Approximate Solution Research Articles

Monitoring the thermal state of an ingot with beveled and rounded corners in a slab crystallizer of a CCM

The organization of ingot cooling in the continuous caster mold affects the reliability, productivity and quality of the resulting ingots. Insufficient heat exchange between the solidifying metal and the wall of the mold leads to a decrease in the thickness of the solid shell of the ingot, which causes the formation of surface defects, and when the ingot is pulled out of the mold, emergency breakthroughs occur. Excessive heat transfer leads to an increase in the thickness of the solid shell, which entails excessive shrinkage of the ingot surface with the formation of a gap between the solid shell of the ingot and the walls of the mold. The appearance of an air gap leads to increased wear of the copper walls of the mold. There is a design of copper walls that makes it possible to reduce the normal component of forces that occurs during wear, reduce the temperature gradient and stress concentration in the corner of the ingot, and also bring the thickness of the hard crust stocking in the corner closer to the perimeter of the mold. However, the thermal state of the formed ingot with such a crystallizer design remains poorly understood. The goal of the work is to create a virtual tool for monitoring the operation of the mold of a slab continuous caster based on the development of a mathematical and computer model of the thermal state of an ingot with beveled and rounded corners. The temperature distribution in the ingot was calculated using the quasi-equilibrium theory of solidification. The numerical implementation of the mathematical model and obtaining an approximate solution using a computer were carried out using the finite dif-ference method and an implicit difference scheme. A computer program was developed in the Matlab development environment. Using computer modeling, the temperature distribution over the volume of the slab at the outlet of the mold with beveled and rounded corners was obtained. The program for the selected mold design allows you to analyze the solidification of the ingot skin and monitor this process for the selected steel grade and specified casting technological parameters

Nonlinear dynamic analyses of sandwich porous FG cylindrical shells with dual-layered FG porous cores

This research examines the nonlinear vibrations and dynamic postbuckling (DPB) analyses of sandwich porous functionally graded (SPFG) cylindrical shells with dual-layered FG porous (FGP) cores exposed to external excitation, using a semi-analytical method. The study investigates two types of FG layers: one with evenly distributed porosities (FG-EPD) and another with unevenly distributed porosities (FG-UEPD). It also studies the FGP cores in four configurations: one featuring symmetric porosity distribution with stiffening in the surface areas (SPD-Stiff), another with softening in the surface areas (SPD-Soft), a third with nonsymmetric porosity distribution (NSPD), and a fourth with uniform porosity distribution (UPD). Therefore, the SPFG cylindrical shells with dual-layered FGP cores with eight different configurations are investigated. Utilizing the classical shell theory (CST) alongside the geometrical nonlinearity in von Kármán–Donnell framework, and Galerkin’s method, this study addresses the nonlinear dynamic problem. An approximate solution for the deflection shape using three terms is selected, and the relationship between frequency and amplitude in nonlinear vibration is clearly defined. The nonlinear dynamic behaviors including vibration and DPB responses are examined using the P-T method, which is named for its use of the piecewise constant argument in conjunction with the Taylor series expansion. The critical dynamic buckling load of SPFG cylindrical shells with dual-layered FGP cores is investigated via the Budiansky–Roth criterion.

A Comparison of Interpolation and Least Squares Methods for the Approximate Solution of Differential Equation

A Comparison of Interpolation and Least Squares Methods for the Approximate Solution of Differential Equation

Approximate and Exact Solution of Linear and Nonlinear Schrödinger Equation using Sawi Transform coupled with Homotopy Perturbation Method

This study presents an approximate solution using the Sawi Transform Method for the Schrödinger equation. Four problems were considered to illustrate the Sawi Transform Homotopy perturbation Method’s (STHPM) effectiveness and capabilities. The equation below is subjected to the Sawi Transform combined with Homotopy Perturbation Method (STHPM). $i\frac{\partial}{\partial t}\\psi(z,t)+\Delta[\psi(z,t)]+N\psi(z,t)=0$ The results obtained are represented in a series of rapidly convergent terms.

