Search Results (9,457)

The current manuscript presents an application of the Sumudu decomposition method (SDM) in efficiently tackling the systems of coupled nonlinear partial fractional differential equations. The technique combines the strengths of the Adomian decomposition method and the Sumudu transform, enabling the transformation of complex systems into rapidly converging series solutions. The efficacy of the technique is then portrayed on various nonlinear coupled fractional models, where approximate solutions are successfully obtained. Furthermore, the computational results indicate efficient numerical performance of the proposed approach for the cases considered. Certainly, the study’s results demonstrate that SDM is an effective and reliable technique for solving the examined class of fractional-order systems. Full article

This study develops an integrated mathematical model to investigate the interaction between tuberculosis (TB) transmission and childhood stunting, which is aligned with the United Nations Sustainable Development Goals (SDG 3). The population is structured into two age groups (0–5 years and ≥5 years), with stunting explicitly incorporated into the pediatric population to capture its potential influence on TB dynamics. The model is formulated as a system of ordinary differential equations and analyzed using equilibrium and stability analysis, with the basic reproduction number, R₀. The disease-free equilibrium is locally asymptotically stable when R₀ < 1, while an endemic equilibrium exists when R₀ > 1. Sensitivity analysis indicates that the transmission rate (β), progression rate from latent to active infection (σ), and recovery rate (γ) are the most influential parameters affecting R₀. These parameters are therefore selected as control variables in an optimal control framework to design effective intervention strategies. Numerical simulations show that the combined control strategy significantly reduces TB transmission, resulting in a reduction of more than 80% in active TB cases within a relatively short intervention period. The results suggest that integrated interventions targeting transmission, disease progression, and recovery are substantially more effective than single-measure strategies. This study provides a quantitative framework to support integrated public health policies addressing TB and childhood stunting simultaneously. Full article

This paper develops a structured analytical framework and a robust numerical methodology for the spectral stability of time-varying bilateral quaternion differential equations of the form $\dot{q}=A\left(t\right)q+qB\left(t\right)$. By systematically extending classical real matrix theory to non-commutative dynamical systems via exact isometric real representations, this study utilizes the Kronecker product of real adjoint matrices to rigorously elucidate the underlying tensor structure of the bilateral evolution operator. This tensor-based reformulation proves that the Floquet multipliers of the bilaterally coupled system can be strictly decoupled into the product of the spectra corresponding to the left and right unilateral subsystems. Second, a “Scalar-Vector Stability Separation Principle” based on logarithmic norms is proposed, demonstrating that the transient energy evolution of the system is governed exclusively by the Hermitian real parts of the coefficient matrices, remaining entirely independent of the anti-Hermitian imaginary parts (rotation terms). Furthermore, for constant-coefficient and slowly varying systems, the Riesz projection from holomorphic functional calculus is introduced to establish algebraic criteria for exponential dichotomies, thereby revealing a cubic scaling law that relates the robustness threshold to the spectral gap (${\epsilon }_0\propto {\beta }^3$). Numerically, a Quaternion Chebyshev Spectral Collocation Method (Q-CSCM) is embedded within this exact vectorization framework to ensure that the algebraic symmetries of the bilateral system are strictly preserved through the isomorphic mapping. By explicitly constructing the fully discrete Kronecker product matrix via the exact real vectorization isomorphism, discrete energy estimates are utilized to rigorously prove that the numerical scheme successfully inherits the intrinsic spectral accuracy of the Chebyshev approximation. Comprehensive numerical experiments demonstrate that, within the low-dimensional regime, this methodology exhibits substantial temporal approximation efficiency advantages and superior numerical robustness compared to an alternative Legendre spectral baseline, as well as traditional explicit and state-of-the-art implicit symplectic Runge–Kutta methods, particularly when solving stiff and critically stable problems such as nonlinear Riccati oscillators. Full article

