Methods For Differential Equations Research Articles

Scientific Training Model of Sports Tourism Based on Nonlinear Differential Equation

Abstract This paper establishes a scientific training model for sports tourism based on the nonlinear differential equation. This paper takes sports tourism as the research object. Then the nonlinear differential equation method is used to model communication, access, risk assessment and transparency. Then this paper analyzes and discusses how to improve residents' participation in sports tourism sports projects. Finally, this paper finds that the method has a good application prospect in the analysis and modeling sports tourism basketball projects by experimental simulation.

Differential equation fault location algorithm with harmonic effects in power system

About 80% of faults in the power system distribution are earth faults. Studies to find effective methods to identify and locate faults in distribution networks are still relevant, in addition to the presence of harmonic signals that distort waves and create deviations in the power system that can cause many problems to the protection relay. This study focuses on a single line-to-ground (SLG) fault location algorithm in a power system distribution network based on fundamental frequency measured using the differential equation method. The developed algorithm considers the presence of harmonics components in the simulation network. In this study, several filters were tested to obtain the lowest fault location error to reduce the effect of harmonic components on the developed fault location algorithm. The network model is simulated using the alternate transients program (ATP)Draw simulation program. Several fault scenarios have been implemented during the simulation, such as fault resistance, fault distance, and fault inception angle. The final results show that the proposed algorithm can estimate the fault distance successfully with an acceptable fault location error. Based on the simulation results, the differential equation continuous wavelet technique (CWT) filter-based algorithm produced an accurate fault location result with a mean average error (MAE) of less than 5%.

Convergence and Stability of an Explicit Method for Autonomous Time-Changed Stochastic Differential Equations with Super-Linear Coefficients

Convergence and Stability of an Explicit Method for Autonomous Time-Changed Stochastic Differential Equations with Super-Linear Coefficients

The Convergence of Truncated Euler-Maruyama Method for Stochastic Differential Equations with Piecewise Continuous Arguments Under Generalized One-Sided Lipschitz Condition

The Convergence of Truncated Euler-Maruyama Method for Stochastic Differential Equations with Piecewise Continuous Arguments Under Generalized One-Sided Lipschitz Condition

High-Order Multi-Resolution Weno Schemes with Lax-Wendroff Method for Fractional Differential Equations

High-Order Multi-Resolution Weno Schemes with Lax-Wendroff Method for Fractional Differential Equations

A fast Markovian method for modeling channel noise in neurons

Channel noise results from rapid transitions of protein channels from closed to open state and is generally considered as the most dominant source of electrical noise causing membrane-potential fluctuations even in the absence of synaptic inputs. The simulation of a realistic channel noise remains a source of possible error. Although the Markovian method is considered as the golden standard for appropriate description of channel noise, its computation time increasing exponentially with the number of channels, it is poorly suitable to simulate realistic features. We describe here a novel algorithm at discrete time unit for simulating ion channel noise based on Markov chains (MC). Although this new algorithm refers to a Monte-Carlo process, it only needs few random numbers whatever the number of channels involved. Our fast MC (FMC) model does not exhibit the drawbacks due to approximations based on stochastic differential equations and the values of spike jitter are comparable to those obtained with the true Markovian method. In fact, we show here, that these drawbacks can be highlighted in the approximation based on stochastic differential equation methods even for a high number of channels (standard deviation of the 5th spike is about two-fold larger than that of MCF or true Markovian method for 5000 sodium channels). The FMC model appears therefore as the most accurate method to simulate channel noise with a fast execution time that does not depend on the channel number.

Hermite‐Shannon‐Cosine spectral method for fractional partial differential equations

AbstractShannon‐Cosine wavelet function possesses almost all excellent characteristics such as interpolation, compact support, and smoothness. As an interpolation wavelet function, it could be applied in fractional partial differential equations effectively. However, when solving engineering problems in a finite interval, the treatment of the boundary is still not smooth enough. So, the Hermite‐Shannon‐Cosine interval wavelet is constructed using the Hermite interpolation function to achieve smoother transitions at the boundary of the interval, thereby reducing boundary effects. Based on this, a method for solving Fractional PDEs is proposed, the method's performance and its processing capability at the interval boundary are verified by taking the Fractional Fokker‐Planck equation and the Time‐Fractional Korteweg‐de Vries equation as examples. Compared with the multi‐scale Faber‐Schauder wavelet collocation method, Point‐Symmetric interval wavelet spectral method, Dynamic interval wavelet spectral method, and so forth, the experimental results show that the method performs better in terms of numerical accuracy and effectiveness.

