Numerical Equations Research Articles

A robust numerical method based on a deep-learning operator for solving the 2D acoustic wave equation

The numerical solution of a wave equation plays a crucial role in computational geophysics problems, which forms the foundation of inverse problems and directly impacts the high-precision imaging results of earth models. However, common numerical methods often lead to significant computational and storage requirements. Due to the heavy reliance on forward-modeling methods in inversion techniques, particularly full-waveform inversion, enhancing the computational efficiency and reducing the storage demands of traditional numerical methods have become key issues in computational geophysics. We develop the deep Lax-Wendroff correction (DeepLWC) method, a deep-learning-based numerical format for solving 2D hyperbolic wave equations. DeepLWC combines the advantages of traditional numerical schemes with a deep neural network. We provide a detailed comparison of this method with the representative traditional Lax-Wendroff correction method. Our numerical results indicate that the DeepLWC significantly improves the calculation speed (by more than 10 times) and reduces the space needed for storage by more than 10,000 times compared with traditional numerical methods. In contrast to the more popular physics-informed neural network method, DeepLWC maximizes the advantages of traditional mathematical methods in solving partial differential equations and uses a new sampling approach, leading to improved accuracy and faster computations. It is particularly worth pointing out that DeepLWC introduces a novel research paradigm for numerical equation solving, which can be combined with various traditional numerical methods, enabling acceleration and reduction in the storage requirements of conventional approaches.

Average Run Length Computations of Autoregressive and Moving Average Process using the Extended EWMA Procedure

In the past, the control chart served as a statistical tool for detecting process changes. The Exponentially Weighted Moving Average (EWMA) control chart is highly effective for detecting small changes. This research introduces the Extended Exponentially Weighted Moving Average (Extended EWMA) control chart for the Autoregressive and Moving average process with order p = 1 and q = 1 (ARMA(1,1)) The Extended EWMA control chart incorporates two smoothing parameters ( λ1 and λ2 ) derived from the EWMA control chart. A comparative analysis of the performance of the EWMA control chart. The Average Run Length (ARL) value as determined by the explicit formulas in this research, serves as a metric for evaluating the performance of the Extended EWMA control chart and the EWMA control chart. The Numerical Integral Equation (NIE) method is used to verify the accuracy of the ARL for the explicit formulas of the two control charts which has not been before discovered. The effectiveness of control charts can also be evaluated by analyzing SDRL, ARL, MRL, RMI, AEQL, and PCI values as metrics for various design parameter values. After analyzing the results of the ARL and all five performance meters, it was determined that the Extended EWMA control chart is better than the EWMA control chart at all shift sizes of process changes. Finally, the assessment of the ARMA process is being conducted to evaluate the ARL using a dataset on PM2.5 dust levels in Bangkok, Thailand during January and February of 2024.

Inclined magnetic field and joule heat on unsteady ternary nanofluidic flow impinging over a convectively heated cylinder

The present analysis considered the condition of unsteady stagnation point flow on ternary nanofluid [Formula: see text] through the regularly affecting and convective heated stretchable cylinder by the effect on inclining Lorentz force. This influence on thermal radiation, velocity slip, viscous dissipation, and Joule dissipation were again integrated by the analysis. The suitable thermo-physical relationship in the hybrid nanofluid is cultivated into followed Xue form. With the help of appropriate comparison alterations, the controlling dimensional numerical equations were transformed by the dimensionless models. The governing equations are transformed through comparison transformation and mathematically tackled in MATLAB with a boundary value problem algorithm. These mathematical solutions were validated with the presented material. Tabular and graphical descriptions of mathematical information were utilized to analyze the physical effect on different relevant parameters in the ternary nanofluid temperature and velocity. This thermal buoyancy force hikes the fluid flow although the opposite direction was noted in magnetic parameters and velocity slips. This heat transport rate in the surface was enhanced by an improvement in thermal radiation, Biot number, and solid fraction of nanoparticles. Moreover, a 44.9754% enhanced decreased skin friction is observed by triple nanoparticle nanofluid it signifies its best behavior as related to both other nanofluids.

