Simple Initial Value Problem Research Articles

Sparse successive approximation for nonlinear [formula omitted] and [formula omitted] optimal control problems under residual errors

Successive approximation techniques are effective approaches to solve the Hamilton–Jacobi–Bellman (HJB)/Hamilton–Jacobi–Isaacs (HJI) equations in nonlinear H2 and H∞ optimal control problems (OCPs), but residual errors in the solving process may destroy its convergence property, and related numerical methods also pose computational burden and difficulties. In this paper, the HJB/HJI partial differential equations (PDEs) for infinite-horizon nonlinear H2 and H∞ OCPs are handled in a unified formulation, and a sparse successive approximation method is proposed. Taking advantage of successive approximation techniques, the nonlinear HJB/HJI PDEs are transformed into sequences of easily solvable linear PDEs, to which the solutions can be computed point-wise by handling simple initial value problems. Extra constraints are also incorporated in the solving process to guarantee the convergence under residual errors. The sparse grid based collocation points and basis functions are then employed to enable efficient numerical implementation. The performance of the proposed method is also numerically demonstrated in simulations.

Intrinsic Discontinuities in Solutions of Evolution Equations Involving Fractional Caputo–Fabrizio and Atangana–Baleanu Operators

There has been considerable recent interest in certain integral transform operators with non-singular kernels and their ability to be considered as fractional derivatives. Two such operators are the Caputo–Fabrizio operator and the Atangana–Baleanu operator. Here we present solutions to simple initial value problems involving these two operators and show that, apart from some special cases, the solutions have an intrinsic discontinuity at the origin. The intrinsic nature of the discontinuity in the solution raises concerns about using such operators in modelling. Solutions to initial value problems involving the traditional Caputo operator, which has a singularity inits kernel, do not have these intrinsic discontinuities.

Numerical investigation of the strength of collapse of a harmonically excited bubble

The nonlinear dynamics of an acoustically excited spherical gas bubble in water is being investigated numerically. The applied model to describe the motion of the bubble radius is the Keller–Miksis equation, a second order ordinary differential equation, which takes into account the compressibility of the liquid. During the radial oscillations of the bubble, it may enlarge and collapse violently causing high temperature and pressure or even launch a strong pressure wave at the collapse site. These extreme conditions are exploited by many applications, for instance, in sonochemistry to generate oxidising free radicals. The recorded properties, such as the very high bubble wall velocity, and maximum bubble radius of the periodic and chaotic solutions are good indicators for the strength of the collapse. The main aim is to determine the domains of the collapse-like behaviour in the excitation pressure amplitude–frequency parameter space. Results show that at lower driving frequencies the collapse is stronger than at higher frequencies, which is in good agreement with many experimental observations (Kanthale et al., 2007, Tatake and Pandit, 2002). To find all the co-existing stable solutions, at each parameter pair the model was solved numerically with a simple initial value problem solver (4th order Runge–Kutta scheme with 5th order embedded error estimation) by applying 5 randomly chosen initial conditions. These co-existing attractors have different behaviour in the sense of the collapse strength.

A Singular Initial-Value Problem for Second-Order Differential Equations

We are interested in the existence of solutions to initial-value problems for second-order nonlinear singular differential equations. We show that the existence of a solution can be explained in terms of a more simple initial-value problem. Local existence and uniqueness of solutions are proven under conditions which are considerably weaker than previously known conditions.

Modal analysis of magneto-electro-elastic plates using the state-vector approach

The state-vector approach is proposed to analyze the free vibration of magneto-electro-elastic laminate plates. The extended displacements and stresses can be divided into the so-called in-plane and out-of-plane variables. Once the state equation for the out-of-plane variables is obtained, a complex boundary value problem is converted into an equivalent simple initial value problem. Through the state equation, the propagator matrix between the top and bottom interfaces of every layer can be easily derived. The global propagator matrix can also be assembled using the continuity conditions. It is obvious that the order of global propagator matrix is not related to the number of layers. Consequently, this approach possesses certain virtues including simple formulation, less expensive computation, etc. To test the formulation, the developed solution is then applied to a simply supported multilayered plate constructed of piezoelectric and/or piezomagnetic materials. The natural frequencies and corresponding mode shapes are computed and compared with existing results. Furthermore, the fundamental modes along with a couple of other higher modes, which have never been reported in previous literature, are presented. Therefore, this completed set of frequencies and mode shapes can be used as benchmarks for future research in this field. It is also believed that the approach could be useful in the analysis and design of smart structures constructed from piezoelectric/piezomagnetic composites.