Undetermined Coefficients Method for Sequential Fractional Differential Equations

The undetermined coefficients method is presented for nonhomogeneous sequential fractional differential equations involving Caputo fractional derivative of order n\alpha where n-1 n\alpha\le\ n and n\in\mathbb{N}. By employing proposed method, a particular solution of the considered equation is obtained. Some details about estimating the particular solution required to apply this method are explained. This method is shown to be particularly effective for nonhomogeneous fractional differential equations when the fractional differential equations involve some specific right-hand side functions.

Numerical Method for Multi-Dimensional Coupled Forward-Backward Stochastic Differential Equations Based on Fractional Fourier Fast Transform

Forward-backward stochastic differential equations (FBSDEs) have received more and more attention in the past two decades. FBSDEs can be applied to many fields, such as economics and finance, engineering control, population dynamics analysis, and so on. In most cases, FBSDEs are nonlinear and high-dimensional and cannot be obtained as analytic solutions. Therefore, it is necessary and important to design their numerical approximation methods. In this paper, a novel numerical method of multi-dimensional coupled FBSDEs is proposed based on a fractional Fourier fast transform (FrFFT) algorithm, which is used to compute the Fourier and inverse Fourier transforms. For the forward component of FBSDEs, time discretization is used as well as the backward equation to yield a recursive system with terminal conditions. For the numerical experiments to be successful, three types of numerical methods were used to solve the problem, which ensured the efficiency and speed of computation. Finally, the numerical methods for different examples are verified.

Nonsymmetric interior penalty Galerkin method for nonlinear time-fractional integro-partial differential equations

This article discusses an efficient numerical method for solving nonlinear time-fractional integro-partial differential initial–boundary-value problems. To tackle the non-linearity of the problem, we first use the Newton linearization process. Then we apply the non-symmetric interior penalty Galerkin method for the spatial variable, and obtain a semi-discrete problem in the time variable. Finally, to obtain the fully-discrete scheme, we use both L1-scheme and L2-scheme for time-fractional derivative, and trapezoidal rule for integral term in the semi-discrete problem. We derive L2-norm stability and error estimates. The validity of the error analysis is supported by numerical experiments.

Optical Solitons and Modulation Instability Analysis with Lakshmanan–Porsezian–Daniel Model Having Parabolic Law of Self-Phase Modulation

This paper seeks to find optical soliton solutions for Lakshmanan–Porsezian–Daniel (LPD) model with the parabolic law of nonlinearity. The spatiotemporal dispersion is included to the model, as it can contribute to handling the problem of internet bottleneck. This study was performed analytically using the traveling wave hypothesis to reduce the model to an integrable form. Then, the resulting equation was handled with two approaches, namely, the auxiliary equation method and the Bernoulli subordinary differential equation (sub-ODE) method. With an intentional focus on hyperbolic function solutions, abundant optical soliton waves including W-shaped, bright, dark, kink-dark, singular, kink, and antikink solitons were derived with the existing conditions. Furthermore, the behaviors of some optical solitons are illustrated. The spatiotemporal dispersion was found to significantly affect the pulse propagation dynamics. Finally, the modulation instability (MI) of the LPD model is explained in detail along with the extraction of the expression of MI gain.