Unsupervised random quantum networks for PDEs

Classical Physics-informed neural networks (PINNs) approximate solutions to PDEs with the help of deep neural networks trained to satisfy the differential operator and the relevant boundary conditions. We revisit this idea in the quantum computing realm, using parameterised random quantum circuits as trial solutions. We further adapt recent PINN-based techniques to our quantum setting, in particular Gaussian smoothing. Our analysis concentrates on the Poisson, the Heat and the Hamilton–Jacobi–Bellman equations, which are ubiquitous in most areas of science. On the theoretical side, we develop a complexity analysis of this approach, and show numerically that random quantum networks can outperform more traditional quantum networks as well as random classical networks.

Fixed-point methodologies and new investments for fuzzy fractional differential equations with approximation results

The application of Caputo-Katugampola gH−differentiability to solve systems of fractional partial differential equations is investigated in this work. The existence and uniqueness of two types of gH−weak solutions for fuzzy fractional coupled partial differential equations are established. Lipschitz conditions are employed, and the Banach fixed-point theorem and mathematical induction are used in our approach. To address the challenges of the initial value problems, a matrix-form Cornwall’s inequality is developed. Additionally, a novel analysis of the continuous dependence of the coupled system’s solutions on the given conditions and approximate solutions is provided. An illustrative example is presented to validate the findings.

Complex Multi-nonlinearity for piezoelectric energy harvesting systems based on an accurate higher-order perturbation method

A theoretical model of a complex nonlinear coupling energy harvester under hybrid excitations is established with analytical and numerical solutions. Different orders of the method of multiple scales are derived to approximate and analyze the equation of motion for the electromechanical coupling system, thus obtaining higher-order approximate analytical solutions for its operating status and output performance indicators. The significant effects of external excitation, linear and nonlinear damping coefficients, as well as impedance on the transverse displacement, voltage, and power of higher-order and lower-order methods are analyzed. The results show that higher-order solution methodology are more sensitive to parameter excitation. For a large damping, the higher-order analysis has more excellent performance in capturing energy. The higher order method shows more pronounced nonlinear softening phenomenon and it is more suitable for low frequency environments. The higher order methods improves solution accuracy of piezoelectric energy harvesting systems. Consequently, it is easier to derive the nonlinear performance of the harvesting systems, and thus improves the average output power.

An Improved Driving Training-Based Optimization Algorithm for the Minimum Gap Graph Connected Partition Problem

In this paper, we consider the minimum gap partition problem in an undirected connected graph with nonnegative vertex weights. This problem involves in dividing the given graph into a specified number of connected subgraphs such that the total difference between the largest and the smallest vertex weights in each subgraph is minimized. Namely, let G = (V, E, W) be a given undirected connected graph with vertex set V , edge set E, and nonnegative vertex weight function W : V → R+, and let k ≥ 2 be a given positive integer. The minimum gap partition problem consists of constructing a partition P = {V1, V2, . . . , Vk} of non-empty and pairwise disjoint subsets of V such that each vertex set Vi, i = 1, 2, . . . , k, induces a connected subgraph G[Vi] of G. The objective of the problem is to find such a partition of V that minimizes the total difference 1≤i≤k maxv∈Viw(v) − minv∈Viw(v). This problem, as many other variants of graph partition problems, is known to be NP-hard problem. We designed an efficient metaheuristic algorithm for finding approximate solution of large-scale instances of this problem. The quality of the proposed algorithm is assessed comparing with the previous algorithms proposed for the problem.

Energy harvesting from transverse galloping using a two-degree-of-freedom flow energy harvester

Abstract A two degree-of-freedom (2-DOF) galloping piezoelectric energy harvesting system where both oscillators are excited by the incoming flow is studied in this paper. A coupled lumped-parameter model is developed to simulate the electro-fluid-structural coupling behaviour assuming a quasi-steady aerodynamic flow field. The differential equations governing the dynamics of the lumped system are converted into nonlinear algebraic equations employing the harmonic balance method (HBM) and then solved using Newton’s method. The approximate analytical solutions are compared to the solutions obtained by numerical integration, and the results agree, both qualitatively and quantitatively. The solutions were subsequently used to investigate the effects of the bluff body dimension, mechanical, electrical, aerodynamic, and electromechanical parameters on the performance of the harvester and to illustrate how to optimise some of these parameters to reduce the cut-in wind speed and maximise the power output of the energy harvester. It will be shown that the energy harvesting performance could be significantly improved by introducing piezoelectric transducers on both oscillators and including both masses in the incoming flow. A comparative study between the proposed 2-DOF flow energy harvesting system, a single-degree-of-freedom (SDOF) galloping piezoelectric energy harvester (GPEH), and other configurations of 2-DOF GPEH shows that the present 2-DOF GPEH generates the largest power peak.

