Partial Differential Equation Model Research Articles

Merits and Demerits of Machine Learning of Ferroelectric, Flexoelectric, and Electrolytic Properties of Ceramic Materials

In the present review, the merits and demerits of machine learning (ML) in materials science are discussed, compared with first principles calculations (PDE (partial differential equations) model) and physical or phenomenological ODE (ordinary differential equations) model calculations. ML is basically a fitting procedure of pre-existing (experimental) data as a function of various factors called descriptors. If excellent descriptors can be selected and the training data contain negligible error, the predictive power of a ML model is relatively high. However, it is currently very difficult for a ML model to predict experimental results beyond the parameter space of the training experimental data. For example, it is pointed out that all-dislocation-ceramics, which could be a new type of solid electrolyte filled with appropriate dislocations for high ionic conductivity without dendrite formation, could not be predicted by ML. The merits and demerits of first principles calculations and physical or phenomenological ODE model calculations are also discussed with some examples of the flexoelectric effect, dielectric constant, and ionic conductivity in solid electrolytes.

Inverse Scheme to Locally Determine Nonlinear Magnetic Material Properties: Numerical Case Study

We are interested in the determination of the local nonlinear magnetic material behaviour in electrical steel sheets due to cutting and punching effects. For this purpose, the inverse problem has to be solved, where the objective function, which penalises the difference between the measured and the simulated magnetic flux density, has to be minimised under a constraint defined according to the corresponding partial differential equation model. We use the adjoint method to efficiently obtain the gradients of the objective function with respect to the material parameters. The optimisation algorithm is low-memory Broyden–Fletcher–Goldfarb–Shanno (BFGS), the forward and adjoint formulations are solved using the finite element (FE) method and the ill-posedness is handled via Tikhonov regularisation, in combination with the discrepancy principle. Realistic numerical case studies show promising results.

The analytical solutions of squeezing flow and heat transfer in an horizontal filtering channel

Flow and heat transfer dynamics inside different industrial chambers play a vital role in understanding momentum and heat transfer during industrial operations. Analysis of such flow and heat transfer helps operators to know how the system works. To better understand the filtration process, the current study analytically investigated the internal flow and heat transfer through a porous horizontal filter channel under the influence of a uniform magnetic field while removing unwanted contaminations from the fluid inside the filter chamber. Thus, the novelty of the current study is to advance the filtration process by investigating how the top moving wall, which aims to minimise filter cake formation, affects permeates outflow (the current design). The partial differential equations (PDEs) model representing the internal flow and heat transfer during filtration is transformed into a system of ordinary differential equations (ODEs) using Lie point symmetry reduction without changing the dynamics under investigation. After that, the transformed equations are solved analytically using the perturbation approximation technique. The comparative analysis between the obtained semi-analytical and NDSolve numerical solutions was used to show the correlation of internal flow, temperature, and pressure profiles. The obtained solutions are used to find the best combination of dimensionless parameters that arise from flow and heat transfer dynamics that lead to optimum outflow. Graphical analysis of flow and heat transfer dynamics are used to indicate the combination of parameters that leads to maximum outflow based on scientific evidence. It is observed that, on average, the velocity increases significantly by 122.55% across all cases of Re ranging from 0 to 3.2 at the top permeable wall y=1. This significant increase is due to the motion of the moving wall and fluid injection happening at y=1. The results show that more chamber volume, small pores size, weak injection, weak magnetic force, and less Joule heating are crucial to increase internal temperature and permeate outflow velocity during filtration.

