Solution Of Equation Research Articles

The effect of vertical temperature gradient on the equivalent depth in thin atmospheric layers

AbstractThe equivalent depth of an atmospheric layer is of importance in determining the phase speed of gravity waves and characterizing wave phenomena. The value of the equivalent depth can be obtained from the eigenvalues of the vertical structure equation (the vertical part of the primitive equations) where the mean temperature profile is a coefficient. Both numerical solutions of the vertical structure equation and analytical considerations are employed to calculate the equivalent depth, , as a function of the atmospheric layer's thickness, . Our solutions for layers of thickness 100 2000 m show that for baroclinic modes, can be over two orders of magnitudes smaller than . Analytic expressions are derived for in layers of uniform temperature and numerical solutions are derived for layers in which the temperature changes linearly with height. A comparison between the two cases shows that a slight temperature gradient (of say 0.65 K across a 100 m layer) decreases by a factor of 3 (but can reach a factor of 10 for larger gradients) compared with its value in a layer of uniform temperature, while a change of 10 K in the layer's uniform temperature hardly changes . The baroclinic mode exists in all combinations of boundary conditions top and bottom while the barotropic mode only exists when the vertical velocity vanishes at both boundaries of the layer.

Power series solution for fractal differential equations

Power series solution for fractal differential equations

β–Ulam–Hyers Stability and Existence of Solutions for Non-Instantaneous Impulsive Fractional Integral Equations

In this paper, we investigate a class of non-instantaneous impulsive fractional integral equations. Utilizing the Banach contraction mapping principle, we establish the existence and uniqueness of solutions for the considered problem. Additionally, employing Schauder’s fixed-point theorem, we demonstrate the existence of solutions within the framework of β-Banach spaces. Moreover, we examine the β–Ulam–Hyers stability of the solutions, providing insights into the stability behavior under small perturbations. An illustrative example is presented to demonstrate the practical applicability and effectiveness of the theoretical results obtained.

Homoclinic solutions for discrete fractional p-Laplacian equation via the Nehari manifold method

Homoclinic solutions for discrete fractional p-Laplacian equation via the Nehari manifold method

РАЗРАБОТКА МОБИЛЬНОГО КОЛЕСНОГО РОБОТА ДЛЯ ДОСТАВКИ ПОСЫЛОК

The study objective is to develop approaches to the design of a mobile wheeled robot for transporting small-sized goods, operating autonomously without the participation of a human operator. At the same time, the main task is to develop a mathematical model of a wheeled robot that takes into account the moments of rolling friction and sliding of the driving wheels on the support surface, and which allows to study the starting operation modes of the electromechanical drive of a mobile device. The increase in the speed of the mobile robot is achieved by minimizing the acceleration time to the required speed values through the use of a 2-stage algorithm for changing the control voltage. The development of a mathematical model of a robotic system is carried out on the basis of methods of analytical mechanics and the theory of automatic control. Using the Laplace transforms, an analytical solution of differential equations describing the dynamics of a mobile robot is obtained. Numerical solution of the obtained equations is carried out in Mathcad interface. The novelty of the work consists in obtaining time dependencies for the speed and current in the circuit of the electric motor of the mobile robot wheeled drive and finding out the basic patterns of movement at its launch. The results of the conducted research can be used in the development of motion control systems for mobile robots with feedback, ensuring the stabilization of a set value of the movement speed in steady-state modes. Conclusions: a mathematical model of a mobile robotic system is developed, the starting modes of its operation and the method of combined motion control are studied, the speed of a mobile wheeled device is increased when moving along a given trajectory.

Pathway to Fractional Integrals, Fractional Differential Equations, and Role of the H-Function

In this paper, the pathway model for the real scalar variable case is re-explored and its connections to fractional integrals, solutions of fractional differential equations, Tsallis statistics and superstatistics in statistical mechanics, the reaction-rate probability integral, Krätzel transform, pathway transform, etc., are explored. It is shown that the common thread in these connections is their H-function representations. The pathway parameter is shown to be connected to the fractional order in fractional integrals and fractional differential equations.

