Introduction:
Type 2 Diabetes Mellitus (T2DM) is a heterogeneous metabolic disorder characterized by insulin resistance, impaired insulin secretion, and altered hepatic glucose regulation, resulting in complex and patient-specific glucose dynamics. Accurate mathematical modeling of these processes is essential for understanding disease mechanisms and supporting personalized therapeutic strategies.

Methods:
We present a hybrid modeling framework that integrates a physiologically interpretable system of ordinary differential equations (ODEs) with Neural ODE residual dynamics and a probabilistic output mechanism. The physiological component explicitly models key glucose–insulin processes relevant to T2DM, including gastric emptying, intestinal glucose absorption, plasma glucose and insulin dynamics, and insulin action. Each state variable corresponds to a measurable or identifiable biological process, ensuring physiological plausibility. A Neural ODE component is introduced to learn residual dynamics from patient-specific glucose time-series data. A Gaussian Mixture Model (GMM) output layer enables probabilistic prediction.

Results:
The proposed model admits a well-defined and stable fasting equilibrium and generates physiologically consistent postprandial glucose trajectories. The Neural ODE residual component enhances adaptability to inter-patient variability, while the GMM output captures uncertainty and multimodal glucose responses.

Conclusions:
This hybrid Physiological–Neural ODE framework combines mechanistic interpretability, data-driven adaptability, and probabilistic forecasting, providing a robust mathematical foundation for personalized glucose dynamics modeling in Type 2 Diabetes.

We deal with the obstacle problem related to an operator with a drift-type lower-order term that in the linear case represents the one related to the Fokker–Plank equation, whose (normalized) solution describes the evolution of the probability density for a stochastic process. As the simplest possible model, we can consider the operator

The main novelty is the presence in the coefficient of the lower-order term of a singularity in the spatial variable. More precisely, we assume that the coefficient of the drift term lies in the Marcinkiewicz class weak-for a.e. time and satisfies the minimal time integrability assumption. The obstacle function is assumed to be time-continuous. Despite the lack of coercivity, we prove the well-posedness of a global solution to the obstacle problem and we describe the asymptotic behavior of such a solution. Moreover, we give quantitative asymptotic, stability estimates for the solutions to different problems. More precisely, we measure the distance in time of a solution to a parabolic obstacle problem from a solution to a stationary one. Fundamental tools in proving our results are a regularizing-in-time procedure and a suitable application of a Gronwall’s type lemma. A bound on the distance of from bounded functions is needed. However, this restriction holds whenever is in the Lebesgue space .

In this paper, we examine fixed point theory in the context of complete b-metric spaces, specifically for a class of generalized Reich-type rational contractions. Fixed point theory is a prominent branch of mathematical analysis, with numerous applications in functional analysis, differential and integral equations, optimization problems, and applied mathematics. A b-metric space is a generalization of a standard metric space, achieved through the modification of the classical triangle inequality by a coefficient, facilitating the examination of a broader class of spaces for which convergence and fixed point theorems can be proven. This flexibility in analysis is especially useful when dealing with complex mathematical models and real-world applications, where strict metric properties may not be applicable.

We prove the existence and uniqueness of fixed points for mappings that satisfy a rational Reich-type contractive condition, which generalizes classical contraction principles by taking into account a rational function of distances between points and their images. Our method of proof involves the construction of an iterative sequence in the b-metric space and proving that it is a Cauchy sequence. By making use of the relaxed triangle inequality that corresponds to the b-metric constant, we show that this sequence converges to a unique fixed point, thus proving both existence and uniqueness.

Moreover, our results generalize several classical fixed point theorems. Specifically, Banach-type, Kannan-type, and classical Reich-type contraction mappings can be obtained by selecting appropriate parameters in the rational contractive condition. This shows the generality of our method of proof, which combines several strands of fixed point theory and generalizes them to generalized b-metric spaces. The work lays a basis for further research in the field of nonlinear analysis.

Thermal analysis of extended surfaces (fins) is critical for preventing overheating in a wide range of industrial systems, from microelectronics to aerospace engineering. Governing differential equations for fin models, particularly those incorporating temperature-dependent thermal conductivity, surface heat flux, or radiative effects, are inherently strongly nonlinear. Such complexity resists exact analytical solution and demands the development of accurate, efficient, and stable numerical methods.

