This research investigates the effectiveness of using Padé approximation in improving the accuracy of analytical solutions for a mathematical model describing the transmission dynamics of the African Swine Fever (ASF) virus. The ASF model consists of a Susceptible–Latent–Infectious–Susceptible–Contaminated (SLI–SC) compartmental framework that incorporates the role of contaminated human vectors. An analytical solution for the model is first constructed using the classical Adomian Decomposition Method (ADM) up to the eighth-order approximation. A comparison between the classical ADM solution and the Runge–Kutta–Fehlberg fourth–fifth-order (RKF45) numerical method reveals that, although ADM provides reliable approximations near the initial conditions, its accuracy deteriorates over larger time intervals due to slow convergence and a limited radius of convergence limitations. The key novelty of this study is the use of Padé approximations of orders of [4,4] and [3,5] to rationalize the ADM series solution. In contrast with traditional methods that solely use truncated series expansions, the hybrid ADM–Padé framework improves convergence behavior and extends the validity of the analytical solution over a wider time domain. By converting polynomial series into a rational form, the method reduces the divergence and instability that are common in long-term nonlinear system simulations. A systematic comparison among the classical ADM, the hybrid ADM–Padé technique, and the RKF45 numerical method shows that the ADM–Padé results closely agree with RKF45 solutions across an extended interval. Improved accuracy, stability and a wider convergence region were made available by Pade acceleration and further demonstrated via graphical simulations created in Maple. Overall, this work presents a strong and solid semi-analytical approach for solving nonlinear differential equation models in mathematical epidemiology. The hybrid technique improves solution accuracy for larger domains while guaranteeing quicker convergence. Consequently, ADM–Padé can be considered a robust, precise mathematical technique to solve epidemiological models arising in mathematical biology.

Advection–diffusion equations play a crucial role in modeling the transport of heat, mass, pollutants, and contaminants in porous and heterogeneous media. These models arise in many practical applications, including environmental engineering, hydrology, geophysics, and material sciences. However, classical integer-order advection–diffusion models are often unable to accurately describe anomalous transport processes observed in complex systems. Such systems frequently exhibit long-range interactions, nonlocal dynamics, and memory effects. As a result, the transport behavior may deviate from classical Fickian diffusion and display sub-diffusive or super-diffusive characteristics that cannot be captured using traditional formulations. To address these limitations, this study investigates a variable-order fractional advection–diffusion integro-differential equation that incorporates fractional derivatives in both time and space under appropriate initial and boundary conditions. Unlike many existing studies that primarily focus on constant-order fractional models, the present formulation allows the fractional order to vary with time and space. This feature provides greater flexibility in modeling evolving memory effects and spatial heterogeneity of the medium. A finite difference numerical scheme is developed to approximate the variable-order fractional derivatives together with the convolution-type integral term. The proposed method achieves an accuracy of order (2−α ) on a nonuniform temporal mesh and reduces to first-order accuracy on a uniform mesh. Furthermore, the stability of the numerical scheme is established through theoretical analysis, ensuring the reliability of the computational results. To illustrate the effectiveness of the proposed approach, numerical simulations are carried out for three representative cases: constant fractional order (α=constant), time-dependent order α(t), and fully variable order α(x,t). The numerical results demonstrate good agreement with the analytical solutions and confirm that the proposed method effectively captures complex anomalous transport dynamics with high flexibility and accuracy.

In this study, we examined a continuous-time Markov chain (CTMC) model, conducted a five-type branching process, and studied the influence of seasonal forcing to analyze the different facets of the transmission of Cystic Echinoccocosis among definitive, intermediate, and accidental hosts. First, we studied the deterministic model related to the disease, derived the basic reproduction number, and analyzed the stability of the equilibria to understand the long-term behavior of the infected classes. We then aligned with the CTMC formulation that captured the stochastic amplification of infected definitive hosts, accompanied by a lower burden of infected carcasses than the corresponding deterministic model. The branching process analysis near the disease-free equilibrium quantifies outbreak and extinction probabilities and shows that early epidemic outcomes are primarily driven by the initial numbers of infected definitive hosts and environmental egg burden rather than accidental host infection. Sensitivity analysis identified the death rate of intermediate hosts and environmental parameters as the most critical factors for CE transmission. Moreover, the results of incorporating seasonal forcing highlighted the importance of temporally adaptive control strategies in this context due to the influence of meteorological conditions on parasite egg viability and host behavioral patterns. Overall, the results of our multimodal approach provide rigorous guidance for robust modeling and effective control.