A kind regularization method with Poisson kernel to solve Cauchy problem of Schrödinger equation

In this work, a precise regularization technique based on the mollification method is proposed to solve the ill-posed Schr o ¨ dinger Cauchy problem with potential-free field. In the sense of Hadamard, this problem is severely ill-posed. Using the mollification regularization method based on Poisson kernel, some new error estimates are obtained under the priori and posteriori parameter selection rules. The regular approximate solutions are created, and the convergence evidences are provided. The good performance and high accuracy of this technique are demonstrated through various examples. The solution works well and is resistant to data disruption noise.

Unsteady Two-Dimensional Drift of a Viscoelastic Fluid with Magnetic and Thermal Effects: A Perturbation Approach

This study explores the unsteady, two-dimensional flow of an electrically conductive, viscoelastic, and incompressible fluid past a uniformly moving, infinite, non-conducting vertical plate, in the presence of a uniform first-order chemical process involving mass and heat transfer. A uniform magnetic field is applied perpendicular to the flow, with the magnetic Reynolds number assumed small to neglect the induced magnetic field. The governing equations are solved using a regular perturbation method, yielding approximate solutions for concentration, velocity, temperature, and shear stress at the plate. Additionally, mass and heat transfer rates are determined, with the influence of viscoelasticity and other physical parameters emphasized. Where applicable, results are graphically represented.

A novel approach for the numerical solution of nonlinear Fredholm integral equations using Hosoya polynomial method

Abstract In this paper, we study the graph theoretical polynomial known as the Hosoya polynomial obtained from one of the standard classes of graphs called path. Using this polynomial applied for the numerical solution of the nonlinear Fredholm integral equation, which reduces in the algebraic system of equation with collocation points, then solving this system using Newton’s iterative with the help of MATLAB, we obtain the required approximate solution. The desired results in terms of a set of continuous polynomials over a closed interval [0, 1]. Illustrative applications show the efficiency, accuracy and validity of the proposed technique.

Sensitivity Matrix Update Method for Electrical Resistance Tomography Based on Error-Constrained Cross Fusion Residual Attention Network

Abstract Electrical resistance tomography (ERT) is an imaging technique of conductivity distribution. The image reconstruction algorithm based on the empty field sensitivity matrix is widely used because it provides an approximate solution to the complex ERT inverse problems. However, due to the soft field effect, the sensitivity matrix changes with the media distribution, leading to significant errors in image reconstruction. To address this issue, an error-constrained deep learning scheme for bubble flow, namely cross fusion residual attention network (CFRA-Net) is designed to update the sensitivity matrix during gas-liquid-solid fluidized bed operation. CFRA-Net employs a multi-head self-attention mechanism to enhance bubble features in boundary measurements, a multilevel cross fusion module to fuse empty field strength and boundary measurement information, and a novel serial-channel residual spatial attention block to boost the feature extraction capability of the model. A weighted loss function is designed to give more weight to the gas phase distribution region. Additionally, this paper establishes a dataset for gas-liquid-solid three-phase flow, considering background field conductivity variations. Validation and reliability of the proposed method are assessed through ablation, comparison, and noise experiments. The experimental results demonstrate that CFRA-Net achieves rapid updates of the sensitivity matrix, high imaging accuracy, and robust noise immunity.

A robust septic hermite collocation technique for dirichlet boundary condition Heat conduction equation

Abstract In the current manuscript, approximate solution for 1D heat conduction equation will be sought with the Septic Hermite Collocation Method (SHCM). To achieve this goal, by means of the roots of both Chebyschev and Legendre polynomials used at the inner collocation points, the pseudo code of the method is found out and applied using Matlab which is one of the widely utilized symbolic programming platforms. The unconditional stability of the scheme is shown by the traditional von-Neumann stability technique. To illustrate the accuracy and effectiveness of this newly current numerical scheme, a comparison among analytical and the computed numerical results is presented in tabular forms. It has been illustrated that the scheme is both accurate and effective one and at the same time can be used in a successful way for finding out numerical solutions of several nonlinear problems as well as linear ones.

