Differential Equation Approach Research Articles

Reformulation and Enhancement of Distributed Robust Optimization Framework Incorporating Decision-Adaptive Uncertainty Sets

Distributionally robust optimization (DRO) is an advanced framework within the realm of optimization theory that addresses scenarios where the underlying probability distribution governing the data is uncertain or ambiguous. In this paper, we introduce a novel class of DRO challenges where the probability distribution of random variables is contingent upon the decision variables, and the ambiguity set is defined through parameterization involving the mean and a covariance matrix, which also depend on the decision variables. This dependency makes DRO difficult to solve directly; therefore, first, we demonstrate that under the condition of a full-space support set, the original problem can be reduced to a second-order cone programming (SOCP) problem. Subsequently, we solve this second-order cone programming problem using a projection differential equation approach. Compared with the traditional methods, the differential equation method offers advantages in providing continuous and smooth solutions, offering inherent stability analysis, and possessing a rich mathematical toolbox, which make the differential equation a powerful and versatile tool for addressing complex optimization challenges.

An analog of the pendulum/rotor system with elementary solutions

Abstract We explore a novel one-dimensional model potential, V(x), which mimics many aspects of the classic pendulum/rotor system, including low- and high-energy limits which are exactly solvable, as well as a divergent period separating the two. Using energy methods, the classical period, T(E), is evaluated in closed form for all values of the energy (0 < E < ∞) in terms of simple logarithmic functions, in contrast to the pendulum/rotor system where the results require elliptic integrals. We find an unexpected duality between the exact expressions for the period on either side of the divergence. Using both energy methods, and a Newton’s law differential equation approach, we present closed form expressions for trajectory solutions, x(t) and p(t) = mv(t) (or the linear momentum), in terms of hyperbolic functions, which explicitly give the same values for T(E).

Terminal Phase Guidance Law Against Maneuvering Targets: Adaptive State-Dependent Differential Riccati Equation Approach

This paper provides a methodology to address the terminal guidance issue for a highly maneuverable target by formulating the problem as a nonlinear autonomous system with matched term uncertainty. This method involves the utilization of the State-Dependent Differential Riccati Equation (SDDRE) technique to develop an optimal adaptive strategy. Since the SDDRE is vulnerable to variations in the model, the adaptive framework has been devised to address the issue of system uncertainty. The utilization of Laguerre Functions (LFs) is employed to characterize the target acceleration due to its arbitrary shape, while the adaptation law computes the coefficients associated with this representation. The effectiveness of the proposed guidance law has been demonstrated in numerical simulation for various target maneuvers. In addition, the effect of the type and number of basis functions on the estimation had been studied. Ultimately, a sensitivity analysis of design parameters is presented and a comparison is provided between the effectiveness of the proposed guidance law and some of the methods that have been proposed in the literature.

Modeling Population Dynamics in the Indian Context: A Differential Difference Equation Approach

The study of population dynamics is of paramount importance in comprehending the evolving demographics and socioeconomic structure of a country. The task of modelling population dynamics in the Indian context poses distinct challenges and opportunities owing to the extensive and heterogeneous nature of the country's population. This research paper utilises a differential difference equation (DDE) methodology to comprehensively model population dynamics in India. The study seeks to enhance comprehension of population dynamics by considering various factors, including birth rates, death rates, migration patterns, and seasonal variations. The study employs Delay Differential Equations (DDEs) as a mathematical framework to effectively model the dynamic characteristics of population systems. These equations account for time delays, historical events, and discrete factors that play a significant role in shaping population dynamics, whether it be growth or decline. Numerical techniques, such as Euler's method and the Runge-Kutta method, are utilised in order to solve Delay Differential Equation (DDE) models and effectively simulate the dynamics of populations over a given time period. The evaluation of each method involves a comparative analysis of its accuracy, smoothness, convergence properties, and computational efficiency. The research findings presented in this study enhance our comprehension of population dynamics in India and offer significant insights that can be utilised by policymakers, urban planners, and researchers in the fields of demography, public health, and social sciences. The results have the potential to provide valuable insights for evidence-based decision-making and the development of effective policies aimed at addressing the various challenges and opportunities related to population dynamics in the Indian context.

