Search Results (1,756)

This article presents a novel approach to looking at steady-state thermoelastic boundary value problems in isotropic elastic plates with curvilinear holes using a complex variable approach and rational conformal mappings. The physical domain with a non-circular opening is mapped conformally to the unit disk. A thermoelastic potential combines the temperature distribution, which is determined by the Laplace equation with Neumann boundary conditions. Gaursat functions, which are shown as truncated power series, show the complicated stress and displacement fields. They are found by putting boundary constraints at certain collocation points. This procedure presents us with a linear system that can be solved using the least squares method. The method is applied in an annular shape that is exposed to a radial temperature gradient. This experiment shows how changes at the boundary affect the distribution of stress. According to numerical simulations, stress distributions are more uniform when boundaries are smoother, but stress concentrations increase with the size of geometric disturbances. The suggested approach remarkably captures the way geometry and thermal effects interact in two-dimensional thermoelasticity. It is a reliable tool for researching intricate, heated elastic domains. Full article

The uneven expansion of renewable energy generation in different regions highlights the necessity of accurately assessing the available transfer capability (ATC) in power systems. This paper proposes a distributed probabilistic inter-regional ATC assessment framework. First, a spatiotemporally correlated wind power output model is established using wind speed forecast data and correlation matrices, enhancing the accuracy of wind power forecasting. Second, a two-stage probabilistic ATC assessment optimization model is proposed. The first stage minimizes both generation cost and risk-related costs by incorporating conditional value-at-risk (CVaR), while the second stage maximizes the power transaction amount. Thirdly, a privacy-preserving two-level iterative alternating direction method of multipliers (I-ADMM) algorithm is designed to solve this mixed-integer optimization problem, requiring only the exchange of boundary voltage phase angles between regions. Case studies are performed on the 12-bus, the IEEE 39-bus and the IEEE 118-bus systems to validate the proposed framework. Hence, the proposed framework enables more reliable and risk-aware intraday ATC evaluation for inter-regional power transactions. Moreover, the impacts of risk parameters and wind farm output correlations on ATC and generation cost are further investigated. Full article

In this paper, the boundary value problem of wave diffraction on a semi-infinite circular pin is solved using the Wiener–Hopf method with compensation of eigenmodes. The solution to the problem is presented as an infinite series defined by a recurrence formula. The reliability and accuracy of the solution are verified numerically in terms of fulfillment of the law of energy conservation. Sufficiently reliable results are obtained at the first iteration. The method used for solving this problem can be applied to solving diffraction problems on axisymmetric volumetric structures. Full article

Fractional differential equations have emerged as a prominent focus of modern scientific research due to their advantages in describing the complexity and nonlinear behavior of many physical phenomena. In particular, when considering problems with initial-boundary value conditions, the solution of nonlinear fractional differential equations becomes particularly important. This paper aims to explore the fractional soliton solutions for the three-component fractional nonlinear Schrödinger (TFNLS) equation under the zero background. According to the Lax pair and fractional recursion operator, we obtain fractional nonlinear equations with Riesz fractional derivatives, which ensure the integrability of these equations. In particular, by the completeness relation of squared eigenfunctions, we derive the explicit form of the TFNLS equation. Subsequently, in the reflectionless case, we construct the fractional N-soliton solutions via the Riemann–Hilbert (RH) method. The analysis results indicate that as the order of the Riesz fractional derivative increases, the widths of both one-soliton and two-soliton solutions gradually decrease. However, the absolute values of wave velocity, phase velocity, and group velocity of one component of the vector soliton exhibit an increasing trend, and show power-law relationships with the amplitude. Full article

