Cauchy Initial Value Problem Research Articles

Singular Cauchy Initial Value Problem for Certain Classes of Integro-Differential Equations

The existence and uniqueness of solutions and asymptotic estimate of solution formulas are studied for the following initial value problem: , , , where is a constant and . An approach which combines topological method of T. Wazewski and Schauder's fixed point theorem is used.

Singular Cauchy Initial Value Problem for Certain Classes of Integro-Differential Equations

Singular Cauchy Initial Value Problem for Certain Classes of Integro-Differential Equations

Time-asymptotic behaviour of a solution of the Cauchy initial-value problem for a conservation law with non-linear divergent viscosity

We study the time-asymptotic behaviour of a solution of the Cauchy initial-value problem for a conservation law with non-linear divergent viscosity. We shall prove that when a bounded measurable initial function has limits at , a solution of the Cauchy initial-value problem converges uniformly to a system of waves consisting of travelling waves and rarefaction waves, where the phase shifts of the travelling waves are allowed to depend on time. The rate of convergence is estimated under additional conditions on the initial function.

On the use of transfer and dispersion coefficient concepts in low-dimensional diffusion–convection-reaction models

The two most important concepts in transport phenomena in the past century are that of transfer and dispersion coefficient. While both concepts reduce the local degrees of freedom and provide a coarse-grained low-dimensional description of the transport process at the macroscale, their physical interpretations and mathematical descriptions differ. The transfer coefficient concept uses two concentration or temperature modes and upscales the microscale molecular effects as an exchange between the local and global scales, leading to hyperbolic models and Cauchy (initial value) problems. The dispersion coefficient concept uses a single mode and incorporates the microscale effects as an effective diffusion or dispersion term in the macroscale transport equation and leads to parabolic models and requires additional boundary conditions or physical constraints. We examine here the application of these concepts for reacting systems and provide some guidelines for their proper use in the low-dimensional description of reacting as well as non-reacting systems.

Sensitivity Analysis of Wave-equation Tomography: A Multi-scale Approach

Earthquakes, viewed as passive sources, or controlled sources, like explosions, excite seismic body waves in the earth. One detects these waves at seismic stations distributed over the earth’s surface. Wave-equation tomography is derived from cross correlating, at each station, data simulated in a reference model with the observed data, for a (large) set of seismic events. The times corresponding with the maxima of these cross correlations replace the notion of residual travel times used as data in traditional tomography. Using first-order perturbation, we develop an analysis of the mapping from a wavespeed contrast (between the “true” and reference models) to these maxima. We develop a construction using curvelets, while establishing a connection with the geodesic X-ray transform. We then introduce the adjoint mapping, which defines the imaging of wavespeed variations from “finite-frequency travel time” residuals. The key underlying component is the construction of the Frechet derivative of the solution to the seismic Cauchy initial value problem in wavespeed models of limited smoothness. The construction developed in this paper essentially clarifies how a wavespeed model is probed by the method of wave-equation tomography.

Einstein-Maxwell system in3+1form and initial data for multiple charged black holes

We consider the Einstein-Maxwell system as a Cauchy initial value problem taking the electric and magnetic fields as independent variables. Maxwell's equations in curved spacetimes are derived in detail using a $3+1$ formalism and their hyperbolic properties are analyzed, showing that the resulting system is symmetric hyperbolic. We also focus on the problem of finding initial data for multiple charged black holes assuming time-symmetric initial data and using a puncturelike method to solve the Hamiltonian and the Gauss constraints. We study the behavior of the resulting initial data families, and show that previous results in this direction can be obtained as particular cases of our approach.

Hamiltonian description of radiation phenomena: Trautman-bondi energy and corner conditions

Cauchy initial value problem on a hyperboloid is proved to define a Hamiltonian system, provided the radiation data at null infinity are also taken into account, as a part of Cauchy data. The "Trautman-Bondi mass", supplemented by the "already radiated energy" assigned to radiation data, plays role of the Hamiltonian function. This approach leads to correct description of the corner conditions.