Partial differential equation modelling of the solid–liquid phase process in iron sintering and interface control with backstepping method

Due to the highly continuous temporal and spatial distribution characteristics of the sintering process, the solid–liquid phase interface during sintering is uneven and discontinuous, which seriously affects the quality of sintered ore. In order to achieve real-time and effective control of the solid–liquid interface in the sintering system, this study proposes a spatiotemporally distributed partial differential equation (PDE) thermal system with Stefan interface boundaries. Slice the 3D sintering mixing space and establish a discrete 1D space multi-distributed system to achieve distributed control of the sintering thermal system. Based on this system, a distributed boundary controller is designed using the backstepping method to drive the solid–liquid phase interface to the desired position. This method fully adapts to the nonlinear problems of the sintering thermal system and can quickly respond to system changes. Based on real data from the sintering site, a multi-physics simulation model was designed to verify the correctness of the theoretical research. Comparing the simulation results with the actual sintering state shows that this control method can effectively control the position of the sintering solid–liquid interface, which plays a vital role in improving the quality of sinter and sintering efficiency.

Well‐posedness and error analysis of wave equations with Markovian switching

Compared to traditional partial differential equation modeling methods, Markov switching models can accurately capture the abrupt changes or jumps that complex systems often experience in the real world. In this paper, we propose a novel wave equation model with Markovian switching to represent complex systems with state‐jumping phenomena better, and the well‐posedness of the model is proved. In addition, a numerical method with non‐uniform grids is also proposed for the proposed model to simulate the data in realistic situations, which is based on the use of finite element discretization in space and central difference discretization in time. Finally, we conduct several experiments to analyze the errors and stability of the proposed model and the traditional model. The results show that the Markov switching model proposed in this paper has a smaller error than the traditional models while ensuring stability and can more accurately simulate the state jump phenomena of real‐world systems.

Numerical modeling of mixed convective nanofluid flow with fractal stochastic heat and mass transfer using finite differences

This study presents the first comprehensive numerical simulation of heat and mass transfer in fractal-like mixed convective nanofluid flows. The flow of non-Newtonian nanofluids over flat and oscillating sheets is modelled mathematically, and a finite difference scheme is used to solve this model. The two-stage scheme can tackle fractal and fractal stochastic mathematical models of partial differential equations. The consistency in the mean square is proved, and Fourier series stability analysis is adopted to find stability conditions for fractal stochastic partial differential equation. The scheme is applied to solve the unsteady Casson nanofluid flow over the flat and oscillatory sheet, which affects thermal radiation, heat source, and chemical reaction. The existence of the solution is also provided for the Navier-Stokes equation of the considered flow model using fractal time derivative. The graph illustrates that the proposed fractal technique achieves faster convergence than the Crank-Nicolson approach. Applications in energy systems, materials science, and environmental engineering are just a few of the domains that could benefit from a better understanding of mixed convective nanofluid flows with fractal features, and that is what this research study hopes to accomplish. Scientists and engineers may better develop efficient and environmentally friendly systems by simulating and analyzing these complicated processes with the suggested finite difference technique.

Qualitative analysis of solutions for a degenerate partial differential equations model of epidemic spread dynamics

Compartmental models are widely used in mathematical epidemiology to describe the dynamics of infectious diseases or in mathematical models of population genetics. In this study, we study a time-dependent Susceptible-Infectious-Susceptible (SIS) Partial Differential Equation (PDE) model that is based on a diffusion-drift approximation of a probability density from a well-known discrete-time Markov chain model. This SIS-PDE model is conservative due to the degeneracy of the diffusion term at the origin. The main results of this article are the qualitative behavior of weak solutions, the dependence of the local asymptotic property of these solutions on initial data, and the existence of Dirac delta function type solutions. Moreover, we study the long-term behavior of solutions and confirm our analysis with numerical computations.

A numerical B-spline Galerkin method with proper generalized decomposition for reduced order modeling of partial differential equations

A numerical study is presented on reduced order modeling of low and high-dimensional partial differential equations using a new B-spline Galerkin proper generalized decomposition (PGD) method. Galerkin-PGD schemes based on B-splines are numerically investigated for various benchmark problems, including the Poisson equation, linear/nonlinear advection-diffusion equations, and the Navier–Stokes equations. The treatment of inhomogeneous boundary conditions, non-separable terms and nonlinearities is numerically investigated. Numerical errors for high-fidelity B-spline solutions and reduced order solutions are provided to demonstrate the effectiveness and accuracy of the proposed method. Computational costs are reported to illustrate the efficiency of the developed approaches.