Local-basis difference potentials method for elliptic PDEs in complex geometry

We develop efficient and high-order accurate finite difference methods for elliptic partial differential equations in complex geometry in the difference potentials framework. The main novelty of the developed schemes is the use of local basis functions defined at near-boundary grid points. The use of local basis functions allows unified numerical treatment of (i) explicitly and implicitly defined geometry; (ii) geometry of more complicated shapes, such as those with corners, multi-connected domain, etc; and (iii) different types of boundary conditions. This geometrically flexible approach is complementary to the classical difference potentials method using global basis functions, especially in the case where a large number of global basis functions are needed to resolve the boundary, or where the optimal global basis functions are difficult to obtain. Fast Poisson solvers based on FFT are employed for standard, centered finite difference stencils regardless of the designed order of accuracy. Proofs of convergence of difference potentials in maximum norm are outlined both theoretically and numerically.

Strong convergence of an adaptive time-stepping Milstein method for SDEs with monotone coefficients

We introduce an explicit adaptive Milstein method for stochastic differential equations with no commutativity condition. The drift and diffusion are separately locally Lipschitz and together satisfy a monotone condition. This method relies on a class of path-bounded time-stepping strategies which work by reducing the stepsize as solutions approach the boundary of a sphere, invoking a backstop method in the event that the timestep becomes too small. We prove that such schemes are strongly L_2 convergent of order one. This order is inherited by an explicit adaptive Euler–Maruyama scheme in the additive noise case. Moreover we show that the probability of using the backstop method at any step can be made arbitrarily small. We compare our method to other fixed-step Milstein variants on a range of test problems.

Investment decisions and strategies of China's energy storage technology under policy uncertainty: A real options approach

Energy storage technology is one of the critical supporting technologies to achieve carbon neutrality target. However, the investment in energy storage technology in China faces policy and other uncertain factors. Based on the characteristics of China's energy storage technology development and considering the uncertainties in policy, technological innovation, and market, this study proposes a sequential investment decision model under two investment strategies and uses the differential equation method to solve the investment threshold and investment opportunity value of the energy storage technology. Then, taking energy storage participation in peaking auxiliary services in China as an example, we verify the model validity and analyze the impact of uncertainty factors and investment strategies. The findings of this study are as follows: 1) The frequency of policy adjustments and the magnitude of subsidy adjustments can both influence energy storage technology investments, but the magnitude of subsidy adjustments is more significant. 2) Technological innovation can offset the adverse effects of policy uncertainty 3) In the case of a relatively certain policy, the subsidy is at least 0.0246 USD/kWh to effectively incentivize investment. In the case of uncertainty about the policy adjustment frequency and the subsidy reduction intensity, the subsidy must reach 0.0311 USD/kWh.

Modified Extended Lie-Group Method for Hessenberg Differential Algebraic Equations with Index-3

Hessenberg differential algebraic equations (Hessenberg-DAEs) with a high index play a critical role in the modeling of mechanical systems and multibody dynamics. Motivated by the widely used Lie-group differential algebraic equation (LGDAE) method, which handles index-2 systems, we first propose a modified extended Lie-group differential algebraic equation (MELGDAE) method for solving index-3 Hessenberg-DAEs and then provide a theoretical analysis to deepen the foundation of the MELGDAE method. Moreover, the performance of the MELGDAE method is compared with the standard methods RADAU and MEBDF on index-2 and -3 DAE systems, and it is demonstrated that the MELGDAE integrator exhibits a competitive performance in terms of high accuracy and the preservation of algebraic constraints. In particular, all differential variables in index-3 Hessenberg-DAEs achieve second-order convergence using the MELGDAE method, which suggests the potential for extension to Hessenberg-DAEs with an index of 4 or higher.

Auxiliary differential equation (ADE) method based complying-divergence implicit FDTD method for simulating the general dispersive anisotropic material.

The preceding works introduced the leapfrog complying divergence implicit finite-difference time-domain (CDI-FDTD) method, which exhibits high accuracy and unconditional stability. In this study, the method is reformulated to simulate general electrically anisotropic and dispersive media. The auxiliary differential equation (ADE) method is employed to solve the equivalent polarization currents, which are then integrated into the CDI-FDTD method. The iterative formulae are presented, and the calculation method is similar to that of the traditional CDI-FDTD method. Additionally, the Von Neumann method is utilized to analyze the unconditional stability of the proposed method. To evaluate the performance of the proposed method, three numerical cases are conducted. These include calculating the transmission and reflection coefficients of a monolayer graphene sheet and a monolayer magnetized plasma, as well as the scattering properties of a cubic block plasma. The numerical results obtained by the proposed method demonstrate its accuracy and efficiency in simulating general anisotropic dispersive media, compared to both the analytical method and the traditional FDTD method.