Computing optical bistability in one-dimensional nonlinear structures

Efficient numerical methods are developed to analyze optical bistability in a multilayer structure with Kerr nonlinearity. It corresponds to computing multiple solutions of the one-dimensional nonlinear Helmholtz equation for a range of frequencies and incident intensities. The input–output characteristics is calculated as a simple initial value problem without any iterations. Bistability intervals of the incident intensity are determined directly without using a continuation scheme. The complete transmission spectrum at a fixed incident intensity and a varying frequency is obtained including both stable and unstable solutions.

Fluctuation effects in disordered Peierls systems

We review the density of states (DOS) and related quantities of quasi one-dimensional disordered Peierls systems in which fluctuation effects of a backscattering potential play a crucial role. The low-energy behavior of non-interacting fermions which are subject to a static random backscattering potential will be described by the strictly one-dimensional fluctuating gap model (FGM). Recently, the FGM has also been used to explain the pseudogap phenomenon in high-Tc superconductors. We develop a non-perturbative method which allows for a simultaneous calculation of the DOS and inverse localization length for an arbitrary given disorder potential by solving a simple initial value problem. In the white noise limit, we recover all known results by solving a Fokker-Planck equation. For the physically interesting case of finite correlation lengths, we use analytical and numerical methods to show that a complex order parameter leads to a suppression of the DOS, i.e. a pseudogap, and that for a real order parameter this pseudogap is overshadowed by a singularity in the DOS. We will also consider the case of classical phase fluctuations which applies to low temperatures where amplitude fluctuations are frozen out. For this regime we present analytic results for the DOS, the inverse localization length, the specific heat, and the Pauli susceptibility.

Fluctuation effects in disordered Peierls systems

We review the density of states (DOS) and related quantities of quasi one-dimensional disordered Peierls systems in which fluctuation effects of a backscattering potential play a crucial role. The low-energy behavior of non-interacting fermions which are subject to a static random backscattering potential will be described by the strictly one-dimensional fluctuating gap model (FGM). Recently, the FGM has also been used to explain the pseudogap phenomenon in high-Tc superconductors. We develop a non-perturbative method which allows for a simultaneous calculation of the DOS and inverse localization length for an arbitrary given disorder potential by solving a simple initial value problem. In the white noise limit, we recover all known results by solving a Fokker-Planck equation. For the physically interesting case of finite correlation lengths, we use analytical and numerical methods to show that a complex order parameter leads to a suppression of the DOS, i.e. a pseudogap, and that for a real order parameter this pseudogap is overshadowed by a singularity in the DOS. We will also consider the case of classical phase fluctuations which applies to low temperatures where amplitude fluctuations are frozen out. For this regime we present analytic results for the DOS, the inverse localization length, the specific heat, and the Pauli susceptibility.

Fluctuation effects in disordered Peierls systems

AbstractWe review the density of states (DOS) and related quantities of quasi one‐dimensional disordered Peierls systems in which fluctuation effects of a backscattering potential play a crucial role. The low‐energy behavior of non‐interacting fermions which are subject to a static random backscattering potential will be described by the strictly one‐dimensional fluctuating gap model (FGM). Recently, the FGM has also been used to explain the pseudogap phenomenon in high‐Tc superconductors. We develop a non‐perturbative method which allows for a simultaneous calculation of the DOS and inverse localization length for an arbitrary given disorder potential by solving a simple initial value problem. In the white noise limit, we recover all known results by solving a Fokker‐Planck equation. For the physically interesting case of finite correlation lengths, we use analytical and numerical methods to show that a complex order parameter leads to a suppression of the DOS, i.e. a pseudogap, and that for a real order parameter this pseudogap is overshadowed by a singularity in the DOS. We will also consider the case of classical phase fluctuations which applies to low temperatures where amplitude fluctuations are frozen out. For this regime we present analytic results for the DOS, the inverse localization length, the specific heat, and the Pauli susceptibility.

