Differential Equation Approach Research Articles

On the Two Approaches to Incorporate Wave‐Particle Resonant Effects Into Global Test Particle Simulations

AbstractEnergetic electron dynamics in the Earth's radiation belts and near‐Earth plasma sheet are controlled by multiple processes operating on very different time scales: from storm‐time magnetic field reconfiguration on a timescale of hours to individual resonant wave‐particle interactions on a timescale of milliseconds. The most advanced models for such dynamics either include test particle simulations in electromagnetic fields from global magnetospheric models, or those that solve the Fokker‐Plank equation for long‐term effects of wave‐particle resonant interactions. The most prospective method, however, would be to combine these two classes of models, to allow the inclusion of resonant electron scattering into simulations of electron motion in global magnetospheric fields. However, there are still significant outstanding challenges that remain regarding how to incorporate the long term effects of wave‐particle interactions in test‐particle simulations. In this paper, we describe in details two approaches that incorporate electron scattering in test particle simulations: stochastic differential equation (SDE) approach and the mapping technique. Both approaches assume that wave‐particle interactions can be described as a probabilistic process that changes electron energy, pitch‐angle, and thus modifies the test particle dynamics. To compare these approaches, we model electron resonant interactions with field‐aligned whistler‐mode waves in dipole magnetic fields. This comparison shows advantages of the mapping technique in simulating the nonlinear resonant effects, but also underlines that more significant computational resources are needed for this technique in comparison with the SDE approach. We further discuss applications of both approaches in improving existing models of energetic electron dynamics.

Study of Current Distribution over a Power Cable Presenting Non-Uniform Geometry using the Partial Differential Equations Approach

This paper presents a theoretical study of the currents and voltages characteristics transmission over an energy cable (Power Line Transmission PLT). The geometry of line is non-uniform. The proposed approach is based on the Distributed Network model, where the lumped parameters vary as the line's geometry. Most of the numerical methods applied use a frequency approach. However, other techniques are available in the time domain that are more applicable when the perturbation is transitory or an impulse. The partial differential equations (PDE) of telegraphers with variable coefficients are used in order to model the transmission over non-uniform lines. This paper presents the adaptation and application of the FDTD method to solve the problem of transmission of an electric wave over a transmission line whose electromagnetic topology is non-uniform. As an example, the final part of the paper proposes the modeling of the transmission of a transitory wave over an electric power cable. We present an application in the case of the propagation of lightning on an energy cable composed of three twisted wires. The validation of the proposed method will be set by comparing the obtained results with those obtained from a simulation in the frequency domain followed by an IFFT.

Modelling the propagation of invasive tree species: A coupled differential equation approach

Invasive tree species pose a grave global threat to biodiversity, particularly impacting vulnerable grassland ecosystems. This paper presents a novel approach to simulate the invasion of ecosystems by these aggressive species. Our innovative coupled ODE‐PDE model offers explicit invasion simulations, a significant leap beyond traditional integro‐difference matrix models. With robust theoretical properties, including solution positivity, and drawing upon PDE theory, our model offers deeper insights into invasion behavior. Our model reveals that invasions undergo a prolonged quiescent phase before rapidly transitioning to a consistent rate of advancement after a critical point. This study not only sheds light on invasion dynamics but also charts a course for future exploration and potential model extensions.

A Comparative Study of Hybrid Analytical and Laplace Transform Approaches for Solving Partial Differential Equations in Python

A Comparative Study of Hybrid Analytical and Laplace Transform Approaches for Solving Partial Differential Equations in Python

Innovative approaches to fractional modeling: Aboodh transform for the Keller-Segel equation

<abstract><p>This study focuses on developing efficient numerical techniques for solving the fractional Keller-Segel (KS) model, which is critical in explaining chemotaxis events. Within the Caputo operator framework, the study applied two unique methodologies: The Aboodh residual power series method (ARPSM) and the Aboodh transform iteration method (ATIM). These approaches were used to find precise solutions to the fractional KS equation, resulting in a better understanding of chemotactic behavior in biological systems. The comparative examination of the ARPSM and ATIM revealed their distinct strengths and applications in solving complicated fractional models. The work advances numerical approaches for fractional differential equations and improves our understanding of chemotaxis dynamics using a precise modeling approach.</p></abstract>

Pricing Vulnerable Options in Fractional Brownian Markets: a Partial Differential Equations Approach

Pricing Vulnerable Options in Fractional Brownian Markets: a Partial Differential Equations Approach

Capturing natural position relationships: A neural differential equation approach