The subject of this paper is the computational homogenisation and identification of heterogeneous materials in the form of auxetic structures made of materials with nonlinear characteristics. It is assumed that some of the material and topological parameters of the auxetic structures are uncertain and are modelled as interval numbers. Directed interval arithmetic is used to minimise the width of the resulting intervals. The finite element method is employed to solve the boundary value problem, and artificial neural network response surfaces are utilised to reduce the computational effort. In order to solve the identification task, the Pareto approach is adopted, and a multi-objective evolutionary algorithm is used as the global optimisation method. The results obtained from computational homogenisation under uncertainty demonstrate the efficacy of the proposed methodology in capturing material behaviour, thereby underscoring the significance of incorporating uncertainty into material properties. The identification results demonstrate the successful identification of material parameters at the microscopic scale from macroscopic data involving the interval description of the process of deformation of auxetic structures in a nonlinear regime. Full article

Under stringent flight constraint conditions, trajectory optimization of Hypersonic Vehicles (HV) is crucial for guidance. Therefore, this paper proposes an hp-adaptive dynamic trust region SCP method to address the problems of initial value sensitivity and low computational efficiency that arise in traditional Sequential Convex Programming (SCP) methods during HV trajectory optimization. First, the discrete symmetry principle is utilized to simplify the boundary conditions and constraint handling of the trajectory model while combining scale transformation invariance to achieve nondimensionalization of the model’s physical quantities, thereby constructing the trajectory optimization model. On this basis, the hp-adaptive method is adopted to dynamically adjust the discretization step size and polynomial approximation order, and combined with an adaptive adjustment strategy for the trust region radius, to improve computational efficiency while ensuring optimization accuracy. Finally, the effectiveness of the algorithm is validated through optimizing the gliding phase of HV, and compared with GPOPS-II and fixed trust region SCP methods. The experimental results show that the algorithm has superiority in improving convergence speed, computational efficiency and solution accuracy, with its performance significantly outperforming traditional trajectory optimization schemes. Full article

This article presents the results of a study on the possibility of using a single-speed multiphase model with free surface allowance for simulating boiling and condensation processes. The simulation is based on the VOF method, which allows the position of the interphase boundary to be tracked. To increase the stability of the iterative procedure for numerically solving volume fraction transfer equations using a finite volume discretization method on arbitrary unstructured grids, the basic VOF method is been modified by writing these equations in a semi-divergent form. The models of Tanasawa, Lee, and Rohsenow are considered models of interphase mass transfer, in which the evaporated or condensed mass linearly depends on the difference between the local temperature and the saturation temperature with accuracy in empirical parameters. This paper calibrates these empirical parameters for each mass transfer model. The results of our study of the influence of the values of the empirical parameters of models on the intensity of boiling and evaporation, as well as on the dynamics of the interphase boundary, are presented. This research is based on Stefan’s problem of the movement of the interphase boundary due to the evaporation of a liquid and the problem of condensation of vapor bubbles water columns. As a result of a series of numerical experiments, it is shown that the average error in the position of the interfacial boundary for the Tanasawa and Lee models does not exceed 3–6%. For the Rohsenow model, the result is somewhat worse, since the interfacial boundary moves faster than it should move according to calculations based on analytical formulas. To investigate the possibility of condensation modeling, the results of a numerical solution of the problem of an emerging condensing vapor bubble are considered. A numerical assessment of its position in space and the shape and dynamics of changes in its diameter over time is carried out using the VOF method, taking into account the free surface. It is shown herein that the Tanasawa model has the highest accuracy for modeling the condensation process using a VOF method taking into account the free surface, while the Rohsenow model is most unstable and prone to deformation of the bubble shape. At the same time, the dynamics of bubble ascent are modeled by all three models. The results obtained confirm the fundamental possibility of using a VOF method to simulate the processes of boiling and condensation and taking into account the dynamics of the free surface. At the same time, the problem of the studied models of phase transitions is revealed, which consists of the need for individual selection of optimal values of empirical parameters for each specific task. Full article

