Search Results (26)

This paper is concerned with the asymptotic behavior of a viscoelastic wave equation involving distributed delay, logarithmic nonlinearity and dynamic Wentzell boundary conditions. In general, when the memory kernel function $g\left(t\right)$ is monotonically decreasing, the system energy’s decay is similar to that of the kernel function. However, this work addresses the case where the kernel function does not necessarily decay; thus, at this point, whether the system energy can still decay, and especially maintain exponential decay, is a very interesting question. Assuming that the kernel function is not necessarily decreasing, which means that it may oscillate, under some proper conditions, utilizing the Lyapunov functional method and constructing auxiliary functions, an exponential decay result is attained. To some extent, the result extends and improves several earlier related results in the literature. Full article

Within the framework of Somigliana’s displacement and traction identities, we propose an extended equivalent elastic model that enables a BEM that is singularity-free in the primary solution stage for two-dimensional elastostatics. The central idea is to shift the integration boundary from the physical contour $S_1$ to an auxiliary contour $S_2$, introducing a geometric separation that removes boundary-source singularities from the discrete system. When the separation between $S_1$ and $S_2$ is sufficiently large, all integrals in the assembled algebraic equations become regular, and post-processing can be performed in a unified manner using the same nonsingular expressions for both boundary and interior evaluation. We introduce a practical separation measure, the dimensionless parameter $\delta$, and verify that a moderate choice (e.g., $\delta \approx 0.5$) is effective through a rigid-body displacement test. Benchmark examples demonstrate that, at lower computational cost, the proposed method improves accuracy and convergence compared with the conventional direct BEM (displacement boundary integral equation (BIE) with free-term coefficient $c=1/2$) and compares favorably with the finite element method (FEM). In particular, thin structures can be treated directly without invoking plate or shell theories. Full article

Fractional differential equations provide a flexible framework for describing evolutionary processes in complex media, where nonlocality and memory effects play central roles, and classical integer-order models are frequently inadequate to capture these behaviors. In this work, we revisit the time-fractional Harry Dym (HD) evolution equation in the Caputo sense and construct high-precision analytical approximations using the recently developed Tantawy technique (TT). The method generates a rapidly convergent fractional-power series in time without resorting to perturbative assumptions, auxiliary decomposition polynomials, linearization procedures, or integral transforms, and it remains computationally economical even at high approximation orders. Closed, compact expressions are derived up to the fifth-order approximation and can be systematically extended, yielding excellent agreement with the known exact solution of the classical/integer HD model and with approximations obtained via the new iterative method. A detailed error analysis is carried out by computing absolute and maximum residual errors over the entire computational domain, demonstrating the accuracy, stability, and robustness of the TT for the HD-type fractional nonlinear evolution equation. From a physical perspective, the proposed framework offers a reliable tool for modeling nonlinear wave structures in dispersive media with significant memory and, more generally, for treating a broad class of fractional nonlinear wave equations arising in physics and engineering. Full article

This paper focuses on a high-order Korteweg–de Vries wave equation. The extended F-expansion method, a modified form of Kudryashov’s auxiliary equation approach, is employed to construct Jacobi elliptic function solutions for this equation. Three distinct families of solutions are obtained, including solitary waves, breathers, dark/bright solitons, bright–dark interaction solitons, and rogue-like solutions. To better illustrate the complex nonlinear dynamics of the high-order Korteweg–de Vries wave equation, representative solutions are selected, and their moduli are visualized using Maple software through three-dimensional, two-dimensional, and contour plots. Full article

This paper presents a reduced-order Legendre–Galerkin extrapolation (ROLGE) method combined with the scalar auxiliary variable (SAV) approach (ROLGE-SAV) to numerically solve the time-fractional Allen–Cahn equation (tFAC). First, the nonlinear term is linearized via the SAV method, and the linearized system derived from this SAV-based linearization is time-discretized using the shifted fractional trapezoidal rule (SFTR), resulting in a semi-discrete unconditionally stable scheme (SFTR-SAV). The scheme is then fully discretized by incorporating Legendre–Galerkin (LG) spatial discretization. To enhance computational efficiency, a proper orthogonal decomposition (POD) basis is constructed from a small set of snapshots of the fully discrete solutions on an initial short time interval. A reduced-order LG extrapolation SFTR-SAV model (ROLGE-SFTR-SAV) is then implemented over a subsequent extended time interval, thereby avoiding redundant computations. Theoretical analysis establishes the stability of the reduced-order scheme and provides its error estimates. Numerical experiments validate the effectiveness of the proposed method and the correctness of the theoretical results. Full article