To address this challenge, we aim to introduce a spectral collocation method utilizing a basis of Fibonacci polynomials. The technique is designed to solve nonlinear initial and boundary value problems governed by ordinary differential equations, under Dirichlet, Neumann, or mixed-type constraints. The proposed framework systematically transforms the governing differential equation into a tractable system of nonlinear algebraic equations. Solving this system for spectral coefficients yields a highly accurate approximate solution. A rigorous theoretical foundation is established, including a proof of convergence and the derivation of a priori error bounds, both of which are subsequently validated through comprehensive numerical tests.

The method’s practical efficacy and computational performance are demonstrated on two classic yet challenging benchmark problems from the heat transfer literature: (1) heat transfer in a longitudinal fin with temperature-dependent surface heat flux, and (2) the combined convecting–radiating cooling of a lumped system with variable specific heat. A detailed comparative analysis against established methods, including the Adomian decomposition method and variational iteration method, shows that the proposed Fibonacci-based spectral scheme delivers superior accuracy, faster convergence, and notable implementation simplicity. The obtained results confirm the method's potential as an accurate and efficient analytical-numerical tool, suggesting that further exploration and adaptation could extend its applicability to a broader range of strongly nonlinear problems in thermal engineering and applied mathematics.

This paper introduces a new class of systems of additive functional equations

H_K(p_K+ r_K+ f_K+q_K1+ q_K2+ s_K+t_K) = H_K(p_K)+ H_K(r_K)+ H_K(f_K)+ H_K(q_K1)+ H_K(q_K2)+ H_K(s_K)+ H_K(t_K)

including parking availability, restroom facilities, food services, and service-related factors.

This provides a concrete applied framework that connects abstract functional equations with real-world modeling. The primary aim of the study is to establish the Ulam–Hyers (UH) stability of the proposed class of equations in complete normed linear spaces. To achieve this, a general control function is employed in conjunction with sumpowers of norms, product, powers of norms and mixed product-sum expressions involving powers of norms. Using a direct analytical approach, explicit stability conditions are derived for each functional equation considered. One of the significant contributions of this work lies in the introduction of multiple interrelated functional equations and the unified treatment of their stability analysis. Furthermore, an application is presented to demonstrate the practical significance of the theoretical results. A comparative discussion is also included to highlight the consistency of the results with established stability analyses and supporting mathematical computations.

All abbreviations are properly defined, and the equations are expressed in clear mathematical form to ensure readability and precision.

Mathematics can be divided into two main categories: applied mathematics and fundamental or theoretical mathematics. Special functions have contributed and continue to contribute to the advancement of science in general and mathematics in particular. Among these functions are the Laplace transform, the Riemann zeta function, the basic reproduction number $R_{0}$, the Fourier transform, the Stieljest integral, the Lebesgue integral, $\cdots$. All these functions have made it possible, at crucial moments in the history of mathematics, to solve linear and nonlinear partial differential equations, cryptography, number theory, $\cdots$.\\ In this paper, we construct a special function $\Phi_{m}$ depending on the parameter $R_{0}$ and the independent vector $x\in\mathbb{R}^{m}$ of size $m\in\mathbb{N}^{\ast}$. The basic reproduction number $R_{0}$ is by definition the average number of new infections in an environment from an already infected individual. In other words, based on an infected individual, the $R_{0}$ provides information on the average number of people that individual can infect in a healthy environment. This number does not take into account the nature of the disease, the patient's genetic material or immune system, their family tree, or herd immunity. Through this special function $\Phi_{m}$, we propose to take into account additional parameters that are much more realistic and adapted to physical realities. The independent variable $x$ is of dimension $m\in\mathbb{N}^{\ast}$, i.e., $x=(x_{1},x_{2},\cdots,x_{m})\in\mathbb{R}^{m}$, and the dependent variable $\Phi_{m}(R_{0},x)$ is the special function. We first justify that the special function constructed is well defined. We then propose a qualitative analysis (continuity, differentiability, metric space, $\cdots$) of this special function. Finally, we propose a conceptual framework for applying this special function to the mathematical modeling of the interactive dynamics of three entities: the immune system, drugs, and nutrition.

In this paper, we employ the non-commutative (NC) gauge theory of gravity to construct the deformed metric $\hat{g}_{\mu\nu}(r,\Theta)$ corresponding to the Schwarzschild black hole. This construction is performed using the Seiberg–Witten map and the Moyal–Weyl star product, which systematically incorporate the effects of non-commutativity through the parameter $\Theta$. The resulting deformed metric provides a geometric framework to study the geodesic of a massive test particle in the presence of NC gravitational corrections. We begin by deriving the correction to the effective potential of the massive test particle, considering terms up to the second order in the NC parameter $\Theta$. Subsequently, we obtain the geodesic equation for the massive test particle with second order in NC corrections. An analytical solution to this modified geodesic equation is presented using an approximation method suitable for weak-field and slow-motion limits. In addition, we plot several representative orbital trajectories of the massive test particle around the NC-deformed Schwarzschild black hole and analyze their physical behavior. All our results consistently reduce to the well-known commutative case in the limit $\Theta \rightarrow 0$, thereby preserving the coherence and consistency of the theory. Furthermore, we demonstrate that NC effects manifest only at the perihelion of the test particle's orbital motion.

