First-order Integro-differential Equations Research Articles

Identification of Time and Space Dependent Relaxation Kernels for Materials with Memory Related to Cylindrical Domains, I

We consider the problem of identifying a relaxation kernel for a first-order linear abstract integrodifferential equation. Such a problem is related to the theory of stratified materials with memory in cylindrical domains Ω=Ω1×Ω2, Ω1and Ω2being, respectively, an interval inRand a smooth enough domain inRn. The relaxation kernel, in this case, depends on time and one spatial variable. The proofs of the main results will appear in a subsequent paper.

Continuum hydrodynamical equations for multiphase astrophysical flows

We present a formalism for averaging mutually interacting multiphase hydrodynamical flows. Only averaging over space is discussed. The outcome of this formulation is a set of nonlinear first-order integrodifferential equations. We then apply the formalism to the problem of H I regions immersed in H II regions and obtain approximated equations. The H I regions are treated as a dilute suspension of fluid spheres immersed in H II continuum. The equations include terms and coefficients that have to be evaluated from a mesoscopic hydrodynamical theory. The hydrodynamics of the flow past a single cloud, which is defined here as the mesoscale of a fluid particle for the system under consideration, should provide the input physics to the formalism presented here

Direct numerical methods for first-order fredholm integro-differential equations

Numerical methods for Volterra integro-differential equations are adapted for the direct solution of a Fredholm integro-differential equation. A convergence analysis is presented and it is shown for certain test equations, instabilities may arise which may be eliminated with the introduction of a parameter (see Brunner [3]) to modify the methods.

Qualitative analysis of a dynamic model of a resource-limited age distribution

A population model described by a non-linear system of first-order partial integrodifferential equations with an integral boundary condition is considered. The structure and the asymptotic properties of the solution are established and a biological interpretation of the results is proposed.

Estimates of the solutions of two-point boundary-value problems for systems of first-order integrodifferential equations and the method of lines

One proves theorems on the estimates of the solutions of the systems of first-order integrodifferential equations with the boundary conditions On the basis of these theorems, one suggests a method for estimating the norms of integrodifferential equations by the method of the lines for the solutions of the periodic boundary-value problems for second-order integrodifferential equations of parabolic type. On the basis of the established theorem, on the solvability and on the estimate of the solution of the nonlinear equation $$Tx + F\left( x \right) = 0$$ in a Banach space X, where T is a linear unbounded operator, one investigates the convergence of the method of lines for solving the periodic boundary-value problem for a second-order nonlinear integrodifferential equation of parabolic type.

Certain characteristics and solutions of the ablation equation

The change of form of a body due to disintegration (ablation) during aerodynamic or other types of heating is described by an equation (we shall call it the ablation equation); the type of this equation is largely determined by the law of external heating. Such problems in different particular formulations have been investigated in [1–4] and others. Within the realm of simplest assumptions (methods of local similarity for the distribution of convective heat fluxes, absence of preheating) this equation is a first-order integrodifferential equation with significantly nonlinear properties. Below, its characteristic properties are described for two-dimensional problems and a solution is obtained in the neighborhood of the corner points of an initial nonsmooth profile, for which a particular example may be (as will be shown below) a body of stationary form that remains unchanged during the ablation process. It is shown that this solution may belong to one of three types: of these, one, which is discontinuous, retains the corner point, the second smears it, and the third is of mixed character.

The integro-differential equation form of a Hodgkin-Huxley type system of differential equations

A theorem is presented that establishes an equivalence between a Hodgkin-Huxley system of four first-order nonlinear ordinary differential equations (with initial values) and one first-order nonlinear ordinary integrodifferential equation (with initial value). The system was studied theoretically and numerically in [1].

The theory of the passage of Γ-Rays through matter

An analytical treatment of the problem of the passage of γ-rays through a plane-parallel plate is given. It is shown that the integro-differential equation for the scattered γ-ray flux density, taking the boundary conditions into account, reduces in n-th order approximation to a set of 2 n first-order integro-differential equations for the coefficients of the Legendre polynomial expansion of the scattered photon flux density. The 2 n arbitrary constants can be determined from the 2 n boundary conditions. Final results are given only for the case n = 1, which gives the spectral distribution of the scattered γ-ray flux density at a given distance from the source.