Exact fundamental solutions of linear parabolic equations with spatially varying coefficients

The exact general solution is obtained to a linear second order ordinary differential equation which has quite general coefficients depending on an arbitrary function of the independent variable. From this, the exact fundamental solution is derived for the corresponding linear parabolic partial differential equation with coefficients depending on the single space coordinate. In a special case this latter equation reduces to one of the Fokker-Planck type. These coefficients are then generalised and the appropriate fundamental solution is obtained. Extensions are given to linear parabolic equations in two andn space dimensions. The paper provides a collection of basic examples which illustrate and develop the theory for the generation of the exact fundamental solutions. Reduction to, and the corresponding fundamental solutions of the Fokker-Planck equations is presented, where appropriate.     

The Application of Electromagnetic Theory to Electrocardiology: II. Numerical Solution of the Integral Equations

In an earlier paper exact integral equations were derived for the surface potentials resulting from sources within an irregularly shaped inhomogeneous body. These exact equations cannot usually be solved. In this paper a discrete analogue is constructed which is not straightforward to solve, but which can be treated by careful mathematical methods. In particular a deflation procedure greatly facilitates the iterative solution of the problem and overcomes the divergence encountered by other authors. Numerical solutions obtained for simple geometries are compared to the exact analytic solutions available in such cases. The necessary convergence of the solutions of the discrete analog towards the solution of the continuous problem is shown to occur only if the coefficients of the discrete analogue are carefully evaluated. Calculations are then presented for realistic thoracic geometries, typical results being presented as surface potential maps. Finally the important effect of the internal regional inhomogeneities, particularly a realistic cardiac blood mass, is demonstrated by obtaining vector loops with and without these effects.     

Analytical solution to the initial-value problem for traveling bands of chemotactic bacteria

The governing parabolic partial differential equations for the diffusion and chemotactic transport of a distribution of bacteria and for the diffusion and bacterial degradation of a distribution of chemotactic agent are supplemented with boundary and initial conditions that model the recent capillary tube experiments on the formation and propagation of traveling bands of chemotactic bacteria. An iteration procedure that takes the exact solution to the “diffusionless” problem as a first approximation is applied to solve the equations of the complete theoretical model. It is shown that satisfactory agreement with experiment obtains for the analytical results of the first approximation which relate the velocity of propagation and total number of bacteria cells per unit cross-sectional area in a traveling band to the constant parameters in the governing equations and supplementary conditions. The second approximation is shown to yield approximate analytical expressions for the solution functions which are in close correspondence with previously derived traveling band solutions for values of time after the initial period of formation.     

脉冲-扩散周期竞争系统解的渐近行为

研究了在周期变化环境中具有扩散及种群密度可能发生突变的两竞争种群动力系统的数学模型．模型由反应扩散方程组以及初边值及脉冲条件组成．文章建立了研究模型的上下解方法,获得了一些比较原理．利用脉冲常微分方程的比较定理以及利用相应的脉冲常微分方程的解控制和估计所讨论模型的解,研究了系统模型的解的渐近性质．     

A Generalized Simplest Equation Method and Its Application to the Boussinesq-Burgers Equation

In this paper, a generalized simplest equation method is proposed to seek exact solutions of nonlinear evolution equations (NLEEs). In the method, we chose a solution expression with a variable coefficient and a variable coefficient ordinary differential auxiliary equation. This method can yield a Bäcklund transformation between NLEEs and a related constraint equation. By dealing with the constraint equation, we can derive infinite number of exact solutions for NLEEs. These solutions include the traveling wave solutions, non-traveling wave solutions, multi-soliton solutions, rational solutions, and other types of solutions. As applications, we obtained wide classes of exact solutions for the Boussinesq-Burgers equation by using the generalized simplest equation method.     

Analysis of time hierarchy in plant cell volume changes

A simple model of plant cell volume changes is presented. It is based on Kedem-Katchalsky equations for water and solute transport and on linear approximation of the dependence of intracellular hydrostatic pressure on the cell volume. Active transport of solute is also included. The time hierarchy within the system is analyzed by appropriate normalization of variables and by the assessment of the numerical values of model coefficients. The dynamics of the system comprises a slow process of solute exchange and a fast process of water transport. This explains the wellknown biphasic response of the cell volume to a sudden change in external conditions. An approximation of equations describing the system behaviour on the basis of the Tikhonov's theorem is proposed. The approximative solution is compared with the exact numerical solution of the original equations. The approximation is very good under physiological conditions, but it ceases to hold when the solute permeability of the cell membrane increases causing the breakdown of the entire time hierarchy within the system.     

