Differential Equations In Space Research Articles

An infinite dimensional version of the Paley-Wiener-Schwartz isomorphism

In the infinite dimensional case it is known from a Dineen-Nachbin counterexample that the natural generalization of the Paley-Wiener-Schwartz theorem is no longer valid: The image through the Fourier transform F of the space ℰ′(E) is made of entire functions on E′ that, besides the usual inequality, satisfy a more technical condition. We define and study here a dense subspace, denoted by E(E), of ℰ(E) with a proper topology such that F (E′(E)) is made of the entire functions on E′ that satisfy only the classical inequality. For this reason this space E(E) is suited for the study of convolution and partial differential equations in spaces of C∞ functions on locally convex spaces.

Exact Derivation of a Generalized Master Equation for the Motion of Excitons

AbstractStarting from the Nakajima‐Zwanzig equation, by an exact evaluation of its kernel, a generalized master equation, describing the coherent motion of excitons is derived. The time dependence of the kernel is discussed in detail. From this coupled set of integro‐differential equations a set of second order differential equations is obtained. In the continuum approximation, this set results in a partial differential equation in space and time variables, which only for large times assumes the form of a wave equation. Explicit solutions of the various equations of motion are presented and discussed.

Lumped-parameter modelling and identification of distributed-parameter systems. Part 1: Discrete, parallel and cascade models

A method of characterising distributed-parameter systems using lumped discrete models is proposed. These models are classified as parallel and cascade, and they are based on spatial and temporal pulse responses derived from the Green's function associated with the boundary conditions of the system. A technique for representing such models using difference equations in space and time is given and procedures for using the models to derive the outputs in time and space from the system are described. The computer-simulation results are used to confirm the analysis.

Absolute stability of the solutions of linear differential equations with time lags in Banach space

The connection between the exponential stability of the solutions of linear differential equations in space of multidimensional bounded vector sequences and the absolute asymptotic stability of the solutions of differential equations with several time lags is investigated.

Conditions for the existence of clines.

A very general partial differential equation in space and time satisfied by the gene frequency in a monoecious population distributed continuously over an arbitrary habitat is derived. The treatment is restricted to a single diallelic locus in the absence of mutation and random drift, and it is supposed that time is continuous, births and deaths occur at random, and migration is independent of genotype. With the further assumptions that migration is isotropic and homogeneous, the population density is constant and uniform (as permitted by the population regulation mechanism included in the formulation), and Hardy-Weinberg proportions obtain locally, this partial differential equation reduces to the simplest multidimensional generalization of the classical Fisher-Haldane cline model. The efficacy of migration and selection in maintaining genetic variability at equilibrium in this model is investigated by deducing conditions for the existence of clines under various circumstances. The effects of the degree of dominance, a neutral belt between the regions where a particular allele is advantageous and deleterious, finiteness of the habitat, and habitat dimensionality are evaluated. Provided at least one of the alleles is favored only in a finite region, excluding the special case in which its total effective selective coefficient is zero, if conditions for supporting a cline are too unfavorable because migration is too strong, selection is too weak, or both, a cline cannot exist at all. Thus, unless there is overdominance, the population must be monomorphic. It is possible for a cline which can barely exist under the prevailing ecological circumstances to show a large amount of variation in gene frequency.

Use of Functional Approximation Methods in the Computer Solution of Initial Value Partial Differential Equation Problems

Methods of functional approximation for the computer solution of initial value partial differential equation problems provide a device by which these solutions can be approximated by those of initial value problems in sets of ordinary differential equations. A number of ways to achieve this have been suggested, some of them general, some of them utilizing specific properties of the equations at hand. What all these methods have in common is the fact that the solution u(x, t) of a partial differential equation in space x and time t is approximated by a function u*(a1(t), a2(t), . . ., aN(t)), where the dependence upon x is prescribed. In most applications, u* is linear in the ai(t), i.e., u*= =ai(t)fi(x). The ai(t) satisfy a set of ordinary differential equations obtained as the result of the approximation process. This system of ordinary differential equations is then integrated by classical analog or digital computer methods.

Constructing solutions to ordinary differential equations in space

Constructing solutions to ordinary differential equations in space

On the Stability of Linear Differential Equations in Spaces with an Indefinite Metric.

On the Stability of Linear Differential Equations in Spaces with an Indefinite Metric.

Time Variation of the Ginzburg-Landau Order Parameter

We use a nonequilibrium form of the Green's-function formulation of the BCS theory of superconductivity to investigate the circumstances under which differential equations in space and time, i.e., "time-dependent Ginzburg-Landau equations," give a valid description of the space and time variation of the order parameter $\ensuremath{\Psi}$ in superconductors. We find that if the variations are sufficiently slow, time-dependent Ginzburg-Landau equations exist near absolute zero and near the transition temperature. In the former case the equation has wave-like character, and in the latter case it is of diffusion type, with the restriction that either the characteristic frequency of the time variation of $\ensuremath{\Psi}$ is greater than the gap frequency or the ratio of the Fermi velocity to the product of the characteristic wavelength and frequency of the space-time variation of $\ensuremath{\Psi}$ is greater than unity. Under all other circumstances and at general temperatures, there are no differential equations to describe the variations of $\ensuremath{\Psi}$. We discuss also the influence of slowly varying time-dependent fields and derive the dependence of charge and current densities on the variations of $\ensuremath{\Psi}$. Local electrodynamics are assumed. The necessary modifications for the case of dilute superconducting alloys are described. Applications are made to the questions of collective modes, nucleation, and London's theory near absolute zero.

Lösungen des Dirichletschen Problems für elliptische Differentialgleichungen in Räumen mit Belegungsfunktionen

Lösungen des Dirichletschen Problems für elliptische Differentialgleichungen in Räumen mit Belegungsfunktionen