The framework of the research whose part of results are published in this work is the category of real vector bundles over finite dimensional differentiable manifolds. The objects of studies are gauge structures on these vector bundles. We are interested in the dynamical properties of the holonomy groups of Koszul connections as well as on their topological properties, i.e., properties that are of homological nature. For the most part the context is the subcategory of Lie algebroids. In addition to other investigations, three open problems are studied in detail. (P1-Affine Geometry): When is a Koszul connection an affine connection? (P2-Riemannian Geometry): When is a Koszul connection a metric connection? (P3-Fedosov Geometry): When is a Koszul connection a symplectic connection? In the category of tangent Lie algebroids our homological approach leads to deep relations of our homological ingredients with the open problem of how to produce labeled foliations the most studied of which are Riemannian foliations. On a Lie algebroid we define two families of differential equations, the family of differential Hessian equations and the family of differential gauge equations. The solutions of these differential equations are implemented to construct homological ingredients which are key tools for our studies of open problems we are concerned with. We introduce Koszul Homological Series. This notion is a machine for converting obstructions whose nature is vector space into obstructions whose nature is homological class. We define the property of Degeneracy and the property Nondegeneracy of Koszul homological Series. The property of Degeneracy is implemented to solve problems (P1), (P2), and (P3). In the abundant literature on Riemannian foliations, we have only cited references directly related to the open problems which are studied using the tools which are introduced in this work. Thus, the property of nondegeneracy is implemented to give a complete solution of the problem posed by E. Ghys, (P4-Differential Topology): How does one produce Riemannian foliations? See our Theorems 12 and 13, which are fruits of a happy conjunction between gauge geometry and differential topology. Full article

Differential equations with fractional order play an important role in modeling some natural phenomena. This paper investigates the dynamics of the fractional-order commensal symbiosis model with the Allee effect. This model describes the relationship between prey and predator populations. The piecewise-constant approximation technique is applied to discretize this model. Equilibrium points are established, and local stability conditions are calculated using fractional-order linearization and eigenvalue-based arguments. Moreover, the bifurcation theory is successfully invoked to discuss the period-doubling bifurcation. In particular, sufficient conditions are effectively determined for the emergence of the period-doubling bifurcation. We utilize the hybrid control approach to control the behavior of the considered system. Then, some numerical examples are presented to demonstrate the accuracy and validity of the theoretical results. The findings indicate that fractional order and Allee effects improve system dynamics and substantially improve stability limits and bifurcation structures, providing new insights into how to handle the dynamics of ecological systems. Full article

This paper investigates a nonlinear Bitsadze–Samarskii type boundary value problem for an elliptic-hyperbolic operator. First, we establish necessary and sufficient conditions for the unique solvability of the corresponding linear mixed-type problem using the abstract theory of correct restrictions and extensions of operators. Explicit solution formulas are obtained via Green’s function for the elliptic part and integral representations for the hyperbolic part. These results are then extended to a nonlinear mixed operator of power-law type by applying a bijective transformation that reduces the nonlinear problem to the linear case. The key condition for solvability is the continuity and invertibility of a functional coefficient in the boundary condition. Our work provides a systematic framework for handling non-standard boundary conditions in mixed-type problems and highlights the role of operator-theoretic methods in nonlinear settings. Full article

The article considers nonlinear fractional differential equations with cotangent derivative. The boundary conditions are multipoint and integral specified, and the nonlinear terms are in Orlicz function spaces. Several existence theorems for solutions of such boundary value problems are obtained by different fixed-point methods. Illustrative examples serve to illustrate the theoretical parts. Full article

In this study, we investigate a class of mixed fractional partial integro-differential equations (FrPI-DE) involving symmetric singular kernels. The considered model problem involves Caputo fractional derivatives and integral operators that describe spatial interactions in a bounded domain. For the purpose of analysis, the original problem is reformulated in the form of a nonlinear Volterra–Fredholm integral equation (NV-FIE). The existence and uniqueness of the solution are established by the Banach fixed point theorem. To compute numerical solutions, a modified Toeplitz matrix method (TMM) is proposed to handle the singular kernel efficiently. The method transforms the integral equation to a system of nonlinear algebraic equations, which can be solved numerically. The convergence properties of the resulting numerical scheme are analyzed and illustrate the effectiveness of the method by providing numerical examples involving logarithmic, Cauchy-type, and weakly singular kernels. Numerical results indicate that the proposed method provides highly accurate approximations and exhibits stable convergence behavior for different parameter values. Furthermore, these results confirm the effectiveness and reliability of the proposed method for solving fractional integro-differential equations that include symmetric singular kernels. Full article