Mathematical Modeling of Oxygen Transfer Using a Bubble Generator at a High Reynolds Number: A Partial Differential Equation Approach for Air-to-Water Transfer

Mathematical Modeling of Oxygen Transfer Using a Bubble Generator at a High Reynolds Number: A Partial Differential Equation Approach for Air-to-Water Transfer

Dimensionless dynamics: Multipeak and envelope solitons in perturbed nonlinear Schrödinger equation with Kerr law nonlinearity

In this work, the dimensionless form of the improved perturbed nonlinear Schrödinger equation with Kerr law of fiber nonlinearity is solved for distinct exact soliton solutions. We examined the multi-wave solitons and rational solitons of the governing equation using the logarithmic transformation and symbolic computation using an ansatz functions approach. Multi-wave solitons in fluid dynamics describe the situation in which a fluid flow shows several different regions (or peaks) of high concentration or intensity of a particular variable (e.g., velocity, pressure, or vorticity). Multi-wave solitons in turbulent flows might indicate the existence of several coherent structures, like eddies or vortices. These formations are areas of concentrated energy or vorticity in the turbulent flow. Understanding how these peaks interact and change is essential to comprehending the energy cascade and dissipation in turbulent systems. Furthermore, a sub-ordinary differential equation approach is used to create solutions for the Weierstrass elliptic function, periodic function, hyperbolic function, Chirped free, dark-bright (envelope solitons), and rational solitons, as well as the Jacobian elliptic function, periodic function, and rational solitons. Also, as the Jacobian elliptic function's' modulus m approaches values of 1 and 0, we find trigonometric function solutions, solitons-like solutions, and computed chirp free-solitons. Envelope solitons can arise in stratified fluids and spread over the interface between layers, such as layers in the ocean with varying densities. Their research aids in the management and prediction of wave events in artificial and natural fluid settings. In fluids, periodic solitons are persistent, confined wave structures that repeat on a regular basis, retaining their form and velocity over extended distances. These structures occur in a variety of settings, including internal waves in stratified fluids, shallow water waves, and even plasma physics.

Applying Neural ODEs to Derive a Mechanism-Based Model for Characterizing Maturation-Related Serum Creatinine Dynamics in Preterm Newborns.

Serum creatinine in neonates follows complex dynamics due to maturation processes, most pronounced in the first few weeks of life. The development of a mechanism-based model describing complex dynamics requires high expertise in pharmacometric (PMX) modeling and substantial model development time. A recently published machine learning (ML) approach of low-dimensional neural ordinary differential equations (NODEs) is capable of modeling such data from newborns automatically. However, this efficient data-driven approach in itself does not result in a clinically interpretable model. In this work, an approach to deriving an interpretable model with reasonable PMX-type functions is presented. This "translation" was applied to derive a PMX model for serum creatinine in neonates considering maturation processes and covariates. The developed model was compared to a previously published mechanism-based PMX model whereas both models had similar mechanistic structures. The developed model was then utilized to simulate serum creatinine concentrations in the first few weeks of life considering different covariate values for gestational age and birth weight. The reference serum creatinine values derived from these simulations are consistent with observed serum creatinine values and previously published reference values. Thus, the presented NODE-based ML approach to model complex serum creatinine dynamics in newborns and derive interpretable, mathematical-statistical components similar to those in a conventional PMX model demonstrates a novel, viable approach to facilitate the modeling of complex dynamics in clinical settings and pediatric drug development.

Robust asset-liability management games for n players under multivariate stochastic covariance models

This paper investigates a non-zero-sum stochastic differential game among n competitive CARA asset-liability managers, who are concerned about the potential model ambiguity and aim to seek the robust investment strategies. The ambiguity-averse managers are subject to uncontrollable and idiosyncratic random liabilities driven by generalized drifted Brownian motions and have access to an incomplete financial market consisting of a risk-free asset, a market index and a stock under a multivariate stochastic covariance model. The market dynamics permit not only stochastic correlation between the risky assets but also path-dependent and time-varying risk premium and volatility, depending on two affine-diffusion factor processes. The objective of each manager is to maximize the expected exponential utility of his terminal surplus relative to the average among his competitors under the worst-case scenario of the alternative measures. We manage to solve this robust non-Markovian stochastic differential game by using a backward stochastic differential equation approach. Explicit expressions for the robust Nash equilibrium investment policies, the density generator processes under the well-defined worst-case probability measures and the corresponding value functions are derived. Conditions for the admissibility of the robust equilibrium strategies are provided. Finally, we perform some numerical examples to illustrate the influence of model parameters on the equilibrium investment strategies and draw some economic interpretations from these results.

Exploring the effects of temperature on demersal fish communities in the Central Mediterranean Sea using INLA-SPDE modeling approach

Climate change significantly impacts marine ecosystems worldwide, leading to alterations in the composition and structure of marine communities. In this study, we aim to explore the effects of temperature on demersal fish communities in the Central Mediterranean Sea, using data collected from a standardized monitoring program over 23 years. Computationally efficient Bayesian inference is performed using the integrated nested Laplace approximation and the stochastic partial differential equation approach to model the spatial and temporal dynamics of the fish communities. We focused on the mean temperature of the catch (MTC) as an indicator of the response of fish communities to changes in temperature. Our results showed that MTC decreased significantly with increasing depth, indicating that deeper fish communities may be composed of colder affinity species, more vulnerable to future warming. We also found that MTC had a step-wise rather than linear increase with increasing water temperature, suggesting that fish communities may be able to adapt to gradual changes in temperature up to a certain threshold before undergoing abrupt changes. Our findings highlight the importance of considering the non-linear dynamics of fish communities when assessing the impacts of temperature on marine ecosystems and provide important insights into the potential impacts of climate change on demersal fish communities in the Central Mediterranean Sea.

