Search Results (219)

This paper explores the complex dynamics of mixed convective flow in a Casson fluid saturated in a non-Darcy porous medium, focusing on the influence of gyrotactic microorganisms, internal heat generation, and multiple convective mechanisms. Casson fluids, known for their non-Newtonian behavior, are relevant in various industrial and biological contexts where traditional fluid models are insufficient. This study addresses the limitations of the standard Darcy’s law by examining non-Darcy flow, which accounts for nonlinear inertial effects in porous media. The governing equations, derived from conservation laws, are transformed into a system of no linear ordinary differential equations (ODEs) using similarity transformations. These ODEs are solved numerically using a finite differencing method that incorporates central differencing, tridiagonal matrix manipulation, and iterative procedures to ensure accuracy across various convective regimes. The reliability of this method is confirmed through validation with the MATLAB (R2024b) bvp4c scheme. The investigation analyzes the impact of key parameters (such as the Casson fluid parameter, Darcy number, Biot numbers, and heat generation) on velocity, temperature, and microorganism concentration profiles. This study reveals that the Casson fluid parameter significantly improves the velocity, concentration, and motile microorganism profiles while decreasing the temperature profile. Additionally, the Biot number is shown to considerably increase the concentration and dispersion of motile microorganisms, as well as the heat transfer rate. The findings provide valuable insights into non-Newtonian fluid behavior in porous environments, with applications in bioengineering, environmental remediation, and energy systems, such as bioreactor design and geothermal energy extraction. Full article

The objective of the present study is to detect chirped optical solitons of the perturbed Chen–Lee–Liu equation with full nonlinearity. By virtue of the traveling wave hypothesis, the discussed model is reduced to a simple form known as an elliptic equation. The latter equation, which is a second-order ordinary differential equation, is handled by the undetermined coefficient method of two forms expressed in terms of the hyperbolic secant and tangent functions. Additionally, the auxiliary equation method is applied to derive several miscellaneous solutions. Various types of chirped solitons are revealed such as W-shaped, bright, dark, gray, kink and anti-kink waves. Taking into consideration the existence conditions, the dynamical behaviors of optical solitons and their corresponding chirp are illustrated. The modulation instability of the perturbed CLL equation is examined by means of the linear stability analysis. It is found that all solutions are stable against small perturbations. These entirely new results, compared to previous works, can be employed to understand pulse propagation in optical fiber mediums and dynamic characteristics of waves in plasma. Full article

Darboux transformations are relations between the eigenfunctions and coefficients of a pair of linear differential operators, while Painlevé equations are nonlinear ordinary differential equations whose solutions arise in diverse areas of applied mathematics and mathematical physics. Here, we describe the use of confluent Darboux transformations for Schrödinger operators, and how they give rise to explicit Wronskian formulae for certain algebraic solutions of Painlevé equations. As a preliminary illustration, we briefly describe how the Yablonskii–Vorob’ev polynomials arise in this way, thus providing well-known expressions for the tau functions of the rational solutions of the Painlevé II equation. We then proceed to apply the method to obtain the main result, namely, a new Wronskian representation for the Ohyama polynomials, which correspond to the algebraic solutions of the Painlevé III equation of type $D_7$. Full article

The incorporation of advanced smart materials, such as shape memory alloys (SMAs), in civil engineering presents significant challenges, particularly in modeling their complex behavior. Traditional numerical SMA models often require material parameters that are difficult to estimate and validate. The objective of this paper is to introduce an equation-based approach to modeling the superelastic behavior of SMAs based on rheological models. The proposed phenomenological model accurately captures SMA superelasticity under isothermal conditions, with each material parameter directly correlated to data from standard mechanical experiments. Four modifications to the baseline rheological model are proposed, highlighting their impact on superelastic characteristics. The resulting constitutive relationships are expressed as non-linear ordinary differential equations, making them compatible with commercial finite element method (FEM) software through user-defined subroutines. The practical application of this modeling approach is demonstrated through the strengthening of a historical masonry wall subjected to seismic activity. Comparative analysis shows that ties incorporating SMA segments outperform traditional steel ties by reducing the potential damage and enhancing the structural performance. Additionally, the energy dissipation during the SMA phase transformation improves the damping of vibrations, further contributing to the stability of the structure. This study underscores the potential of SMA-based solutions in seismic retrofitting and highlights the advantages of equation-based modeling for practical engineering applications. Full article