Robust Direction Estimation of Terrestrial Signal via Sparse Non-Uniform Array Reconfiguration under Perturbations

DOA (Direction of Arrival), as an important observation parameter for accurately locating the Signals of Opportunity (SOP), is vital for navigation in GNSS-challenged environments and can be effectively obtained through sparse arrays. In practical application, array perturbations affect the estimation accuracy and stability of DOA, thereby adversely affecting the positioning performance of SOP. Against this backdrop, we propose an approach to reconstruct non-uniform arrays under perturbation conditions, aiming to improve the robustness of DOA estimation in sparse arrays. Firstly, we theoretically derive the mathematical expressions of the Cramér–Rao Bound (CRB) and Spatial Correlation Coefficient (SCC) for the uniform linear array (ULA) with perturbation. Then, we minimize CRB as the objective function to mitigate the adverse effects of array perturbations on DOA estimation, and use SCC as a constraint to suppress sidelobes. By doing this, the non-uniform array reconstruction model is formulated as a high-order 0–1 optimization problem. To effectively solve this nonconvex model, we propose a polynomial-time algorithm, which can converge to the optimal approximate solution of the original model. Finally, through a series of simulation experiments utilizing frequency modulation (FM) signal as an example, the exceptional performance of this method in array reconstruction has been thoroughly validated. Experimental data show that the reconstructed non-uniform array excels in DOA estimation accuracy compared to other sparse arrays, making it particularly suitable for estimating the direction of terrestrial SOP in perturbed environments.

Ensuring the Safety of an Extraction Well from an Upgradient Point Source of Pollution in a Computationally Constrained Setting

Shallow groundwater is an important resource, especially in low- and middle-income countries; however, shallow groundwater is particularly vulnerable to point sources of pollution such as latrines or unlined waste disposal ponds. The objective of this paper is to derive a quantitative criterion for siting an extraction well and an upgradient point source of pollution to ensure that they are hydraulically disconnected, i.e., that no water flows from the point source to the well. To achieve this objective, we modeled the flow of shallow groundwater considering uniform regional flow, a single point source of pollution, and a single extraction well. For any set of flow rates and upgradient point source distance, we sought the minimum “off-center distance” ymin (i.e., the distance in the direction perpendicular to regional flow) that ensures the well and the point source are hydraulically disconnected. For constituencies with access to computing resources and coding expertise, we used a computer-based method for determining ymin that is exact to within the accuracy of a root-finding algorithm; this approach is recommended when computer access is available. For constituencies lacking these resources, we determined a simple, closed-form, approximate solution for ymin that has an average error of less than 3% for the conditions we tested. For a subset of scenarios in which the point source is sufficiently far upgradient of the well (n = 77), the root mean square relative error of the approximate solution is only 0.52%. We found that ymin depends on a length parameter (Qw + Qps)/QR, where Qw is the extraction rate of the well, Qps is the injection rate of the point source, and QR is the regional groundwater flow rate per unit of perpendicular length. Either the exact solution or the closed-form approximation can help to site wells near point sources of pollution, or to site point sources near wells, in a manner that protects the health of the well user. The approximate solution is valuable because many constituencies that rely on shallow wells for water supply and latrines for sanitation also lack access to the computer resources necessary to apply the exact solution.

SCATTERING OF THE E-POLARIZED ELECTROMAGNETIC WAVE BY SOLITUDE CONDUCTIVE STRIP WITH IMPEDANCE

The key problem of scattering of a plane E - polarized electromagnetic wave by a solitude electrically conductive impedance strip is studied in detail. A well-posed full-wave mathematical model of scattering in the form of the first order singular integral equation was obtained by applying a modifica- tion of a classical method if integral equations. To examine this singular integral equation two basic methods were used. The first one is remarkable Lord Rayleigh’s asymptotic method, which delivers an explicit analytical solution in a narrow frequency range. It deals with the long-wave case of the plane E - polarized electromagnetic wave scattering problem. The second method is a direct numerical method, dealing with approximate computa- tion of definite integrals, which gives an approximate solution as well, but in the full-frequency range. A series of numerical experiments on scattering a E-polarized electromagnetic wave by impedance strip were carried out and are presented in this paper. Comparing of computation results shows good agreement in long-wave case of the plane E - polarized electromagnetic wave scattering problem, which indicates the correctness of numeric compu- tation in full frequency range as well.

