Search Results (543)

Calculating the pH values of carbonic acid solutions is an important task in studies of chemical equilibria in freshwater systems, with applications to environmental chemistry, geology, and hydrology. These pH values are also highly relevant in the context of climate change, since increasing atmospheric CO₂ affects the concentration of dissolved carbon dioxide and carbonic acid, collectively denoted as [H₂CO₃*] = [H₂CO_3(aq)] + [CO_2(aq)]. Solving equilibrium systems to obtain analytical functions is particularly useful when such functions are required, for example, in data fitting. We show here that, although exact or near-exact solutions typically result in third- to fourth-order equations that must be solved numerically, reasonable approximations can be derived that lead to analytical second-order equations. In this framework, the chosen approximations need to meet the boundary conditions of the systems, particularly for c_T → 0 and for high c_T values (where c_T = [H₂CO₃*] + [HCO₃⁻] + [CO₃²⁻]). Finally, we provide exact solutions for a closed system containing both H₂CO₃* and alkalinity, which enables the description of virtually any aquatic environment without assuming equilibrium with atmospheric CO₂. Implications for pH calculations in natural waters are also briefly discussed. Full article

Background: Integrated medical information systems process large volumes of sensitive clinical data and are exposed to persistent cyber threats. Artificial intelligence (AI) is increasingly used for anomaly detection and incident response, yet its systemic effect on the dynamics of security indicators is not fully quantified. Aim: To develop and numerically study a nonlinear dynamical model describing the joint evolution of system vulnerability, threat activity, compromise level, AI detection quality, and response resources in a medical data protection context. Method: A five-dimensional system of ordinary differential equations was formulated for variables $V$, $T$, $C$, $D$, $R$. Parameters characterize appearance and elimination of vulnerabilities, attack intensity, AI learning and degradation, and resource consumption. The corresponding Cauchy problem $V\left(0\right)=0.5$, $T\left(0\right)=0.6$, $C\left(0\right)=0.1$, $D\left(0\right)=0.4$, $R\left(0\right)=0.8$ was solved on $\left[0,200\right]$ numerically using a fourth-order Runge–Kutta method. Results: Numerical modelling showed convergence to a favourable steady regime. On the interval t ∈ [195, 200] the mean values were $V=0.0073$, $T=0.3044$, $C=7.7·{10}^{-5}$, $D=0.575$, $R=19.99$. Thus, the initial 10% compromise is reduced by more than 99.9%, while AI detection quality stabilizes at around 0.58, and response capacity increases 25-fold. Conclusions: The model quantitatively confirms that the integration of AI detection and a managed response capacity enables the system to reach a stable state with virtually zero compromised medical data even with non-zero threat activity. Full article

Severe drill string vibrations, particularly stick–slip, significantly compromise drilling efficiency and tool longevity in deep hard formations. Compound percussive drilling (CPD) has emerged as a promising technique to mitigate these vibrations and enhance the rate of penetration (ROP). However, the complex coupling mechanisms between impact loads and bit dynamics remain insufficiently understood. This study aims to elucidate the axial–torsional vibration characteristics of the drill bit and the underlying vibration reduction mechanisms under CPD conditions. A multi-degree-of-freedom (MDOF) dynamic model was first established, integrating both the dynamics of the CPD tool and the regenerative cutting effects inherent in bit–rock interactions. The governing equations were then solved numerically using the fourth-order Runge–Kutta method, followed by a systematic parametric sensitivity analysis to quantify the influence of impact parameters on vibration mitigation. The results show that while CPD induces detrimental axial–torsional vibrations in soft rock formations, it effectively suppresses stick–slip and enhances ROP in hard rock formations. Notably, coupled axial–torsional impact loading exhibits superior vibration suppression capabilities compared to singular axial or torsional impacts. A critical proportional relationship for parameter optimization was identified; specifically, maximizing vibration mitigation requires scaling the axial impact load proportionally with the torsional impact load. For example, when the axial impact load amplitudes are 5 kN and 10 kN, the corresponding optimal torsional impact load amplitudes are approximately 500 N·m and 1000 N·m, respectively. Furthermore, maintaining the impact frequency within the range of 10–30 Hz yields optimal vibration reduction effects. The benefits of CPD become increasingly pronounced with higher rock strength and longer drill strings. These findings confirm the suitability of CPD technology for deep hard rock environments and provide theoretical guidelines for the optimal selection of impact parameters in engineering applications. Full article

