offer a generalized modeling approach that enables the analysis of nonlocality and memory effects. However, the deterministic framework for studying FDEs neglects the random nature of real-life events. In this regard, four continuous probability distributions with bounded support (uniform, Beta, triangle and Bates) are used to analyze the variability of the solutions depending on the random order, which is defined to vary between [0.65, 0.95] according to these probability distributions with identical expected values. Monte-Carlo simulations with $N=2^{15}$ repetitions are performed to investigate the random-order FDEs using a predictor-corrector approach. Results show that the decrease in the deviation from the uniform distribution to the Bates distribution is reflected in the random characteristics of the solutions and the order of differentiation to almost the same extent. The findings indicate that the average solutions obtained with random orders from each distribution show almost identical results, whereas the variability changes significantly based on the distribution of the order of differentiation. This information provides useful guidance in working with probabilistic models instead of deterministic systems for uncertainty quantification or sensitivity analysis. Full article

The biological interfaces encountered by ultrasound are rarely welded in the ideal elastic sense. Around tissue–implant contacts, fibrotic capsules, thin membranes, hydrated layers, and tissue-mimicking phantoms, a weak boundary may involve finite mechanical compliance, viscoelastic memory, and local thermo-diffusive exchange. Here, we develop a forward scattering model for a plane P-wave incident from an elastic half-space onto a fractional bio-thermo-diffusive viscoelastic half-space through such an imperfect interface. Caputo-type memory is used in the viscoelastic moduli and the thermal and diffusive relaxation terms, while normal and tangential spring-layer laws describe the mechanical weakness of the contact. The formulation gives a coupled longitudinal dispersion matrix and a reduced six-amplitude interface system. In the revised flux calculation, mechanical reflection and transmission are obtained from the signed total stress–velocity work of the complete reflected and transmitted fields, so modal cross-contributions are retained. The accepted computational population contains 1326 paths and 131,361 points from sub-kilohertz frequencies to ten megahertz, with high-precision recomputation and reliability grades used where conditioning requires caution. Thermochemical quantities remain separate diagnostic channels because a physical absorption coefficient cannot be identified from the present source model. The results show parameter-dependent associations with fractional order, interface stiffness, and frequency, but they do not establish single-parameter causation. The model is therefore intended as a verification-oriented framework for future calibrated studies of weak biological interfaces, not as an experimentally validated or patient-specific predictor. Full article

This investigation is concerned with a time-fractional diffusion equation governed by the Hilfer fractional derivative of order $1<{\rho }_1<2$, subject to homogeneous Neumann boundary data and generalized initial conditions of the Hilfer type. This paper addresses three interconnected problems: a direct problem and two inverse source problems (ISPs). In the first ISP, the objective is to recover an unknown space-dependent source function from measurements taken at a specified final time. In the second ISP, the goal is to determine an unknown time-dependent coefficient through an integral-type over-specification condition. By employing eigenfunction expansions in conjunction with the LT technique, we derive explicit series representations of the solutions in terms of the Mittag-Leffler function. Rigorous existence and uniqueness results for classical solutions are established for all three problems. The second ISP is reformulated as a Volterra integral equation, whose unique solvability is demonstrated via the Banach fixed point theorem. Both ISPs are shown to be ill-posed in the Hadamard sense, indicating instability with respect to data perturbations. Numerical experiments are also presented to validate the theoretical findings and to illustrate the performance of the proposed reconstruction methods. As limiting cases, the formulations corresponding to the Riemann–Liouville and Caputo fractional derivatives are recovered, illustrating the generality of the proposed framework. Full article