RBF neural network disturbance observer-based backstepping boundary vibration control for Euler–Bernoulli beam model with input saturation

The main objective of this paper is to address the issue of vibration control for a class of Euler–Bernoulli beam systems that are subject to external disturbances and input saturation. The proposed controller differs from other backstepping methods in that it employs a radial basis function (RBF) neural network to accurately estimate boundary disturbances and incorporates the hyperbolic tangent function to ensure input constraints. The nonlinear partial differential equation (PDE) model is initially derived based on Hamilton’s principle to capture the dominant dynamic characteristics of the flexible beam. In the framework of the Lyapunov direct approach, an adaptive RBF neural network-based law is subsequently designed to estimate the state-related boundary disturbances. The backstepping approach is then developed to propose sufficient conditions for ensuring the stability and convergence of closed-loop systems subject to input saturation. Finally, the effectiveness and superiority of the proposed methodology are further demonstrated by comparing the simulation results of constrained backstepping controllers.

Discrete and continuous mathematical models of sharp-fronted collective cell migration and invasion

Mathematical models describing the spatial spreading and invasion of populations of biological cells are often developed in a continuum modelling framework using reaction–diffusion equations. While continuum models based on linear diffusion are routinely employed and known to capture key experimental observations, linear diffusion fails to predict well-defined sharp fronts that are often observed experimentally. This observation has motivated the use of nonlinear degenerate diffusion; however, these nonlinear models and the associated parameters lack a clear biological motivation and interpretation. Here, we take a different approach by developing a stochastic discrete lattice-based model incorporating biologically inspired mechanisms and then deriving the reaction–diffusion continuum limit. Inspired by experimental observations, agents in the simulation deposit extracellular material, which we call a substrate , locally onto the lattice, and the motility of agents is taken to be proportional to the substrate density. Discrete simulations that mimic a two-dimensional circular barrier assay illustrate how the discrete model supports both smooth and sharp-fronted density profiles depending on the rate of substrate deposition. Coarse-graining the discrete model leads to a novel partial differential equation (PDE) model whose solution accurately approximates averaged data from the discrete model. The new discrete model and PDE approximation provide a simple, biologically motivated framework for modelling the spreading, growth and invasion of cell populations with well-defined sharp fronts. Open-source Julia code to replicate all results in this work is available on GitHub .

Modeling the transmission dynamics of a two-strain dengue disease with infection age

In this paper, we introduce a partial differential equation (PDE) model to describe the transmission dynamics of dengue with two viral strains and possible secondary infection for humans. The model features the variable infectiousness during the infectious period, which we call the infection age of the infectious host. We define two thresholds [Formula: see text] and [Formula: see text], and show that the strain [Formula: see text] can not invade the system if [Formula: see text]. Further, the disease-free equilibrium of the system is globally asymptotically stable if [Formula: see text]. When [Formula: see text], strain [Formula: see text] dominance equilibrium [Formula: see text] exists, and is locally asymptotically stable when [Formula: see text], [Formula: see text], [Formula: see text]. Then, by applying Lyapunov–LaSalle techniques, we establish the global asymptotical stability of the dominance equilibrium corresponding to some strain [Formula: see text]. This implies strain [Formula: see text] eliminates the other strain as long as [Formula: see text], where [Formula: see text] denotes the probability of a given susceptible mosquito being transmitted by a primarily infected human with strain [Formula: see text]. Finally, we study the existence of the coexistence equilibria under some conditions.