Functionally graded porous beams are increasingly used in lightweight engineering structures, where thermal effects and material inhomogeneity significantly influence vibration behavior. In this study, the thermoelastic free vibration of functionally graded porous Euler–Bernoulli beams with temperature-dependent material properties is investigated by considering uniform and symmetric porosity distributions, together with uniform, linear, and nonlinear temperature fields. The governing equations are derived based on classical Euler–Bernoulli beam theory and solved using the Differential Transformation Method, while the accuracy of the semi-analytical formulation is verified through a Hermite-based finite element model. The results show that increasing temperature reduces the bending stiffness due to thermal axial forces and leads to a rapid decrease in natural frequency as the critical buckling temperature is approached. Increasing porosity generally decreases the natural frequency, although a slight increase may occur in symmetric distributions because of the accompanying reduction in mass density. The present study provides a computational framework for the thermo-vibration analysis of functionally graded porous beams in lightweight structural applications. Full article

A symbolic method is presented for examining the solvability and constructing the exact solution to boundary value problems for systems of linear ordinary loaded differential equations and loaded integro-differential equations with nonlocal boundary conditions. The method uses the inverse of the differential operator involved in the system of loaded differential or integro-differential equations. A solvability criterion based on the determinant of a matrix and an exact analytical matrix-form solution formula are presented. For the implementation of the method into computer algebra system software, two algorithms are provided. The effectiveness of the method is demonstrated by solving several problems. The theoretical and practical results obtained complement the existing literature on the subject. Full article

Integral-form nonlocal elasticity provides a mechanically meaningful approach to describing size effects, yet it leads to Volterra-type integro-differential equations that are difficult to solve analytically and numerically challenging for boundary layers and high-order modes. In this work, we developed a symplectic numerical integration framework for Eringen’s two-phase (local/nonlocal mixture) integral model by embedding the constitutive operator into a Hamiltonian formulation and discretizing the influence domain in a belt-wise manner. A step-increase strategy was incorporated to allow flexible spatial marching while preserving the geometric (symplectic) structure of the transfer operation. In addition, a symmetry-explicit, element-level stiffness representation was derived for the discretized integral operator; it exposes a mirrored long-range coupling pattern and enables symmetric, energy-consistent assembly. The resulting kernel-agnostic algorithm accommodates both smooth and finite-range kernels. Static benchmarks and longitudinal vibrations are investigated for exponential, Gaussian, and triangular kernels over representative length ratios and mixture parameters. Comparisons with available analytical and asymptotic solutions show good agreement within their validity ranges, and the method yields stable higher-order eigenfrequencies when asymptotic expansions may be unreliable. The current study is limited to a linear one-dimensional rod setting, and validation is restricted to published analytical/asymptotic solutions rather than experimental calibration. Full article

Numerous applications from electrical engineering and mechanical structures are mathematically modeled using dynamical systems theory. Our paper concerns the behaviors of a 3D dynamic system in terms of damped or periodical oscillations and asymptotic representation, considering the dependence on three physical parameters. This system is explicitly integrated via a smooth-function solution of a third–order nonlinear differential equation, which means that the obtained exact parametric solutions describe a heteroclinical orbit. The modified Optimal Parametric Iteration Method (mOPIM) is used to study the influence of the physical parameters. The advantages of the applied method include the small number of iterations due to due to the appropriate choice of auxiliary convergence control functions. The mOPIM solutions are in good agreement with the corresponding numerical results and this aspect is highlighted qualitatively by figures and quantitatively by tables, respectively, in this work. The accuracy of the obtained solutions is assessed via a comparison with the OPIM method and the iterative solutions using 5–8 iterations, via an iterative method. A qualitative analysis of errors is performed. Full article

