The research developed a nonlinear seismic-response framework for single- and multiple-degree-of-freedom building models using Caputo fractional damping. Its memory term describes hereditary, frequency-dependent dissipation, while the restoring law is restricted to convex, monotone cubic hardening. That restriction gives the restoring force a well-defined potential and makes rigorous statements about existence, uniqueness, continuation, and energy possible. It also marks an important boundary: the framework does not reproduce pinching, stiffness or strength degradation, hysteretic internal variables, or residual drift, so its mathematical guarantees cannot automatically be transferred to more realistic building models.

The authors separated two questions that are often conflated in fractional structural analysis. The continuous model has a mechanical energy identity in which accumulated hereditary work is nonnegative, allowing a forcing bound and continuation of solutions over finite time under a coercive potential. The numerical method then has its own discrete energy property. It combines midpoint L1 product integration for the Caputo kernel, average-acceleration Newmark kinematics, and a discrete gradient of the nonlinear potential. Under the stated assumptions, every nonlinear time step has exactly one solution, and the resulting scheme satisfies a cumulative discrete energy balance without introducing positive net numerical energy over any prefix of the time grid.

The discrete proof relies on a positive quadratic form associated with the midpoint L1 history weights. The method’s Jacobian is positive definite, making the nonlinear residual strongly monotone and coercive. In an unforced free-vibration test of a three-story system, the maximum relative defect in the energy balance was at the level of floating-point roundoff. This numerical check was consistent with the theorem, although the authors stress that it verifies the implementation rather than replacing the analytical argument. The direct treatment of the memory history also makes every contribution auditable, an advantage for verification but one that may become costly for long records or high-dimensional models.

Convergence tests reveal a sharp distinction between smooth and singular early-time behaviour. For a smooth manufactured solution, the maximum displacement error decreased from about 5.60 times 10 to the power of minus 3 on the coarsest grid to 1.39 times 10 to the power of minus 5 on the finest grid. The measured rate fell from approximately 1.86 to 1.60 as the grid was refined, approaching the expected fractional consistency regime rather than a uniform second-order limit. A separate solution with a weak initial layer reduced the uniform-grid convergence rate to about 0.40, showing that the smooth-solution convergence theorem cannot simply be applied to every fractional initial-value problem.

Concentrating resolution near the singular start recovered much of the lost accuracy. A deliberately graded mesh reduced the error in the weakly singular test by more than two orders of magnitude on the finest grid and produced finite-grid slopes between 1.54 and 1.57. The researchers interpret those slopes as pre-asymptotic behaviour arising from competition among midpoint, Newmark, and first-cell errors, not as proof of an asymptotic order above the method’s smooth consistency ceiling. The experiment demonstrates the practical value of mesh grading while retaining the mathematical limits of the convergence result.

The damping comparison was redesigned after the authors recognised that varying the fractional order while keeping one numerical coefficient fixed is not an equal-damping test. Fractional coefficients and orders have dimensional and frequency-dependent interpretations, so an apparently large response change could reflect unequal dissipation rather than a distinct seismic effect. The revised study instead calculated scalar fractional modal damping from the complex dynamic stiffness and calibrated each fractional matrix, together with a classical Rayleigh comparator, to 4% damping in the first and third modes. The second mode was not fitted directly; its equivalent damping emerged at approximately 3.4% to 3.6%.

This physical normalization largely removed the earlier apparent advantage of increasing fractional order. In a representative three-story response scaled to 0.35 g, peak roof displacement ranged from 3.998 to 4.245 centimetres across the fractional and viscous models, compared with much larger separation in the earlier fixed-coefficient comparison. More importantly, the 12-motion ensemble showed only small between-model differences relative to the record-to-record standard deviations. Paired confidence intervals relative to the viscous model included zero for roof displacement, interstory drift ratio, and peak floor acceleration. The evidence therefore supports changes in phase and frequency dependence under a chosen calibration, not a universal improvement in seismic performance from a larger fractional order.