Vektorfeldalgebren, Gruppenaktionen und das Keplerproblem

AbstractWe supply an analytic proof of the theorem of Palais about the existence of a Lie group action on a compact manifold M with a Lie algebra 𝔳 of vector fields. Every compact connected integral manifold N of 𝔳 is then diffeomorphic to a homogeneous manifold. In the special case of a Lie algebra of Hamiltonian vector fields, the group action consists of symplectic diffeomorphisms. Applications are the non‐commutative version of the Cauchy initial value problem for Hamilton‐Jacobi systems and of Liouville's theorem. As examples with non‐compact M, we determine group actions for the harmonic oscillator and for Kepler's problem.

Existence and uniqueness for fractional neutral differential equations with infinite delay

In this paper, the Cauchy initial value problem is discussed for the fractional neutral functional differential equations with infinite delay and various criteria on existence and uniqueness are obtained.

Existence and uniqueness for [formula omitted]-type fractional neutral differential equations

In this paper, the Cauchy initial value problem is discussed for the p -type fractional neutral functional differential equations and various criteria on existence and uniqueness are obtained.

A highly unstable initial value boundary value problem

In this paper we consider a two-dimensional diffusion equation on the closed right halfspace R + 2 ¯ satisfying a boundary condition for which the minimum principle fails. As a consequence, the associated Cauchy initial value problem fails to be well-posed. In particular, solutions need not exist and, when they do exist, they may do so for only a finite length of time. Among other things, we provide a necessary and sufficient condition on the initial data in order that the solution exist for all time and remain non-negative.

Integration of the Continual System of Nonlinear Interaction Waves

We derive the continual system of nonlinear interaction waves from the compatibility of the transport equation on the whole line and the equation governing the time-evolution of the eigenfunctions of the transport operator. The transport equation represents the continual generalization from the n-component system of first-order ordinary differential equations. The continual system describes a nonlinear interaction of waves. We prove that the continual system can be integrated by the inverse scattering method. The method is based on applying the results of the inverse scattering problem for the transport equation to finding the solution of the Cauchy initial-value problem for the continual system. Indeed, the transition operator for the scattering problem admits right and left Volterra factorizations. The intermediate operator for this problem determines the one-to-one correspondence between the preimages of a solution of the transport equation. This operator is related to the transition operator and admits not only right and left Volterra factorizations but also the analytic factorization. By virtue of this fact the potential in the transport equation is uniquely reconstructed in terms of the solutions of the fundamental equations in inverse problem.

Solution of the Cauchy problem for a time-dependent Schrödinger equation

We construct an explicit solution of the Cauchy initial value problem for the n-dimensional Schrödinger equation with certain time-dependent Hamiltonian operator of a modified oscillator. The dynamical SU(1,1) symmetry of the harmonic oscillator wave functions, Bargmann’s functions for the discrete positive series of the irreducible representations of this group, the Fourier integral of a weighted product of the Meixner–Pollaczek polynomials, a Hankel-type integral transform, and the hyperspherical harmonics are utilized in order to derive the corresponding Green function. It is then generalized to a case of the forced modified oscillator. The propagators for two models of the relativistic oscillator are also found. An expansion formula of a plane wave in terms of the hyperspherical harmonics and solution of certain infinite system of ordinary differential equations are derived as by-products.

Nonlinear Generalization of Schrödinger's Equation Uniting Quantum Mechanics and Thermodynamics

The nonlinear equation of motion that accomplishes a self-consistent unification of quantum mechanics (QM) and thermodynamics conceptually different from the (von Neumann) foundations of quantum statistical mechanics (QSM) and (Jaynes) quantum information theory (QIT), but which reduces to the same mathematics for the thermodynamic equilibrium (TE) states, and contains standard QM in that it reduces to the time-dependent Schrodinger equation for zero entropy states is discussed in full mathematical detail. By restricting the discussion to a strictly isolated system (noninteracting, disentangled and uncorrelated), we show how the theory departs from the conventional QSM/QIT rationalization of the second law of thermodynamics, which instead emerges in QT (quantum thermodynamics) as a theorem of existence and uniqueness of a stable equilibrium state for each set of mean values of the energy and the number of constituent particles. To achieve this, the theory assumes Trρ lnρ B −k for the physical entropy and is designed to implement two fundamental ansatzs: (1) that in addition to the standard QM states described by idempotent density operators (zero entropy), a strictly isolated and uncorrelated system admits also states that must be described by non-idempotent density operators (nonzero entropy); (2) that for such additional states the law of causal evolution is determined by the simultaneous action of a Schrodinger-von Neumann-type Hamiltonian generator and a nonlinear dissipative generator which conserves the mean values of the energy and the number of constituent particles, and in forward time drives the density operator in the 'direction' of steepest entropy ascent (maximal entropy generation). The resulting dynamics is well defined for all non-equilibrium states, no matter how far from TE. Existence and uniqueness of solutions of the Cauchy initial value problem for all density operators implies that the equation of motion can be solved not only in forward time, to describe relaxation towards TE, but also in backward time, to reconstruct the 'ancestral' or primordial lowest entropy state or limit cycle from which the system originates. Zero entropy states as well as a well defined family of non-dissipative states evolve unitarily according to pure Hamiltonian dynamics and can be viewed as unstable limit cycles of the general nonlinear dynamics.