Optimal avoidance strategy based on nonlinear approximate analytic solution of non-cooperative differential game

Abstract This study examines the pursuit-evasion game involving unmanned aerial vehicles (UAVs), with a specific focus on the scenario of N-pursuers-one-escapee. The primary objective is to develop an optimal strategy for the escapee when the pursuers possess superior capabilities. To obtain this objective, we conduct the following study. Firstly, to enhance realism, a non-cooperative differential game model is formulated, incorporating multiple motion characteristics, including aerodynamics, overloading, and imposed constraints. Secondly, the end-value performance index is subsequently converted to an integral one, simplifying the solution process of the Hamilton-Jacobi-Bellman (HJB) equation. An iterative method is utilised to determine the covariates using the Cauchy initial value problem, and its convergence and uniqueness are established. The optimal avoidance strategy is subsequently derived from the covariates. Finally, the superiority of the proposed strategy is validated through simulation experiments and compared to three advanced optimal avoidance strategies. A total of 1,000 anti-jamming simulation experiments are conducted to verify the robustness of the proposed strategy.

Nonlinear scenario‐based model predictive control for quadrotors with bidirectional thrust

AbstractThe control of quadrotor vehicles under state and parameter uncertainty is a well studied problem that is vitally important to the deployment of these systems under real world conditions. In this article, we propose a linearization‐based extension to nonlinear systems of the existing scenario model predictive control (MPC) framework, which quantifies the impact of uncertainty on the vehicle dynamics through repeated sampling of the uncertainty space. Given the computational costs of such an approach, we also propose two simplifications of the scenario MPC algorithm that are significantly more tractable. In order to evaluate the performance of the algorithms, the specific problem of the control of a bidirectionally actuated quadrotor vehicle is considered. Simulations are carried out for each scenario MPC scheme as well as for a reference deterministic MPC scheme. When a sufficiently large sample count is considered, each of the scenario MPC algorithms achieves safer performance than the deterministic formulation without sacrificing any optimality. Additionally, the approximate solution techniques conclusively outperform the original nonlinear scenario MPC formulation for the same computational cost.

Derivation of analytic formulae for several resonance frequencies of the SparkJet actuator

Building on previous research that emphasized the importance of focusing on resonance frequencies for a fundamental understanding of the thrust and complex internal flow phenomena of the SparkJet actuators, this study theoretically derives analytic formulae for several important resonance frequencies. The research addresses the typical configuration of the SparkJet actuators, which consists of a cylindrical cavity and orifice connected by a conical converging nozzle. While the resonance frequencies of the SparkJet actuators can be obtained by solving the eigenvalue problem presented in previous studies, this eigenvalue problem, despite being a typical boundary value problem in the form of an elliptic partial differential equation, is challenging to solve using conventional numerical methods such as iterative methods, because the eigenvalue is included as an unknown in the operator. Consequently, Boundary Element Method (BEM) or methods using the Green functions have been proposed to obtain numerical solutions, but these still require handling large matrix data, resulting in significant computational costs and memory consumption. To overcome this, the current study first simplifies the eigenvalue problem using the conformal mapping presented in previous research. Then, referencing prior studies that claim that three types of resonance frequencies (Helmholtz resonance frequency and resonance frequencies representing streamwise or radial directional oscillation) significantly affect the thrust of the SparkJet actuators, characteristics of these frequencies are mathematically defined. Using these mathematical characteristics, the study derives the asymptotic and approximate solutions of the simplified eigenvalue problem, from which the resonance frequencies are obtained. The analytic formulae proposed in this study theoretically explain the already known geometric tendencies of the resonance frequencies and reveal new geometric factors influencing resonance frequencies, which were previously unknown. This approach is expected to facilitate obtaining several important resonance frequencies of the SparkJet actuator more promptly and accurately and to provide a deeper understanding of the nature of the complex oscillation phenomena inside the actuator.