Numerical method and error estimate for stochastic Landau–Lifshitz–Bloch equation

Abstract In this paper we study numerical methods for solving a system of quasilinear stochastic partial differential equations known as the stochastic Landau–Lifshitz–Bloch (LLB) equation on a bounded domain in ${\mathbb{R}}^{d}$ for $d=1,2$. Our main results are estimates of the rate of convergence of the Finite Element Method to the solutions of stochastic LLB. To overcome the lack of regularity of the solution in the case $d=2$, we propose a Finite Element scheme for a regularized version of the equation. We then obtain error estimates of numerical solutions and for the solution of the regularized equation as well as the rate of convergence of this solution to the solution of the stochastic LLB equation. As a consequence, the convergence in probability of the approximate solutions to the solution of the stochastic LLB equation is derived. A stronger result is obtained in the case $d=1$ due to a new regularity result for the LLB equation which allows us to avoid regularization.

Bifurcation analysis and solitary wave solution of fractional longitudinal wave equation in magneto-electro-elastic (MEE) circular rod

In this study, we investigate the longitudinal wave equation (LWE) with the M−fractional derivative, which describes the propagation of longitudinal waves along a rod while incorporating interactions between mechanical, electrical, and magnetic fields within the material. Initially, we apply bifurcation analysis to examine the critical points or phase portraits where the system transitions to new behaviors, such as stability shifts or the emergence of chaos, and observe the mechanism of static soliton formation through a saddle-node bifurcation. Subsequently, we utilize a modified simple equation (MSE) technique to find solitary wave solutions. Depending on the relationships between free parameters, the solutions are expressed as hyperbolic, trigonometric, and exponential functions. The numerical form of the obtained solutions reveals complex phenomena, including dark and bright bell waves, kink periodic lump waves, kink periodic waves, periodic lump waves, interaction waves between kink and periodic waves, linked lump waves, and interactions of periodic and lump waves. Additionally, we compare our results with previously published work, demonstrating that the discussed methods are valuable tools for providing distinct, accurate soliton solutions relevant to nonlinear science and technology applications.

A multilevel Monte Carlo algorithm for stochastic differential equations driven by countably dimensional Wiener process and Poisson random measure

In this paper, we investigate properties of standard and multilevel Monte Carlo methods for weak approximation of solutions of stochastic differential equations (SDEs) driven by infinite-dimensional Wiener process and Poisson random measure with Lipschitz payoff function. The error of the truncated dimension randomized numerical scheme, which depends on two parameters i.e., grid density n∈N and truncation dimension parameter M∈N, is of the order n−1/2+δ(M) such that δ(⋅) is positive and decreasing to 0. We derive a complexity model and provide proof for the complexity upper bound of the multilevel Monte Carlo method which depends on two increasing sequences of parameters for both n and M. The complexity is measured in terms of upper bound for mean-squared error and is compared with the complexity of the standard Monte Carlo algorithm. The results from numerical experiments as well as Python and CUDA C implementation details are also reported.

The role of conjugacy in the dynamics of time of arrival operators

The construction of time of arrival (TOA) operators canonically conjugate to the system Hamiltonian entails finding the solution of a specific second-order partial differential equation called the time kernel equation (TKE). In this paper, we provide an exact analytic solution of the TKE for a special class of potentials satisfying a specific separability condition. The solution enables us to investigate the time evolution of the eigenfunctions of the conjugacy-preserving TOA operators (CPTOA) and show that they exhibit unitary arrival at the intended arrival point at a time equal to their corresponding eigenvalues. We also compare the dynamics between the TOA operators constructed by quantization and those independent of quantization for specific interaction potentials. We find that the CPTOA operator possesses smoother and sharper unitary dynamics over the Weyl-quantized one within numerical accuracy.

Polynomial solutions to the first order difference equations in the bivariate difference field

The bivariate difference field provides an algebraic framework to study sequences satisfying a recurrence of order two, and it can be used to transform summations involving such sequences into first order difference equations over the bivariate difference field. Based on this, we present an algorithm for finding a series of polynomial solutions of such equations in the bivariate difference field, and show an upper bound on the degree of any possible polynomial solutions, which in turn is sufficient to compute all polynomial solutions by using the method of undetermined coefficients.

An analytic method to investigate hemodynamics of the cardiovascular system: Biventricular system.