In this paper, we develop an HIV/AIDS epidemic model that incorporates a saturated incidence rate to reflect the limited transmission capacity and the impact of behavioral saturation in contact patterns. The model is formulated as a system of seven non-linear ordinary differential equations representing key population compartments. In addition to model formulation, we introduce an optimal control problem involving three control measures: educational campaigns, screening of unaware infected individuals, and antiretroviral treatment for aware infected individuals. We begin by establishing the positivity and boundedness of the model solutions under constant control inputs. The existence and local and global stability of both the disease-free and endemic equilibrium points are analyzed, depending on the effective reproduction number (${\mathcal{R}}_e$). Bifurcation analysis reveals that the model undergoes a forward bifurcation at ${\mathcal{R}}_e=1$. A local sensitivity analysis of ${\mathcal{R}}_e$ identifies the disease transmission rate as the most sensitive parameter. The optimal control problem is then formulated by incorporating the dynamics of infected subpopulations, control costs, and time-dependent controls. The existence of optimal control solutions is proven, and the necessary conditions for optimality are derived using Pontryagin’s Maximum Principle. Numerical simulations support the theoretical analysis and confirm the stability of the equilibrium points. The optimal control strategies, evaluated using the Incremental Cost-Effectiveness Ratio (ICER), indicate that implementing both screening and treatment (Strategy D) is the most cost-effective intervention. These results provide important insights for designing effective and economically sustainable HIV/AIDS intervention policies. Full article

A mathematical model of the anomalous diffusion of surfactant and the process of adsorption–desorption on an interface is analyzed using a fractional calculus approach. The model is based on time-fractional partial differential equations in the bulk phases and the corresponding time-fractional description of the flux bulk–interface. The general case, when the surfactant is soluble in both phases, is considered under the assumption that the adsorption–desorption process is diffusion-controlled. Some of the most popular kinetic models of Henry, Langmuir, and Volmer are considered. Applying the Laplace transform, the partial differential model is transformed into a single multi-term time-fractional nonlinear ordinary differential equation for the surfactant concentration on the interface. Based on existing analytical solutions of linear time-fractional differential equations, the exact solution in the case of the Henry model is derived in terms of multinomial Mittag–Leffler functions, and its asymptotic behavior is studied. Further, the fractional differential model in the general nonlinear case is rewritten as an integral equation, which is a generalization of the well-known Ward–Tordai equation. For computer simulations, based on the obtained integral equation, a predictor–corrector numerical technique is developed. Numerical results are presented and analyzed. Full article

Air pockets in water distribution networks can cause various operational issues, as their expansion during drainage operations leads to sub-atmospheric conditions that may result in pipeline collapse depending on soil conditions and pipe stiffness. This study presents an analytical solution for calculating air pocket pressure, water column length, and water velocity during drainage operations in a pipeline with an entrapped air pocket and a closed upstream end. The existing system of three differential equations is reduced to two first-order nonlinear differential equations, enabling a rigorous analysis of the existence and uniqueness of solutions. The system is then further reduced to a single secondorder nonlinear ordinary differential equation (ODE), providing an intuitive framework for examining the physical behaviour of the hydraulic and thermodynamic variables. Furthermore, through a change of variables, the second-order ODE is transformed into a first-order linear ODE, facilitating the derivation of an analytical solution. The analytical solution is validated by comparing it with a numerical solution. Additionally, a practical application demonstrates the effectiveness of the developed tool in predicting the extreme pressure values in the air pocket during the water drainage process in a pipe, within a controlled environment. Full article