The Transformer has emerged as the predominant model in Natural Language Processing due to its exceptional performance in various sequence modeling tasks, particularly in handling long-term dependencies. However, the traditional absolute and relative position encoding methods, which do not learn from data, tend to ignore the inherent structure of natural language sequences due to the position embedding layer at the input end of the Transformer model. This paper introduces a novel learnable neural Ordinary Differential Equation Position Encoding (ODEPE) method that can implicitly capture the natural position relationships within a sequence without requiring additional position embeddings. ODEPE can model continuous sequences and leverage differential equations to simulate the evolution of position information along the sequence, enabling position information to flow seamlessly between sequences. Additionally, a highly effective recurrent attention framework is proposed, which hybridizes attention with the ODEPE method to improve model performance. Compared to the Transformer-based sequence modeling network, our framework demonstrates a performance improvement of 24.0 points on the WikiText-103 dataset, while also achieving a performance improvement of 1.06 points on the Enwik8 dataset. This corresponds to an improvement of 4.9% and 0.17%, respectively.

An intuitionistic fuzzy differential equation approach for the lake water and sediment phosphorus model

Intuitionistic fuzzy sets cannot consider the degree of indeterminacy (i.e., the degree of hesitation). This study presents an intuitionistic fuzzy differential equation approach for the lake water and sediment phosphorus model. We examine the proposed model by assuming generalized trapezoidal intuitionistic fuzzy numbers for the initial condition. Feasible equilibrium points, along with their stability criteria, are evaluated. We describe the characteristics of intuitionistic fuzzy solutions and clarify the difference between strong and weak intuitionistic fuzzy solutions. Numerical simulations are performed in MATLAB to validate the model results.

Pricing a resettable convertible bond based on decomposition method and PDE models

In this paper, a partial differential equation approach based on the underlying stock price path decomposition is developed to price an American-style resettable convertible bond. The American-style resettable convertible bond is viewed as a mixture of three simple securities, which can be used to replicate the feature of payoffs of the resettable convertible bond completely. The partial differential equations under the Black–Scholes framework are established to price these simple securities. An implicit Euler method is used to discretize the first-order time derivative while a central finite difference method on a piecewise uniform mesh is used to discretize the spatial derivatives. The error estimates are developed by using the maximum principle in two mesh sets both for the time semi-discretization scheme and the spatial discretization scheme, respectively. It is proved that the scheme is first-order convergent for the time variable and second-order convergent for the spatial variable. Numerical experiments support these theoretical results.

Carbon Sequestration Through Forestry: A Differential Equation Approach

Carbon sequestration through forestry represents a promising approach to partially counteract anthropogenic greenhouse gas emissions driving climate change. Tree growth naturally removes carbon dioxide from the atmosphere, storing it as biomass. Sustainably managed forests can effectively function as carbon sinks. However, determining optimal forestry policies involves balancing complex ecological dynamics with economic constraints. This study develops differential equation models quantitatively capturing forest growth, timber harvesting, and carbon sequestration dynamics. Logistic models are first adapted to simulate biomass accumulation of representative tree species. Lifecycle growth patterns spanning juvenile to mature phases are incorporated, along with climate effects. Biomass levels are proportionally related to carbon dioxide removal rates from the atmosphere. Deforestation impacts are analyzed by incorporating harvesting-induced biomass reductions. Sustainability constraints are implemented to ensure minimum viable tree densities across harvest rotations. Optimization techniques then identify management guidelines maximizing economic returns given ecological stability considerations. The goal is providing quantitative insights into rotation lengths, planting densities, and allowable cuts upholding both climate change mitigation and commercial demands. Findings can inform science-based forestry policies to leverage forests as sustainable natural carbon sinks.

NODE-ImgNet: A PDE-informed effective and robust model for image denoising

Inspired by the traditional partial differential equation (PDE) approach for image denoising, we propose a novel neural network architecture, referred as NODE-ImgNet, that combines neural ordinary differential equations (NODEs) with convolutional neural network (CNN) blocks. NODE-ImgNet is intrinsically a PDE model, where the dynamic system is learned implicitly without the explicit specification of the PDE. This naturally circumvents the typical issues associated with introducing artifacts during the learning process. By invoking such a NODE structure, which can also be viewed as a continuous variant of a residual network (ResNet) and inherits its advantage in image denoising, our model achieves enhanced accuracy and parameter efficiency. In particular, our model exhibits consistent effectiveness in different scenarios, including denoising gray and color images perturbed by Gaussian noise, as well as real-noisy images, and demonstrates superiority in learning from small image datasets.

