Partial Differential Equation Approach Research Articles

A Novel Approach for Time-Local Fractional Solutions of Certain Nonlinear Partial Differential Equations in Fractal Dimension

Time-local fractional approaches for nonlinear partial differential equations in fractal dimensions are essential for capturing the complex, irregular behaviors found in fractal systems. In this paper, a new modification of the local fractional Laplace variational iteration method (MLFLVIM) for obtaining analytical approximate solutions to the fractional gas dynamics equation, fractional Stefan equation, and fractional Newell-Whitehead-Segel equation within the context of fractal time space is presented. The proposed method (MLFLVIM) elegantly combines the local fractional Laplace transform (LFLT) with modified variational iteration method. Specifically, we first apply the (LFLT) to the given local fractional PDEs, yielding a transformed system of equations. We then apply modified variational iteration to this system. Finally, we use the inverse of (LFLT) to obtain the desired solution. To demonstrate the effectiveness of this approach, we implement it on three numerical physical problems. The results show that the (MLFLVIM) can successfully handle these nonlinear LFPDEs and provide accurate analytical approximation solutions.

Dynamic optimal energy flow of integrated electricity and gas systems in continuous space

The intrinsic flexibility of integrated electricity and gas systems (IEGSs) hinges on the gas flow dynamics dictated by partial differential equations (PDEs). However, conventional numerical approaches for PDEs grapple with inefficiencies and inaccuracies for operational analysis, chiefly due to the discretization of PDE invoked. For the first time, this study pioneers an analytical methodology for IEGS's analysis, effectively eliminating the pitfalls of discretization in solving gas flow dynamics. Based on this, the dynamic optimal energy flow problem in IEGS is reformulated in continuous space, significantly simplifying complex models while bolstering analytical precision. Case studies affirm the remarkable advantages of the proposed analytical approach. Under comparable scenarios, the proposed method improves computational efficiency dozens of times and exhibits concurrently heightened levels of accuracy.

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

Lagrangian, game theoretic, and PDE methods for averaging G-equations in turbulent combustion: existence and beyond

G-equations are popular level set Hamilton–Jacobi nonlinear partial differential equations (PDEs) of first or second order arising in turbulent combustion. Characterizing the effective burning velocity (also known as the turbulent burning velocity) is a fundamental problem there. We review relevant studies of the G-equation models with a focus on both the existence of effective burning velocity (homogenization), and its dependence on physical and geometric parameters (flow intensity and curvature effect) through representative examples. The corresponding physical background is also presented to provide motivations for mathematical problems of interest. The lack of coercivity of Hamiltonian is a hallmark of G-equations. When either the curvature of the level set or the strain effect of fluid flows is accounted for, the Hamiltonian becomes highly nonconvex and nonlinear. In the absence of coercivity and convexity, the PDE (Eulerian) approach suffers from insufficient compactness to establish averaging (homogenization). We review and illustrate a suite of Lagrangian tools, most notably min-max (max-min) game representations of curvature and strain G-equations, working in tandem with analysis of streamline structures of fluid flows and PDEs. We discuss open problems for future development in this emerging area of dynamic game analysis for averaging noncoercive, nonconvex, and nonlinear PDEs such as geometric (curvature-dependent) PDEs with advection.

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.

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.

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.

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.

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.

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

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

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.

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.

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.

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 PDE Approach to the Problems of Optimality of Expectations

Let (X, Z) be a bivariate random vector. A predictor of X based on Z is just a Borel function g(Z). The problem of "least squares prediction" of X given the observation Z is to find the global minimum point of the functional E[(X − g(Z))2] with respect to all random variables g(Z), where g is a Borel function. It is well known that the solution of this problem is the conditional expectation E(X|Z). We also know that, if for a nonnegative smooth function F: R×R → R, arg ming(Z)E[F(X, g(Z))] = E[X|Z], for all X and Z, then F(x, y) is a Bregmann loss function. It is also of interest, for a fixed ϕ to find F (x, y), satisfying, arg ming(Z)E[F(X, g(Z))] = ϕ(E[X|Z]), for all X and Z. In more general setting, a stronger problem is to find F (x, y) satisfying arg miny∈RE[F (X, y)] = ϕ(E[X]), ∀X. We study this problem and develop a partial differential equation (PDE) approach to solution of these problems.

Mapping the risk of Rift Valley fever in Uganda using national seroprevalence data from cattle, sheep and goats.

Uganda has had repeated outbreaks of Rift Valley fever (RVF) since March 2016 when human and livestock cases were reported in Kabale after a long interval. The disease has a complex and poorly described transmission patterns which involves several mosquito vectors and mammalian hosts (including humans). We conducted a national serosurvey in livestock to determine RVF virus (RVFV) seroprevalence, risk factors, and to develop a risk map that could be used to guide risk-based surveillance and control measures. A total of 3,253 animals from 175 herds were sampled. Serum samples collected were screened at the National Animal Disease Diagnostics and Epidemiology Centre (NADDEC) using a competition multispecies anti-RVF IgG ELISA kit. Data obtained were analyzed using a Bayesian model that utilizes integrated nested Laplace approximation (INLA) and stochastic partial differential equation (SPDE) approaches to estimate posterior distributions of model parameters, and account for spatial autocorrelation. Variables considered included animal level factors (age, sex, species) and multiple environmental data including meteorological factors, soil types, and altitude. A risk map was generated by projecting fitted (mean) values, from a final model that had environmental factors onto a spatial grid that covered the entire domain. The overall RVFV seroprevalence was 11.3% (95% confidence interval: 10.2-12.3%). Higher RVFV seroprevalences were observed in older animals compared to the young, and cattle compared to sheep and goats. RVFV seroprevalence was also higher in areas that had (i) lower precipitation seasonality, (ii) haplic planosols, and (iii) lower cattle density. The risk map generated demonstrated that RVF virus was endemic in several regions including those that have not reported clinical outbreaks in the northeastern part of the country. This work has improved our understanding on spatial distribution of RVFV risk in the country as well as the expected burden of the disease in livestock.