Strength design of built-up radially battened columns subject to axial compression and arbitrary directional bending moment

Built-up radially battened columns (RBCs), comprising several identical circular steel tubes interconnected by multiple radial-shaped battens, represent an innovative column design that combines aesthetic appeal with exceptional load-bearing capacity. This makes them well-suited for architectural applications in large public buildings such as stations and airports. The axial compressive strength design methods for the built-up RBC have been extensively explored in the authors' prior studies. However, in real-world engineering applications, built-up RBCs may experience simultaneous axial compression and varying directional bending moments, particularly when considering potential eccentric compression or wind and earthquake loads in public buildings. Currently, strength design methods for this loading condition are still lacking. To address this deficiency, this paper primarily delves into the failure mechanism and strength design methods for built-up RBCs experiencing combined axial compression and arbitrary directional bending moments. Initially, the sectional strength of the built-up RBC is investigated by using a shell finite element model (FEM) for the built-up RBC. It is revealed that the compression-bending sectional strength of the built-up RBC exhibits significant differences with varying bending directions and tube numbers. Moreover, in some bending directions, the M-N sectional strength curve of the built-up RBC exhibits convexity. By integrating theoretically derived equations grounded in the principles of materials mechanics with FEM numerical calculations, concise and practical design equations are proposed for accurately predicting the sectional strength of the built-up RBC under combined axial compression and arbitrary directional bending moments. Subsequently, The buckling strength of the built-up RBC is studied using the FEM, considering initial geometrical imperfections consistent with first-order flexural buckling and accounting for geometric nonlinearity. It is noted that the bending direction of the built-up RBC may deviate during loading. Besides, the M-N buckling strength curve of the built-up RBC is not as convex as its M-N sectional strength curve. Based on calculations from numerous examples, practical design equations for predicting the buckling strength for the built-up RBC under axial compression and any directional bending moments are introduced. The key findings of this paper contribute to the compression-bending strength design of RBCs, enriching RBC design theory and promoting the widespread application of built-up RBCs in actual engineering projects.

A practical way to apply a technique that accelerates time history analysis of structures under digitised excitations

Time history analysis using direct time integration is a versatile and widely accepted tool for analysing the dynamic behaviour of structures. In 2008, a technique was proposed to accelerate thetime history analysis of structural systems subjected to digitised excitations. Recently, this techniquehas been named as the SEB THAAT* (Step-Enlargement-Based Time-History-Analysis-Acceleration-Technique), and the determination of appropriate values for its parameter is introduced as the mainchallenge. To overcome this challenge, a procedure is proposed in this paper. The basis of the procedure is the comments on accuracy control in structural dynamics and numerical analysis ofordinary differential equations, legalised in the New Zealand Seismic Code, NZS 1170.5:2004. As themain achievement, by using the proposed procedure, we can apply the SEB THAAT and carry out thetime history analysis clearly and with less parameter setting compared to the ordinary time historyanalysis. The proposed procedure is always applicable and, except when the behaviour is very complex, oscillatory and non-linear, the reductions in analysis run-time are considerable while the changes in accuracy are negligible. The performance can be sensitive to the problem, the integration method, the target response, and the severity of the non-linear behaviour. Compared to the previous tests on the SEB THAAT, the efficiency of applying the SEB THAAT using the proposed procedure is better, the sensitivity of the performance to the problem is lower, and a measure of accuracy is available. Compared to other techniques for accelerating structural dynamic analyses, the use of the SEB THAAT according to the proposed procedure has several positive points, including the simplicity of implementation.

LFT: Neural Ordinary Differential Equations With Learnable Final-Time.