Anomalous diffusion is a well-documented phenomenon in biological tissues, where complex microstructural features such as cellular crowding, membrane barriers, and heterogeneous extracellular matrices give rise to non-Gaussian transport and long-range memory effects. Classical diffusion models and constant-order fractional equations are often insufficient to capture the spatial variability of diffusion mechanisms observed across different tissue regions. Variable-order fractional diffusion models provide a natural and physiologically meaningful framework to describe such heterogeneous transport processes.

In this work, we propose an efficient and energy-preserving numerical method for a class of time–space fractional diffusion equations with spatially variable fractional order, motivated by anomalous transport in heterogeneous biological tissues. The model incorporates a Caputo time-fractional derivative and a piecewise-defined spatial fractional diffusion operator, enabling the coexistence of distinct diffusion regimes associated with local microstructural properties.

The numerical scheme combines a memory-efficient temporal discretisation of the fractional derivative with a stable spatial approximation of the variable-order operator. A discrete energy functional is constructed, and unconditional stability of the fully discrete scheme is rigorously established through preservation of the dissipative energy structure of the continuous problem. Convergence is proven under mild regularity assumptions.

Numerical experiments in two-dimensional heterogeneous tissue-like domains validate the theoretical results and demonstrate biologically relevant diffusion phenomena, including spatially dependent propagation speeds and effective diffusion barriers induced by microstructural heterogeneity. These results illustrate the limitations of constant-order models and highlight the relevance of variable-order fractional formulations for quantitative modelling of anomalous transport in complex biological tissues.

The electrical capacitance technique is used as a non-destructive method and serves to evaluate the root system of plants. Capacitance allows for an indirect interpretation of a plant's physiological state and root development. Therefore, the aim of this work was the study of root capacitance in strawberry plants under physicochemical variations, pH changes and conductivity, using the RC model. The methodology of this research work consisted of measuring the electrical response of the root-substrate system using an LCR meter, where the plant is in a distilled water solution, varying the pH and electrical conductivity of the medium in a controlled manner. The results obtained will be analyzed using an equivalent RC model, which allows for the separation of the resistive effects associated with ionic flow and the capacitive effects related to the structure and state of the membranes. The temporal evolution of capacitance will allow the identification of changes in root activity associated with environmental variations, since this parameter is related to the active surface and the state of cell membranes. In this way, the analysis of the capacitive signal over time allows the detection of conditions of stress, adaptation or growth of the root system. The results include the evaluation of the sensitivity of electrical parameters to changes in pH and conductivity, as well as the identification of nonlinear behaviors in the system response.

In this work, the Laplace–Residual Power Series Method (L-RPSM) is employed to obtain accurate analytical–numerical solutions for a class of nonlinear time-fractional evolution equations involving the Caputo fractional derivative. Such equations play an important role in applied mathematics, as they are widely used to model complex physical and biological phenomena characterized by memory and hereditary effects. The proposed methodology is first formulated for a general class of nonlinear time-fractional evolution problems, where the main steps of the L-RPSM are clearly outlined. In addition, a detailed convergence analysis is carried out to ensure the existence, uniqueness, and validity of the resulting series solutions, thereby confirming the reliability of the proposed approach from a theoretical perspective.

To demonstrate the effectiveness and practical applicability of the method, the nonlinear time-fractional Newell–Whitehead equation is considered as a test model. This equation arises naturally in the mathematical modeling of pattern formation, reaction–diffusion systems, and related processes in applied sciences. Numerical simulations are performed for different values of the fractional order, and the corresponding results are presented in tabular form as well as illustrated through two- and three-dimensional graphical representations. The influence of the fractional-order parameter on the qualitative behavior of the solutions is examined and discussed in detail. Moreover, the use of the Laplace transform within the proposed framework enhances the convergence properties and computational efficiency of the method, leading to improved accuracy of the obtained results. The results clearly indicate that the L-RPSM provides highly accurate, efficient, and reliable solutions, making it a powerful and effective tool for solving nonlinear time-fractional evolution models encountered in applied mathematics and related scientific fields.