Partial differential equations (PDEs) with nonlocal initial or boundary conditions arise in a wide range of scientific and engineering applications, including heat conduction with integral constraints, population dynamics, anomalous diffusion, materials with memory, and models of fluid flow and transport influenced by global conservation laws. These nonlocal formulations often provide a more accurate description of physical processes than classical local conditions, but they also introduce additional computational challenges.

In recent years, Physics-Informed Neural Networks (PINNs) have been proposed as a flexible framework for solving PDEs without explicit discretization of the domain. Despite their growing popularity, the computational efficiency of PINNs for PDE problems, especially those involving nonlocal constraints, remains insufficiently explored. This work investigates optimization-based approaches for solving parabolic and elliptic PDEs with nonlocal conditions.

Two numerical approaches are investigated. The first method employs finite difference discretization of the PDEs, from which a discrete residual-based loss function is constructed. This loss is minimized using three distinct optimization algorithms: ADAM, AMSGrad and L-BFGS. The second method uses PINNs, where the PDE operators and nonlocal constraints are embedded directly into the training loss. The same three optimizers and stopping criteria are applied to both approaches to ensure a consistent comparison, and in both cases gradients are computed using automatic differentiation.

The methods are evaluated on representative parabolic and elliptic PDE test problems with nonlocal conditions in regular domains. Performance is assessed primarily in terms of execution time required to achieve a given accuracy. The comparison is extended by fixing the spatial domain and increasing the density of discretization points. This allows for a consistent evaluation of how execution time scales with spatial resolution for both approaches.

Relation-theoretic fixed point theory extends classical contraction principles by allowing contractive conditions to be imposed only on pairs of elements that satisfy a prescribed binary relation. This framework has proved useful for studying nonlinear operators that do not satisfy global contraction conditions in metric or normed spaces.

Let H be a Hilbert space admitting a finite orthogonal decomposition H = H₁ ⊕ H₂ ⊕ ··· ⊕ H_k. Motivated by this structure, a binary relation R is introduced on H by defining xRy whenever the difference x − y belongs to one of the component subspaces H_i. This relation reflects the block structure of the space and restricts attention to pairs of elements that differ along a single orthogonal direction.

Nonlinear mappings T : H \to H that preserve this relation and satisfy a Banach-type contraction condition on related pairs, ‖Tx − Ty‖ ≤ α‖x − y‖, with 0 < α < 1, for all x, y with xRy, are considered. Under suitable relational admissibility conditions, the convergence behaviour of the associated Picard iteration defined by xₙ₊₁ = T(xₙ) is analyzed. It is shown that the generated sequence converges strongly to a fixed point of the operator.

The results illustrate how orthogonal decompositions of Hilbert spaces naturally induce relational structures that support relation-theoretic fixed point arguments for nonlinear operators whose behaviour may not be contractive in the global sense.

In this study, we investigate a mathematical model describing the competition between two microbial species in a chemostat environment. One of the species is capable of producing a toxin that inhibits the growth of its competitor, while its own growth is negatively affected by substrate inhibition. The model is reduced to a planar system, and the existence and stability of all steady states are analyzed in detail with respect to the operating parameters of the chemostat.

When classical Michaelis–Menten or Monod growth functions are considered, the system admits a unique positive equilibrium. However, this equilibrium is shown to be unstable whenever it exists. By extending the model to include both monotone and non-monotone growth functions, we demonstrate the emergence of an additional positive equilibrium that can become stable for certain parameter values.

The resulting general model exhibits a wide range of dynamical behaviors, including stable coexistence of the two microbial species, multistability phenomena, and the appearance of stable limit cycles generated through supercritical Hopf bifurcations. Furthermore, homoclinic bifurcations are identified, highlighting the complexity of the system’s dynamics. An operating diagram is constructed to describe the long-term behavior of the system as the operating parameters vary, and to illustrate how substrate inhibition influences the formation and persistence of the species coexistence region.

Heat conduction problems in multilayered media are widely used to model processes in heterogeneous structures, including composites, semiconductors, biomaterials, and nanostructures. In this context, the study of models that describe the evolution of the temperature field under periodic forcing is relevant. In the case, the initial conditions lose their physical meaning and are naturally replaced by time-periodic conditions.