Fractional-order models are useful for describing systems with memory, anomalous relaxation, and non-classical dynamic behavior. However, parameter estimation in these models may be affected by strong covariance between the kinetic coefficient and the fractional order, reducing the independent interpretability of the estimated parameters. This work proposes a reference-time scaling strategy for an unforced fractional-order decay model and a forced fractional-order step response model, aiming to improve the local conditioning of the estimation problem by making the sensitivity vectors of the dimensionless coefficient and the fractional order locally orthogonal. The covariance structure is analyzed through the local sensitivity matrix, and a sensitivity-based expression for the reference time is derived to set the off-diagonal term of the approximate Gauss–Newton covariance matrix to zero, thereby reducing first-order linear dependencies. The methodology is evaluated using previously reported experimental data from Amiodarone plasma concentration-time profiles for fractional pharmacokinetic modeling and from the temperature response of a didactic thermal system to a step change in the manipulated variable for fractional-order system identification. The results show that the fitted trajectories, residual sums of squares, dimensional kinetic coefficients, and fractional orders remain invariant under reference-time scaling. Nevertheless, the choice of reference time strongly affects the covariance and correlation between the dimensionless parameter $\mu$ or $\kappa$ and $\beta$, while the recovered dimensional coefficients m and k remain invariant. Conventional choices, such as the maximum, arithmetic mean, geometric mean, and harmonic mean of the experimental times, did not systematically reduce parameter correlation. In contrast, the proposed reference time, selected from the local sensitivity structure, reduced the first-order local correlation in both applications. The results indicate that reference-time scaling is a simple reparameterization tool that improves local statistical interpretability under the Gauss–Newton approximation of fractional-order parameter estimates without modifying the physical model or the quality of the fit. Full article

This work constructs high-order numerical methods for a class of nonlinear fractional differential equations featuring the Caputo fractional derivative with respect to another function. Through variable transformation of the fractional differential operator, the original governing problem is converted into a weakly singular Volterra integral equation. After conducting the variable substitution $x=\varphi \left(t\right)$, the fractional integral term relative to $\varphi$ is restated into a standard fractional integral in the transformed variable. We then construct quadratic and cubic Lagrange interpolation approximations for the fractional integral operator on graded meshes. Special treatments are introduced near the initial point and on the last subinterval so that the resulting schemes can be implemented explicitly. Under suitable regularity assumptions allowing weak singularities, detailed error estimates are derived. The theoretical results show that, by choosing an appropriate grading parameter, the proposed methods can recover the expected convergence orders $\mathcal{O}\left(N^{-(3+\alpha )}\right)$ and $\mathcal{O}\left(N^{-4}\right)$, where $\alpha \in (0,1)$ is the order of the fractional derivative. Several numerical test cases are presented to validate the theoretical error estimates and demonstrate the effectiveness of the constructed schemes. Full article

We present and analyze a time-fractional, partly diffusive model of two electrically coupled Hindmarsh–Rose neurons in which only the membrane potentials undergo spatial transport, through a nonlinear Neumann m-Laplacian, while the temporal evolution is governed by a Caputo derivative of order $\rho \in (0,1]$. The fractional operator incorporates hereditary relaxation, whereas the m-Laplacian represents gradient-dependent degenerate transport and recovers ordinary diffusion when $m=2$. In a Gelfand triple adapted to the no-flux boundary condition, we derive a fractional energy inequality, a uniform dissipative estimate, and the existence of a global weak solution by a Faedo–Galerkin approximation, fractional compactness, and Minty’s method. Uniqueness and continuous dependence are obtained in the stated bounded solution class. We prove global Mittag–Leffler synchronization above an explicit coupling threshold and establish a practical synchronization bound under parameter mismatch. A fully implicit L1 finite-volume method is then constructed; every time step is solvable, uniqueness follows under an explicit monotonicity condition, and the scheme is unconditionally energy dissipative and convergent. Manufactured-solution tests recover the expected $2-\rho$ temporal and second-order spatial rates. In the neuronal simulations, reducing $\rho$ from 1 to $0.90$ lengthens the mean bursting period from about 379 to 565 time units, an increase of roughly one half, and raises the number of spikes per burst from about $31.7$ to $38.8$. Over $2\le m\le 4$ the temporal rhythm is essentially unchanged, the burst period staying near 362 time units, while the diffusion exponent reshapes the peak amplitude and the spatial gradient profiles of the traveling fronts. The empirical synchronization threshold for the canonical parameter set is approximately $11.0$, far below the global sufficient bound $2.3917\times {10}^4$, which quantifies the conservatism of the analytical certificates. Full article