Synthetic inverse identification showed a similar split between controlled numerical recoverability and real-world limitations. The researchers generated 600 single-degree-of-freedom parameter pairs and trained Extra Trees, Random Forest, and standardized Ridge regressors using five-fold shuffled cross-validation. Each input combined 40 displacement samples with seven global descriptors. Under clean, within-family conditions, all methods recovered the known fractional parameters accurately, and the simple Ridge baseline outperformed Extra Trees in mean absolute error, root mean square error, and squared coefficient of determination for both targets. That result indicates that the synthetic feature relationship contains a comparatively simple low-dimensional signature rather than demonstrating that a sophisticated artificial-intelligence system is required.

Stress tests weakened the inverse estimates substantially. With 1% Gaussian noise, mean absolute errors rose to 0.0285 for fractional order and 0.0604 for the fractional coefficient; at 10% noise, they reached 0.0643 and 0.1613. Perturbing structural stiffness and nonlinear properties produced errors of 0.1003 and 0.1955, while training on one synthetic excitation and testing on another produced the largest failures, 0.1105 and 0.4645. These results establish numerical identifiability only within a controlled model family, not reliable parameter recovery from actual buildings. Timings also showed a modest overhead for the fractional solver at small problem sizes, while leaving scalability to longer histories and larger structural models unresolved.

Subject of Research: Numerical analysis of fractional damping, nonlinear seismic response, and synthetic parameter identification in building models

Article Title: Fractional seismic simulation and synthetic inverse identification of nonlinear building models: discrete energy balance, convergence, and equivalent-damping calibration

Article References: Salas, A. H., Acosta M, C. D., & Solórzano, E. (2026). Fractional seismic simulation and synthetic inverse identification of nonlinear building models: discrete energy balance, convergence, and equivalent-damping calibration. Frontiers in Built Environment, 12, Article 1956502. https://doi.org/10.3389/fbuil.2026.1956502

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DOI: 10.3389/fbuil.2026.1956502

Keywords: fractional damping, nonlinear seismic response, Caputo derivative, discrete energy balance, equivalent damping, synthetic motion, inverse identification, structural dynamics, convergence analysis, numerical identifiability, Fractional, seismic

The model represents the computer population as a normalized total. Uninfected computers can enter the system, and computers can leave it or be removed. New infections enter the latent group and may progress to the active “seizing” group, while latent and active computers can also be cured. Transmission depends on both latent and active infected computers and is evaluated at a delayed time, with an exponential attenuation factor associated with the delay. The analytical results apply to the deterministic fractional-order delay system obtained by setting the stochastic intensities to zero.

For that deterministic system, the researchers used a fractional-integral formulation and the Banach fixed-point theorem to establish a unique local solution. They then extended the argument through continuation on the feasible region. The authors also proved that solutions starting from nonnegative histories remain nonnegative, with the uninfected population remaining positive under the stated initial condition, and that every nonnegative solution is uniformly bounded. These proofs concern the deterministic model rather than the fully stochastic system.

The analysis identified two equilibria: a computer-virus-free equilibrium and a computer-virus-present equilibrium. A next-generation matrix calculation produced a basic reproduction number, with the virus-free equilibrium associated with the subcritical case below a threshold of one and the virus-present equilibrium appearing when the threshold is exceeded. The researchers derived local and global stability conditions for both equilibria in the deterministic fractional-order delay model. They explicitly did not claim almost-sure, mean-square or other probabilistic stochastic stability from these analytical results.

Sensitivity analysis of the reproduction number indicated that contact rates and infection probabilities had the largest positive normalized forward sensitivity indices. Parameters representing intervention measures and recovery had negative indices in the analysis, while the fractional order also affected sensitivity. These findings are properties of the model under its specified assumptions and parameter choices; the study did not estimate intervention effects from observed malware outcomes.

To simulate the stochastic model, the authors developed a stochastic non-standard finite difference scheme. The method retains the model’s fractional-memory term, evaluates transmission at the delayed time level and applies the same Brownian increment to the three equations. Positivity and boundedness were proved for the deterministic version of the scheme. Because Brownian increments can be negative, the researchers did not claim unconditional positivity or boundedness for the fully stochastic discretization.