Stochastic models for an open biochemical system

This paper uses the theory of Markov processes to derive stochastic models for a single open biochemical system at st?ady state under 3 sets of assumptions. The system is a one substrate, one product reaction. Each set of assumptions results in a separate solution for the probability functions. A system of linear equations in the probability function as well as an equivalent differential equation in its generating function are derived. The assumption of no flux leads to the first (exact) solution of the linear equations. The form agrees with that of the closed systems. Making assumptions that simplify the system to model active transport results in the second (exact) solution to the linear equations. Assuming the presence of a large number of molecules in the system facilitates obtaining the third (approximate) solution to the differential equations.     

Kadomtsev−Petviashvili equation for a flow of highly nonisothermal collisionless plasma

It is shown that the equations of two-fluid electrodynamics for a cold ions flow and Boltzmann electrons in the vicinity of the ion-sound point can be reduced to the Kadomtsev?Petviashvili equation. Examples of two-dimensional equilibria with pole singularities obtained by exactly solving the equations are presented. An exact self-similar solution describing a two-dimensional transonic flow and having no pole singularities is found.     

The effect of solution non-ideality on membrane transport in three-dimensional models of the renal concentrating mechanism

Previous models of the renal concentrating mechanism employ ideal approximations of solution thermodynamics for membrane transport calculation. In three-dimensional models of the renal medulla, predicted urine concentrations reach levels where there idealized approximations begin to break down. In this paper we derive equations that govern membrane transport for non-dilute solutions and use these equations in a three-dimensional model of the concentrating mechanism. New numerical methods were employed that are more stable than those employed previously. Compared to ideal solution models, the urea non-ideality tends to increase predicted osmolarities, whereas NaCl non-ideality decreases predictions.     

Properties of the transfer functions of compartmental models

This paper is concerned with the nonlinear system of algebraic equations relating the positive parameters of a linear time-invariant compartmental model to its transfer function coefficients. The general form that these equations must take is shown, and simple necessary conditions for the existence of positive solutions are given. An immediate use of these conditions is the development of necessary conditions for a polynomial with positive coefficients to have negative roots. A method is then outlined which triangularizes the system and reduces the complete solution problem to one of finding and counting roots of a polynomial. Sufficient conditions for the existence of real and positive solutions are demonstrated.     

An integral method for the analysis of blood flow

The existing methods to solve the problems of pulsatile flow in the cardiovascular system are based on either linear axisymmetric equations or non-linear one-dimensional equations. The solutions thus obtained give only a mediocre comparison with measurements. In this paper, a non-linear axisymmetric theory is proposed. The starting point of the present theory is a third degree polynomial representation of the velocity profile. Integral methods are then applied to obtain the governing equations. To ascertain the accuracy of the theory proposed above, the calculations for a simple case involving pulsatile flow in a long rigid tube were performed. The results are: (a) the average velocities compare very well with exact solutions and (b) the velocity profiles for a given frequency agree very well with exact solutions for flow in small tubes, but tend to differ as tube size is increased.     

Asymptotic method for peristaltic transport

The purpose of this paper is to justify an asymptotic method developed for the study of peristaltic transport in a tube of arbitrary cross section. Within the framework of long wave approximation, the three-dimensional nonlinear Navier-Stokes equations are reduced to a sequence of two-dimensional linear boundary value problems of Laplace and biharmonic operators. It is shown that, if a Reynolds number is less than some constant, the solution of the approximate equations is indeed an asymptotic approximation to the exact solution of the problem as the ratio of the maximum radius of the tube to the wave length of the peristaltic motion of the wall tends to zero, and the error estimates are expressed inL ₂ norms. Furthermore, under the same condition the exact solution is shown to be unique and stable under arbitrary perturbation of spatially periodic disturbance. Application of the stability condition to peristaltic transport in a tube of circular cross section is given.     

