Search Results (255)

This extensive study introduces the Rosenbrock method (RosM) for numerically integrating the chaotic Lorenz system, with a focus on its ability to preserve the system’s intrinsic dynamical and structural symmetries. The Lorenz system exhibits significant symmetry, most notably an inversion symmetry $(x,y,z)⟶(-x,-y,z)$, which is a fundamental feature of its chaotic attractor. We lay forth the algorithm and, after systematic comparisons to explicit Runge–Kutta higher-order schemes and semi-analytically obtained solutions, show that the second-order Rosenbrock method performs with excellent accuracy and stability. Crucially, we demonstrate that RosM reliably preserves the system’s symmetry over long-term integration, a property where some explicit methods can exhibit subtle drift. We give a formal error characterization, assess the computational efficiency, and verify the method via bifurcation analysis to support that RosM is a robust and symmetry-aware tool for simulating chaotic systems. Full article

A variety of methods that provide approximate piecewise- analytical solutions to initial-value problems governed by scalar, nonlinear, first-order, ordinary differential equations is presented. The methods are based on fixing the independent variable in the right-hand side of these equations and approximating the resulting term by either its first- or second-order Taylor series expansion. It is shown that the second-order Taylor series approximation results in Riccati equations with constant coefficients, whereas the first-order one results in first-order, linear, ordinary differential equations. Both approximations are shown to result in explicit finite difference equations that are unconditionally linearly stable, and their local truncation errors are determined. It is shown that, for three of the nonlinear, first-order, ordinary differential equations studied in this paper that are characterized by growing or decaying solutions, as well as by solutions that first grow and then decrease, a second-order Taylor series expansion of the right-hand side of the differential equation evaluated at each interval’s midpoint results in the most accurate method; however, the accuracy of this method degrades substantially for problems that exhibit either blowup in finite time or quadratic approximations characterized by a negative radicand. It is also shown that methods based on either first- or second-order Taylor series expansion of the right-hand side of the differential equation evaluated at either the left or the right points of each interval have similar accuracy, except for one of the examples that exhibits blowup in finite time. It is also shown that both the linear and the quadratic approximation methods that use the midpoint for the independent variable in each interval exhibits the same trends as and have errors comparable to the second-order trapezoidal technique. Full article

We present an analytical and numerical study of nonlinear wave localization in a one-dimensional rhombic (diamond) waveguide array that combines forward- and backward-propagating channels. This mixed-index configuration, realizable through Bragg-type couplers or corrugated waveguides, produces a tunable spectral gap and supports nonlinear self-localized states in both transmission and forbidden-band regimes. Starting from the full set of coupled-mode equations, we derive the effective evolution model, identify the role of coupling asymmetry and nonlinear coefficients, and obtain explicit soliton solutions using the method of multiple scales. The resulting envelopes satisfy a nonlinear Schrödinger equation with an effective nonlinear parameter $\theta$, which determines the conditions for soliton existence ($\theta >0$) for various combinations of focusing and defocusing nonlinearities. We distinguish solitons formed outside and inside the bandgap and analyze their dependence on the dispersion curvature and nonlinear response. Direct numerical simulations confirm the analytical predictions and reveal robust propagation and interactions of counter-propagating soliton modes. Order-of-magnitude estimates show that the predicted effects are accessible in realistic integrated photonic platforms. These results provide a unified theoretical framework for soliton formation in mixed-index lattices and suggest feasible routes for realizing controllable nonlinear localization in Bragg-type photonic structures. Full article

To address the difficulty in obtaining analytical solutions for the torsional vibration response of pile foundations in orthotropic layered soil foundations subjected to torsional excitation at the pile top, this study investigates a layered recursive algorithm based on the Hankel transform. An integral transformation method is employed to reduce the dimensionality of the coupled pile–soil torsional vibration equations, converting the three-dimensional system of partial differential equations into a set of ordinary differential equations. Combining the constitutive properties of transversely anisotropic strata with interlayer contact conditions, a transfer matrix model is established. Employing inverse transformation coupled with the Gauss–Kronrod integration method, an explicit frequency-domain solution for the torsional dynamic impedance at the pile top is derived. The research findings indicate that the anisotropy coefficient of the foundation significantly influences both the real and imaginary parts of the impedance magnitude. The sequence of soil layer distribution and the bonding state at interfaces jointly affect the nonlinear transmission characteristics of torque along the pile shaft. Full article

This study employs a finite element approach to investigate wellbore stability in interbedded weak formations, such as unconsolidated layers, with a focus on the failure-tendency method, which is derived according to the principle of Mohr–Coulomb theory. The numerical model is successfully verified through analytical solutions for stress distributions around a borehole. Through finite element modeling, the method captures critical shear failure thresholds, exemplifying how variations in horizontal stress anisotropy, orientation of interbedded weak layers, and mechanical properties of layered geological formations impact wellbore stability in stratified formations. Results indicate that the potential unstable regions, aligned in the direction of minimum principal stress, and the range of unstable regions gradually enlarge as the internal cohesive strength decreases. By modeling heterogeneous rock sequences with explicit representation of interbedded weak layers and stress anisotropy, the analysis reveals that interbedded weak layers are prone to shear-driven borehole breakouts due to stress redistribution and relatively lower internal cohesive strength. As compressive stresses concentrate at interfaces between stiff and compliant layers, breakouts are induced at those weak layers along the interfaces; this type of failure is also manifested through a field borehole breakout observation. Simulation results reveal the significant influences of the mechanical properties of layered formations and in situ stress on the distribution of instability regions around a borehole. The study underscores the necessity of layer-specific geomechanical models to predict shear failure in complex layered geological formations and offers insights for optimizing drilling parameters to enhance wellbore stability in anisotropic, stratified subsurface environments. Full article

