Search Results (223)

The object of the study is a single heated carbonaceous particle of relatively small size, 0.003 to 0.01 m. Main hypothesis: The formation of soot particles and black carbon particles is caused by the thermochemical destruction of dry organic matter of forest fuel and the mechanical fragmentation of coke residue. The aim of the study is to conduct numerical simulations of heat and mass transfer in a single heated carbonaceous particle, taking into account the soot formation process and assessing its fragmentation with regard to heat exchange with the external environment in a 2D setting. As part of this study, a new model of heat and mass transfer in a pyrolyzed carbonaceous particle was developed, taking into account its step-by-step fragmentation (fragmentation tree model with four secondary particle formations from the initial particle). The calculations resulted in the distributions of temperature and volume fractions of phases in the carbonaceous particle across various scenarios. Scenarios of surface fires (initial temperatures of 900 K and 1000 K), crown fires (1100 K), and a firestorm (1200 K) for typical vegetation (pine, spruce, birch) are considered. Cubic carbonaceous particles are considered in the approximation of a 2D mathematical model. To describe heat and mass transfer in the structure of the carbonaceous particle, a differential equation of thermal conductivity with corresponding initial and boundary conditions of the third type is used, taking into account the gross reaction in the kinetic scheme of pyrolysis and soot formation. Differential analogues of partial differential equations are solved using the finite difference method of second-order approximation. Options for using the developed mathematical model and probabilistic fragmentation criterion for assessing aerosol emissions are proposed. Recommendations: The suggested mathematical model must be incorporated with mathematical models of forest fire plume and aerosol transport in the upper layers of the atmosphere. Moreover, probabilistic criteria for health assessment must be developed for the practical use of the suggested mathematical model. 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

This study introduces a modified computational scheme for handling linear and nonlinear fractal time-dependent partial differential equations. The method is constructed using three different stages that provide third-order accuracy in the fractal time variable. The stability of the approach is examined using scalar fractal models and Fourier analysis, while convergence is established for coupled convection–diffusion systems. The numerical algorithm is applied to analyze the mixed convective flow of a Carreau–Yasuda non-Newtonian fluid over stationary and oscillating plates under the influence of viscous dissipation and magnetic field effects. For spatial discretization, the incompressible continuity equation is handled by a first-order difference scheme, whereas higher-order compact schemes are implemented for the momentum, thermal, and concentration equations. The numerical findings show that increasing the Weissenberg number and magnetic field inclination reduces the velocity distribution. An accuracy assessment against existing numerical techniques demonstrates that the present method yields smaller computational errors, particularly when central difference schemes are used in space. In addition, a surrogate machine learning model is developed to predict the skin friction coefficient and local Nusselt number using Reynolds, Weissenberg, Prandtl, and Eckert numbers as input features. The predictive capability of the model is validated through Parity plots, bar charts for sensitivity analysis, scatter visualization, and Taylor Diagrams, confirming strong agreement with the numerical results. Full article

This paper introduces a new explicit integration method for second-order ordinary differential equations (ODEs) commonly encountered in engineering applications. Traditionally, these problems are solved either by reformulating them as first-order systems to apply one-step methods such as Runge–Kutta schemes, or by using direct second-order approaches widely adopted in linear dynamics, including the generalized-α, central difference, and Newmark methods. The proposed method is derived from a Taylor series expansion truncated at the third derivative, resulting in a fully explicit algorithm that requires only one function evaluation per time step. Similar to Newmark’s formulation, it includes adjustable parameters that allow the user to balance accuracy and stability. For a specific parameter choice, the method exhibits convergence and stability properties comparable to those of the central difference scheme. An important advantage is that it remains explicit even when nonlinearities depend on first-derivative terms. The paper presents a theoretical analysis covering stability, local truncation error, spectral properties, numerical damping, and period elongation. The method is validated through four test cases from multibody dynamics, including linear and nonlinear problems. Results demonstrate that the Explicit Integration Grade 3 (EIG-3) method achieves accuracy comparable to existing explicit second-order integrators while significantly reducing computational cost, particularly in nonlinear applications. Full article

This paper presents a systematic application of the Homotopy Perturbation Method (HPM) to the nonlinear static analysis of cantilever beams subjected simultaneously to three coplanar follower loads: an axial force H, a transverse force V, and a bending moment M₁. The studied configuration introduces complex mathematical self-coupling, as the bending moment depends on the solution of the differential equation even in its boundary conditions (γ₁), transforming the problem into a nonlinear one that is resistant to standard analytical methods. The primary methodological contribution of this work is the successful extension of the HPM framework to treat, within a unified mathematical formalism, this complete loading case, which has practical applications in compliant mechanisms, micro-electromechanical systems (MEMSs), and auxetic structures. The paper provides a complete mathematical formulation and explicit derivation of the HPM solution terms up to the third order and a rigorous demonstration of the method’s convergence, with quantitative error estimates and the establishment of a practical domain of validity, γ₁ < 30°, for an accuracy below 0.5%. As a direct consequence of this analytical advancement, we derive a series of practical engineering tools: nomograms, simplified empirical formulas, interaction diagrams, and a systematic six-step design procedure, which includes an adaptive algorithm for selecting the auxiliary parameter η to optimize convergence. The solution’s structure also lends itself to AI-based optimization frameworks, demonstrating how HPM solutions can serve as a foundation for machine learning surrogates and automated multi-objective optimizations. HPM proves to be a robust and efficient alternative, providing semi-analytical solutions in the form of convergent series without requiring an explicitly small physical parameter. This enables a direct parametric understanding of the structural response and offers rapid tools for the conceptual and preliminary sizing phases, thereby complementing the intensive numerical methods used in the final design stages. Full article