An analytical study of the nonlinear response of imperfect stiffened doubly curved shells made of functionally graded material (FGM) is presented. The formulation of the problem is based on the first-order shear deformation shell theory in conjunction with the von Kármán geometrical nonlinear strain–displacement relationships. The nonlinear equations of the motion of stiffened double-curved shells based on the extended Sanders’s theory were derived using Galerkin’s method. The material properties vary in the direction of thickness according to the linear rule of mixture. The effect of both longitudinal and transverse stiffeners was considered using Lekhnitsky’s technique. The fundamental frequencies of the stiffened shell are compared with the FE solutions obtained by using the ABAQUS 6.14 software. A stepwise approximation technique is applied to model the functionally graded shell. The resulting nonlinear ordinary differential equations were solved numerically by using the fourth-order Runge–Kutta method. Closed-form solutions for nonlinear frequency–amplitude responses were obtained using He’s energy method. The effect of power index, functionally graded stiffeners, geometrical parameters, and initial imperfection on the nonlinear response of the stiffened shell are considered and discussed. Full article

The study offers a comprehensive investigation of periodic solutions in highly nonlinear oscillator systems, employing advanced analytical and numerical techniques. The motivation stems from the urgent need to understand complex dynamical behaviors in physics and engineering, where traditional linear approximations fall short. This work precisely applies He’s Frequency Formula (HFF) to provide theoretical insights into certain classes of strongly nonlinear oscillators, as illustrated through five broad examples drawn from various scientific and engineering disciplines. Additionally, the novelty of the present work lies in reducing the required time compared to the classical perturbation techniques that are widely employed in this field. The proposed non-perturbative approach (NPA) effectively converts nonlinear ordinary differential equations (ODEs) into linear ones, equivalent to simple harmonic motion. This method yields a new frequency approximation that aligns closely with the numerical results, often outperforming existing approximation techniques in terms of accuracy. To aid readers, the NPA is thoroughly explained, and its theoretical predictions are validated through numerical simulations using Mathematica Software (MS). An excellent agreement between the theoretical and numerical responses highlights the robustness of this method. Furthermore, the NPA enables a detailed stability analysis, an area where traditional methods frequently underperform. Due to its flexibility and effectiveness, the NPA presents a powerful and efficient tool for analyzing highly nonlinear oscillators across various fields of engineering and applied science. Full article

The weakly nonlinear stability analysis of a convective flow in a planar vertical fluid layer is performed in this paper. The base flow in the vertical direction is generated by internal heat sources distributed within the fluid. The system of Navier–Stokes equations under the Boussinesq approximation and small-Prandtl-number approximation is transformed to one equation containing a stream function. Linear stability calculations with and without a small-Prandtl-number approximation lead to the range of the Prantdl numbers for which the approximation is valid. The method of multiple scales in the neighborhood of the critical point is used to construct amplitude evolution equation for the most unstable mode. It is shown that the amplitude equation is the complex Ginzburg–Landau equation. The coefficients of the equation are expressed in terms of integrals containing the linear stability characteristics and the solutions of three boundary value problems for ordinary differential equations. The results of numerical calculations are presented. The type of bifurcation (supercritical bifurcation) predicted by weakly nonlinear calculations is in agreement with experimental data. Full article

This study presents Galerkin and collocation algorithms based on Telephone polynomials (TelPs) for effectively solving high-order linear and non-linear ordinary differential equations (ODEs) and ODE systems, including those with homogeneous and nonhomogeneous initial conditions (ICs). The suggested approach also handles partial differential equations (PDEs), emphasizing hyperbolic PDEs. The primary contribution is to use suitable combinations of the TelPs, which significantly streamlines the numerical implementation. A comprehensive study has been conducted on the convergence of the utilized telephone expansions. Compared to the current spectral approaches, the proposed algorithms exhibit greater accuracy and convergence, as demonstrated by several illustrative examples that prove their applicability and efficiency. Full article

Most European Union governments and numerous railway operators have announced plans to replace most of their diesel units by 2030–2040. However, a significant portion of the rail network remains non-electrified. In some cases, the proposed solution has been to close certain tracks, but this approach entails considerable societal costs for small cities and represents a loss of prior railway investments. Consequently, hybrid locomotives and multiple units (either new or refurbished) emerge as a viable solution during this transitional period to enhance energy efficiency and preserve services on these lines, particularly for freight operations. These hybrid units can operate on both electrified and non-electrified tracks and can also serve as “railway prosumers”, contributing to both storage and generation in fully or partially electrified areas. However, implementing these “prosumer tasks” faces challenges, such as the rapid power demand fluctuations during acceleration and the loss of energy recovery potential during braking in hybrid or fully electric units. These losses may also impact the overall power system. This paper presents an alternative approach to modeling double-layer capacitors (supercapacitors) combined with electrical equivalent models for lithium-ion batteries. The Differential Transformation Method (DTM) is used to solve the non-linear ordinary differential equations governing the supercapacitor model, while parameter optimization is achieved through a grid search approach, demonstrating high accuracy compared with laboratory trials. This framework highlights the potential of hybrid units, as illustrated through simulations that analyze storage sizing, energy management, increased energy recovery, and changes in unit performance. These models facilitate a pre-feasibility evaluation of energy storage systems for hybrid railway applications. Full article