Neural Networks (NNs) have been used in many areas with great success. When an NN’s structure (model) is given, during the training steps, the parameters of the model are determined using an appropriate criterion and an optimization algorithm (training). Then, the trained model can be used for the prediction or inference step (testing). As there are also many hyperparameters related to optimization criteria and optimization algorithms, a validation step is necessary before the NN’s final use. One of the great difficulties is the choice of NN structure. Even if there are many “on the shelf” networks, selecting or proposing a new appropriate network for a given data signal or image processing task, is still an open problem. In this work, we consider this problem using model-based signal and image processing and inverse problems methods. We classify the methods into five classes: (i) explicit analytical solutions, (ii) transform domain decomposition, (iii) operator decomposition, (iv) unfolding optimization algorithms, (v) physics-informed NN methods (PINNs). A few examples in each category are explained. Full article

We derive an analytical solution to the one-dimensional linearized Boussinesq equation with mixed boundary conditions (Dirichlet–Neumann), formulated to describe drainage in porous media. The solution is obtained via the unified transform method (Fokas method), extending its previous applications in infiltration problems and illustrating its utility in soil hydrology. An explicit integral representation is constructed, considering different types of initial conditions. Numerical examples are presented to demonstrate the accuracy of the solution, with direct comparisons to the classical Fourier series approach. Full article

This work investigates the use of physics-informed neural networks (PINNs) for solving representative classes of differential and integro-differential equations, including the Burgers, Poisson, and Volterra equations. The examples presented are chosen to address both symmetric and asymmetric domains. PINNs integrate prior physical knowledge with the approximation capabilities of neural networks, allowing the modeling of physical phenomena without explicit domain discretization. In addition to evaluating accuracy against analytical solutions (where available) and established numerical methods, the study systematically examines the impact of key hyperparameters—such as the number of hidden layers, neurons per layer, and training points—on solution quality and stability. The impact of a symmetric domain on solution speed is also analyzed. The experimental results highlight the strengths and limitations of PINNs and provide practical guidelines for their effective application as an alternative or complement to traditional computational approaches. Full article

This paper investigates the global dynamics of a broad class of nonlinear rational difference equations given by $x_{n+1}={\displaystyle \frac{a{x}_{n+1-2k}}{b+c{x}_{n+1-k}{x}_{n+1-2k}}},n=0,1,\dots ,$ which generalizes several known models in the literature. We establish the existence of exactly three equilibrium points and show that the trivial equilibrium is globally asymptotically stable when the parameter ratio $\alpha =(b/a)$ lies in $(-1,1)$. The nontrivial equilibria are shown to be always unstable. An explicit general solution is derived, enabling a detailed analysis of solution behavior in terms of initial conditions and parameters. Furthermore, we identify and classify minimal period $2k$ and $4k$ solutions, providing necessary and sufficient conditions for the occurrence of constant and periodic behaviors. These analytical results are supported by numerical simulations, confirming the theoretical predictions. The findings generalize and refine existing results by offering a unified framework for analyzing a wide class of rational difference equations. Full article

This article investigates a mathematical model with the Caputo derivative for the transient unidirectional flow of an incompressible viscous fluid with pressure-dependent viscosity. The fluid flows in the spatial domain bounded by two parallel plates extended to infinity. The plates translate in their planes with time-dependent velocities, and the fluid adheres to the solid boundaries. The generalization of the model consists of formulating a fractional constitutive equation to introduce the memory effect into the mathematical model. In addition, the fluid’s viscosity is assumed to be pressure-dependent. More precisely, in this article, the viscosity is considered a power function of the vertical coordinate of the channel. Analytic solutions of the dimensionless initial and boundary value problems have been determined using the Laplace transform and Bessel equations. The inversion of Laplace transforms is conducted using both the methods of complex analysis and the Stehfest numerical algorithm. In addition, we discuss the explicit solution in some meaningful particular cases. Using numerical simulations and graphical representations, the results of the ordinary model $(\alpha =1)$ are compared with those of the fractional model $(0<\alpha <1)$, highlighting the influence of the memory parameter on fluid behavior. Full article