This paper presents a nonlinear vertical dynamic model of an electromagnetic suspension (EMS) system in maglev trains regulated by a dual nonlinear saturation controller (DNSC) under simultaneous resonance ($\Omega \cong {\omega }_s, {\omega }_s\cong 2{\omega }_c$). The governing nonlinear differential equations of the system are addressed analytically utilizing the multiple time-scale technique (MTST), concentrating on resonance situations obtained from first-order approximations. The suggested controller incorporates two nonlinear saturation functions in the feedback and feedforward paths to improve system stability, decrease vibration levels, and enhance passenger comfort amidst external disturbances and parameter changes. The dynamic bifurcations caused by DNSC parameters are examined through phase portraits and time history diagrams. The goal of control is to minimize vibration amplitude through the implementation of a dual nonlinear saturation control law based on displacement and velocity feedback signals. A comparative analysis is performed on different controllers such as integral resonance control (IRC), positive position feedback (PPF), nonlinear integrated PPF (NIPPF), proportional integral derivative (PID), and DNSC to determine the best approach for vibration reduction in maglev trains. DNSC serves as an effective control approach designed to minimize vibrations and enhance the stability of suspension systems in maglev trains. Stability evaluation under concurrent resonance is conducted utilizing the Routh–Hurwitz criterion. MATLAB 18.2 numerical simulations (fourth-order Runge–Kutta) are employed to analyze time-history responses, the effects of system parameters, and the performance of controllers. The evaluation of all the derived solutions was conducted to verify the findings. Additionally, quadratic velocity feedback leads to intricate bifurcation dynamics. In the time domain, higher displacement and quadratic velocity feedback may destabilize the system, leading to shifts between periodic and chaotic movements. These results emphasize the substantial impact of DNSC on the dynamic performance of electromagnetic suspension systems. Frequency response, bifurcation, and time-domain evaluations demonstrate that the DNSC successfully reduces nonlinear oscillations and chaotic dynamics in the EMS system while attaining enhanced transient performance and resilience. Full article

Non-linear systems of equations play a fundamental role in various engineering and data science models, where accurate solutions are essential for both theoretical research and practical applications. However, solving such systems is highly challenging due to their inherent non-linearity and computational complexity. This study proposes a novel hybrid iterative method with fourth-order convergence. The foundation of the proposed scheme combines the Walrus Optimization Algorithm and a fourth-order iterative technique. The objective of this hybrid approach is to enhance global search capability, reduce the likelihood of convergence to local optima, accelerate convergence, and improve solution accuracy in solving non-linear problems. The effectiveness of the proposed method is checked on standard benchmark problems and two real-world case studies, hydrocarbon combustion and electronic circuit design, and one non-linear boundary value problem. In addition, a comparative analysis is conducted with several well-established optimization algorithms, based on the optimal solution, average fitness value, and convergence rate. Furthermore, the proposed scheme effectively addresses key limitations of traditional iterative techniques, such as sensitivity to initial point selection, divergence issues, and premature convergence. These findings demonstrate that the proposed hybrid method is a robust and efficient approach for solving non-linear problems. Full article

This paper presents a Crank–Nicolson mixed finite element method along with its reduced-order extrapolation model for a fourth-order nonlinear diffusion equation with Caputo temporal fractional derivative. By introducing the auxiliary variable $v=-{\epsilon }^2\Delta u+f(u)$, the equation is reformulated as a second-order coupled system. A Crank–Nicolson mixed finite element scheme is established, and its stability is proven using a discrete fractional Gronwall inequality. Error estimates for the variables u and v are derived. Furthermore, a reduced-order extrapolation model is constructed by applying proper orthogonal decomposition to the coefficient vectors of the first several finite element solutions. This scheme is also proven to be stable, and its error estimates are provided. Theoretical analysis shows that the reduced-order extrapolation Crank–Nicolson mixed finite approach reduces the degrees of freedom from tens of thousands to just a few, significantly cutting computational time and storage requirements. Numerical experiments demonstrate that both schemes achieve spatial second-order convergence accuracy. Under identical conditions, the CPU time required by the reduced-order extrapolation Crank–Nicolson mixed finite model is only $1/60$ of that required by the Crank–Nicolson mixed finite scheme. These results validate the theoretical analysis and highlight the effectiveness of the methods. Full article