The thermal vibrations of a thick-walled functionally graded material (FGM) plate–cylindrical shells in unsteady supersonic flow with a Navier–Stokes equation analytical–numerical flow model and third-order shear deformation theory (TSDT) displacement models are investigated. The aerodynamic pressure load can be provided by using the Navier–Stokes equation analytical–numerical flow model. The data regarding the effect of the aerodynamic pressure load and TSDT model of the motion equation on the thermal stress and displacement of the FGM plate–cylindrical shells in unsteady supersonic flow are calculated with the generalized differential quadrature (GDQ) method. The Navier–Stokes equation analytical–numerical flow model, TSDT model, and advanced shear correction coefficient provide an additional effect on the values of displacement and stress. Full article

The Burgers–Huxley equation is a nonlinear partial differential equation that incorporates convective, diffusive and reactive effects and arises in various reaction–diffusion and fluid flow models. In this paper, a numerical method based on the method of lines is proposed for its solution. The spatial derivatives are approximated using a third-order finite difference scheme, which converts the governing partial differential equation into a system of ordinary differential equations. The resulting semi-discrete system is solved in time using the classical fourth-order Runge–Kutta method. The stability and convergence properties of the proposed scheme are analyzed to establish its numerical reliability. Several numerical experiments are presented to illustrate the accuracy and efficiency of the method. The computed results confirm that the proposed approach provides accurate and stable solutions for the Burgers–Huxley equation. Full article

With the rapid advancement of deepfake generation technologies, forged videos have become increasingly realistic in visual quality and temporal consistency, posing serious threats to multimedia security. Existing detection methods often struggle to effectively model temporal dynamics and capture subtle inter-frame anomalies. To address these challenges, we propose a Stochastic Differential Equation and Quantum Uncertainty Network (SDEQ-Net), a novel deepfake video anomaly detection framework that integrates continuous time stochastic modeling with quantum uncertainty mechanisms. First, a Continuous Time Neural Stochastic Differential Filtering Module (CNSDFM) is introduced to characterize the continuous evolution of latent inter-frame states using neural stochastic differential equations, enabling robust temporal filtering and uncertainty estimation. Second, a Quantum Uncertainty Aware Fusion Module (QUAFM) incorporates Hermitian-symmetric density matrix representations and von Neumann entropy to enhance feature fusion under uncertainty, leveraging the mathematical symmetry properties of quantum state representations for principled uncertainty quantification. Third, a Fractional Order Temporal Anomaly Detection Module (FOTADM) is proposed to generate fine grained temporal anomaly scores based on fractional order residuals, which are used as dynamic weights to guide attention toward anomalous frames. Extensive experiments on three benchmark datasets, including FaceForensics++, Celeb-DF, and DFDC, demonstrate the effectiveness of the proposed method. SDEQ-Net achieves AUC scores of 99.81% on FF++ (c23) and 97.91% on FF++ (c40). In cross dataset evaluations, it obtains 89.55% AUC on Celeb-DF and 86.21% AUC on DFDC, consistently outperforming existing state-of-the-art methods in both detection accuracy and generalization capability. Full article

In this work, we develop a disturbance suppression-oriented fuzzy sliding mode secured sampled-data controller for third-order parabolic partial differential equations that ought to cope with nonlinearities, hybrid cyber attacks, and modeled disturbances. This endeavor is mainly driven by formulating an observer model with a T–S fuzzy mode of execution that retrieves the latent state variables of the perceived system. Progressing onward, the disturbance observers are formulated to estimate the modeled disturbances emerging from the exogenous systems. In due course, the information received from the system and disturbance estimators, coupled with the sliding surface, is compiled to fabricate the developed controller. Furthermore, in the realm of security, hybrid cyber attacks are scrutinized through the use of stochastic variables that abide by the Bernoulli distributed white sequence, which combat their unpredictability. Proceeding further in this framework, a set of linear matrix inequality conditions is established that relies on the Lyapunov stability theory. Precisely, the refined looped Lyapunov–Krasovskii functional paradigm, which reflects in the sampling period that is intricately split into non-uniform intervals by leveraging a fractional-order parameter, is deployed. In line with this pursuit, a strictly $({\Phi }_1,{\Phi }_2,{\Phi }_3)-\varrho$ dissipative framework is crafted with the intent to curb norm-bounded disturbances. A simulation-backed numerical example is unveiled in the closing segment to underscore the potency and efficacy of the developed control design technique. Full article