The conditional Feynman integral provides solutions to integral equations equivalent to heat and Schrödinger equations. The Cameron–Martin translation theorem illustrates how the Wiener measure changes under translation via Cameron–Martin space elements in abstract Wiener space. Translation theorems for analytic Feynman integrals have been established in many research articles. This study aims to present a translation theorem for the conditional function space integral of functionals on the generalized Wiener space $C_{a,b}[0,T]$ induced via a generalized Brownian motion process determined using continuous functions $a\left(t\right)$ and $b\left(t\right)$. As an application, we establish a translation theorem for the conditional generalized analytic Feynman integral of functionals on $C_{a,b}[0,T]$. We then provide explicit examples of functionals on $C_{a,b}[0,T]$ to which the conditional translation theorem on $C_{a,b}[0,T]$ can be applied. Our formulas and results are more complicated than the corresponding formulas and results in the previous research on the Wiener space $C_0[0,T]$ because the generalized Brownian motion process used in this study is neither stationary in time nor centered. In this study, the stochastic process used is subject to a drift function. Full article

In this paper, we develop the explicit finite difference method (FDM) to solve an ill-posed Cauchy problem for the 3D acoustic wave equation in a time domain with the data on a part of the boundary given (continuation problem) in a cube. FDM is one of the numerical methods used to compute the solutions of hyperbolic partial differential equations (PDEs) by discretizing the given domain into a finite number of regions and a consequent reduction in given PDEs into a system of linear algebraic equations (SLAE). We present a theory, and through Matlab Version: 9.14.0.2286388 (R2023a), we find an efficient solution of a dense system of equations by implementing the numerical solution of this approach using several iterative techniques. We extend the formulation of the Jacobi, Gauss–Seidel, and successive over-relaxation (SOR) iterative methods in solving the linear system for computational efficiency and for the properties of the convergence of the proposed method. Numerical experiments are conducted, and we compare the analytical solution and numerical solution for different time phenomena. Full article

Padé approximants are computational tools customarily employed for resumming divergent Stieltjes series. However, they become ineffective or even fail when applied to Stieltjes series whose moments do not satisfy the Carleman condition. Differently from Padé, Levin-type transformations incorporate important structural information on the converging factors of a typical Stieltjes series. For example, the computational superiority of Weniger’s $\delta$-transformation over Wynn’s epsilon algorithm is ultimately based on the fact that Stieltjes series converging factors can always be represented as inverse factorial series. In the present paper, the converging factors of an important class of superfactorially divergent Stieltjes series are investigated via an algorithm developed one year ago from the first-order difference equation satisfied by the Stieltjes series converging factors. Our analysis includes the analytical derivation of the inverse factorial representation of the moment ratio sequence of the series under investigation, and demonstrates the numerical effectiveness of our algorithm, together with its implementation ease. Moreover, a new perspective on the converging factor representation problem is also proposed by reducing the recurrence relation to a linear Cauchy problem whose explicit solution is provided via Faà di Bruno’s formula and Bell’s polynomials. Full article

The initial part of this study fills a notable research gap by investigating the substantial impact of numerical diffusion errors from different schemes on sloshing tank models. Multiple numerical models were developed: first- and higher-order upwind schemes equipped with precise wall treatment using ghost nodes, MacCormack and central methods that are explicit second-order finite difference methods, and Preissmann and staggered methods employed in full-implicit and semi-implicit modes. Furthermore, the separation of variables technique was proposed for simulating sloshing tanks and deriving an analytical equation for the tank’s natural period. An analytical solution to the perturbation was employed to examine the numerical diffusion of the schemes. Subsequently, two sloshing tests, resonant and near-resonant excitations, were employed to determine the numerical diffusion and calibrate the physical diffusion coefficients, respectively. Finally, an efficient and accurate numerical scheme was applied to a linear shallow water model including physical diffusion and coupled with a single degree of freedom (SDOF), to simulate tuned liquid dampers (TLDs). It shows that the efficiency of TLD is associated with a compact domain around resonance excitation. Contrary to SDOF alone, when SDOF interacts with TLD the impact of structural damping on reducing the response is minimal in resonance excitation. Full article

In the era of big-data-driven multi-platform and multimodal health information dissemination, the rapid spread of false and misleading content poses a critical threat to public health awareness and decision making. To address this issue, a dual-stream Transformer-based multimodal health misinformation detection framework is presented, incorporating a symbol drift detection module, a symbol-aware text graph neural network, and a crossmodal alignment fusion module. The framework enables precise identification of implicit misleading health-related symbols, comprehensive modeling of textual dependency structures, and robust detection of crossmodal semantic conflicts. A domain-specific health-symbol-sensitive lexicon is constructed, and contextual drift intensity is quantitatively measured and embedded as explicit features into the text GNN. Bidirectional cross-attention and contrastive learning are further employed to enhance crossmodal semantic alignment. Extensive experiments on a large-scale real-world multimodal health information dataset, encompassing heterogeneous data sources typical of big data environments, demonstrate that the proposed method consistently outperforms state-of-the-art baselines in CTR prediction, multimodal recommendation, and ranking tasks. The results indicate substantial improvements in both accuracy and ranking quality, while ablation studies further verify the contributions of symbol drift modeling, graph-structured representation, and crossmodal fusion. Overall, the proposed approach advances big data analytics for multimodal misinformation detection and provides an interpretable and scalable solution for public health communication governance. Full article