The dynamic response of light bridges to moving loads presents significant challenges in controlling vibrations that can impact on the structural integrity and the user comfort. This study investigates the effectiveness of nonlinear semi-active absorbers in mitigating these vibrations on light bridges that are particularly susceptible to human-induced vibrations, due to their inherent low damping and flexibility, especially under near-resonance conditions. Traditional passive vibration control methods, such as dynamic vibration absorbers (DVAs), may not be entirely adequate for mitigating vibrations, as they require adjustments in damping and stiffness when operating conditions change over time. Therefore, suitable strategies are needed to dynamically adapt DVA parameters and ensure optimal performance. This paper explores the effectiveness of linear and nonlinear DVAs in reducing vertical vibrations of lightweight beams subjected to moving loads. Using the Bubnov-Galerkin method, the governing partial differential equations are reduced to a set of ordinary differential equations and a novel nonlinear DVA with a variable damping dashpot is investigated, showing better performances compared to traditional constant-parameter DVAs. The nonlinear viscous damping device enables real-time adjustments, making the DVA semi-active and more effective. A footbridge case study demonstrates significant vibration reductions using optimized nonlinear DVAs for lightweight bridges, showing broader frequency effectiveness than linear ones. The quadratic nonlinear DVA is the most efficient, achieving a 92% deflection reduction in the 1.5–2.5 Hz range, and under running and jumping reduces deflection by 42%. Full article

This paper introduces an Evolutionary Computing Control Strategy (ECCS) for the motion control of nonholonomic robots, and integrates an ordinary differential equation (ODE)-based kinematics model with a nonlinear model predictive control (NMPC) strategy and a particle-based evolutionary computing (PEC) algorithm. The ECCS addresses the key challenges of traditional NMPC controllers, such as their tendency to fall into local optima when solving nonlinear optimization problems, by leveraging the global optimization capabilities of evolutionary computation. Experiment results on the MATLAB Simulink platform demonstrate that the proposed ECCS significantly improves motion control accuracy and reduces control errors compared to linearized MPC (LMPC) strategies. Specifically, the ECCS reduces the maximum error by 90.6% and 94.5%, the mean square error by 67.8% and 92.6%, and the root mean square error by 43.5% and 70.3% in velocity control and steering angle control, respectively. Furthermore, experiments are separately implemented on the CarSim platform and the physical environment to verify the availability of the proposed ECCS. Furthermore, experiments are separately implemented on the CarSim platform and the physical environment to verify the availability of the proposed ECCS. These results validate the effectiveness of embedding ODE kinematics into the evolutionary computing framework for robust and efficient motion control of nonholonomic robots. Full article

This work introduces a systematic method for identifying analytical and semi-analytical solutions of force-free magnetic fields with plane-parallel and axial symmetry. The method of separation of variables is used, allowing the transformation of the non-linear partial differential equation, corresponding to force-free magnetic fields, to a system of decoupled ordinary differential equations, which nevertheless, are in general non-linear. It is then shown that such solutions are feasible for configurations where the electric current has a logarithmic dependence to the magnetic field flux. The properties of the magnetic fields are studied for a variety of physical parameters, through solution of the systems of the ordinary differential equations for various values of the parameters. It is demonstrated that this new logarithmic family of solutions has properties that are highly distinct from the known linear and non-linear equations, as it allows for bounded solutions of magnetic fields, for periodic solutions and for solutions that extend to infinity. Possible applications to astrophysical fields and plasmas are discussed as well as their use in numerical studies, and the overall enrichment of our understanding of force-free configurations. Full article