Jost function analysis of elastic and inelastic nucleon–nucleon scattering with a new model potential

The analysis of nucleon–nucleon elastic and inelastic scattering data is presented by considering the combined interaction of nuclear and electromagnetic potentials of equal range. The exact analytical off-shell solutions of the nonrelativistic Schrödinger equation for the effective potential with the centrifugal term are investigated using ordinary differential equation approach. Numerical results of scattering phase shifts, differential cross-sections and Transition matrices are in sensible conformity with previous works.

Robust optimal asset-liability management with mispricing and stochastic factor market dynamics

This paper investigates a robust optimal asset-liability management problem under an expected utility maximization criterion. More specifically, the manager is concerned about the potential model uncertainty and aims to seek the robust optimal investment strategies. We incorporate an uncontrollable random liability described by a generalized drifted Brownian motion. Also, the manager has access to an incomplete financial market consisting of a risk-free asset, a market index with potentially path-dependent, time-varying risk premium and volatility, and a pair of mispriced stocks. The market dynamics are assumed to rely on an affine-form, square-root factor process and the price error is modeled by a co-integrated system. We adopt a backward stochastic differential equation approach hinging on the martingale optimality principle to solve this non-Markovian robust control problem. Closed-form expressions for the robust optimal investment strategies, the probability perturbation process under the well-defined worst-case scenario and the corresponding value function are derived. The admissibility of the robust optimal controls is verified under some technical conditions. Finally, we perform some numerical examples to illustrate the effects of model parameters on the robust investment strategies and draw some economic interpretations from these results.

An analytical approach to the pricing of an exchange option with default risk under a stochastic volatility model

The exchange option, which has two correlated underlying assets, is one of the most popular exotic options in the over-the-counter markets. This paper studies the valuation of exchange options with default risk of option issuer, where default is allowed only at maturity. Moreover, we consider three underlying assets with stochastic volatilities and assume that fast mean-reverting processes determine the stochastic volatilities. Based on the partial differential equation approach, we derive the analytical pricing formula of the exchange option price with default risk using the asymptotic expansion. To verify the accuracy and efficiency of our pricing formula, we compare the results by our pricing formula with those by Monte Carlo simulation, which is considered a benchmark. In addition, we provide several graphs to illustrate the properties of the option for significant parameters.

A Modified Approach for Forecasting Relative Humidity in Indoor Premises

In order to forecast certain microclimatic parameters in a room, such as relative humidity, it is necessary to collect a large amount of data. For this reason, an electronic system for providing comfortable conditions in a room has been developed, which is based on the becoming largely popular Internet of Things (IoT) technology. On the basis of the obtained data, the future change of the relative humidity in a room can be forecasted, which aims to ensure the health and well-being of its inhabitants. It should be noted that working with large datasets requires more sophisticated methods for prediction, and for this reason, this paper presents results obtained from a modified ordinary differential equation approach for forecasting relative humidity. For the purposes of the study, a program code was developed in Matlab. The resulting predicted relative humidity values are comparable to actual measured values, which would help to improve indoor air quality and provide thermal comfort to occupants.

Monitoring regional soil organic matter content using a spatiotemporal model with time-series synthetic Landsat images

Accurately monitoring soil organic matter (SOM) content is crucial for food and soil security. Current methods of monitoring are expensive and existing sparse data cannot provide detailed spatial information about SOM content changes in an area. This study proposes using a spatiotemporal model with time-series synthetic Landsat images to monitor SOM content dynamics at the regional scale. The approach was implemented in Google Earth Engine (GEE) platform and tested in Jiangsu province, China, using the soil survey data from 2006 to 2007 and synthetic Landsat images from 1986 to 2007. The model generated SOM maps every three years between 1986 and 2007 and was evaluated using another soil survey from 2000 in southern Jiangsu. The model associated with 20 covariates derived from the synthetic Landsat image explained 70% of the variation in SOM content with root mean squared error (RMSE) and Lin's concordance correlation coefficient (CCC) of 5.17 g/kg and 0.57, respectively. The results showed that the model could reveal regional spatial and temporal differences in SOM content distribution. The SOM content in Jiangsu increased in the north, while decreasing in the central and southern areas. Temporally, the mean SOM contents increased from 1986 to 1992 by 0.17 g/kg, decreased in 1995, and increased again from 1998 to 2000 before decreasing from 2004 to 2007 by 0.14 g/kg. The validation based on the soil data in 2000 showed that the approach generated an RMSE of 5.97 g/kg, accounting for 22.77% of the average SOM content of the data. The study concluded that this approach could be used for monitoring SOM content and other soil properties. This approach had a relatively better accuracy than a previous study using the Integrated Nested Laplace Approximation with the Stochastic Partial Differential Equation approach with the same soil data but 4 times more samples.