Since the last decade, deep neural networks have shown remarkable capability in learning representations. The recently proposed neural ordinary differential equations (NODEs) can be viewed as the continuous-time equivalence of residual neural networks. It has been shown that NODEs have a tremendous advantage over the conventional counterparts in terms of spatial complexity for modeling continuous-time processes. However, existing NODEs methods entail their final time to be specified in advance, precluding the models from choosing a desirable final time and limiting their expressive capabilities. In this article, we propose learnable final-time (LFT) NODEs to overcome this limitation. LFT rebuilds the NODEs learning process as a final-time-free optimal control problem and employs the calculus of variations to derive the learning algorithm of NODEs. In contrast to existing NODEs methods, the new approach empowers the NODEs models to choose their suitable final time, thus being more flexible in adjusting the model depth for given tasks. Additionally, we analyze the gradient estimation errors caused by numerical ordinary differential equations (ODEs) solvers and employ checkpoint-based methods to obtain accurate gradients. We demonstrate the effectiveness of the proposed method with experimental results on continuous normalizing flows (CNFs) and feedforward models.

Biased motility-induced phase separation: from chemotaxis to traffic jams

We propose a one-dimensional model of active particles interpolating between quorum sensing models used in the study of motility-induced phase separation (MIPS) and models of congestion of traffic flow on a single-lane highway. Particles have a target velocity with a density-dependent magnitude and a direction that flips with a finite rate that is biased toward moving right. Two key parameters are the bias and the speed relaxation time. MIPS is known to occur in such models at zero bias and zero relaxation time (overdamped dynamics), while a fully biased motion with no velocity reversal models traffic flow on a highway. Using both numerical simulations and continuum equations derived from the microscopic dynamics, we show that a single phase-separated state extends from the usual MIPS to congested traffic flow in the phase diagram defined by the bias and the speed relaxation time. However, in the fully biased case, inertia is essential to observe phase separation, making MIPS and congested traffic flow seemingly different phenomena if not simultaneously considering inertia and tumbling. We characterize the velocity of the dense phase, which is static for usual MIPS and moves backward in traffic congestion. We also find that in presence of bias, the phase diagram becomes richer, with an additional transition between phase separation and a microphase separation that is seen above a threshold bias or relaxation rate.

Superconvergence of a new energy dissipation finite element scheme for nonlinear Schrödinger equation with wave operator

This paper is devoted to develop a new energy dissipation scheme by finite element method (FEM) for a nonlinear Schrödinger equation with wave operator (NLSW) and investigate its superconvergence behavior through introducing a new second order discrete approximation with respect to time. The new scheme not only guarantees the mass and energy stable but also is efficient in achieving the numerical solution at the last point. Also, an application to other general nonlinear equations with wave operator is discussed. On the one hand, the boundedness of the solution about the numerical equation is deduced and as a result, the existence and uniqueness of the solution are obtained. On the other hand, the superconvergence error estimate is derived and some numerical results are given to confirm the theoretical predictions.

Efficient Monitoring of Autoregressive and Moving Average Process using HWMA Control Chart

Quality control is an essential process for manufacturing and industry because it enhances product quality, consumer satisfaction, and overall profitability. Among many other statistical process control tools, quality practitioners typically employ control charts to monitor the industrial process and detect production changes. Control charts are widely used to detect flaws in many applications, such as distributed circuits and systems, electronic devices, and systems and signals. In this study, we derived an explicit formula for Average Run Length (ARL) of the Homogenously Weighted Moving Average control chart (HWMA) under the ARMA (p,q) process. The accuracy was checked using the numerical integral equation (NIE) technique. The finding showed that the explicit formulas and numerical solutions presented an outstanding level of agreement. However, the computational time for the explicit formulas was approximately one second, which was less than that required for the NIE. Moreover, the performance efficiency of the HWMA control chart is compared with the cumulative sum control chart for ARMA (p, q) processes including ARMA (2,1), ARMA (2,3), and ARMA (1,1) processes. The results found that the HWMA control chart performance is found to be preferable to the CUSUM control chart performance. Additionally, the explicit formula of the HWMA control chart was implemented in a practical application of the count of nonconformities in printed circuit boards (PCBs).