Efficient Multigrid Reduction-in-Time for Method-of-Lines Discretizations of Linear Advection

Parallel-in-time methods for partial differential equations (PDEs) have been the subject of intense development over recent decades, particularly for diffusion-dominated problems. It has been widely reported in the literature, however, that many of these methods perform quite poorly for advection-dominated problems. Here we analyze the particular iterative parallel-in-time algorithm of multigrid reduction-in-time (MGRIT) for discretizations of constant-wave-speed linear advection problems. We focus on common method-of-lines discretizations that employ upwind finite differences in space and Runge-Kutta methods in time. Using a convergence framework we developed in previous work, we prove for a subclass of these discretizations that, if using the standard approach of rediscretizing the fine-grid problem on the coarse grid, robust MGRIT convergence with respect to CFL number and coarsening factor is not possible. This poor convergence and non-robustness is caused, at least in part, by an inadequate coarse-grid correction for smooth Fourier modes in space-time known as characteristic components. We propose an alternative coarse-grid operator that provides a better correction of these modes. This coarse-grid operator is related to previous work and uses a semi-Lagrangian discretization combined with an implicitly treated truncation error correction. Theory and numerical experiments show the proposed coarse-grid operator yields fast MGRIT convergence for many of the method-of-lines discretizations considered, including for both implicit and explicit discretizations of high order. Parallel results demonstrate speed-up over sequential time-stepping.

Estimation of fine-resolution PM2.5 concentrations using the INLA-SPDE method

Estimating spatio-temporal PM2.5 concentrations is a method that can be used to impute missing observations; which means that it may have important applications in the assessment of health effects. Based on observations gathered from 119 monitoring stations located within Henan Province and its surrounding areas, China, this study constructed a spatio-temporal dynamic model and used the Integrated Nested Laplace Approximation-Stochastic Partial Differential Equation (INLA-SPDE) method to estimate daily PM2.5 concentrations in Henan Province. Local geographical variables, such as elevation, road density, and land use, were integrated as spatial covariates. Meanwhile, meteorological variables, such as precipitation, air pressure, relative humidity, temperature, and wind speed, were integrated as spatio-temporal covariates. In addition, a first-order autoregressive process and a spatially correlated random effect were explicitly specified to capture the spatio-temporal dependence of PM2.5 concentrations. The validation results showed that the predictions were in good agreement with the observations (10-fold cross validated R2: 0.9407; Root mean square error: 10.7135 μg/m3), and the coverage probability of the 95% confidence interval was 96.04%. This study confirmed that the INLA-SPDE method can effectively model the spatio-temporal variability in PM2.5 concentrations. We also derived estimations of PM2.5 concentrations with a high spatio-temporal resolution, which should improve the assessment of related health effects.

Piecewise Polynomial Least Squares Method for Differential Equations of Fractional Order

Abstract In this paper a new method to compute approximate analytical solutions for differential equations of fractional order is presented. The proposed computational method, called the Piecewise Polynomial Least Squares Method (PWPLSM), is a combination of the Polynomial Least Squares Method and of piecewise-defined functions. Numerical results for differential equations of different fractional orders are discussed. The approximate solutions obtained with PWPLSM are compared with other existing analytical and numerical solutions. The tables and figures included demonstrate the accuracy of the new method.

The virtual element method

The present review paper has several objectives. Its primary aim is to give an idea of the general features of virtual element methods (VEMs), which were introduced about a decade ago in the field of numerical methods for partial differential equations, in order to allow decompositions of the computational domain into polygons or polyhedra of a very general shape.Nonetheless, the paper is also addressed to readers who have already heard (and possibly read) about VEMs and are interested in gaining more precise information, in particular concerning their application in specific subfields such as ${C}^1$ -approximations of plate bending problems or approximations to problems in solid and fluid mechanics.