Simulation experiments varied the fractional order while holding other settings fixed, and the resulting trajectories showed that memory changed the transient evolution of the uninfected, latent and active computer populations. Separate experiments varied the transmission delay with the fractional order fixed, showing that delay altered both the timing of the infection input and its effective magnitude through the attenuation factor. Combined simulations varied both parameters. Under the tested settings, trajectories remained bounded over the simulation interval.

Increasing the stochastic intensity produced greater pathwise variation in all three modeled populations, while the deterministic case followed a smoother trajectory. In a common-path comparison, the stochastic non-standard finite difference scheme and the fractional Euler–Maruyama method generated qualitatively similar trajectories. The Euler–Maruyama method had slightly smaller maximum absolute error and root-mean-square error in the particular experiment analyzed. The non-standard scheme nevertheless showed comparatively controlled behavior as the time step increased, an observation that the authors characterized as time-step sensitivity rather than proof of convergence.

The study used parameter values drawn from fitted or estimated sources and values adopted from prior literature. It also cited global malware and cyberattack statistics from 2020 through 2025 as context, including reported growth in malware samples and cyberattacks, but did not use those time series to calibrate or statistically validate the model. The principal limitations were the absence of a complete real-world time-series calibration, numerical rather than rigorous treatment of stochastic stability and the absence of a general convergence proof for the proposed scheme. The authors identified data-driven estimation, rigorous stochastic analysis, network heterogeneity and optimal-control extensions as possible future work.

Subject of Research: Stochastic fractional-order delay modeling and numerical simulation of computer-virus propagation

Article Title: A stochastic fractional model for computer virus propagation with memory effects and exponential delay

Article References: Raza, A., Lampart, M., Shafique, U., Baleanu, D., & Abd Elwahab, M. E. (2026). A stochastic fractional model for computer virus propagation with memory effects and exponential delay. PLOS One, 21(10), e0360254. https://doi.org/10.1371/journal.pone.0360254

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DOI: 10.1371/journal.pone.0360254

Keywords: computer-virus model, fractional differential equations, delay dynamics, stochastic modeling, Brownian perturbations, numerical simulation, reproduction number, sensitivity analysis, cybersecurity, stochastic, fractional, model

The central objects of the study are curvilinear fractional integral cost functionals that are path-independent and built on Riemann–Liouville fractional integrals. In classical calculus, integrals and derivatives describe instantaneous rates of change and accumulated quantities over well-defined intervals. Fractional calculus generalizes these notions to arbitrary, non-integer orders, which allows a mathematical model to retain a weighted memory of the entire past trajectory of a system rather than only its current state. This memory property is precisely what makes fractional operators attractive for describing viscoelastic materials, anomalous diffusion, biological processes, and learning dynamics in neural networks, where present behavior is shaped by a long tail of previous states.

What distinguishes this research is the combination of three difficult ingredients in a single framework. First, the cost functionals are fractional, meaning that the objective being minimized or maximized integrates information over the system’s history through Riemann–Liouville operators. Second, the objective functions are interval-valued, so instead of assigning an exact payoff to each decision, the model works with ranges that capture uncertainty, imprecision, or inherent variability in the data. Third, the optimization is constrained: the admissible trajectories must satisfy a system of partial differential equations and partial differential inequalities, which encode the physical or computational laws governing the system under study. Solving such problems requires optimality conditions that respect all three layers simultaneously.

To handle the interval-valued character of the objectives, the authors employ the concept of LR-optimality, a solution notion widely used in interval optimization. In interval-valued optimization, comparing two intervals is not straightforward, because one interval may be better in terms of its lower bound while worse in terms of its upper bound. The LR ordering addresses this by treating the left and right endpoints of each interval through a partial order, and a point is called locally LR-optimal when no nearby feasible alternative produces an interval objective that dominates it in this ordering. The paper’s key theoretical contribution is a condition, rooted in fractional calculus, under which local LR-optimality can be upgraded to global LR-optimality for the associated variational control problem governed by the PDE and PDI constraints.