The integral equation formulation of Maxwell’s equations proposed by Brandt provides an alternative to the H and T-A formulations for modelling high-temperature superconducting (HTS) tapes. A modified version of Brandt’s method in the literature models ferromagnetic domains near the tapes by considering the ferromagnetic domains as equivalent surface current. This paper extends this method by including the effect of external magnetic field acting on the ferromagnetic and HTS domains. The proposed method is used on a benchmark problem, which considers an HTS tape with a ferromagnetic substrate under an external time-varying magnetic field. The results agree closely (error in average ac loss less than 3%) with the widely-used T-A formulation implemented in COMSOL down to 2 mT. In addition, the proposed method is also applied to HTS tapes carrying transport ac current in a slot of a machine’s stator iron core, and HTS tapes in a stator iron slot in a machine under working conditions. It is found that ac loss calculated by the proposed method increases as the discretization size of the ferromagnetic material’s boundary decreases, and overshoots the value calculated by the T-A formulation in COMSOL when using very fine discretization. Full article

This study presents an innovative numerical method for solving linear fractional differential equations (LFDEs) using modified Bernstein polynomial bases. The proposed approach effectively addresses the challenges posed by the nonlocal nature of fractional derivatives, providing a robust framework for handling multiple initial and boundary value constraints. By integrating the LFDEs and approximating the solutions with modified fractional-order Bernstein polynomials, we derive operational matrices to solve the resulting system numerically. The method’s accuracy is validated through several examples, showing excellent agreement between numerical and exact solutions. Comparative analysis with existing data further confirms the reliability of the approach, with absolute errors ranging from 10⁻¹⁸ to 10⁻⁴. The results highlight the method’s efficiency and versatility in modeling complex systems governed by fractional dynamics. This work offers a computationally efficient and accurate tool for fractional calculus applications in science and engineering, helping to bridge existing gaps in numerical techniques. Full article

In this study, we investigate implicit fractional differential equations subject to anti-periodic boundary conditions. The fractional operator incorporates two distinct generalizations: the Caputo tempered fractional derivative and the Caputo fractional derivative with respect to a smooth function. We investigate the existence and uniqueness of solutions using fixed-point theorems. Stability in the sense of Ulam–Hyers and Ulam–Hyers–Rassias is also considered. Three detailed examples are presented to illustrate the applicability and scope of the theoretical results. Several existing results in the literature can be recovered as particular cases of the framework developed in this work. Full article

This study extends the Theory of Functional Connections, previously applied to constraints specified at discrete points, to encompass continuous integral constraints of the form ${\displaystyle {\int }_{x_0}^{x_f}f(x,t)\mathrm{d}x=\mathcal{I}\left(t\right)}$, where $\mathcal{I}\left(t\right)$ can be a constant, a prescribed function, or an unknown function to be estimated through optimization. The framework of continuous integral constraints is developed within the context of initial value problems (IVP) and boundary value problems (BVP). To demonstrate the effectiveness of this analytical approach, examples validate the method and highlight distinctions between satisfying continuous integral constraints via simple interpolation versus functional interpolation. A limitation of the proposed approach is the inability to inherently enforce inequality constraints, such as the positivity constraint $f(x,t)\ge 0$, for modeling probability density functions in classical mechanics. Despite this, numerical experiments on boundary-value problems rarely result in negative values, indicating that the issue occurs infrequently. However, a mitigation strategy based on non-negative least-squares methods combined with Bernstein polynomials is proposed to address these rare cases. This approach is validated through an additional numerical test, demonstrating its efficacy in ensuring nonnegativity when required. Full article

This paper presents a reformulation of classical existence and uniqueness results for second-order boundary value problems (BVPs) using the Kannan fixed point theorem, extending beyond the Banach contraction principle. We shift focus from the nonlinearity j to the solution operator T defined via Green’s function and establish a sufficient condition under which T satisfies the Kannan contraction criterion. Specifically, if the derivative of j is bounded by K and $K·{(\eta -\zeta )}^2/8<1/3$, then T is a Kannan contraction, ensuring a unique solution. This condition applies even when the Banach contraction principle fails. We also explore the plausibility of applying the Chatterjea contraction, though rigorous verification remains open. Examples illustrate the applicability of the results. This work highlights the utility of generalized contractions in differential equations. Full article