Soil salinization poses a severe threat to global wheat production, yet the physiological mechanisms underlying photosynthetic responses to neutral, alkaline, and combined salt stress remain poorly understood. This study systematically evaluated the photosynthetic physiology and salt tolerance of six spring wheat genotypes under three types of salt stress at varying concentrations. By integrating phenotypic data, gas exchange parameters, chlorophyll fluorescence indices, and biomass measurements, and applying structural equation modeling and multivariate analysis, key traits regulating biomass were identified. The results revealed significant interactions among salt stress type, genotype, and concentration on photosynthetic parameters. Structural equation modeling analysis revealed that under neutral salt stress, both gas exchange parameters and chlorophyll content had significant direct effects on seedling biomass, with standardized path coefficients of 0.421 and 0.400, respectively. Under alkaline and combined salt stresses, only chlorophyll content showed a significant direct effect on biomass, with standardized path coefficients of 0.873 and 0.790, respectively. Multiple regression analysis further identified key photosynthetic factors influencing growth under different stress types. Under neutral salt stress, phi (Ro) and E significantly affected biomass, whereas under alkaline and combined salt stresses, biomass was primarily co-regulated by phi (Ro) and phi (Eo). Based on a comprehensive evaluation of salt tolerance index, damage index, and biomass response, genotypes W06 and W02 exhibited the strongest overall salt tolerance. This study systematically elucidates the differential response mechanisms of photosynthesis in spring wheat under distinct salt stress types, providing an important theoretical basis and elite germplasm resources for breeding salt-tolerant wheat varieties. Full article

We revisit a hyperbolic wildfire model based on reaction–diffusion dynamics with relaxation effects and extend it by incorporating an advection transport term that accounts for wind-driven fire spread. After a planar two-dimensional reformulation and non-dimensionalization of the model, the analysis is restricted to the minimal ignition regime characterized by the presence of a logistic reaction term governing the evolution of the fire-affected tree fraction. The focus of the study is to assess the influence of the effective wind velocity on the propagation dynamics of the fire-affected tree fraction. For this purpose, analytical solutions of the extended wildfire model are derived by applying the Simple Equations Method (SEsM) in its (1,1) variant using a Riccati-type ordinary differential equation as a simple equation. The obtained families of exact solutions describe physically relevant transition fronts connecting fire-unaffected and fully fire-affected states, or vice versa. Numerical simulations of the derived analytical solutions are performed to demonstrate how the internal front thickness and the profile morphology depend on the specific variant of the Riccati-type solution and on the magnitude of the effective wind velocity. A phase-plane stability and bifurcation analysis of the reduced traveling wave system is carried out. Hopf bifurcation thresholds with respect to the effective wind velocity parameter are identified, revealing transitions between monotone front propagation and oscillatory regimes. A regime map is constructed in the parameter plane spanned by the effective wind velocity and the traveling wave speed. This regime diagram delineates regions of qualitatively different propagation behavior, including monotone advancing fronts, possible oscillatory regimes, and regimes in which traveling wave fronts cease to exist. Full article

Finger seals operate over extended periods under complex conditions involving high-pressure differentials, elevated rotational speeds, and rotor radial runout. Intense convective heat transfer arises within the seal, significantly impacting its structural deformation. To elucidate the influence of temperature on finger-seal deformation during convective heat transfer, the present study derives heat transfer energy equations for finger seals based on the Local Thermal Non-Equilibrium (LTNE) model. A three-dimensional porous-media flow-field model incorporating the LTNE framework, along with a solid thermal-deformation model, is developed. The effects of pressure differential and interference-fit magnitude on the structural deformation and average contact pressure of finger seals are analyzed under both the Local Thermal Equilibrium (LTE) and LTNE models. The results indicate that the LTNE model predicts a higher maximum seal temperature and a lower leakage rate compared to the LTE model. In both models, the deformation of individual seal-blade layers increases with rising pressure differentials and interference-fit magnitudes. Furthermore, the overall blade deformation is more pronounced under the LTNE model, suggesting a substantial thermal influence on sealing performance. The effects of pressure difference and interference fit on the thermal deformation of the seal plate are similar: both have the greatest impact on radial deformation, followed by circumferential deformation and axial deformation. Within the pressure difference range, the radial deformation of the third-layer seal plate in the LTNE model increases by 14.55%. When the interference fit increases from 0.05 mm to 0.2 mm, the radial deformation of each layer of the seal plate in the LTNE model increases by 0.18 mm. The average contact pressure increases with both pressure differential and interference-fit magnitude across both models. At a given pressure differential, the LTNE model yields a higher average contact pressure than the LTE model, with a maximum observed difference of 0.01 MPa. When the interference-fit magnitude is small, the pressure difference between the models remains minimal; however, at the maximum interference-fit, the difference reaches 0.08 MPa. Full article