Previously, we found analytic solutions for single ventricular system based on the lumped parameter model (LPM). In this study, we generalized the method to biventricular system and derived its analytic solutions. LPM is just a set of differential equations, but it is difficult to solve due to time-varying ventricular elastance and high order. Mathematically, there exist no elementary solutions for time-varying equations. It turns out that instead of differential equations, according to volume conservation, a set of algebraic equations can be carried out. The solutions of the set of equations are just physiological states at end of systolic and diastolic phases such as end systolic/diastolic pressure/volume of left ventricle. As a preliminary application, the method is utilized to deduce the hemodynamic effects of VA ECMO. Left ventricular (LV) distension, a serious complication of VA ECMO, is usually attributed to factors such as increased afterload, inadequate LV unloading, reduced myocardial contractility or aortic valve regurgitation (AR), bronchial and Thebesian return in the absence of aortic valve (AoV) opening. Among these, reduced contractility and AR are strongly associated with LV distension. However, in the absence of reduced contractility or AR, it is less clear whether increased afterload or inadequate LV unloading alone can cause LV distension. This leads to the critical question: under what conditions does LV distension occur in the absence of reduced contractility or AR? The analytic formulas derived in this study give conditions for LV distension. Furthermore, the results show that the analytic hemodynamics are coincident with simulated results.

Neural operators for robust output regulation of hyperbolic PDEs

The recently introduced neural operator (NO) has been employed as a gain approximator in the backstepping stabilization control of first-order hyperbolic and parabolic partial differential equation (PDE) systems. Due to the global approximation ability of the DeepONet, the NO provides approximate spatial gain function with arbitrary accuracy. The closed-loop system stability can be ensured by the backstepping controller involving the approximate gain with sufficiently small error. In this paper, the NO theory is leveraged to solve the robust output regulation problem for a class of uncertain hyperbolic PDE systems under the design framework of backstepping-based regulator. The NO is trained offline on a dataset containing a sufficient number of system parameters and corresponding prior solutions of the kernel equation, so as to generate feedback gain for the robust regulator. Once the NO is trained, the kernel equation does not need to be solved ever again, for any new system parameters that do not exceed the range of the training set. Based on the internal model principle, the regulator is inherently robust to a degree of parameter uncertainty and error in approximate gain. Therefore, the tracking error can still converge to 0 if the extended regulator equations are solvable and the parameter uncertainty leads to an asymptotically stable origin. We provide a series of theory proofs and a numerical test under the approximate control and observation gains to demonstrate the robust regulation problem. It is shown that the NO is almost three orders of magnitude faster than PDE solver in generating kernel function, and the loss remains on the order of 10−4 in the test. This provides an opportunity to use the NO methodology for accelerated gain scheduling regulation for PDEs with time-varying system parameters.

Four extremal solutions of discrete-time algebraic Riccati equations: existence theorems and computation

Four extremal solutions of discrete-time algebraic Riccati equations: existence theorems and computation

Nonlinear Poisson-Boltzmann solutions for charged parallel plates: When opposite charges repel.

I present an exact solution of the Poisson-Boltzmann equation for two parallel plates and discuss the solution properties. I discuss in more detail plates with opposite charges: In this case, there are two critical separations, Lc,1 < Lc,2. For separations less than Lc,1, the force between plates is repulsive. It switches to attractive at Lc,1, but with the electric potential having the same signon both plates. For L > Lc,2, the force remains attractive, and the potential at the plates has the same signas the charge on each plate. I also describe charge regulation, determined by pKa, and provide formulas for both the critical distance where oppositely charged plates repel and their charging process. The implications of these results for the nanoparticle assembly, as driven by electrostatic interactions, are also discussed.

Stability of equilibrium states for a stochastic difference equation of exponential form