Comparison of the applicability of seven calculation equations of glomerular filtration rate among elderly people in China

BackgroundAt present, estimated glomerular filtration rate (eGFR) remains the most frequently utilized parameter in the evaluation of kidney injury severity. Numerous equations have been formulated based on serum creatinine (Scr) or serum cystatin C (Cysc) levels. However, there is a lack of consensus regarding the efficacy of these equations in assessing eGFR, particularly for elderly individuals in China. This study aimed to evaluate the applicability of the MDRD, MDRDc, CKD-EPI series, BIS1, and FAS equations within the Chinese elderly population.MethodsA cohort of 298 elderly patients with measured GFR (mGFR) was enrolled. The patients were categorized into three subgroups based on their mGFR levels. The eGFR performance was examined, taking into account bias, interquartile range (IQR), accuracy P30, and root-mean-square error (RMSE). Bland–Altman plots were employed to verify the validity of eGFR.ResultsThe participants had a median age of 71 years, with 167 (56.0%) being male. Overall, no significant differences in bias were observed among the seven equations (P > 0.05). In terms of IQR, P30, and RMSE, the BIS1 equation demonstrated superior accuracy (14.61, 72.1%, and 13.53, respectively). When mGFR < 30 ml/min/1.73 m2, all equations underestimated the true GFR, with the highest accuracy reaching only 59%. Bland–Altman plots indicated that the BIS1 equation exhibited the highest accuracy, featuring a 95% confidence interval (CI) width of 52.37.ConclusionsThis study suggested that the BIS1 equation stands out as the most applicable for estimating GFR in Chinese elderly patients with normal renal function or only moderate decline. 2020NL-085-03, 2020.08.10, retrospectively registered.

Cu Pumping Analysis during Cu/SiO2 Hybrid Bonding using In-situ SPM Imaging

The assessment of Cu pumping (also known as Cu extrusion) during thermal annealing is vital information for the successful execution of hybrid bonding as well as defect-free processing of the through silicon vias (TSV). Unpredicted Cu pumping can pose major reliability issues. Correspondingly, in this study, high-temperature analysis of Cu pumping was conducted utilizing in-situ scanning probe microscopic (SPM) imaging. Cu/SiO2 surfaces with recessed and protruded Cu topographies were produced by chemical mechanical polishing (CMP) and used for Cu pumping investigations. The amount of Cu pumping upon thermal annealing up to 400°C and cooling down to room temperature was precisely quantified. The SPM results were compared with FEM simulation results, and a numerical equation for Cu pumping was proposed, accordingly. It was shown that by using in-Situ SMP imaging, valuable information on the behavior of hybrid Cu/dielectric surfaces can be generated.

Bifurcation analysis and exact wave solutions of the nano-ionic currents equation: Via two analytical techniques

In this work, numerous nonlinear evolution equations, namely the nano-ionic current equation, have wide applications in various fields of science and technology. To achieve this, we employ the generalized auxiliary equation and extended hyperbolic function techniques to investigate the exact wave solutions of the model under consideration. We obtained a range of exact wave solutions, including bright, dark, singular-periodic, periodic, and w-shape solutions, for the nonlinear model under examination, as well as rational and exponential function solutions with various structures, which are explored in our manuscript. In this paper, we compared the solutions obtained using the GAE and EHF approaches and also compared the obtained solutions using the GAE scheme with the solutions reported in Aljahdaly et al. [43]. Moreover, the three-dimensional and contour graphs are included to illustrate the structure of some obtained solutions by selecting specific parameter values. Bifurcation analysis is illustrated by phase plane analysis; phase plots are employed to depict the dynamic structure and are also analyzed for the existence of other solutions. These techniques serve as reliable, simple, and potent tools to analyze various nonlinear evolution equations found in physics, applied mathematics, and engineering.

