The lack of exponential stability for a class of second-order systems with memory

We analyse the decay properties of the solution semigroup S(t) generated by the linear integrodifferential equationwhere the operator A is strictly positive self-adjoint with A–1 not necessarily compact. The asymptotic stability of S(t) is investigated in terms of the dependence of the parameter γ ∈ ℝ. In particular, we show that S(t) is not exponentially stable when γ ≠ 1.

Sherman-Morrison-Woodbury Formula for Linear Integrodifferential Equations

The well-known Sherman-Morrison-Woodbury formula is a powerful device for calculating the inverse of a square matrix. The paper finds that the Sherman-Morrison-Woodbury formula can be extended to the linear integrodifferential equation, which results in an unified scheme to decompose the linear integrodifferential equation into sets of differential equations and one integral equation. Two examples are presented to illustrate the Sherman-Morrison-Woodbury formula for the linear integrodifferential equation.

POLYNOMIAL DECAY FOR SOLUTIONS OF HYPERBOLIC INTEGRODIFFERENTIAL EQUATIONS

We consider a linear integrodifferential equation of second order in a Hilbert space and show that the solution tends to zero polynomially if the decay of the convolution kernel is polynomial. Both polynomials are of the same order.

Early stages of infiltration in two- and three-dimensional systems

The paper establishes the series solution of problems of infiltration from cylindrical and spherical cavities. The leading term is the fundamental horizontal absorption solution. The second term, representing linearly additive interactions of gravity and m-dimensionality (m = 2, 3), follows at once from known solutions for one-dimensional infiltration and for m-dimensional absorption. The third term is evaluated by solution of an ordinary linear integrodifferential equation. Higher terms involve circumferential flow components, and partial equations must be solved to evaluate them. For small enough times these higher terms are negligible. Physically, this implies that flow is essentially radial at such times; mathematically, that, taken no further than the third term, the solution applies indifferently to infiltration from cavities and from semicircular furrows (m = 2) and hemispherical basins (m = 3). The relation between this solution and the linearized and delta-function approximations is explored. The practical implications for sorptivity determinations based on short-time infiltration from furrows and basins are examined.