In this study, we presented a novel fractional nonlocal thermoelastic heat conduction model that extends the Guyer–Krumhansl framework by incorporating size-dependent nonlocal thermal effects and non-Fourier heat conduction characteristics. The model extends the traditional approach using the single-phase-lag (SPL) method derived from Moore–Gibson–Thompson (MGT) heat theory. By employing the Atangana–Baleanu (AB) fractional derivative with a non-singular kernel, we integrated nonlocal features through fractional derivatives, enhancing its applicability to complex thermal behaviors in materials exhibiting combined nonlocal and fractional dynamics. To validate the model, thermoelastic interactions were examined in a long, hollow cylinder subjected to a uniform electromagnetic field. The outer surface was thermally insulated and traction-free, while the inner surface, also traction-free, experienced thermal shock. Governing equations were solved using the Laplace transform method, and numerical solutions were obtained via the Dubner–Abate algorithm. The results were compared with conventional and generalized thermoelastic models to assess accuracy and effectiveness. Additional analysis explored material properties through graphical data, considering various fractional orders and operators, thereby enriching the understanding of system behavior under different conditions. The findings demonstrated the advantages of the fractional nonlocal thermoelastic model in capturing complex thermal interactions within advanced materials, contributing significantly to heat conduction theory.

This study primarily aimed to apply a novel, advanced stochastic volatility model incorporating leverage and an asymmetrically heavy-tailed distribution (ASV-GHskw) to 18 financial asset returns, including 10 Asia-Pacific and six European stock market returns and two crude oil market returns. The study employed a generalized hyperbolic skew student's t-distribution using the Bayesian approach. The study's secondary aim was to concentrate on one-day-ahead downside market risk measurement of the related financial asset returns under Basel Ⅳ regulations based on the ASV-GHskw model. For comparison, an asymmetric stochastic volatility model with a symmetric student's t-distribution (ASV-st) was also considered. The findings indicated that the proposed Bayesian Markov chain Monte Carlo (MCMC) approach was sufficiently efficient, especially when estimating the ASV-GHskw model parameters. Further, the ASV-GHskw model provided a better fit for all financial asset returns examined than its competing ASV-st model. Moreover, the results revealed that the financial institutions taking positions in the examined financial markets will be required to hold additional regulatory capital for market risk exposures under the latest Basel capital regulations. Lastly, the findings give valuable information for risk-averse investors interested in the examined financial markets, as they expect to receive higher-than-average returns by holding less-risky assets.

Since some viruses share transmission routes, coinfection can occur. While most models assume one target cell, many viruses infect and replicate in multiple cell types. The purpose of this study is to develop and analyze a model describing coinfection by two viruses that grow and compete within two distinct target-cell populations. We prove that the proposed model is mathematically well-defined and admits unique, biologically meaningful solutions. Using the next-generation matrix method, we derive expressions for the basic reproduction numbers corresponding to virus type 1 single infection ($ R_{1} $), virus type 2 single infection ($ R_{2} $), and two-virus coinfection ($ R_{0} $). The model usually admits an infection-free equilibrium. The existence conditions for the virus type 1 single-infection, virus type 2 single-infection, and two-virus coexistence equilibria are also established. Applying the Lyapunov direct method, we demonstrate the global stability of all steady states. The obtained results reveal new insights into the factors that allow two viruses to coexist in a stable state, thereby enabling the possibility of chronic coinfections. The model is further extended to examine the influence of two reverse transcriptase (RT) inhibitors and to explore the role of a second target-cell population in two-virus codynamics. We find that neglecting the second target-cell population leads to underestimation of $ R_{1} $ and $ R_{2} $; consequently, drug levels determined from a one-target-cell model may be insufficient to clear the viruses. The model is extended to incorporate antiviral drug therapy and to determine the minimum drug efficacies required to eliminate viral coinfection. The results provide deeper insight into the dynamics of dual infections involving viruses that compete for distinct target-cell populations.

This paper discusses the effects of applying different non-linear transformations in combination to well-known fractal structures and aggregates, to obtain novel fractal images. Five different non-linear transformations are selected to be combined, and applied to existing aggregates, and the new fractal structures obtained are discussed. We show how different non-linear transformations effects (bending effects, diffusion effects, etc.) can be combined just by function composition processes, and their final results as new fractal images. We analyze the effects of the combination of non-linear transformations over classical fractals such as the Sierpinski triangle and Sierpinski carpet, in terms of the fractal dimension of the new structures created. Finally, we will also show the effect of the non-linear transformation on other fractal structures, such as diffusion limited aggregation and strange attractors.

Runge–Kutta (RK) pairs remain among the most effective tools for the numerical integration of ordinary differential equations; however in the scalar autonomous setting, their structure allows further efficiency. In this case, the system of order conditions simplifies, with fewer equations needing to be satisfied, which in turn enables the construction of embedded pairs of higher accuracy than in the general case. Notably, one may design pairs of order 7(5) requiring only eight stages per step, whereas conventional RK pairs with the same number of stages are limited to order 6(5). In the present work, we derived the modified set of equations of condition up to seventh order, and by means of differential evolution techniques, constructed a new embedded pair of order 7(5) specifically adapted to scalar autonomous problems. The performance of the method was assessed in the context of system identification through neural ODEs, where it was used to approximate governing dynamics from data. The logistic growth and saturating cubic models were employed as a representative test cases, illustrating both the efficiency advantages of the proposed scheme. Numerical experiments confirmed that the new pair provides a valuable bridge between high–order RK methodology and modern machine learning approaches to dynamical systems.

This work investigates the stochastic dynamics of the nonlinear Maccari system by incorporating Brownian motion into its evolution equations, modeling random fluctuations relevant to physical settings such as optical fibers, plasma waves, and fluid interfaces, where noise significantly influences wave propagation. To obtain solitary wave solutions under stochastic effects, we use the refined Riccati–Bernoulli subsidiary ordinary differential equation (RB sub-ODE) method to reduce the stochastic partial differential equations (PDEs) to a manageable system of ordinary differential equations. This methodology enables the construction of a diverse family of stochastic wave solutions, including rational profiles, periodic structures, and solitary waves with noise-dependent modulation. The obtained solutions reveal significant influences of stochastic forcing on amplitude, phase, stability, and interaction dynamics. The results demonstrate that even weak noise can induce substantial modulation, phase drift, and deformation of coherent structures in the Maccari system. To illustrate the propagation of solitary waves for the stochastic nonlinear Maccari system, 2D charts of selected solutions are created using Matlab. Overall, this work offers a robust framework for analyzing noise-driven wave phenomena and highlights the effectiveness of the RB sub-ODE technique in deriving exact stochastic solutions for complex integrable models.

Let $ R $ be a ring. An element $ a \in R $ is called a quasi-idempotent if there exists a central unit $ k $ in $ R $ such that $ a^2 = ka $. The quasi-idempotent graph of $ R $, denoted by $ G_{Qid}(R) $, is the simple undirected graph with vertex set $ R $ itself, where two distinct vertices $ a $ and $ b $ are adjacent if and only if $ a+b $ is a quasi-idempotent. This paper presents a systematic study of the graph $ G_{Qid}(R) $. We examine its basic structural properties, including connectivity and girth. We introduce a new invariant of the ring, termed the quasi-idempotent sum number, and establish the precise relationship between this invariant and the graph diameter. Furthermore, a complete classification is obtained for all finite commutative rings $ R $ according to the genus of $ G_{Qid}(R) $, thereby characterizing the rings for which this graph has genus $ 0 $, $ 1 $, or $ 2 $.