Impact of Anisotropic Cosmic-Ray Transport on the Gamma-Ray Signatures in the Galactic Center

The very high energy (VHE) emission of the central molecular zone (CMZ) is rarely modeled in 3D. Most approaches describe the morphology in 1D or simplify the diffusion to the isotropic case. In this work, we show the impact of a realistic 3D magnetic field configuration and gas distribution on the VHE γ-ray distribution of the CMZ. We solve the 3D cosmic-ray transport equation with an anisotropic diffusion tensor using the approach of stochastic differential equations as implemented in the CRPropa framework. We test two different source distributions for five different anisotropies of the diffusion tensor, covering the range of effectively fieldline-parallel diffusion to isotropic diffusion. Within the tested magnetic field configuration, the anisotropy of the diffusion tensor is close to the isotropic case, and three point sources within the CMZ are favored. Future missions such as the upcoming CTA will reveal more small-scale structures that are not yet included in the model. Therefore, a more detailed 3D gas distribution and magnetic field structure will be needed.

Fractional Spectral Approaches for Some Fractional Partial Differential Equations

The multi-order linear and nonlinear partial differential equations of fractional order have been solved numerically. A computational strategy based on fractional spectral operational matrices (OM) and generalized fractional Lageurre (GFL), generalized fractional shifted Legendre (GFSL), generalized fractional modified Bernstein (GFMB) are provided. Our fractional spectral methods have been used to solve nonlinear fractional partial differential equations and fractional Korteweg-de Vries-Burgers (KdVB) equation. Making use of a variety of test and application examples, the effectiveness of the numerical solution have been satisfied.

Comparing spatiotemporal species distribution models: A case study of a Scotian Shelf sea cucumber (Cucumaria frondosa)

AbstractNumerous spatiotemporal species distribution modeling frameworks are now available to the ecological practitioner. This study compared three such frameworks accessible in the R programming language: generalized additive models with spatiotemporal smooths as implemented by mgcv, spatiotemporal generalized linear mixed models based on nearest neighbor Gaussian processes as implemented by starve, and spatiotemporal generalized linear mixed models based on the stochastic partial differential equations approach as implemented by sdmTMB. The primary focus was to compare the inferences obtained from applying these frameworks to the case study of the orange‐footed sea cucumber, Cucumaria frondosa, on the Scotian Shelf off Nova Scotia, Canada. Each model was fit to catch data (2000–2019) from Fisheries and Oceans Canada's annual Research Vessel and Snow Crab surveys. Environmental covariates were sourced from high‐resolution data layers, including physical oceanographic, bathymetric, and seafloor morphometric datasets. The three models captured variability in sea cucumber distribution that would have been overlooked without a spatiotemporal approach. Although their predictions were similar, including within C. frondosa spatial reserves, the models provided different inferences regarding covariate effects. This suggests that while practitioners primarily interested in mapping species distributions need only apply the most familiar framework, those most concerned with identifying predictive environmental covariates may benefit from comparing the output from multiple approaches. Employing multiple approaches can also serve as a validation technique.

Effective Measures of Thickness Evolution of the Solid Electrolyte Interphase of Graphite Anodes for Li-Ion Batteries.

Upon forming, the intensity or thickness of the solid electrolyte interphase (SEI) in a Li-ion battery (LIB) evolves to various states depending on the cell materials and operation conditions. Despite a crucial role in comprehending the behaviors of an LIB, its quantitative measure is far from satisfactory mainly because of the undue complexity of the concentration profiles of the comprising chemical species. Here, we calculate the depth profiles of atomic mole fractions of C and F and their ratio as RC/F = C/F of graphite anodes for LIBs in comparison to an X-ray photoelectron spectroscopy (XPS) experiment. To this end, we take a differential equation approach to dC/dt*, where t* is the reduced XPS etching time for depth. As a result, the respective analytical expression derived for C, F, and RC/F(t*) is verified to accurately account for the experiment. Moreover, we show that RC/F(t*) in the j state can be practically expressed in , where γ is a constant for a given anode. Based on this, we suggest ξj* = (αi + βi - βj)/αj as a measure of the SEI thickness evolution from the i to j state in terms of the cycle number. As an intriguing finding, the SEI thickness evolves up to about 3 times that of its initial state, beyond which it does not appear to grow any more.