In this contribution, we propose and study long-time behaviors of a new class of N-dimensional delayed Kirchhoff–Carrier-type problems with variable transfer coefficients involving a logarithmic nonlinearity. We take into account the dependence of diffusion and damping coefficients on the position and direction, as well as the presence of different types of delays. This class of nonlocal anisotropic and nonlinear wave-type equations with multiple time-varying mixed delays and dampings, of a fairly general form, containing several arbitrary functions and free parameters, is of the following form: ${\displaystyle \frac{{\partial }^{2}u}{\partial t^2}-{div(\mathcal{K}(\Vert \sigma \nabla u\Vert }_{L^2(\Omega )}^2)A_{\sigma }{(\mathbf{x})\nabla u)+\mathcal{M}(\Vert u\Vert }_{L^2(\Omega )}^2)u-div(\zeta (t)A_{\sigma }(\mathbf{x})\nabla \frac{\partial u}{\partial t})+d_0(t)\frac{\partial u}{\partial t}+{\mathbb{D}}_r(\mathbf{x},t;\frac{\partial u}{\partial t})=\mathbb{G}(u),}$ where $u(\mathbf{x},t)$ is the state function, $\mathcal{M}$ and $\mathcal{K}$ are the nonlocal Kirchhoff operators and the nonlinear operator $\mathbb{G}(u)$ corresponds to a logarithmic source term. The symmetric tensor $A_{\sigma }$ describes the anisotropic behavior and processes of the system, and the operator ${\mathbb{D}}_r$ represents the multiple time-varying mixed delays related to velocity ${\displaystyle \frac{\partial u}{\partial t}}$. Our problem, which encompasses numerous equations already studied in the literature, is relevant to a wide range of practical and concrete applications. It not only considers anisotropy in diffusion, but it also assumes that the strong damping can be totally anisotropic (a phenomenon that has received very little mathematical attention in the literature). We begin with the reformulation of the problem into a nonlinear system coupling a nonlocal wave-type equation with ordinary differential equations, with the help of auxiliary functions. Afterward, we study the local existence and some necessary regularity results of the solutions by using the Faedo–Galerkin approximation, combining some energy estimates and the logarithmic Sobolev inequality. Next, by virtue of the potential well method combined with the Nehari manifold, conditions for global in-time existence are given. Finally, subject to certain conditions, the exponential decay of global solutions is established by applying a perturbed energy method. Many of the obtained results can be extended to the case of other nonlinear source terms. Full article

Periodic solutions of high-order nonlinear differential equations are fundamental in dynamical systems, yet they remain challenging to establish with traditional methods. This paper addresses the existence of periodic solutions in general $2n$-order autonomous and nonautonomous ordinary differential equations. By extending Carathéodory’s variational technique from the calculus of variations, we reformulate the original periodic solution problem as an equivalent higher-order variational problem. The approach constructs a convex function and introduces an auxiliary transformation to enforce convexity in the highest-order term, enabling a tractable operator-theoretic analysis. Within this framework, we prove two main theorems that provide sufficient conditions for periodic solutions in both autonomous and nonautonomous cases. These results generalize the known theory for second-order equations to arbitrary higher-order systems and highlight a connection to the Hamilton–Jacobi theory, offering new insights into the underlying variational structure. Finally, numerical examples validate our theoretical results by confirming the periodic solutions predicted by the theory and demonstrating the approach’s practical applicability. Full article

Empirically based approaches, like the Universal Soil Loss Equation (USLE), are appropriate for estimating mass movement attributed to rill erosion. USLE and its associates become widespread even in spatially extended studies in spite of its original plot-level concept, as well as with certain constraints on the supply of suitable input spatial data. At the same time, there is a continuously expanding opportunity and offer for the application of remote sensing (RS) imagery together with machine learning (ML) techniques to model and monitor various environmental processes utilizing their versatile benefits. The present study focused on the applicability of data-driven geospatial models for predicting soil erosion in three vineyards in the Upper Pannon Wine Region, Central Europe, considering the seasonal variation in influencing factors. Soil loss was formerly modeled by USLE, thus providing non-observation-based reference datasets for the calibration of parcel-specific prediction models using various ML methods (Random Forest, eXtreme Gradient Boosting, Regularized Support Vector Machine with Linear Kernel), which is a well-established approach in digital soil mapping (DSM). Predictions used spatially exhaustive, auxiliary, and environmental covariables. RS data were represented by multi-temporal Sentinel-2 satellite imagery data, which were supplemented by (i) topographic covariates derived from a UAV-based digital surface model and (ii) digital primary soil property maps. In addition to spatially quantifying soil erosion, the feasibility of transferring the inferred models between nearby vineyards was tested with ambiguous outcomes. Our results indicate that ML models can feasibly replace the empirical USLE model for erosion prediction. However, further research is needed to assess model transferability even to nearby parcels. Full article