Simulation of radiative nonlinear heat dynamism on Buongiorno-modeled nanoliquid through porous inclined plate with adjustable chemical response

Nanofluids are the fluid suspensions in nanoparticles. A considerable enhancement in their features is less nanoparticle concentrations. Various studies on nanofluids focused on representing their performance with respect to the functions — here enhancing straight heat transfer was critical, like that in nuclear reactors, transportation, different industrial settings, biology, food and electronics. Hence, this consideration analyzes the utilization of the novel mathematical method, called the bvp4c method by viscous heat energy research in Buongiorno-modeled nanoliquid confined by the apt permeable plate along with slip mechanism. The thermophoresis and Brownian dispersion affects are again assumed. This transfer of solutal and thermal energy was dependent on the appreciable effect on heat source, variable chemical reactions and nonlinear thermal radiation. The dimensional model of partial differential equations (PDEs), applied to precise related applications, had been adapted into ordinary differential equations (ODEs). This modified Nusselt number decreases with increasing viscous heating, thermal radiation, thermophoresis parameter and Brownian motion, always it rises due to increasing temperature ratio parameter. The validation of the outcomes was attained with past solutions by free convectional flow and non-magnetic research. There are many functions in petroleum industries and engineering like electroplating, chemical processing of substantial metals and solar water heaters.

Partial differential equation models for invasive species spread in the presence of spatial heterogeneity.

Models of invasive species spread often assume that landscapes are spatially homogeneous; thus simplifying analysis but potentially reducing accuracy. We extend a recently developed partial differential equation model for invasive conifer spread to account for spatial heterogeneity in parameter values and introduce a method to obtain key outputs (e.g. spread rates) from computational simulations. Simulations produce patterns of spatial spread which appear qualitatively similar to observed patterns in grassland ecosystems invaded by exotic conifers, validating our spatially explicit strategy. We find that incorporating spatial variation in different parameters does not significantly affect the evolution of invasions (which are characterised by a long quiescent period followed by rapid evolution towards to a constant rate of invasion) but that distributional assumptions can have a significant impact on the spread rate of invasions. Our work demonstrates that spatial variation in site-suitability or other parameters can have a significant impact on invasions and must be considered when designing models of invasive species spread.

Event-triggered adaptive vibration control for a flexible satellite with time-varying actuator faults

In this study, vibration control is studied for the flexible satellite through wireless communication, and given the fact that the actuators are prone to unknown faults, the input signal event-triggering and time-varying actuator faults are considered for the flexible satellite simultaneously. The satellite system is composed of two symmetrical flexible panels attached to a centrebody, which is a distributed parameter system and the established model is represented by partial differential equations (PDEs). Based on the PDEs model of the flexible satellite, the input event-triggered mechanism using the relative threshold strategy is designed firstly to lower the communication burden, and then the event-triggered-based adaptive fault-tolerant control laws are developed with the aid of the designed auxiliary signals for the system with time-varying actuator faults. With the proposed control scheme, the vibration of the system is controlled to the small neighborhood of zero, and the angle of the panels is regulated to the desired position. Finally, numerical simulation results are given to further show the effectiveness of the designed control scheme.

Efficient Learning of PDEs via Taylor Expansion and Sparse Decomposition into Value and Fourier Domains

Accelerating the learning of Partial Differential Equations (PDEs) from experimental data will speed up the pace of scientific discovery. Previous randomized algorithms exploit sparsity in PDE updates for acceleration. However such methods are applicable to a limited class of decomposable PDEs, which have sparse features in the value domain. We propose Reel, which accelerates the learning of PDEs via random projection and has much broader applicability. Reel exploits the sparsity by decomposing dense updates into sparse ones in both the value and frequency domains. This decomposition enables efficient learning when the source of the updates consists of gradually changing terms across large areas (sparse in the frequency domain) in addition to a few rapid updates concentrated in a small set of “interfacial” regions (sparse in the value domain). Random projection is then applied to compress the sparse signals for learning. To expand the model applicability, Taylor series expansion is used in Reel to approximate the nonlinear PDE updates with polynomials in the decomposable form. Theoretically, we derive a constant factor approximation between the projected loss function and the original one with poly-logarithmic number of projected dimensions. Experimentally, we provide empirical evidence that our proposed Reel can lead to faster learning of PDE models (70-98% reduction in training time when the data is compressed to 1% of its original size) with comparable quality as the non-compressed models.

