W. J. Padgett

Random integral equations with applications to stochastic systems

General introduction.- Preliminaries.- A random integral equation of the volterra type.- Approximate solutions of the random volterra integral equation.- A stochastic integral equation of the fredholm type with application to systems theory.- Random discrete fredholm and volterra equations.- The stochastic differential systems.- The stochastic differential systems.- The stochastic differential systems with lag time.

A Random Differential Equation Approach to the Probability Distribution of Bod and Do in Streams

In this paper a stochastic model for stream pollution is given which involves a random differential equation of the form \[( * )\qquad \dot {\bf X}( t ) = {\bf A}{\bf X}( t ) + {\bf Y},\quad t\geqq 0,\] where

Existence of a Solution of a Stochastic Integral Equation in Turbulence Theory

{\bf X}( t )

On a semi-stochastic model arising in a biological system

is a two-dimensional vector-valued stochastic process with the first component giving the biochemical oxygen demand (BOD) and the second component representing the dissolved oxygen (DO) at distance t downstream from the source of pollution. The fundamental Liouville’s theorem is utilized to obtain the probability distribution of the solution of

A stochastic model for chemotherapy: computer simulation

( * ),{\bf X}( t )

On the Existence and Uniqueness of a Random Solution to a Perturbed Random Integral Equation of the Fredholm Type

, at each t with various distributional assumptions on the random initial conditions and random inhomogeneous term. Computer simulations of the trajectories of the BOD and DO processes as well as the mean and variance functions are given for several initial distributions and are compared with the deterministic results.

On Bayes Estimation of Reliability for Mixtures of Life Distributions

In the theory of turbulence, the random position of a tagged point in a continuous fluid in turbulent motion, r(t; ω), is a vector‐valued random function of time t ≥ 0, ω e Ω, where Ω is the supporting set of the underlying probability space (Ω, B, P). If u(r, t; ω) is the Eulerian velocity field, then r(t; ω) satisfies the stochastic integral equation r(t;ω)=∫0tu(r(Υ;ω),Υ;ω)dΥ, t≥0. General conditions under which a random solution of this stochastic integral equation exists are given in the form of a theorem, and the theorem is proved using the concepts of admissibility with respect to an operator on a Banach space and fixed‐point methods of functional analysis.

Bayes Estimation of Reliability Using an Estimated Prior Distribution

Abstract A stochastic version of a mathematical model for chemotherapy, which was developed deterministically by Bellman et al. [5, 6], is given. After an injection at the heart entrance, the concentration of drug in the blood plasma of a biological system with one organ and a simplified heart results in the semistochastic integral equation u L (t;ω) = − c V ∗ ∫ 0 t [u L (y;ω) − u R (y;ω)] dy, t ⩽ 0, where u L ( t ; ω ) is the concentration of drug plasma leaving the heart and u R ( t ; ω ) is the concentration of drug in plasma entering the heart at time t . The function u L ( t ; ω ) is a deterministic function of time for 0 ⩽ t T , where T is the blood recirculation time lag, and a random function of t , T ⩽ t ⩽ M , for some M ∈ ( T ∞). It is shown that the integral equation has a solution for both t ∈ [0, T ), using the method of successive approximations, and t ∈ [ T , M ], using some methods of probabilistic functional analysis.

A stochastic model for chemotherapy: two-organ systems.

Abstract A stochastic version of a deterministic mathematical model for chemotherapy of Bellman et al. [4,5] was formulated by the authors (Math. Biosci.9, 105–117) for a one-organ biological system and, due to the random nature of diffusion of a drug from the blood plasma into the extracellular space in a capillary bed, resulted in a semi-stochastic integral equation with the concentration of drug in plasma leaving the heart at time t as the unknown function. In this article the diffusion of drug into the extracellular space of the organ is simulated using a two-dimensional diffusion process. Values of the concentration of drug entering the heart at time t are obtained and, using numerical integration, a realization of the semirandom solution of the semistochastic integral equation is found. The solution compares favorably with experimental results of Bassingthwaighte [1] and Bellman et al. [6].

The origins and applications of stochastic integral equations

A study of a random or stochastic integral equation of the Fredholm type of the form\[ x( {t;\,\omega } ) = h( t,x( t;\omega ) ) + \int_0^\infty {k_0 ( t,\tau ;\omega )e( \tau ,x( \tau ;\omega ) )d\tau },\qquad t\geqq 0, \]is presented, where