This paper aims to explore the numerical solution of non-linear fractional-order Burgers’-Huxley equation based on Caputo’s formulation of fractional derivatives. The equation serves as a versatile tool for analyzing a wide range of physical, biological, and engineering systems, facilitating valuable insights into nonlinear dynamic phenomena. The fractional operator provides a comprehensive mathematical framework that effectively captures the non-locality, hereditary characteristics, and memory effects of various complex systems. The approximation of temporal differential operator is carried out through finite difference based L1 scheme, while spatial discretization is performed using modified cubic B-spline basis functions. The stability as well as convergence analysis of the approach are also presented. Additionally, some numerical test experiments are conducted to evaluate the computational efficiency of a modified fourth-order cubic B-spline (M43BS) approach. Finally, the results presented in the form of tables and graphs highlight the applicability and robustness of M43BS technique in solving fractional-order differential equations. The proposed methodology is preferred for its flexible nature, high accuracy, ease of implementation and the fact that it does not require unnecessary integration of weight functions, unlike other numerical methods such as Galerkin and spectral methods. Full article

This paper presents a finite difference approach for solving the time-fractional Burgers’ equation, which is a model for nonlinear flow with memory effects. The method leverages the $L1-2$ formula for the fractional derivative and provides a novel linearization strategy to efficiently transform the system into a stable linear problem. Rigorous analysis establishes the existence, uniqueness, and pointwise-in-time convergence of the numerical solution in the $L_2$ norm. The proposed formulation achieves second-order time accuracy and fourth-order spatial accuracy under smooth initial conditions, with numerically verified temporal convergence rates of $O({\tau }^{1+\alpha }+{\tau }^2{t}_n^{\alpha -2})$ for solutions with weak singularities. Critically, numerical findings demonstrate that the method is robust and highly efficient, offering high-resolution solutions at a substantially lower computational cost than equivalent graded-mesh formulations. Full article

This study investigates lump wave structures that arise from the interplay of dispersion and nonlinearity in a generalized Calogero–Bogoyavlenskii–Schiff-like model with spatially symmetric nonlinearity in (2+1) dimensions. A generalized bilinear representation of the governing equation is formulated using extended bilinear derivatives of the fourth order, providing a convenient framework for analytic treatment. Through symbolic computation, we construct positive quadratic wave solutions, which give rise to rationally localized lump wave tructures that decay algebraically in all spatial directions at fixed time. Analysis shows that the critical points of these quadratic waves lie along a straight line in the spatial plane and propagate at a constant velocity. Along this characteristic trajectory, the amplitudes of the lump waves remain essentially unchanged, reflecting the stability of these coherent structures. The emergence of these lumps is primarily driven by the combined influence of five dispersive terms in the model, highlighting the crucial role of higher-order dispersion in balancing the nonlinear interactions and shaping the resulting localized waveforms. Full article

The classic second-order Rayleigh equation governs the linear stability of single-phase inviscid plane flows, and its extension to two-phase inviscid plane flows with a crossflow of another fluid remains to be investigated. The present work studies the linear stability of steady uniform inviscid two-phase flow in a horizontal channel with gas bubbles injected from the lower wall and removed from the upper wall. An extended fourth-order Rayleigh equation with constant coefficients is derived for the linear stability of the two-phase uniform inviscid plane flow with the bubble crossflow injected at the bubble terminal velocity. Our analytical results show that the uniform inviscid plane flow driven by the bubble crossflow is linearly unstable with rapidly growing disturbances in the absence of the lift force. On the other hand, when the positive lift force coefficient is nearly equal to the added mass coefficient, the uniform inviscid plane flow driven by the bubble crossflow is linearly stable to the admissible disturbances consistent with the bubble-injection boundary conditions. These analytical results reveal the destabilizing effect of the bubble crossflow and confirm the stabilizing effect of the positive lift force on the inviscid plane flows, which could stimulate further research interest in the qualitatively different roles of the bubble crossflow and the lift force in the stability of inviscid plane flows as compared to viscous plane flows. Full article