Propagator of a Charged Particle with a Spin in Uniform Magnetic and Perpendicular Electric Fields

We construct an explicit solution of the Cauchy initial value problem for the time-dependent Schrödinger equation for a charged particle with a spin moving in a uniform magnetic field and a perpendicular electric field varying with time. The corresponding Green function (propagator) is given in terms of elementary functions and certain integrals of the fields with a characteristic function, which should be found as an analytic or numerical solution of the equation of motion for the classical oscillator with a time-dependent frequency. We discuss a particular solution of a related nonlinear Schrödinger equation and some special and limiting cases are outlined.

The fundamental solution of the time-dependent system of crystal optics

We set up the electromagnetic system and its plane-wave solutions with the associated slowness and wave surfaces. We treat the Cauchy initial-value problem for the electric vector and make explicit the quantities necessary for numerical evaluation. We use the Herglotz-Petrovskii representation as an integral around loops which, for each position and time form the intersection of a plane in the space of slownesses with the slowness surface. The field and especially its singularities are strongly dependent on the varying geometry of these loops; we use a level set numerical technique to compute those real loops which essentially give us second order accuracy. We give the static term corresponding to the mode with zero wave speed. Numerical evaluation of the solution is presented graphically followed by some concluding remarks.

On the thermodynamics, stability and hierarchy of entropy functions in fluid flow

Symmetric forms of systems of equations describing physical phenomena present exceptional desirable properties such as symmetric coefficient matrices, well posedness of the Cauchy initial value problem, finite wave propagation speeds and a link to nonlinear stability principles. Therefore, they should be used when possible to design better numerical methods. In this paper such symmetric forms are reviewed for various models of fluid flow, and a hierarchy of generalized entropy functions is presented. Standard concepts are extended to the diffusive terms.

A nonlinear boundary problem by the inventory’s optimization of a productive system with not wholly quadratic costs

In this paper a costs functional is assumed asymmetrical for the production and the inventory charges: As a state variable, the deviation u(t) = Iid(t)− Ire(t) between the inventories has been chosen. Obviously the manufacturer wishes to keep the cost-functional C(u) as a minimum. The relevant Euler-Lagrange time-equation ü(t) =α2u(t) + 2u(t)3, is faced with the initial condition u(0) = u0 ≠ 0 ; and with the final condition u(T) = 0 meaning that after a time T > 0 the deviation u(t) is re-absorbed at all. In the article the above nonlinear boundary value problem is solved through a much more tractable nonlinear (Cauchy) initial value problem, plus a “shooting”. The integration is carried out by means of the Jacobi elliptic functions, which do here their appearance in a micro-economic context for the first time. After having so detected the “optimal” deviation time law u(t) , a new production plan is then designed for driving the deviation till to its extinction within T .

Fundamental Solutions for Zero-Coupon Bond Pricing Models

The transformation approach is employed to reduce one-factor bond-pricing equations into the heat equation for which the fundamental solution is wellknown. These transformations are subsequently used to construct the fundamental solutions of two zero-coupon bond-pricing equations. The closed-form analytical solutions of the Cauchy initial value problems of the two bond-pricing model equations are then obtained.

Ansatz solutions to the Cauchy initial value problem of the nonlinear particle transport theory

Explicit analytical solutions to a nonlinear spatially homogeneous model Boltzmann equation are presented for a physical situation of reference, characterized by constant collision frequencies and scattering probability distributions depending only on the velocity after collision. It is shown that these rigorous solutions can, alternatively, be expressed in terms of the incomplete beta-function, the Gaussian hypergeometric function or the Fox H-function.