Simple Initial Value Problems and Mechanisms for Baroclinic Growth

Abstract In this paper a simple 2D linear quasigeostrophic model is used to investigate how the development of local confined cyclonic perturbations is dependent on the perturbation scale, location, and tilt in Eady-type basic states. It is found that the initial growth of the perturbation can be maximized by reducing both the vertical and horizontal scale and using a “midtropospheric” vertical location. “Potential vorticity (PV) thinking” suggests the concept of “PV unshielding” to explain this result. Adding a meridional gradient of basic-state PV lowers the vertical location of perturbations that optimally excite sustained growth. This can be understood by considering the behavior of the upward and downward propagating parts of the initial perturbation. It is found that the importance of the initial perturbation tilt is diminished for confined perturbations. It is shown that diabatic heating in a vertically confined region can lead to a perturbation that exhibits rapid growth. The findings in this pape...

Exact numerical calculation of the density of states of the fluctuating gap model

We develop a powerful numerical algorithm for calculating the density of states rho(omega) of the fluctuating gap model, which describes the low-energy physics of disordered Peierls and spin-Peierls chains. We obtain rho(omega) with unprecedented accuracy from the solution of a simple initial value problem for a single Riccati equation. Generating Gaussian disorder with large correlation length xi by means of a simple Markov process, we present a quantitative study of the behavior of rho (omega) in the pseudogap regime. In particular, we show that in the commensurate case and in the absence of forward scattering the pseudogap is overshadowed by a Dyson singularity below a certain energy scale omega^{ast}, which we explicitly calculate as a function of xi.

PROGRESS IN THE GLOBAL ANALYSIS TO THE ATMOSPHERIC DYNAMICAL EQUATIONS

The initial value problem weather prediction is a classical deterministic problem in classical physics,there are a lot of difficulties in the method of initial value problem to predict long range weather and short range climate.Three characteristics of atmospheric motion are very important:nonlinearity,being forced and dissipation. It is necessary to combine the dynamical and statistical methods.The prediction problem could be looked as the inverse problem of the initial value problem,i.e. an evolution problem,so the multiple time data in the past could be used to modify initial values and/or numerical prediction model.The inverse problems are divided into there kinds,there are corresponding solutions. After the inverse problem of the initial value problem is introduced,the global analysis to the atmospheric dynamical equations(ADE) are reviewed. The global analysis is another method of climate research different from the simple initial value problem. The global analysis is also called global asymptotic analysis.It is used to study the long range behavior of deterministic systems.It is unnecessary to solve the equations,the properties of ADE are researched by means of studying the equations themselves.With the aid of the direct geometric pictures(the concept of phase space is introduced),the qualitative global analysis to the limit solution set of ADE could be done.It is proved that ADE is a special operator equation in Hilbert Space.The properties of operators must be maintained when ADE is simplified or discretized.It is revealed that the nonlinear adaptation process of a system to external source,so the system could be described by a small number of degrees of freedom sustaining attractors because of dissipative process.Cell to cell mapping is a powerful tool to study nonlinear systems,by use of the concept and method of cell to cell mapping,the global analysis to the general characteristics of numerical model of atmospheric system could be done.The global analysis to round off errors and observational errors could be done.With the aid of the language of probability theory,the statistical description of the characteristics could be given.It is revealed that the indeterminacy of individual state and the determinacy of global state.The restrict mathematical definition of the vague concept of “CLIMATE”is:climate could be defined as probability distribution function of state variables in chaotic attractors determined by controlling variables,and it is proved that it exists chaotic state in climate system really.The approach of quantitative studies to the predictability of the atmosphere could be given,the predictability is the time before the state of a system arrive the probability distribution of chaotic attractors.

