Equations In Spaces Research Articles

High-order stability for a stochastic reaction-diffusion equation under random fluctuation on ℝ N

In this article, we systematically study the high-order stability for a stochastic reaction-diffusion equation under random fluctuation. We obtain the convergence of solutions and upper semi-continuity of random attractors for the fluctuation equations in the initial space L 2(ℝ N ), without an additional assumption on the nonlinearity. By means of a splitting method, we technically prove that the L p ∩H 1-difference of solutions is controlled by the L 2-difference of initial values near the initial time, where p , p > 2 , is the dissipation exponent of the nonlinearity. This along with the L 2-stability is sufficient for us to establish the L p ∩H 1-stability of random attractors. By utilizing some iteration arguments, we show that the solution is uniformly bounded on a bounded interval near the initial time in L δ (ℝ N ) for arbitrary δ > p, in which space the convergence of solutions and the upper semi-continuity of random attractors are also established.

A local meshless method for the numerical solution of multi-term time-fractional generalized Oldroyd-B fluid model

This work presents an accurate and efficient method, for solving a two dimensional time-fractional Oldroyd-B fluid model. The proposed method couples the Laplace transform (LT) with a radial basis functions based local meshless method (LRBFM). The suggested numerical scheme first uses the LT which transform the given equation to an elliptic equation in LT space, and then it utilizes the LRBFM to solve transformed equation in LT space, and then the solution is converted back into the time domain via the improved Talbot's scheme. The local meshless methods are widely recognized for scattered data interpolation and for solving PDEs in complex shaped domains. The adaptability, simplicity, and ease of use are features that have led to the popularity of local meshless methods. The local meshless methods are easy and straightforward, they only requires to solve linear system of equations. The main objective of using the LT is to avoid the computation of costly convolution integral in time-fractional derivative and the effect of time stepping on accuracy and stability of numerical solution. The stability and the convergence of the proposed numerical scheme are discussed. Further, the Ulam-Hyers (UH) stability of the proposed model is discussed. The accuracy and efficiency of the suggested numerical approach have been demonstrated using numerical experiments on five different domains with regular nodes distribution.

Isometries to Analyze the Stability of Norm-Based Functional Equations in p-Uniformly Convex Spaces

Over the past two decades, significant advancements have been made in understanding the stability according to Hyers–Ulam involving different functional equations (FEs). This study investigates the generalized stability of norm-based (norm-additive) FEs within the framework of arbitrary (noncommutative) groups and p-uniformly convex spaces. Specifically, we analyze two key functional equations, ∥η(gh)∥=∥η(g)+η(h)∥ and ∥η(gh−1)∥=∥η(g)−η(h)∥foreveryg,h∈G, where (G,·) denotes an arbitrary group and B is considered to be a p-uniformly convex space. The surjectivity of the function η:G→B is a critical assumption in our analysis. Drawing upon the foundational works of L. Cheng and M. Sarfraz, this paper applies the large perturbation method tailored for p-uniformly convex spaces, where p≥1. This study extends previous research by offering a deeper exploration of the conditions under which these functional equations demonstrate Hyers–Ulam stability. In this study, the additive functional equation demonstrates a fundamental form of symmetry, where the order of operands does not affect the results. This symmetry under permutation of arguments is crucial for the analysis of stability. In the context of norm-additive FEs, this stability criterion investigates how small changes in the inputs of a functional equation affect the outputs, especially when the function is expected to follow an additive form.

Derivative free processes of high order for nondifferentiable equations in Banach spaces

In this article, we consider two parametric classes of derivative free iterative methods of high order of convergence in Banach spaces. We study local and semilocal convergence results in Banach spaces for both families of iterative processes. We carry out a numerical application to solve a nonlinear and nondifferentiable Fredholm integral equation in the Banach space of continuous functions with the maximum norm.