Dynamic interactions of HSV-2 and HIV/AIDS: A mathematical modeling approach

To enhance my research, I would like to share my knowledge of Herpes simplex virus-2 (HSV-2) and HIV/AIDS (Acquired Immunodeficiency Syndrome) through a mathematical model. The study’s objective is to develop and examine the co-disease model for the modern era using mathematical modeling. Based on their modes of transmission, HSV-2 and HIV/AIDS are currently the leading causes of death from infectious and severe chronic diseases. Depending on the severity of HIV-2’s chronic disease, the model is divided into five phases: the first two HIV-2 stages and the remaining three HIV stages. Ordinary differential equations (ODEs) are arranged differently by each individual. Investigations into the mathematical equation model have revealed the points of equilibrium between the free and endemic models. A study of the developed model was conducted using the basic reproduction numbers [R0] of HSV-2 and HIV. The results demonstrate that if R0&amp;lt;1, the free equilibrium of disease is asymptotically locally stable. When R0&amp;gt;1, equilibrium endemic states are regarded as existing. Finally, MATLAB software was used to simulate the numerical equations of the model.

An analytical approach for Yang transform on fractional-order heat and wave equation

A novel approach to locate the approximate analytical solutions for non-linear partial differential equations is presented in this paper: the Yang transformation method combined with the Caputo derivative. In the current work, we determine the fractional Heat and Wave equation’s approximate analytical solutions. This current work addresses the Yang transformation approach in addition with the Caputo derivative. The suggested method yields approximately analytical solutions in the form of series with a simple, straightforward mechanics and a proportionality dependent on values of the fractional-order derivative. A few numerical heat equation and wave equation problems are solved to show the usefulness and reliability of the method. The tabular form [tables 7–12] makes the claim that the absolute error decreased as the number of terms in the series increased. It is also confirmed that the results are graphical compatible.

Performance Evaluation of HWMA Control Chart based on AR(p) with Trend Model to Detect Shift Process Mean

The main goal of this study is to establish explicit solutions for the average run length (ARL) of the Homogenously Weighted Moving Average control chart when subjected to autoregressive with trend process. The accuracy of the explicit formula for the ARL is evaluated in comparison to the numerical integral equation method. To evaluate the two approaches, the accuracy percentage was employed. A determination is carried out of the HWMA control chart’s effectiveness using the median run length (MRL), the standard deviation of run length (SDRL), and the average run length (ARL). A comprehensive comparison is performed between the HWMA control chart, the Extended Exponentially Weighted Moving Average (EEWMA), and the cumulative sum (CUSUM) control charts with mean process shifts to illustrate the design and implementation of the HWMA control chart. As criteria for various values of design parameters, the performance of these control charts can also be evaluated using the relative mean index (RMI), the average extra quadratic loss (AEQL), and the performance comparison index (PCI). To evaluate the effectiveness of our explicit formula approach, we employ this formula on copper price data.

Activated Carbons as Effective Adsorbents of Non-Steroidal Anti-Inflammatory Drugs

In this study, the adsorption of naproxen sodium, ibuprofen sodium, and diclofenac sodium on activated carbon is investigated. Comprehensive studies of adsorption equilibrium and kinetics were performed using UV-Vis spectrophotometry. Thermal analysis and zeta potential measurements were also performed for pure activated carbon and hybrid materials (activated carbon–drug) obtained after adsorption of naproxen sodium, ibuprofen sodium, and diclofenac sodium. The largest amount and rate of adsorption was demonstrated for naproxen sodium. A significant impact of temperature on the adsorption of the tested salts of non-steroidal anti-inflammatory drugs was also indicated. Faster kinetics and larger amounts of adsorption were recorded at higher temperatures. Thermodynamic parameters were also determined, based on which it was indicated that adsorption in the tested experimental systems is an endothermic, spontaneous, and thermodynamically privileged process of a physical nature. The generalized Langmuir isotherm was used to study the equilibrium data. The adsorption rate data were analyzed using numerous adsorption kinetics equations, including FOE, SOE, MOE, f-FOE-, f-SOE, f-MOE, and m-exp.