Quick large-scale spatiotemporal flood inundation computation using integrated Encoder-Decoder LSTM with time distributed spatial output models

Accurate spatiotemporal flood simulations are essential for making informed decisions regarding flood release in affected regions, such as flood detention areas. Traditional spatiotemporal flood simulation approach uses partial differential equation (PDE) models (or physics-based models), which need high computational time. Although many machine learning (ML) models for inundation are increasingly being used to emulate the PDE models to address this issue, utilizing conventional ML models to achieve large-scale spatiotemporal flood prediction (i.e., simulation output in tens of thousands of grids and time steps over the whole flood event) remains a significant challenge. Therefore, we developed a new inundation model (IM) using encoder-decoder long short-term memory (ED-LSTM) with Time Distributed Spatial output model (ED-LSTM-TDS) that can acquire accurate spatially distributed flood information more rapidly. In the new IM framework, each ED-LSTM-TDS is built to simultaneously generate output for multiple (K) cell grids and multiple ED-LSTM-TDS models are built for prediction at all grids of entire PDE model. This study is the first of its kind to employ the ED-LSTM-TDS method to address spatiotemporal flood inundation simulation problems for flood detention areas. A 1994 km2 flood detention area in northeastern China was used as a case study. ED-LSTM-TDS exhibited better performance in predicting flood characteristics (e.g., water depth, velocity) than alternative methods, including ordinary LSTM, artificial neural network (ANN), and multiple linear regression (MLR). In addition, we investigated the trade-off relationship between the accuracy of flood characteristic prediction and the computation time of the proposed model by considering different numbers of K cell grids in each ED-LSTM-TDS model. The final proposed inundation model could accurately predict the spatiotemporal flood characteristics within 1.5 min to acquire the same information that required approximately 1 h by the PDE models. Such rapid and accurate prediction by the proposed model is important for evacuation planning, upstream flood control decisions and flood damage reduction.

A nonlocal Lagrangian traffic flow model and the zero-filter limit

In this study, we start from a Follow-the-Leaders model for traffic flow that is based on a weighted harmonic mean (in Lagrangian coordinates) of the downstream car density. This results in a nonlocal Lagrangian partial differential equation (PDE) model for traffic flow. We demonstrate the well-posedness of the Lagrangian model in the L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1$$\\end{document} sense. Additionally, we rigorously show that our model coincides with the Lagrangian formulation of the local LWR model in the “zero-filter” (nonlocal-to-local) limit. We present numerical simulations of the new model. One significant advantage of the proposed model is that it allows for simple proofs of (i) estimates that do not depend on the “filter size” and (ii) the dissipation of an arbitrary convex entropy.

Modeling anomalous transport in fractal porous media: A study of fractional diffusion PDEs using numerical method

Abstract Fractional diffusion partial differential equation (PDE) models are used to describe anomalous transport phenomena in fractal porous media, where traditional diffusion models may not be applicable due to the presence of long-range dependencies and non-local behaviors. This study presents an efficient hybrid meshless method to the compute numerical solution of a two-dimensional multiterm time-fractional convection-diffusion equation. The proposed meshless method employs multiquadric-cubic radial basis functions for the spatial derivatives, and the Liouville-Caputo derivative technique is used for the time derivative portion of the model equation. The accuracy of the method is evaluated using error norms, and a comparison is made with the exact solution. The numerical results demonstrate that the suggested approach achieves better accuracy and computationally efficient performance.

Biomarkers and computational models for predicting efficacy to tumor ICI immunotherapy.