Besides the closed-form expansion coefficients of a weak-form numerical differentiator (WFND), we introduce a cubic boundary shape function with the aid of two parameters for reducing the boundary errors of fourth-order numerical derivatives to zero. So that the accuracy of numerical derivatives obtained by the new WFND can be improved significantly. The fourth-order numerical derivation can be modeled as a linear beam equation subjecting to specified boundary conditions and displacements to recover an unknown forcing term. By means of boundary shape functions, two numerical collocation methods automatically satisfying the boundary conditions are developed. For a simply supported linear Euler–Bernoulli beam with an elastic foundation, the unknown spatially–temporally dependent force is retrieved. The displacement at a final time and strain on the right-boundary of the beam are over-specified to recover the external force using the method of superposition of boundary shape functions (MSBSF). When the displacement is determined to satisfy the prescribed right-boundary strain, we can recover an unknown spatially–temporally dependent force by inserting the displacement into the linear beam equation. An embedded method (EM) is developed to transform the linear beam model into a vibrating linear beam equation, and then we can develop a robust technique to compute the fourth-order derivative of noisy data by using the EM and MSBSF. The four proposed methods for evaluating the fourth-order derivatives of noisy data are efficient and accurate. Full article

This study presents an analysis of transverse vibration behavior of a fluid-conveying pipe mounted on an elastic foundation, incorporating both classical analytical techniques and modern physics-informed neural network (PINN) methodologies. A partial differential equation (PDE) architecture is developed to approximate the solution by embedding the physics PDE, initial, and boundary conditions directly into the loss function of a deep neural network. A one-dimensional fourth-order PDE is employed to model governing transverse displacement derived from Euler–Bernoulli beam theory, with additional terms representing fluid inertia, flow-induced excitation, and stochastic force modelled as Gaussian white noise. The governing PDE is decomposed via separation of variables into spatial and temporal components, and modal analysis is employed to determine the natural frequencies and mode shapes under free–free boundary conditions. The influence of varying flow velocities and excitation frequencies on critical buckling behavior and mode shape deformation is analyzed. The network is trained using the Resilient Backpropagation (RProp) optimizer. A preliminary validation study is presented in which a baseline PINN is benchmarked against analytical modal solutions for a fluid-conveying pipe on an elastic foundation under deterministic excitation. The results demonstrate the capability of PINNs to accurately capture complex vibrational phenomena, offering a robust framework for data-driven modelling of fluid–structure interactions in engineering applications. Full article

Many problems in science, engineering, and economics require solving of nonlinear equations, often arising from attempts to model natural systems and predict their behavior. In this context, iterative methods provide an effective approach to approximate the roots of nonlinear functions. This work introduces five new parametric families of multipoint iterative methods specifically designed for solving nonlinear equations. Each family is built upon a two-step scheme: the first step applies the classical Newton method, while the second incorporates a convex mean, a weight function, and a frozen derivative (i.e., the same derivative from the previous step). The careful design of the weight function was essential to ensure fourth-order convergence while allowing arbitrary parameter values. The proposed methods are theoretically analyzed and dynamically characterized using tools such as stability surfaces, parameter planes, and dynamical planes on the Riemann sphere. These analyses reveal regions of stability and divergence, helping identify suitable parameter values that guarantee convergence to the root. Moreover, a general result proves that all the proposed optimal parametric families of iterative methods are topologically equivalent, under conjugation. Numerical experiments confirm the robustness and efficiency of the methods, often surpassing classical approaches in terms of convergence speed and accuracy. Overall, the results demonstrate that convex-mean-based parametric methods offer a flexible and stable framework for the reliable numerical solution of nonlinear equations. Full article

We consider the fourth-order PDE $u_{xxxx}+2u_{xxyy}+u_{yyyy}-u_{tt}=h\left(u\right)$. The Lie and Noether symmetry generators are constructed, and we reduce the PDE to simpler ODEs. Furthermore, we use some well-known methods to compute the conserved vectors associated with the PDE. An analysis of reduced ordinary differential equations (ODEs), invariant solutions, and their physical interpretations is presented. Full article

The problem of nonlinear elastic transverse oscillations of a beam moving along its axis and subjected to an axial compressive or tensile force is considered. A theoretical study is carried out using the asymptotic method of nonlinear mechanics KBM (Krylov–Bogolyubov–Mitropolsky). Using this methods, differential equations were obtained in a standard form, determining the law of variation in amplitude and frequency as functions of kinematic, force, and physico-mechanical parameters in both resonant and non-resonant regimes. The fourth-order Runge–Kutta method was applied for the oscillatory system numerical analysis. The computation of complex mathematical expressions and graphical representation of the results were implemented in the mathematical software Maple 15. The results obtained can be applied for engineering calculations of structures containing moving beams subjected to compressive or tensile forces. Full article