Exact solutions to the Telegraph equation in terms of Airy functions

Two exact different solutions to the Telegraph equation in three-dimensional space are obtained in terms of Airy functions. As a result, these solutions unveil a distinctive propagation pattern along a coordinate that resembles a speed-cone-like coordinate of the system. This unique characteristic leads to effective Schr¨ödinger-like equations, amenable to exact solutions through Airy functions.

A Precise Temperature Control Method for Lithium-ion Battery Pack based on the Nonlinear Model Predictive Control Algorithm

Abstract To utilize the maximum performance of the battery while ensuring its thermal safety, a battery thermal management system is used to control the battery maximum temperature within a safe range. This paper centres on the establishment of a temperature prediction model and the development of the nonlinear-based model predictive control (MPC) strategy. First, to address the need of predicting battery temperature, this paper develops a distributed parameter thermal resistance model to predict battery temperature quickly and accurately. Secondly, the open-loop formulation of the nonlinear-based MPC is derived based on the established state space equations. Then the soft and hard constraints of the model are established based on the actual current conditions, pump conditions, temperature difference and temperature rise indexes, so as to establish the objective function of the MPC algorithm. Finally, the established temperature nonlinear MPC algorithm is embedded on the board and the hardware platform of battery liquid cooling system is established. The experiment test result shows that the maximum error of temperature control is less than 0.1°C, and the effectiveness of the temperature control strategy of lithium-ion battery is verified through experiments.

Predictive control of linear time-varying model for hydraulic erecting systems based on linear expanded state observer

By analyzing the deficiencies of existing hydraulic erecting systems (HESs) control methods, this study proposes a linear time-varying model predictive control (LTV-MPC) method based on the linear extended state observer (LESO) for HESs. First, the working mechanism of HESs is methodically analyzed and the corresponding state space equations are established. Second, the LESO system is designed to estimate the current unknown real-time states. Then, the LTV-MPC is employed to evaluate and output the optimal solution of the servo voltage signal. Finally, through simulation and experiment, the effectiveness of the proposed method is confirmed and discussed. The results show that the displacement error rate of the proposed method is still lower than 0.223% under larger external disturbances, which can effectively improve the control accuracy and stability of the system compared with other methods.

Stability Of 4D alternate additive functional equation

In this paper, we confer the generalized Ulam-Hyers stability of a 4D alternate additive functional equation in Banach and Random Banach space via direct and fixed point methods.

Computational decomposition and composition technique for approximate solution of nonstationary problems

Stable computational algorithms for the approximate solution of the Cauchy problem for nonstationary problems are based on implicit time approximations. Computational costs for boundary value problems for systems of coupled multidimensional equations can be reduced by additive decomposition of the problem operator(s) and composition of the approximate solution using particular explicit–implicit time approximations. Such a technique is currently applied in conditions where the decomposition step is uncomplicated. A general approach is proposed to construct decomposition–composition algorithms for evolution equations in finite-dimensional Hilbert spaces. It is based on two main variants of the decomposition of the unit operator in the corresponding spaces at the decomposition stage and the application of additive operator-difference schemes at the composition stage. The general results are illustrated on the boundary value problem for a second-order parabolic equation by constructing standard splitting schemes on spatial variables and region-additive schemes (domain decomposition schemes).

Deep Learning of Transition Probability Densities for Stochastic Asset Models with Applications in Option Pricing

