Formulation Of Initial Value Problem Research Articles

ON THE NONCHARACTERISTICITY OF A CAUCHY SURFACE AND APPLICATIONS

In mathematical physics, Cauchy surfaces play a crucial role in defining initial data for the evolution of physical systems. The characteristic nature of Cauchy surface means that information about the future evolution of the system can be obtained from the data on the surface itself. However, not all Cauchy surfaces are characteristic. This article aims to investigate the noncharacteristic nature of Cauchy surfaces and their applications in various physical scenarios. We demonstrate that the noncharacteristicity of a Cauchy surface has important implications for the behavior of physical systems, including the existence of singularities and the formulation of initial value problems. We explore the applications of noncharacteristic Cauchy surfaces in classical and quantum field theory, general relativity, and quantum gravity. Our analysis provides a deeper understanding of the mathematical and physical properties of Cauchy surfaces and their significance in the study of fundamental physical phenomena. In this work we show that under the noncharacteristic condition one can prove a local existence result for the Cauchy problem similar to the semi-linear and quasilinear cases. For this, we consider the Eikonal equation and Hamilton-Jacobi equations. Therefore, we solve simultaneously for the characteristics and the solution value, resulting in curves that live in the solution graph.

Black hole perturbations in higher-order scalar–tensor theories: initial value problem and dynamical stability

We propose a physically sensible formulation of initial value problem for black hole perturbations in higher-order scalar–tensor theories. As a first application, we study monopole perturbations around stealth Schwarzschild solutions in a shift- and reflection-symmetric subclass of degenerate higher-order scalar–tensor (DHOST) theories. In particular, we investigate the time evolution of the monopole perturbations by solving a two-dimensional wave equation and analyze the Vishveshwara’s classical scattering experiment, i.e., the time evolution of a Gaussian wave packet. As a result, we confirm that stealth Schwarzschild solutions in the DHOST theory are dynamically stable against the monopole perturbations with the wavelength comparable or shorter than the size of the black hole horizon. We also find that the damped oscillations at the late time do not show up unlike the ringdown phase in the standard case of general relativity. Moreover, we investigate the characteristic curves of the monopole perturbations as well as a static spherically symmetric solution with monopole hair.

Non-conservative variational approximation for nonlinear Schrödinger equations

In the work of Galley (Phys Rev Lett 110:174301, 2013) an initial value problem formulation of Hamilton’s principle was proposed and applied to non-conservative systems. Here, we explore this formulation for complex partial differential equations of the nonlinear Schrodinger (NLS) type, examining the dynamics of the coherent solitary wave structures of such models by means of a non-conservative variational approximation (NCVA). We compare the formalism of the NCVA to two other variational techniques used in dissipative systems, namely the perturbed variational approximation and a generalization of the so-called Kantorovich method. All three variational techniques produce equivalent equations of motion for the perturbed NLS models studied herein. We showcase the relevance of the NCVA method by exploring test case examples within the NLS setting including combinations of linear and density-dependent loss and gain. We also present an example applied to exciton–polariton condensates that intrinsically feature loss and a spatially dependent gain term.

Existence Theorem for Impulsive Differential Equations with Measurable Right Side for Handling Delay Problems

Due to noncontinuous solution, impulsive differential equations with delay may have a measurable right side and not a continuous one. In order to support handling impulsive differential equations with delay like in other chapters of differential equations, we formulated and proved existence and uniqueness theorems for impulsive differential equations with measurable right sides following Caratheodory’s techniques. The new setup had an impact on the formulation of initial value problems (IVP), the continuation of solutions, and the structure of the system of trajectories. (a) We have two impulsive differential equations to solve with one IVP (φσ0=ξ0) which selects one of the impulsive differential equations by the position of σ0 in a,bν. Solving the selected IVP fully determines the solution on the other scale with a possible delay. (b) The solutions can be continued at each point of α,β×Ω0≕Ω by the conditions in the existence theorem. (c) These changes alter the flow of solutions into a directed tree. This tree however is an in-tree which offers a modelling tool to study among other interactions of generations.

Numerical initial data deformation exploiting a gluing construction: I. Exterior asymptotic Schwarzschild

In this work a new numerical technique to prepare Cauchy data for the initial value problem formulation of Einstein’s field equations is presented. Directly inspired by the exterior asymptotic gluing result of Corvino (2000) our (pseudo)-spectral scheme is demonstrated under the assumption of axisymmetry so as to fashion composite Hamiltonian constraint satisfying initial data featuring internal binary black holes as glued to exterior Schwarzschild initial data in isotropic form. The generality of the method is illustrated in a comparison of the ADM mass of gluedinitial data sets featuring internal binary black holes as modelled by Brill–Lindquist and Misner data. In contrast to the recent work of Doulis and Rinne (2016); Pook-Kolb and Giulini (2018) we do not make use of the York–Lichnerowicz conformal framework to reformulate the constraints.