This local-to-global result is more than a technical curiosity. In practical optimization, algorithms typically search for local optima because finding global optima directly is computationally prohibitive, especially in infinite-dimensional settings such as variational control problems. A verified condition guaranteeing that a locally optimal solution is in fact globally optimal removes the risk that a numerical method will settle for a merely adequate solution while a substantially better one exists elsewhere in the feasible set. For engineers and applied mathematicians, such a guarantee transforms the reliability of the solutions they can extract from fractional, interval-valued models with differential constraints.

The machinery behind the result draws on the structure of Riemann–Liouville fractional integrals, which accumulate a weighted history of a function with a power-law kernel. Because these operators are nonlocal, the resulting optimality conditions couple the behavior of the solution at every point of the domain with its values across an extended past, rather than producing purely local conditions as in classical variational calculus. The authors show how this nonlocal structure, combined with the path-independence of the curvilinear cost functionals, yields the pivotal condition ensuring that local and global optimality coincide. The constraints, expressed as coupled partial differential equations and inequalities, are handled within the same variational framework, keeping the analysis self-contained.

To demonstrate that the theory is not merely abstract formalism, the paper includes an illustrative example showing how fractional calculus and memory dynamics can be harnessed to model and optimize real-world dynamic processes, specifically in the control of artificial neural networks. Neural networks are natural candidates for fractional modeling: training and inference both exhibit path dependence, in which the network’s current performance reflects an accumulated history of parameter updates and activations. By formulating the network control problem with interval-valued fractional objectives and differential constraints, the framework allows uncertainty in performance measures to be treated explicitly while respecting the dynamical laws that the network must obey.

The use of interval-valued objectives deserves particular attention in an era when machine learning systems are increasingly deployed under uncertainty. A crisp objective function forces a modeler to commit to exact numerical targets, even when data are noisy, measurements are imprecise, or performance criteria are inherently ambiguous. Interval-valued objectives encode this imprecision directly into the mathematics, and the LR-optimality framework then provides a rigorous way to decide when one uncertain outcome is preferable to another. Combining this with fractional memory effects produces a modeling language that is simultaneously uncertainty-aware and history-aware, a combination that few existing optimization frameworks offer in a constrained, distributed-parameter setting.

The publication appears in Neural Processing Letters, a journal whose scope spans neural computation and the mathematical foundations of learning systems, making the venue a fitting home for work that connects fractional variational analysis with neural network control. The article is fully open access under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, and the authors report no funding and no conflicts of interest. The paper was received on 9 October 2025, accepted on 18 August 2026, and released with a permanent DOI, ensuring that the results are immediately citable and accessible to the research community.

For the broader field, the study signals a maturing of fractional optimization as a practical tool rather than a purely theoretical pursuit. By establishing when local guarantees extend globally, and by doing so in the presence of interval uncertainty and differential constraints, the Bucharest researchers have provided a template that others can follow in areas ranging from control engineering and signal processing to computational neuroscience. As complex systems with memory and uncertainty become central to technology and science, frameworks of this kind are likely to move from the pages of mathematics journals into the toolkits of practitioners designing the next generation of intelligent, adaptive systems.

Subject of Research: Constrained fractional optimization with interval-valued objective functions and Riemann–Liouville fractional integrals

Article Title: A Study of Constrained Fractional Optimization Problems Involving Interval-Valued Functions

Article References: Postavaru, O., Toma, A., & Treanţă, S. (2026). A Study of Constrained Fractional Optimization Problems Involving Interval-Valued Functions. Neural Processing Letters. https://doi.org/10.1007/s11063-026-11884-9

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DOI: 10.1007/s11063-026-11884-9

Keywords: fractional optimization, interval-valued functions, Riemann–Liouville fractional integral, LR-optimality, variational control, PDE constraints, fractional calculus, memory effects, artificial neural networks, constrained optimization, Constrained, Fractional