This paper investigates the asymptotic eventual stability (AE-S) for nonlinear impulsive Caputo fractional differential equations (ICFDEs) with fixed impulse moments, employing auxiliary Lyapunov functions (ALF) which are specifically constructed as analogues of vector Lyapunov functions (VLF). A novel derivative tailored for VLF is introduced, offering a more robust framework than existing approaches based on scalar Lyapunov functions (SLF). Adequate conditions for AE-S involving ICFDEs are provided. We also used the predictor corrector method to implement a numerical solution for a given impulsive Caputo fractional differential equation. These findings extend and improve upon existing results, providing significant advancements in the stability analysis of systems with memory effects and impulsive dynamics. The study holds practical relevance for modeling and analyzing real-world systems, including control processes, biological systems, and economic dynamics where fractional-order behavior and impulses play a crucial role. Full article

Modern mechanical engineering faces high competition in global markets, which requires manufacturers of process equipment to significantly reduce production costs while ensuring high product quality and maximum productivity. Metalworking occupies a significant part of industrial production and consumes a significant share of the world’s energy and natural resources. Improving the technology of manufacturing parts with an emphasis on more efficient use of metalworking machines is necessary to maintain the competitiveness of the domestic machine tool industry. Hybrid metalworking systems based on the principles of multi-purpose integration eliminate the disadvantages of monotechnologies and increase efficiency by reducing time losses and intermediate operations. The purpose of this work is to develop and implement a hybrid machine tool system and an appropriate combined technology for manufacturing machine parts. Theory and methods. Studies of the possible structural composition and layout of hybrid equipment at integration of mechanical and surface-thermal processes were carried out, taking into account the basic provisions of structural synthesis and componentization of metalworking systems. Theoretical studies were carried out using the basic provisions of system analysis, geometric theory of surface formation, design of metalworking machines, methods of finite elements, and mathematical and computer modeling. The mathematical modeling of thermal fields and structural-phase transformations during HEH HFC was carried out in ANSYS (version 19.1) and SYSWELD (version 2010) software packages using numerical methods of solving differential equations of unsteady heat conduction (Fourier equation), carbon diffusion (2nd Fick’s law) and elastic–plastic behavior of the material. The verification of the modeling results was carried out using in situ experiments employing the following: optical and scanning microscopy; and mechanical and X-ray methods of residual stress determination. Formtracer SV-C4500 profilograph profilometer was used in the study for simultaneous measurement of shape deviations and surface roughness. Surface topography was assessed using a Walter UHL VMM 150 V instrumental microscope. The microhardness of the hardened surface layer of the parts was evaluated on a Wolpert Group 402MVD. Results and discussion. The original methodology of structural and kinematic analysis for pre-design studies of hybrid metalworking equipment is presented. Methodological recommendations for the modernization of multi-purpose metal-cutting machine tool are developed, the implementation of which will make it possible to implement high-energy heating with high-frequency currents (HEH HFC) on a standard machine tool system and provide the formation of knowledge-intensive technological equipment with extended functionality. The innovative moment of this work is the development of hybrid metalworking equipment with numerical control and writing a unique postprocessor to it, which allows to realize all functional possibilities of this machine system and the technology of combined processing as a whole. Special tooling and tools providing all the necessary requirements for the process of surface hardening of HEH HFC were designed and manufactured. The conducted complex of works and approbation of the technology of integrated processing in real conditions in comparison with traditional methods of construction of technological process of parts manufacturing allowed to obtain the following results: increase in the productivity of processing by 1.9 times; exclusion of possibility of scrap occurrence at finishing grinding; reduction in auxiliary and preparatory-tasking time; and reduction in inter-operational parts backlogs. Full article