Numerous studies have shown that immune checkpoint inhibitor (ICI) immunotherapy has great potential as a cancer treatment, leading to significant clinical improvements in numerous cases. However, it benefits a minority of patients, underscoring the importance of discovering reliable biomarkers that can be used to screen for potential beneficiaries and ultimately reduce the risk of overtreatment. Our comprehensive review focuses on the latest advancements in predictive biomarkers for ICI therapy, particularly emphasizing those that enhance the efficacy of programmed cell death protein 1 (PD-1)/programmed cell death-ligand 1 (PD-L1) inhibitors and cytotoxic T-lymphocyte antigen-4 (CTLA-4) inhibitors immunotherapies. We explore biomarkers derived from various sources, including tumor cells, the tumor immune microenvironment (TIME), body fluids, gut microbes, and metabolites. Among them, tumor cells-derived biomarkers include tumor mutational burden (TMB) biomarker, tumor neoantigen burden (TNB) biomarker, microsatellite instability (MSI) biomarker, PD-L1 expression biomarker, mutated gene biomarkers in pathways, and epigenetic biomarkers. TIME-derived biomarkers include immune landscape of TIME biomarkers, inhibitory checkpoints biomarkers, and immune repertoire biomarkers. We also discuss various techniques used to detect and assess these biomarkers, detailing their respective datasets, strengths, weaknesses, and evaluative metrics. Furthermore, we present a comprehensive review of computer models for predicting the response to ICI therapy. The computer models include knowledge-based mechanistic models and data-based machine learning (ML)models. Among the knowledge-based mechanistic models are pharmacokinetic/pharmacodynamic (PK/PD) models, partial differential equation (PDE) models, signal networks-based models, quantitative systems pharmacology (QSP) models, and agent-based models (ABMs). ML models include linear regression models, logistic regression models, support vector machine (SVM)/random forest/extra trees/k-nearest neighbors (KNN) models, artificial neural network (ANN) and deep learning models. Additionally, there are hybrid models of systems biology and ML. We summarized the details of these models, outlining the datasets they utilize, their evaluation methods/metrics, and their respective strengths and limitations. By summarizing the major advances in the research on predictive biomarkers and computer models for the therapeutic effect and clinical utility of tumor ICI, we aim to assist researchers in choosing appropriate biomarkers or computer models for research exploration and help clinicians conduct precision medicine by selecting the best biomarkers.

Boundary Layer Stagnation Point Flow and Heat Transfer over a Nonlinear Stretching/Shrinking Sheet in Hybrid Carbon Nanotubes: Numerical Analysis and Response Surface Methodology under the Influence of Magnetohydrodynamics

The present study aims to offer new numerical solutions and optimisation strategies for the fluid flow and heat transfer behaviour at a stagnation point through a nonlinear sheet that is expanding or contracting in water-based hybrid nanofluids. Most hybrid nanofluids typically use metallic nanoparticles. However, we deliver a new approach by combining single- and multi-walled carbon nanotubes (SWCNTs-MWCNTs). The flow is presumptively steady, laminar, and surrounded by a constant temperature of the ambient and body walls. By using similarity variables, a model of partial differential equations (PDEs) with the magnetohydrodynamics (MHD) effect on the momentum equation is converted into a model of non-dimensional ordinary differential equations (ODEs). Then, the dimensionless first-order ODEs are solved numerically using the MATLAB R2022b bvp4C program. In order to explore the range of computational solutions and physical quantities, several dimensionless variables are manipulated, including the magnetic parameter, the stretching/shrinking parameter, and the volume fraction parameters of hybrid and mono carbon nanotubes. To enhance the originality and effectiveness of this study for practical applications, we optimise the heat transfer coefficient via the response surface methodology (RSM). We apply a face-centred central composite design (CCF) and perform the CCF using Minitab. All of our findings are presented and illustrated in tabular and graphic form. We have made notable contributions in the disciplines of mathematical analysis and fluid dynamics. From our observations, we find that multiple solutions appear when the magnetic parameter is less than 1. We also detect double solutions in the shrinking region. Furthermore, the increase in the magnetic parameter and SWCNTs-MWCNTs volume fraction parameter increases both the skin friction coefficient and the local Nusselt number. To compare the performance of hybrid nanofluids and mono nanofluids, we note that hybrid nanofluids work better than single nanofluids both in skin friction and heat transfer coefficients.