Impact of Demographic Developments and PCV13 Vaccination on the Future Burden of Pneumococcal Diseases in Germany—An Integrated Probabilistic Differential Equation Approach

Streptococcus pneumonia is the primary cause of morbidity and mortality in infants and children globally. Invasive pneumococcal disease (IPD) incidence is affected by various risk factors such as age and comorbidities. Additionally, this bacterium is a major cause of community-acquired pneumonia (CAP), leading to higher rates of hospitalization, especially among older adults. Vaccination with pneumococcal conjugate vaccines (PCVs) has proven effective, but the demographic transition in Germany poses a challenge. This study introduces a novel stochastic approach by integrating a population forecast model into a transmission dynamic model to investigate the future burden of pneumococcal diseases in three age groups (0–4, 5–59, and 60 and older). Our simulations, presented through mean predictions and 75% prediction intervals, indicate that implementing PCV13 (13-valent pneumococcal conjugate vaccine) until the year 2050 results in reduced cases of IPD and CAP in all age groups compared to scenarios without infant vaccination. However, cases with non-vaccine serotypes may persist at higher levels compared to scenarios without infant vaccination. Consequently, there may be a need for improvement in the current national vaccine policy, such as implementing the use of higher-valent PCVs and strengthening adult vaccination uptake.

Single-Shot Factorization Approach to Bound States in Quantum Mechanics

Single-Shot Factorization Approach to Bound States in Quantum Mechanics

Optimal investment strategy for asset-liability management with a defaultable bond under stochastic default intensity

This paper studies a continuous-time asset-liability management problem with a defaultable bond under stochastic interest rates and stochastic default intensity. Specifically, an asset-liability manager is allowed to invest in a cash, a treasury bond, a defaultable bond and a stock. Using the stochastic control scheme and partial differential equation approach, we derive closed-form expressions for optimal investment strategies and corresponding value functions under the power and exponential utility functions. In addition, we also provide numerical experiments to illustrate the impact of relevant parameters in the default process and recovery rate on the optimal asset allocation strategy.

The Existence and Uniqueness Conditions for Solving Neutrosophic Differential Equations and Its Consequence on Optimal Order Quantity Strategy

Background: Neutrosophic logic explicitly quantifies indeterminacy while also maintaining the independence of truth, indeterminacy, and falsity membership functions. This characteristic assumes an imperative part in circumstances, where dealing with contradictory or insufficient data is a necessity. The exploration of differential equations within the context of uncertainty has emerged as an evolving area of research. Methods: the solvability conditions for the first-order linear neutrosophic differential equation are proposed in this study. This study also demonstrates both the existence and uniqueness of a solution to the neutrosophic differential equation, followed by a concise expression of the solution using generalized neutrosophic derivative. As an application of the first-order neutrosophic differential equation, we discussed an economic lot sizing model in a neutrosophic environment. Results: This study finds the conditions for the existing solution of a first-order neutrosophic differential equation. Through the numerical simulation, this study also finds that the neutrosophic differential equation approach is much better for handling uncertainty involved in inventory control problems. Conclusions: This article serves as an introductory exploration of differential equation principles and their application within a neutrosophic environment. This approach can be used in any operation research or decision-making scenarios to remove uncertainty and attain better outcomes.

GIFNet: an effective global infection feature network for automatic COVID-19 lung lesions segmentation.

The ongoing COronaVIrus Disease 2019 (COVID-19) pandemic carried by the SARS-CoV-2 virus spread worldwide in early 2019, bringing about an existential health catastrophe. Automatic segmentation of infected lungs from COVID-19 X-ray and computer tomography (CT) images helps to generate a quantitative approach for treatment and diagnosis. The multi-class information about the infected lung is often obtained from the patient's CT dataset. However, the main challenge is the extensive range of infected features and lack of contrast between infected and normal areas. To resolve these issues, a novel Global Infection Feature Network (GIFNet)-based Unet with ResNet50 model is proposed for segmenting the locations of COVID-19 lung infections. The Unet layers have been used to extract the features from input images and select the region of interest (ROI) by using the ResNet50 technique for training it faster. Moreover, integrating the pooling layer into the atrous spatial pyramid pooling (ASPP) mechanism in the bottleneck helps for better feature selection and handles scale variation during training. Furthermore, the partial differential equation (PDE) approach is used to enhance the image quality and intensity value for particular ROI boundary edges in the COVID-19 images. The proposed scheme has been validated on two datasets, namely the SARS-CoV-2 CT scan and COVIDx-19, for detecting infected lung segmentation (ILS). The experimental findings have been subjected to a comprehensive analysis using various evaluation metrics, including accuracy (ACC), area under curve (AUC), recall (REC), specificity (SPE), dice similarity coefficient (DSC), mean absolute error (MAE), precision (PRE), and mean squared error (MSE) to ensure rigorous validation. The results demonstrate the superior performance of the proposed system compared to the state-of-the-art (SOTA) segmentation models on both X-ray and CT datasets.

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.

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.