We investigate the Lagrange formulation for the one-dimensional Saint Venant–Exner system. The system describes shallow-water equations with a bed evolution, for which the bedload sediment flux depends on the velocity, $Q\left(t,x\right)=A_g{u}^m,m\ge 1$. In terms of the Lagrange variables, the nonlinear hyperbolic system is reduced to one master third-order nonlinear partial differential equation. We employ Lie’s theory and find the Lie symmetry algebra of this equation. It was found that for an arbitrary parameter m, the master equation possesses four Lie symmetries. However, for $m=3$, there exists an additional symmetry vector. We calculate a one-dimensional optimal system for the Lie algebra of the equation. We apply the latter for the derivation of invariant functions. The invariants are used to reduce the number of the independent variables and write the master equation into an ordinary differential equation. The latter provides similarity solutions. Finally, we show that the traveling-wave reductions lead to nonlinear maximally symmetric equations which can be linearized. The analytic solution in this case is expressed in closed-form algebraic form. Full article

Several iterative integro-differential formulations for two-point, second- and third-order, nonlinear, boundary-value problems of ordinary differential equations based on Green’s functions and the method of variation of parameters are presented. It is shown that the generalized or dual Lagrange multiplier method (GVIM) previously developed for the iterative solution of nonlinear, boundary-value problems of ordinary differential equations that makes use of modified functionals and two Lagrange multipliers, is nothing but an iterative Green’s function formulation that does not require Lagrange multipliers at all. It is also shown that the two Lagrange multipliers of GVIM are associated with the left and right Green’s functions. The convergence of iterative methods based on both the Green function and the method of variation of parameters is proven for nonlinear functions that depend on the dependent variable and is illustrated by means of two examples. Several new iterative integro-differential formulations based on Green’s functions that use a multiplicative function for convergence acceleration are also presented. Full article

In the current study, we used Chebyshev’s Pseudospectral Method (CPM), a novel numerical technique, to solve nonlinear third-order Emden–Fowler delay differential (EF-DD) equations numerically. Fractional derivatives are defined by the Caputo operator. These kinds of equations are transformed to the linear or nonlinear algebraic equations by the proposed approach. The numerical outcomes demonstrate the precision and efficiency of the suggested approach. The error analysis shows that the current method is more accurate than any other numerical method currently available. The computational analysis fully confirms the compatibility of the suggested strategy, as demonstrated by a few numerical examples. We present the outcome of the offered method in tables form, which confirms the appropriateness at each point. Additionally, the outcomes of the offered method at various non-integer orders are investigated, demonstrating that the result approaches closer to the accurate solution as a value approaches from non-integer order to an integer order. Additionally, the current study proves some helpful theorems about the convergence and error analysis related to the aforementioned technique. A suggested algorithm can effectively be used to solve other physical issues. Full article

This article presents an algorithm for solving the direct and inverse problem for a model consisting of a fractional differential equation with non-integer order derivatives with respect to time and space. The Caputo derivative was taken as the fractional derivative with respect to time, and the Riemann–Liouville derivative in the case of space. On one of the boundaries of the considered domain, a fractional boundary condition of the third kind was adopted. In the case of the direct problem, a differential scheme was presented, and a metaheuristic optimization algorithm, namely the Group Teaching Optimization Algorithm (GTOA), was used to solve the inverse problem. The article presents numerical examples illustrating the operation of the proposed methods. In the case of inverse problem, a function occurring in the fractional boundary condition was identified. The presented approach can be an effective tool for modeling the anomalous diffusion phenomenon. Full article

In this paper, we study the time-space truncated M-fractional model of shallow water waves in a weakly nonlinear dispersive media that characterizes the nano-solitons of ionic wave propagation along microtubules in living cells. A fractional wave transformation is applied, reducing the model to a third-order differential equation formulated as a conservative Hamiltonian system. The stability of the equilibrium points is analyzed, and the corresponding phase portraits are constructed, providing valuable insights into the expected types of solutions. Utilizing the dynamical systems approach, a variety of predicted exact fractional solutions are successfully derived, including solitary, periodic and unbounded singular solutions. One of the most notable features of this approach is its ability to identify the real propagation regions of the desired waves from both physical and mathematical perspectives. The impacts of the fractional order and gravitational force variations on the solution profiles are systematically analyzed and graphically illustrated. Moreover, the quasi-periodic dynamics and chaotic behavior of the model are explored. Full article

The paper deals with the results of a study of a third-order nonlinear differential equation with moving singular points and critical poles. So far, this type of equation cannot be solved in quadratures. The development of the author’s approach in proving the theorem of the existence of moving singular points and solutions in the vicinity of a critical pole, based on a modified Cauchy majorant method, is given. An analytical approximate solution in the vicinity of a moving singular point is obtained, and an expression for the a priori error estimate is presented. A numerical experiment confirming the obtained theoretical results is provided. Full article