Whitham modulation theory for the two-dimensional Benjamin-Ono equation.

Whitham modulation theory for the two-dimensional Benjamin-Ono (2DBO) equation is presented. A system of five quasilinear first-order partial differential equations is derived. The system describes modulations of the traveling wave solutions of the 2DBO equation. These equations are transformed to a singularity-free hydrodynamic-like system referred to here as the 2DBO-Whitham system. Exact reductions of this system are discussed, the formulation of initial value problems is considered, and the system is used to study the transverse stability of traveling wave solutions of the 2DBO equation.

Initial Value Problem Formulation TDBEM with 4-D Domain Decomposition Method and Application to Wake Fields Analysis

Initial Value Problem Formulation TDBEM with 4-D Domain Decomposition Method and Application to Wake Fields Analysis

Initial Value Problem Formulation of 3-D Time Domain Boundary Element Method

A time domain boundary element method (TDBEM) gives us another possibility of time domain microwave simulations in addition to a finite difference time domain (FDTD) method. In particular, the TDBEM has good advantages in analysis of coupling problems with charged particle motion such as in a particle accelerator. However, it is known that time domain microwave simulations in the particle accelerator by the conventional TDBEM often encounter numerical instability and inaccuracy because of its bad matrix property. To avoid the numerical instability and inaccuracy caused by the conventional open boundary problem formulation of the TDBEM, an initial value problem formulation of 3-D TDBEM is presented in this paper.

Variational method for nonconservative field theories: Formulation and twoPT-symmetric case examples

In a recent publication [Galley, Phys. Rev. Lett. 110, 174301 (2013)], Galley proposed an initial value problem formulation of Hamilton's principle that enables consideration of dissipative systems. Here we explore this formulation at the level of field theories with infinite degrees of freedom. In particular, we illustrate that it affords a previously unwarranted and appealing as well as broadly relevant possibility, namely, to generalize the popular collective coordinate or variational method to open systems, i.e., nonconservative ones. To showcase the relevance or validity of the method we explore two case examples from the timely area of $\mathcal{PT}$-symmetric variants of field theories, in this case for a sine-Gordon and for a ${\ensuremath{\phi}}^{4}$ model.

Simflowny: A general-purpose platform for the management of physical models and simulation problems

Simflowny is a software platform which aims to formalize the main elements of a simulation flow. It allows users to manage (i) formal representations of physical models based on Initial Value Problems (hyperbolic, parabolic and mixed-type partial differential equations), (ii) simulation problems based on such models, and (iii) discretization schemes to translate the problem to a finite mesh. Additionally, Simflowny generates automatically code for general-purpose simulation frameworks. This paper first presents an introductory example of such problems. Then, formal representations are explained. Afterwards, it summarizes the platform’s architecture. Finally, validation results are provided. Program summaryProgram title: SimflownyCatalogue identifier: AEPL_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEPL_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Modified BSDNo. of lines in distributed program, including test data, etc.: 5539418No. of bytes in distributed program, including test data, etc.: 418007334Distribution format: tar.gzProgramming language: Java, C++, JavaScript, and Fortran.Computer: Any Linux platform ranging from personal workstations to clusters and parallel supercomputers.Operating system: Linux.Has the code been vectorized or parallelized?: MPI basedRAM: Problem dependentSupplementary material: A virtual machine file is available.Classification: 4.3.External routines: HDF5 [1], MPI [2]Nature of problem: Any problem based on an Initial Value Problem formulation (hyperbolic, parabolic or mixed type partial differential equations).Solution method: Any discretization scheme based on either Finite Volume Methods, Finite Difference Methods, or meshless methods.Additional comments: The distribution file for this program is over 410 MB and therefore is not delivered directly when Download or Email is requested. Instead an html file giving details of how the program can be obtained is sent.Full documentation is provided in the distribution file.Running time: Problem dependent.

Initial value problem formulation of time domain boundary element method for electromagnetic microwave simulations

To aim to obtain more stable solutions and wider area applications for the Time Domain Boundary Element Method (TDBEM), initial value problem formulation of the TDBEM is newly introduced for microwave simulations. The initial value problem formulation of the TDBEM allows us to solve transient microwave phenomena as interior region problems, which gives us well matrix property and interior resonance free solutions. This paper concentrates on applying the initial value problem formulation of the TDBEM to wake field phenomena in particle accelerator cavities.