The stirred tank forced

We examine the response of a continuous stirred tank reactor (CSTR), in which a simple irreversible reaction A → B occurs, to periodic variations of the coolant temperature. The amplitude and the frequency of the forcing are used as control parameters. The remaining operating conditions are chosen so that the unforced CSTR exhibits a single stable oscillation; this means that for small forcing amplitudes the system behaves like a typical forced oscillator: its response consists of alternating entrainment and quasi-periodicity. As the forcing amplitude grows, more characteristic traits of the CSTR and the particular forcing variable become apparent, and a complicated picture develops, involving the co-existence of multiple periodic, quasi-periodic and even chaotic oscillations, period doublings and global bifurcations. We track several of these traits numerically, placing special emphasis on the mechanisms by which such features branch out from the well-defined low-amplitude region. Several algorithms based on shooting methods for boundary value problems are used in this task, and some appropriate ways of tackling the simple initial value problem of simulation are also employed. In particular, the need for efficient algorithms to tackle global bifurcations is stressed. Such algorithms seem to be indispensable tools in the systematic study of the forced CSTR and, more generally, of periodically forced oscillators. The qualitative traits discovered for the forced CSTR are compared with other known results for model systems, and several questions present themselves as possible subjects of further research, both for the CSTR and for a wide class of periodically forced and/or coupled reactor models.

Why teach existence and uniqueness theorems in the first course in ordinary differential equations?

Two simple initial value problems are given which illustrate the need to apply standard existence and uniqueness theorems prior to generating numerical solutions to ordinary differential equations.

Weak pitch angle scattering of energetic protons in the magnetosphere

During the solar particle event of April 12–14, 1973, a flux depression of solar protons in the dawn side pseudotrapping region relative to the flux in the dusk side pseudotrapping region was frequently observed with the low-altitude polar-orbiting satellite Esro 4. This flux decrease is interpreted as being due to weak pitch angle scattering of the solar protons during their drift from dusk to dawn. Treating the pitch angle diffusion as a simple initial value problem, we derive quantitative values of the equatorial bounce-averaged pitch angle diffusion coefficients at small pitch angles for different energies. These values represent lower limits for magnetospheric conditions at the time of the Esro 4 observations. Through the assumption that the pitch angle diffusion is a gyroresonant interaction process of the particles with either Alfven or ion cyclotron waves, the necessary power of the waves at the resonant interaction frequency is calculated.

Toward Abstract Numerical Analysis

This paper deals with a technique for proving that certain problems of numerical analysis are numerically unsolvable. So that only methods which are natural for dealing with analytic problems may be presented, notions from recursive function theory have been avoided. Instead, the number of necessary function evaluations is taken as the measure of computational complexity. The role of topological concepts in the study of computability is examined. Last, a topological result is used to prove that a simple initial-value problem is numerically unsolvable.

Landau poles, branch cuts and supra-luminous waves in a one-dimensional plasma

A simple initial value problem is set up for a one-dimensional, mobile, electron plasma coexisting with a positive charged, immobile background. It is solved relativistically, a la Landau, and it is demonstrated that for a stable plasma the perturbation electric field consists of three contributions: (1) the usual damped Landau poles from waves with phase velocities less than the speed of light; (2) a damped branch cut contribution with branch points at +or-c; (3) undamped supra-luminous poles from waves with phase velocities greater than the speed of light. The modified Nyquist diagrams are sketched and, in the limit of long times, it is shown that the branch cut terms can, for particular plasma distribution functions, exceed the Landau pole terms.

Zur Reversibilität der Vlasov‐Gleichung

AbstractThe difference between Landaus decaying solutions of the linearized Vlasov equation and the damped solutions of a loaded electrical circuit is shown. An integral equation for the field acting in the plasma is investigated and the Laplace transforms of its solutions are considered. Some simple initial value problems for the Cauchy distribution for a perturbation by external fields too, are solved. It is demonstrated that the time arrow in the behaviour of measurable quantities of the Vlasov plasma stems from the accentuation of a non‐probable initial state.