Transition probability density functions (TPDFs) are fundamental to computational finance, including option pricing and hedging. Advancing recent work in deep learning, we develop novel neural TPDF generators through solving backward Kolmogorov equations in parametric space for cumulative probability functions. The generators are ultra-fast, very accurate and can be trained for any asset model described by stochastic differential equations. These are “single solve,” so they do not require retraining when parameters of the stochastic model are changed (e.g., recalibration of volatility). Once trained, the neural TDPF generators can be transferred to less powerful computers where they can be used for e.g. option pricing at speeds as fast as if the TPDF were known in a closed form. We illustrate the computational efficiency of the proposed neural approximations of TPDFs by inserting them into numerical option pricing methods. We demonstrate a wide range of applications including the Black-Scholes-Merton model, the standard Heston model, the SABR model, and jump-diffusion models. These numerical experiments confirm the ultra-fast speed and high accuracy of the developed neural TPDF generators. This paper was accepted by Kay Giesecke, finance. Funding: H. Su received research funding support from Nottingham Business School at Nottingham Trent University. M. V. Tretyakov was supported by the Engineering and Physical Sciences Research Council [Grant EP/X022617/1]. D. P. Newton received research funding support from the School of Management at Bath University. Supplemental Material: The online appendices and data files are available at https://doi.org/10.1287/mnsc.2022.01448 .

<span>The Ulam-Hyers stability of 2-variable radical functional equations in quasi-Banach spaces</span>

The purpose of this study is to prove Ulam-Hyers stability of 2-variable radical functional equations in quasi-Banach spaces. As a consequence of the main result, we get an outcome on the stability of such functional equations in Banach spaces.

Effective action approach to the dynamical map

The dynamical map represents a fundamental concept in quantum field theory, providing a solution of the field equations in the Fock space of asymptotic fields. In this paper, we show how to express the dynamical map of a scalar field in the language of quantum effective action. This grants us new insights into the study of topological defects in quantum field theory by showing a connection between the usual least-action principle and Umezawa’s boson transformation method.

A novel Adomian natural decomposition method with convergence analysis of nonlinear time-fractional differential equations

ABSTRACT The objective of the current article is to use the Banach fixed point theorem to present proofs for the existence and uniqueness theorems for a nonlinear time-fractional differential equation, namely, the linear and nonlinear, homogeneous and nonhomogeneous time-fractional diffusion equations. In addition, we explore exact solutions to nonlinear time fractional partial differential equations using an effective method called the Adomian natural decomposition method (ANDM), which is a combination of the Adomian decomposition method (ADM) and the natural transform method (NTM). To demonstrate the effectiveness of the suggested scheme, three examples are provided along with graphs and numerical tables to support our work. The outcomes demonstrate how effortless and effective the ANDM is for addressing fractional partial differential equations in multi-dimensional spaces, some of which we examined in this study as special cases. AMS Classifications 34A08, 65P99, 49J15, 35R11, 26A33, 74G10.

Qualitative properties of fractional convolution elliptic and parabolic operators in Besov spaces

The maximal Bp,qs\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_{p,q}^{s}$$\\end{document}-regularity properties of a fractional convolution elliptic equation is studied. Particularly, it is proven that the operator generated by this nonlocal elliptic equation is sectorial in Bp,qs\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ B_{p,q}^{s}$$\\end{document} and also is a generator of an analytic semigroup. Moreover, well-posedeness of nonlocal fractional parabolic equation in Besov spaces is obtained. Then by using the Bp,qs\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_{p,q}^{s}$$\\end{document}-regularity properties of linear problem, the existence, uniqueness of maximal regular solution of corresponding fractional nonlinear equation is established.

Research on fuzzy adaptive GPC of average coolant temperature based on regulation of boric acid concentration

In order to keep the balance of power distribution in the core of a nuclear reactor, the method of adjusting the boric acid concentration to regulate the average temperature of the coolant has been adopted to track the demands of power changes in the core. However, the chemical reaction process of regulating the average coolant temperature by adjusting the boric acid concentration is too slow to better meet the demands of power changes. In view of this challenge, this paper proposes an innovative fuzzy adaptive incremental generalized predictive control strategy to optimize the process of regulating the coolant temperature. This strategy employs a dynamic adjustment of control parameters, specifically through the application of incremental factors γ, which is calibrated using a fuzzy logic-based algorithm to enhance system responsiveness and robustness in real-time. The strategy aims to improve the efficiency and responsiveness of the nuclear reactor control mechanism and further enhance the safety and stability of nuclear power generation.Firstly, the system state space equations are established based on the boric acid concentration equation and the nuclear reactor kinetics. Secondly, the step factors β and incremental factors γ are used in improved-GPC to improve control real-time and robustness. The step factor β speeds up the rolling optimization, and the incremental factor γ is adjusted based on the fuzzy algorithm according to the online recognition degree of the system. Finally, the 1000 MW Pressurized Water Reactor is modeled and simulated. The results show that improved-GPC has a shorter regulation time and a stronger ability to cope with the model mismatch in this system than other controllers.