Functional calculus via Laplace transform and equations with infinitely many derivatives

We study nonlocal linear equations of the form f(∂t)ϕ=J(t), t≥0, in which f is an entire function. We develop an appropriate functional calculus via Laplace transform, we solve the aforementioned equation completely in the space of exponentially bounded functions, and we analyze the delicate issue of the formulation of initial value problems for nonlocal equations.

System Solver: an open source tool for mathematically modelling dynamical systems

The following paper presents a freeware modelling tool simulating dynamic systems that can be represented by either an ordinary differential equation (ODE) or a set of differential equations of different orders. The main idea leading to this software development is related to the fact that many physical, biological, ecological, economical, chemical, social and engineering problems can be expressed in this way. Furthermore, the solution to these problems requires some expertise in numerical methods and programming. Such knowledge is uncommon in some of the experts in such scientific domains. A tool to fill in this knowledge gap, increase productivity within modelling-related research and support the teaching of mathematical modelling topics is thus needed. This paper introduces System Solver, a computer application that facilitates the formulation of initial value problems for ODE systems, numerically solves these problems and provides a user with not only the solution but also debugged Visual Basic source code for the application. The obtained code can be easily shared among researchers, which facilitates the replication of numerical experiments even across different operating systems. This software's introduction is accompanied by examples from different domains, including one example from stochastic modelling.

Asymptotic behavior of recursions via fixed point theory

In this paper we provide a formulation of initial value problems for (explicit and implicit) difference equations in terms of abstract equations in sequence spaces. They will be solved using appropriate fixed point theorems and we obtain quantitative attractivity properties.

On the formulation of initial-value problems for systems consisting of relativistic particles

We discuss questions related to the well-posedness of problems on the motion of relativistic many-body systems. For one-dimensional relativistic motion of N similar charges, we prove that an ordinary Cauchy problem usual in Newton mechanics can be stated; this is done in the framework of microscopic Maxwell-Lorentz electrodynamics (including a model with self-action) or Wheeler-Feynman theory.

Application of multiple scales analysis and the fundamental matrix method to rugate filters: initial-value and two-point boundary problem formulations

In this paper, the filtering problem of apodized rugates is solved by deriving first-order, as well as second-order, coupled-mode equations via the perturbation method of multiple scales. The first-order perturbation equations are the same as those of coupled-mode theory. However, the second-order perturbation expansion is more accurate, and permits the use of larger amplitudes of the periodic index variation of the rugate. The coupled-mode equations are solved numerically by using two different formulations. The first approach is a two-point boundary-value problem formulation, based on the fundamental matrix solution, that is essentially the exact solution for the unapodized rugate. The second approach is an initial-value problem formulation, that uses backward integration of the coupled-mode equations. Comparison with the characteristic matrix method is made for the case of unapodized rugate in terms of speed and accuracy, and it is found that the fundamental matrix solution is the fastest. The accuracy of the multiple scales solution is measured in terms of the amplitude error and the phase error of the filter's spectral response, taking the characteristic matrix solution as a reference for the unapodized rugate. The proposed formulations are utilized to calculate the spectral response of apodized rugates.

Efficient Newton-Raphson and implicit Euler methods for solving the HNC equation

In this paper a new Newton-Raphson (NR) method and a new implicit Euler method are developed for numerically solving the HNC and related equations for simple fluids. The new methods are efficient, stable and easy to implement. The implicit Euler method is derived through a reformulation of the usual HNC problem into the form of an initial value problem (IVP). The IVP formulation allows application of specialized numerical techniques developed for the solution of stiff systems of differential equations. The implicit Euler method is shown to have excellent stability and rate of convergence. The relationship between the implicit Euler method and the NR method is elucidated, and it is shown that in the limit of infinite step-size, the implicit Euler method reduces to the NR method.

Numerical solutions to vibroimpact via an initial value problem formulation

The dynamics of vibroimpact is posed in terms of an initial value problem for a vertical impact pair, which consists of a primary mass constrained to move vertically within a massive hollow shell. Harmonic excitation drives the shell, while gravity and shell contact govern the dynamic response of the primary mass. This non-linear dynamics problem is formulated as an initial-value problem, which is numerically solved on a personal computer. The equations of motion for the separate domains controlling the mass are expressed as functional relationships; therefore, no numerical time integration is necessary. A numerical time-stepping method determines which of three domains controls mass motion at a given time and applies the appropriate equations. Also, the exact time of transition from one domain to another is iteratively calculated to within a small tolerance. The physical response of the primary mass is presented for several excitation conditions, and the accuracy of these solutions is investigated through numerical tests. Experimental results for a specific vertical impact pair system testify to the accuracy of the solution technique.