Original Initial Value Problem Research Articles

Large Time Asymptotics in Contaminant Transport in Porous Media

In this paper we derive large time solutions of the partial differential equations modelling contaminant transport in porous media for initial data with bounded support. While the main emphasis is on two space dimensions, for the sake of completeness we give a brief summary of the corresponding results for one space dimension. The philosophy behind the paper is to compare the results of a formal asymptotic analysis of the governing equations as $t \to \infty $ with numerical solutions of the complete initial value problem. The analytic results are obtained using the method of dominant balance which identifies the dominant terms in the model equations determining the behaviour of the solution in the large time limit. These are found in terms of time scaled space similarity variables and the procedure results in a reduction of the number of independent variables in the original partial differential equation. This generates what we call a reduced equation, the solution of which depends crucially on the value of a parameter appearing in the problem. In some cases the reduced equation can be solved explicitly, while others have a particularly intractable structure which inhibits any analytic or numerical progress. However, we can extract a number of global and local properties of the solution, which enables us to form a reasonably complete picture of what the profiles look like. Extensive comparison with numerical solution of the original initial value problem provides convincing confirmation of our analytic solutions. In the final section of the paper, by way of motivation for the work, we give some results concerning the temporal behaviour of certain moments of the two-dimensional profiles commonly used to compute physical parameter characteristics for contaminant transport in porous media.

A Systematic Approach to the Evolution Equation of a Probability Distribution of a Deterministic Dynamical System

A systematic approximation method for time evolutions of distribution functions of deterministic dynamical systems is developed when the distribution functions are not far from Gaussian. For simplicity and definiteness, the procedure of the method is shown by taking a simple dynamical system, that is, one-dimensional autonomous nonlinear oscillator. An essential point of the method is to replace an original initial value problem of the evolution of a distribution function by an appropriate initial value problem of the evolution of a Gaussian distribution function by using the Kullback-Leibler's information. Basic properties of an initial value problem posed by the method are also studied.

Mathematical foundations of the quasistatic approximation in reactor physics

The quasistatic method used in reactor-physics calculations is applied to a model integro-differential equation in a Banach space. The resulting quasistatic equations are shown to be locally consistent with the original initial-value problem. An iterative scheme to solve these equations is presented and shown to be convergent.

Dual extremum principles for the heat equation

SynopsisDual extremum principles characterising the solution of initial value problems for the heat equation are obtained by imbedding the problem in a two-point boundary-value problem for a system in which the original equation is coupled with its adjoint. Bounds on quantities of interest in the original initial value problem are obtained. Such principles are examples of ones which can be obtained for a general class of linear operators on a Hilbert space.