It is well know that many process and problems in different fields of sciences and engineering can be modelled by linear and nonlinear difference equations. Particularly, in case of systems biology, ecology, biochemistry, genetics and physiology dynamics, many population models are governed by exponential difference equations, and lot of papers were published in this matter. The goal of this work is to study a nonlinear second order difference equation of exponential form: Xn+1 = bXn + cXne−σXn−1, n ≥ 0 where the parameters b, c have arbitrary values, σ &gt; 0 and the initial values, X0, X−1 are arbitrary positive numbers. Our equation has an equilibrium points and is exposed to additive stochastic perturbations that are assumed a sequence of independent random variables with zero mean, unit variance and these perturbations are proportional to the deviation of the system state Xn from equilibrium points, and the result is stochastic difference equation. We cannot find the solutions of nonlinear stochastic difference equations of exponential form in all cases. So, one can study the behavior of solutions by asymptotic stability of equilibrium points. Additionally, with the general method of Lyapunov functional constructions for stochastic difference equations with discrete time, we provide the necessary and sufficient conditions for the asymptotic mean square stability of the two equilibrium points (zero and positive) within the linear approximation of the stochastic difference equation. These conditions also serve as sufficient conditions for the stability in probability of the equilibrium points of the initial nonlinear equation. Finally, to demonstrate the validity of the obtained results, some numerical examples and simulations of equations solutions are made and numerous graphical illustrations of stability trajectories of solutions are plotted.

Impact of Multi-Scattered LiDAR Returns in Fog.

In the context of autonomous driving, the augmentation of existing data through simulations provides an elegant solution to the challenge of capturing the full range of adverse weather conditions in training datasets. However, existing physics-based augmentation models typically rely on single scattering approximations to predict light propagation under unfavorable conditions, such as fog. This can prevent the reproduction of important signal characteristics encountered in a real-world environment. Consequently, in this work, Monte Carlo simulations are employed to assess the relevance of multiple-scattered light to the detected LiDAR signal in different types of fog, with scattering phase functions calculated from Mie theory considering real particle size distributions. Bidirectional path tracing is used within the self-developed GPU-accelerated Monte Carlo software to compensate for the unfavorable photon statistics associated with the limited detection aperture of the LiDAR geometry. To validate the Monte Carlo software, an analytical solution of the radiative transfer equation for the time-resolved radiance in terms of scattering orders is derived, thereby providing an explicit representation of the double-scattered contributions. The results of the simulations demonstrate that the shape of the detected signal can be significantly impacted by multiple-scattered light, depending on LiDAR geometry and visibility. In particular, double-scattered light can dominate the overall signal at low visibilities. This indicates that considering higher scattering orders is essential for improving AI-based perception models.

Existence and exponential stability in pth moment of non-autonomous stochastic integro-differential equations

This study is concerned with the existence and exponential stability of solutions of non-autonomous neutral stochastic integro-differential equations with delay; the linear part of this equation is dependent on time and generates a linear evolution system. The obtained results are applied to some neutral stochastic integro-differential equations. These kinds of equations arise in systems related to couple oscillators in a noisy environment or in viscoelastic materials under random or stochastic influences. Finally, we provide an example to illustrate the results.

Uniqueness, regularity, continuity, and stability of solutions of integral equations in irregular domains

This study explores the uniqueness, regularity, continuity, and stability of solutions to integral equations related to elliptical boundary value problems in irregular domains. Traditional methods usually consider soft boundaries, but this work extends these results to include domains with rough boundaries. By employing Sobolev spaces, especially fractional Sobolev spaces, we develop a comprehensive theoretical framework. Under appropriate conditions, the studied integral equation has a unique solution in fractional Sobolev spaces, maintaining the regularity properties of the input function. Two new theorems are introduced to strengthen the findings. The first theorem establishes that solutions are continuous with respect to perturbations in the input data. This means small changes in the input do not cause significant deviations in the solution, ensuring robustness in practical applications. The second theorem demonstrates the stability of the solutions when the domain geometry undergoes small changes. This stability is crucial for real-world scenarios where the exact shapes of domains may not be perfectly known or may vary slightly. These results significantly improve the mathematical understanding of boundary value problems in non-smooth domains. The proposed extended framework allows for more generalized applications, addressing challenges in various fields of physics and engineering. The ability to handle irregular domains with rough boundaries opens up new possibilities for modeling and solving complex problems where traditional soft boundary assumptions are not valid. This research provides a robust basis for future studies and applications, highlighting the importance of considering fractional Sobolev spaces in the analysis of integral equations.

Jump relation for the conormal derivative of parabolic single layer potential

We consider a single layer potential generated by the fundamental solution of a second-order parabolic equation with Dini continuous coefficients. The potential is considered in a semibounded spatial domain with non-smooth lateral boundary from the Dini–Hölder class. We establish a jump relation for the conormal derivative of this potential at the lateral boundary of the domain.