This work investigates a two-layer heat conduction problem for inhomogeneous equations with Dirichlet boundary conditions in the spatial variable and periodic conditions in time. The problem is formulated in the cylindrical domain, defined as the Cartesian product of an interval on the real line and the unit circle. At the interface between the two layers, transmission conditions generalize the classical continuity conditions of the temperature and the heat flux.

The solution is determined as a time-periodic regime, consistent with physical scenarios such as periodic heating or cooling. The existence and uniqueness of the solution in Sobolev spaces of time-periodic functions are proved. The analysis is based on the method of separation of variables, Fourier series expansions, the construction of Green’s functions, and estimates of the determinants related to the problem. A representative model problem is presented along with an approximate solution obtained using a truncated Fourier series expansion.

Among other mathematical difficulties, the variational inequality problem is a broad problem that includes nonlinear equations, optimization problems, complementarity problems, and fixed-point problems. The pricing model of the option, economic equilibrium issues, and specific kinds of partial deferential equations are all studied using variational inequality. In particular classical-variational inequality problem has been broaden to survey a large group of problems arising in optimization theory, physics, economics, financial, structural transportation, and so forth. The study has expanded and epitomized in various fields through different techniques and ideas. Variational Inequality Problem has already defined in various infinite dimensional spaces like manifolds, Housdorff topological vector spaces, Hilbert spaces, Banach space, H-space etc. Also some authors has already introduced vector F-variational inequality problems in ordered topological vector spaces and differentiable ?-manifolds. Also, in one of our previous work we have already defined variational inequality problem on Cesàro sequence space and proved some result like monotonicity and existence theorem on it. The main aim of the work is to introduce the generalized version of variational inequality problems in non-absolute Cesàro sequence spaces Here at first,we have defined generalized variational inequality problem (GVIP) and generalized dual variational inequality problem(GDVIP) on non absolute Cesàro sequence space. We have defined homotopy and functional homotopy for our need.Next we have used KKM mapping to prove existence theorem of generalized Variational inequality problem on a non-absolute Cesàro sequence space to its dual space. Also we have proved some more properties like monotonicity of GVIP on ?- dual of non absolute Cesàro sequence space.

We consider a general-incidence SIRS epidemic model with stochastic dynamics in this paper. The general incidence function, as opposed to traditional methods, takes a more global picture of the dynamics that occur in the real world, such as the change in behavior and the saturation effect during transmission.

On the one hand, through Lyapunov analysis, we determine the existence and uniqueness of a global positive solution, that is, the consistency in the behavior of the system over time. On the other hand, we also obtain a stochastic threshold, an extension of the basic reproduction number, which provides a good condition at which the infection will be expected to fade away.

In order to supplement the theory, we perform numerical simulations. These not only confirm our findings in the analysis but also enable us to answer the question of how the model reacts to variations in important variables such as vaccination rates, incidence functions and noise levels. This assists in determining the real cause of disease dynamics. This flexibility allows the model to readily adjust to a variety of situations and be adjusted using actual data. In the end, this work provides a feasible and versatile instrument for predicting the trends in epidemics and analyzing the effectiveness of control measures in an unpredictable world.

The Hermite–Hadamard inequality is a fundamental result in convex analysis, providing lower and upper bounds for the integral average of a convex function. This research aims to provide a unified treatment of Jensen-type and Hermite–Hadamard-type inequalities by utilizing various classes of Green functions. The study extends classical results to a broader context involving general measures and functional identities.

The core methodology involves the definition of several distinct Green functions. By employing these functions, integral identities are established for functions of the class C². The study utilizes these identities to derive necessary and sufficient conditions for weighted Hermite-Hadamard inequalities to hold, specifically focusing on the behavior of the Green functions under integral operators.

The research establishes several major results. It proves the equivalence between the validity of generalized Jensen-type inequalities and specific integral conditions on Green functions. Furthermore, weighted versions of the Hermite–Hadamard inequality are derived, providing precise error bounds and refinements using L^p norms of the second derivative. New mean-value theorems of Lagrange and Cauchy types are also presented, characterizing the "intermediate point" in these inequalities.

This research shows how the application of Green functions provides a powerful and elegant framework for the systematic study of integral inequalities. The results unify several known generalizations of the Hermite–Hadamard inequality and offer new tools for error estimation in numerical analysis. This approach confirms that the classical Hermite–Hadamard estimates are specific cases of a much broader theory involving functional identities and second-order derivatives.