Mapping the timeliness of routine childhood vaccination in The Gambia: A spatial modelling study

IntroductionTimeliness of routine vaccination shapes childhood infection risk and thus is an important public health metric. Estimates of indicators of the timeliness of vaccination are usually produced at the national or regional level, which may conceal epidemiologically relevant local heterogeneities and makeitdifficultto identify pockets of vulnerabilities that could benefit from targeted interventions. Here, we demonstrate the utility of geospatial modelling techniques in generating high-resolution maps of the prevalence of delayed childhood vaccination in The Gambia. To guide local immunisation policy and prioritize key interventions, we also identified the districts with a combination of high estimated prevalence and a significant population of affected infants. MethodsWe used the birth dose of the hepatitis-B vaccine (HepB0), third-dose of the pentavalent vaccine (PENTA3), and the first dose of measles-containing vaccine (MCV1) as examples to map delayed vaccination nationally at a resolution of 1 × 1-km2 pixel. We utilized cluster-level childhood vaccination data from The Gambia 2019–20 Demographic and Health Survey. We adopted a fully Bayesian geostatistical model incorporating publicly available geospatial covariates to aid predictive accuracy. The model was implemented using the integrated nested Laplace approximation—stochastic partial differential equation (INLA-SPDE) approach. ResultsWe found significant subnational heterogeneity in delayed HepB0, PENTA3 and MCV1 vaccinations. Specificdistricts in the central and eastern regions of The Gambia consistentlyexhibited the highest prevalence of delayed vaccination, while the coastal districts showed alower prevalence forallthree vaccines. We also found that districts in the eastern, central, as well as in coastal parts of The Gambia had a combination of high estimated prevalence of delayed HepB0, PENTA3 and MCV1 and a significant population of affected infants. ConclusionsOur approach provides decision-makers with a valuable tool to better understand local patterns of untimely childhood vaccination and identify districts where strengthening vaccine delivery systems could have the greatest impact.

Load–displacement relation and gap distribution between rough surfaces: Partial differential equations approach

We develop a theoretical model to predict the load–displacement relation and probability density function for gaps between contacting rough surfaces. We derive partial differential equations from the previous model (Joe et al., 2018), and extend them to the non-adhesive contact problem. The predictions of the present theory are compared with numerical results using the Green’s function molecular dynamics algorithm, and a good agreement is obtained.

Covariance–Based Rational Approximations of Fractional SPDEs for Computationally Efficient Bayesian Inference

ABSTRACT The stochastic partial differential equation (SPDE) approach is widely used for modeling large spatial datasets. It is based on representing a Gaussian random field u on R d as the solution of an elliptic SPDE L β u = W where L is a second-order differential operator, 2 β ∈ N is a positive parameter that controls the smoothness of u and W is Gaussian white noise. A few approaches have been suggested in the literature to extend the approach to allow for any smoothness parameter satisfying β > d / 4 . Even though those approaches work well for simulating SPDEs with general smoothness, they are less suitable for Bayesian inference since they do not provide approximations which are Gaussian Markov random fields (GMRFs) as in the original SPDE approach. We address this issue by proposing a new method based on approximating the covariance operator L − 2 β of the Gaussian field u by a finite element method combined with a rational approximation of the fractional power. This results in a numerically stable GMRF approximation which can be combined with the integrated nested Laplace approximation (INLA) method for fast Bayesian inference. A rigorous convergence analysis of the method is performed and the accuracy of the method is investigated with simulated data. Finally, we illustrate the approach and corresponding implementation in the R package rSPDE via an application to precipitation data which is analyzed by combining the rSPDE package with the R-INLA software for full Bayesian inference. Supplementary materials for this article are available online.

A continuous-time Markov chain and stochastic differential equations approach for modeling malaria propagation

Malaria is the world’s most fatal and challenging parasitic disease, caused by the Plasmodium parasite and transmitted to humans by the bites of infected female mosquitos. This study presents a stochastic process representing the evolution in time of an unexpected phenomenon. We use different probability techniques to assume the possible outcome in a Malaria model since state variables or parameters are random in a stochastic model. This study considers a deterministic vector-host malaria model and converts this model into the corresponding stochastic model with a Continuous-time Markov chain (CTMC) and Stochastic differential equation (SDE). We prove the positivity and boundedness of solutions and calculate the equilibrium points. The threshold values are evaluated using different approaches. Moreover, the Latin Hypercube Sampling (LHS) and Partial Rank Correlation Coefficients (PRCC) procedures are used to analyze the sensitivity of each model parameter. Finally, a transient numerical simulation is discussed, and their outcomes strongly correlate with the actual scenario.