Singular backward stochastic Volterra integral equations in infinite dimensional spaces

In this paper, the notion of singular backward stochastic Volterra integral equations (singular BSVIEs for short) in infinite dimensional space is introduced, and the corresponding well-posedness is carefully established. A class of singularity conditions are proposed, which not only cover that of fractional kernel, Volterra Heston model kernel, completely monotone kernels, to mention a few, but also happen to be used in the forward stochastic Volterra integral with new conclusions arising. Motivated by mathematical physics problem such as the viscoelasticity/thermoviscoelasticity of materials, heat conduction in materials with memory, optimal control problems of abstract stochastic Volterra integral equations (including fractional stochastic evolution equations and stochastic evolutionary integral equations) are presented. At last, our BSVIEs are surprisingly used in maximum principle of controlled stochastic delay evolution equations. One advantage of this new standpoint is that the final cost functional can naturally depend on the past state.

A Note on the Convergence of Multigrid Methods for the Riesz–Space Equation and an Application to Image Deblurring

In recent decades, a remarkable amount of research has been carried out regarding fast solvers for large linear systems resulting from various discretizations of fractional differential equations (FDEs). In the current work, we focus on multigrid methods for a Riesz–Space FDE whose theoretical convergence analysis of such multigrid methods is currently limited in the relevant literature to the two-grid method. Here we provide a detailed theoretical convergence study in the multilevel setting. Moreover, we discuss its use combined with a band approximation and we compare the result with both τ and circulant preconditionings. The numerical tests include 2D problems as well as the extension to the case of a Riesz–FDE with variable coefficients. Finally, we investigate the use of a Riesz–Space FDE in a variational model for image deblurring, comparing the performance of specific preconditioning strategies.

A regularization strategy to a backward problem for general nonlinear parabolic equations in Sobolev spaces

This article is devoted to solving a backward problem for general nonlinear parabolic equations in Sobolev spaces. The problem is hardly solved by computation since it is severely ill‐posed in Hadamard's sense. Using the Fourier transform and the truncation of high‐frequency components, we construct a regularized solution for the problem from the data given inexactly. As usual for ill‐posed problems, the rate of convergence can be obtained under additional smoothness assumptions on the regularity of the exact solution, the so‐called source conditions. By using a source condition in a generalized form, we prove a convergence estimate in a Sobolev space . Corresponding to different levels of the source condition, the convergence rates are improved gradually. We apply our theory to the backward problem with a convection–diffusion operator and a cubic nonlinearity. Finally, the numerical results are presented to test the validity of the proposed regularization method.

Interpreting systems of continuity equations in spaces of probability measures through PDE duality

We introduce a notion of duality solution for a single or a system of transport equations in spaces of probability measures reminiscent of the viscosity solution notion for nonlinear parabolic equations. Our notion of solution by duality is, under suitable assumptions, equivalent to gradient flow solutions in case the single/system of equations has this structure. In contrast, we can deal with a quite general system of nonlinear non-local, diffusive or not, system of PDEs without any variational structure.

Weak solutions for the coupled system of Riemann-Stieltjes integral equations

We present an existence theorem for at least one weak solution for a coupled system of Riemann-Stieltjes integral equations in a reflexive Banach space . As an application, we study the existence of weak solutions for the coupled system of Hammerstien-Stieltjes integral equations in