A class of boundary value problems for fractional differential systems involving variable-order derivatives is considered. Such problems can be transformed into some boundary value problems for nonlinear Caputo fractional differential systems. Here, the relations between linear Caputo fractional differential equations and their corresponding linear integral equations are investigated, and the results demonstrate that a proper Lipschitz-type condition is needed for studying nonlinear Caputo fractional differential equations. Then, an existence and uniqueness result is established in some vector subspaces by Banach’s fixed-point theorem and ${\parallel ·\parallel }_e$ norm. In addition, two examples are presented to illustrate the theoretical conclusions. Full article

A class of initial boundary value problems is here considered for a one-dimensional diffusion equation with a general time-fractional derivative with the Sonin kernel. One of the boundary conditions is in a general non-classical form, which includes no-nlocal cases of integral or multi-point boundary conditions. The problem is studied here by applying spectral projection operators to convert it to a system of relaxation equations in generalized eigenspaces. The uniqueness of the solution is established based on the uniqueness property of the spectral expansion. An algorithm is given for constructing the solution in the form of spectral expansion in terms of the generalized eigenfunctions. Estimates for the time-dependent components in this expansion are established and applied to prove the existence of a solution in the classical sense. The obtained results are applied to a particular case in which the specified boundary conditions lead to two sequences of eigenvalues, one of which consists of triple eigenvalues. Full article

Nanotechnology has become a transformative field in modern science and engineering, offering innovative approaches to enhance conventional thermal and fluid systems. Heat and mass transfer phenomena, particularly fluid motion across various geometries, play a crucial role in industrial and engineering processes. The inclusion of nanoparticles in base fluids significantly improves thermal conductivity and enables advanced phase-change technologies. The current work examines Powell–Eyring nanofluid’s heat transmission properties on a stretched Riga plate, considering the effects of magnetic fields, porosity, Darcy–Forchheimer flow, thermal radiation, and activation energy. Using the proper similarity transformations, the pertinent governing boundary-layer equations are converted into a set of ordinary differential equations (ODEs), which are then solved using the boundary value problem fourth-order collocation (BVP4C) technique in the MATLAB program. Tables and graphs are used to display the outcomes. Due to their significance in the industrial domain, the Nusselt number and skin friction are also evaluated. The velocity of the nanofluid is shown to decline with a boost in the Hartmann number, porosity, and Darcy–Forchheimer parameter values. Moreover, its energy curves are increased by boosting the values of thermal radiation and the Biot number. A stronger Hartmann number M decelerates the flow (thickening the momentum boundary layer), whereas increasing the Riga forcing parameter Q can locally enhance the near-wall velocity due to wall-parallel Lorentz forcing. Visual comparisons and numerical simulations are used to validate the results, confirming the durability and reliability of the suggested approach. By using a systematic design technique that includes training, testing, and validation, the fluid dynamics problem is solved. The model’s performance and generalization across many circumstances are assessed. In this work, an artificial neural network (ANN) architecture comprising two hidden layers is employed. The model is trained with the Levenberg–Marquardt scheme on reliable numerical datasets, enabling enhanced prediction capability and computational efficiency. The ANN demonstrates exceptional accuracy, with regression coefficients $R\approx 1.0$ and the best validation mean squared errors of $8.52\times {10}^{-10}$, $7.91\times {10}^{-9}$, and $1.59\times {10}^{-8}$ for the Powell–Eyring, heat radiation, and thermophoresis models, respectively. The ANN-predicted velocity, temperature, and concentration profiles show good agreement with numerical findings, with only minor differences in insignificant areas, establishing the ANN as a credible surrogate for quick parametric assessment and refinement in magnetohydrodynamic (MHD) nanofluid heat transfer systems. Full article