The Shape Parameter in the Shifted Surface Spline—A Sharp and Friendly Approach

This is a continuation of our previous study on the shape parameter contained in the shifted surface spline. We insist that the data points be purely scattered without meshes and the domain can be of any shape when conducting function interpolation by shifted surface splines. We also endeavor to make our approach easily accessible for scientists, not only mathematicians. However, the space of interpolated functions is smaller than that used before, leading to sharper function approximation. This function space has particular significance in numerical partial differential equations, especially for equations whose solutions lie in Sobolev space. Although the Fourier transform is deeply involved, scientists without a background in Fourier analysis can easily understand and use our approach.

Heat Transfer Modelling of Biomass Torrefaction

The depletion of coal reserves and the increase of environmental problems urge the use of renewable energy sources. Biomass is a renewable energy source and is also used as one of the feeds for cofiring, a promising energy generation technology. However, the shortcomings of biomass, such as low calorific value, hygroscopic, and low grindability, limit its usage. Torrefaction is a mild form of pyrolysis of biomass that results in better solid fuel properties. This study aims to model a simple heat transfer numerical equation to predict the torrefaction of biomass in various operating conditions. The temperature distribution in biomass during the torrefaction process is also predicted, as well as the mass yield, energy yield, and HHV behaviour of the torrefied biomass. The modelling was conducted based on Calliandra wood biomass. The pellet biomass diameter was modelled to vary within a range of 1-3 cm and a length of 10 cm. The operating conditions used for this modelling were the temperature of torrefaction with a range of 230- 270 °C. Heat transfer modelling was carried out by using MATLAB. The modelling produces a numerical equation of transient heat transfer with a radial axis base with the kinetic model used as TPR (Three Parallel Reaction). The modelling produces an error of 1.424% compared to the experimental data by Felfli et al. The simulation shows that torrefaction that makes a higher and more uniform temperature distribution will result in a higher HHV value. Based on the simulation, the recommendation of torrefaction operating conditions for Calliandra wood with a diameter of 1-3 cm is at a torrefaction temperature of 270℃ and residence time of 1.5 hours. This will result in a calorific value of approximately 5300 kcal/kg.

MELTING PROCESS SIMULATION OF PHASE CHANGE MATERIAL LAYER USED IN THE BUILDING ENVELOPE

The melting process of phase change material (PCM) for thermal energy storage purposes has been studied and simulated in this paper. Using the ANSYS (Fluent) 2023 R1 software, the melting of a rectangular PCM zone with left and right sides exposed to convection heat and the other sides adiabatic is simulated. The study also discusses the program specifications, numerical modeling equations, and assumptions. For a total melting duration of 360 min, the contours of the liquid fraction, temperature, density, and velocity of phase change material particles are shown at regular intervals of 30 min. The contours aid in identifying how the melting interface changes in shape and motion as melting time increases. It also describes how the melting fraction changes over time, which provides information regarding the amount of melted PCM at various points.

Study of fractional forced KdV equation with Caputo–Fabrizio and Atangana–Baleanu–Caputo differential operators

It is essential for mathematicians, physicists, and engineers to construct fractional mathematical models for specific phenomena and develop numerical or analytical solutions for these models. In this work, we implement the natural decomposition approach with nonsingular kernel derivatives to investigate the solution of nonlinear fractional forced Korteweg–de Vries (FF-KdV) equation. We first investigate the FF-KdV equation under the Caputo–Fabrizio fractional derivative. The similar equations are then examined using the Atangana–Baleanu derivative. This approach combines the decomposition method with the Natural transform method. The series solution of the suggested equations is thus obtained using the natural transform. The key benefit of this novel approximate-analytical approach is that it may provide an analytical solution for the FF-KdV problem in the form of convergent series with simple computations. For each equation, three unique situations are chosen to demonstrate and test the viability of the proposed method. To guarantee the competence and dependability of the proposed method, the nature for various values of the Froude number Fr have been provided. The present approach is also used to calculate solutions at various fractional orders. The approximate series solution’s behavior for various fractional orders has been graphically displayed. The outcomes demonstrate that the methodology is simple to use and reliable when applied to numerous fractional differential equations.