Forest managers need inventory data and information to address sustainability concerns over extended temporal horizons. In situ information is usually derived from field data and computed using appropriate equations. Nonetheless, fieldwork is time-consuming and costly. Thus, new technologies like Light Detection and Ranging (LiDAR) have emerged as an alternative method for forest assessment. In this study, we evaluated the accuracy of geostatistical methods in predicting the Site Index (SI) using LiDAR metrics as auxiliary variables. Since primary variables, which were obtained from forestry inventory data, were used to calculate the SI, secondary variables obtained from LiDAR surveying were considered and multivariate kriging techniques were tested. The ordinary cokriging (CK) method outperformed the simple cokriging (SK) and Inverse Distance Weighted (IDW) methods, which was interpolated using only the primary variable. Aside from having fewer SI sample points, CK was proven to be a trustworthy interpolation method, minimizing interpolation errors due to the highly correlated auxiliary variables, highlighting the significance of the data’s spatial structure and autocorrelation in predicting forest stand attributes, such as the SI. CK increased the SI prediction accuracy by 36.6% for eucalyptus, 62% for maritime pine, 72% for pedunculate oak, and 43% for cork oak compared to IDW, outperforming this interpolation approach. Although cokriging modeling is challenging, it is an appealing alternative to non-spatial statistics for improving forest management sustainability since the results are unbiased and trustworthy, making the effort worthwhile when dense secondary variables are available. Full article

The Boiti–Leon–Mana–Pempinelli Equation (BLMPE) is an essential mathematical model describing wave propagation in incompressible fluid dynamics. In the present manuscript, a novel generalization of the BLMPE is introduced, called herein the functional BLMPE (F-BLMPE), which involves different functions, including exponential, logarithmic and monomaniacal functions. In these cases, the F-BLMPE reduces to an explicit form in the dependent variable. In addition to this, it is worth deriving approximate similarity solutions of the F-BLMPE with constant coefficients using the extended unified method (EUM). In this method, nonlinear partial differential equation (NLPDE) solutions are expressed in polynomial and rational forms through an auxiliary function (AF) with adequate auxiliary equations. Exact solutions are estimated using formal solutions substituted into the NLPDEs, and the coefficients of the AF of all powers are set equal to zero. This approach is valid when the NLPDE is integrable. However, this technique is not valid for non-integrable equations, and only approximate solutions can be found. The maximum error can be controlled by an adequate choice of the parameters in the residue terms (RTs). Multiple similarity solutions are derived, and the ME is depicted in various examples within this work. The results found here confirm that the EUM is an efficient method for solving NLPDEs of the F-BLMPE type. Full article

In this article, we investigate the utilization of Riemann–Liouville’s fractional integral and the Caputo derivative in the application of the Optimal Auxiliary Function Method (OAFM). The extended OAFM is employed to analyze fractional non-linear coupled ITO systems and non-linear KDV systems, which feature equations of a fractional order in time. We compare the results obtained for the ITO system with those derived from the Homotopy Perturbation Method (HPM) and the New Iterative Method (NIM), and for the KDV system with the Laplace Adomian Decomposition Method (LADM). OAFM demonstrates remarkable convergence with a single iteration, rendering it highly effective. In contrast to other existing analytical approaches, OAFM emerges as a dependable and efficient methodology, delivering high-precision solutions for intricate problems while saving both computational resources and time. Our results indicate superior accuracy with OAFM in comparison to HPM, NIM, and LADM. Additionally, we enhance the accuracy of OAFM through the introduction of supplementary auxiliary functions. Full article

The generalized Calogero–Bogoyavlenskii–Schiff equation (GCBSE) is examined and analyzed in this paper. It has several applications in plasma physics and soliton theory, where it forecasts the soliton wave propagation profiles. In order to obtain the analytically exact solitons, the model under consideration is a nonlinear partial differential equation that is turned into an ordinary differential equation by using the next traveling wave transformation. The new extended direct algebraic technique and the modified auxiliary equation method are applied to the generalized Calogero–Bogoyavlenskii–Schiff equation to get new solitary wave profiles. As a result, novel and generalized analytical wave solutions are acquired in which singular solutions, mixed singular solutions, mixed complex solitary shock solutions, mixed shock singular solutions, mixed periodic solutions, mixed trigonometric solutions, mixed hyperbolic solutions, and periodic solutions are included with numerous soliton families. The propagation of the acquired soliton solution is graphically presented in contour, two- and three-dimensional visualization by selecting appropriate parametric values. It is graphically demonstrated how wave number impacts the obtained traveling wave structures. Full article

In this paper, the method of transmutation operators is used to construct an exact solution of the Goursat problem for a fourth-order hyperbolic equation with a singular Bessel operator. We emphasise that in many other papers and monographs the fractional Erdélyi-Kober operators are used as integral operators, but our approach used them as transmutation operators with additional new properties and important applications. Specifically, it extends its properties and applications to singular differential equations, especially with Bessel-type operators. Using this operator, the problem under consideration is reduced to a similar problem without the Bessel operator. The resulting auxiliary problem is solved by the Riemann method. On this basis, an exact solution of the original problem is constructed and analyzed. Full article