Hydrodynamics of bolus flow—An analytical approach to blood flow in capillaries

A model which was used by Prothero and Burton to simulate a particular configuration in capillary blood vessels is investigated from a hydrodynamic point of view. In this model, the erythrocytes are approximated by rigid pistons, and plasma is assumed to be an incompressible Newtonian fluid. An order of magnitude analysis using the physiologically realistic values for various parameters reduces the exact equations of motion to an equation describing the creeping motion of the fluid. An analytical approach to the solution of the equation is proposed and some results are reported here. The solution of the flow field is given in terms of a stream function which is represented by two infinite series composed of known functions. Two coupled infinite systems of algebraic equations determining the coefficients of the two series have been derived. This method of solution is proposed as an alternative to the entirely numerical procedure of solving the similar problem proposed by Bugliarelloet al. A limiting case of large aspect ratio (the ratio of the axial spacing of the two successive erythrocytes to the capillary diameter) is studied and the solution, valid away from the erythrocyte surface, has been obtained in simple form. It resembles the classical Poisenille flow, but the pressure gradient is related to the erythrocyte speed.     

Model transformations to evaluate transient thermal responses at a tissue surface

For a spatially distributed model describing the transient temperature response of a thermistor-tissue system, Wei et al. [J. Biomech. Eng., 117:74-85, 1995] obtained an approximate transformation for fast analysis of the temperature response at the tissue surface. This approximate transformation reduces the model to a single ordinary differential equation. Here, we present an exact transformation that yields a single differential-integral equation. Numerical solutions from the approximate and exact transformations were compared to evaluate the differences with several sets of parameter values. The maximum difference between the exact and approximate solutions did not exceed 15 percent and occurred for only a short time interval. The root-mean-square error of the approximate solution was no more than 5 percent and within the level of experimental noise. Under the experimental conditions used by Wei et al., the approximate transformation is justified for estimating model parameters from transient thermal responses.     

One-dimensional computer analysis of oscillatory flow in rigid tubes.

The dynamic characteristics of catheter-transducer systems using rigid tubes with compliance lumped in the transducer and oscillatory flow of fluid in rigid tubes were analyzed. A digital computer model based on one dimensional laminar oscillatory flow was developed and verified by exact solution of the Navier-Stokes Equation. Experimental results indicated that the damping ratio and resistance is much higher at higher frequencies of oscillation than predicted by the one dimensional model. An empirical correction factor was developed and incorporated into the computer model to correct the model to the experimental data. Amplitude of oscillation was found to have no effect on damping ratio so it was concluded that the increased damping ratio and resistance at higher frequencies was not due to turbulence but to two dimensional flow effects. Graphs and equations were developed to calculate damping ratio and undamped natural frequency of a catheter-transducer system from system parameters. Graphs and equations were also developed to calculate resistance and inertance for oscillatory flow in rigid tubes from system parameters and frequency of oscillation.     

Nonlinear ion acoustic waves in a quantum degenerate warm plasma with dust grains

A study is made of the propagation of ion acoustic waves in a collisionless unmagnetized dusty plasma containing degenerate ion and electron gases at nonzero temperatures. In linear theory, a dispersion relation for isothermal ion acoustic waves is derived and an exact expression for the linear ion acoustic velocity is obtained. The dependence of the linear ion acoustic velocity on the dust density in a plasma is calculated. An analysis of the dispersion relation reveals parameter ranges in which the problem has soliton solutions. In nonlinear theory, an exact solution to the basic equations is found and examined. The analysis is carried out by Bernoulli’s pseudopotential method. The ranges of the phase velocities of periodic ion acoustic waves and the velocities of solitons are determined. It is shown that these ranges do not overlap and that the soliton velocity cannot be lower than the linear ion acoustic velocity. The profiles of the physical quantities in a periodic wave and in a soliton are evaluated, as well as the dependence of the critical velocity of solitons on the dust density in a plasma.     

Toxicokinetics of Metals in Terrestrial Invertebrates: Making Things Straight with the One-Compartment Principle

In this analysis, we first performed a critical review of one-compartment models used to describe metal toxicokinetics in invertebrates and found mathematical or conceptual errors in almost all published studies. In some publications, the models used do not represent the exact solution of the underlying one-compartment differential equations; others use unrealistic assumptions about constant background metal concentration and/or zero metal concentration in uncontaminated medium. Herein we present exact solutions of two differential-equation models, one describing simple two-stage toxicokinetics (metal toxicokinetic follows the experimental phases: the uptake phase and the decontamination phase) and another that can be applied for more complex three-stage patterns (toxicokinetic pattern does not follow two phases determined by an experimenter). Using two case studies for carabids exposed via food, based on previously published data, we discuss and compare our models to those originally used to analyze the data. Our conclusion is that when metal toxicokinetic follows a one-compartment model, the exact solution of a set of differential equations should be used. The proposed models allow assimilation and elimination rates to change between toxicokinetic stages, and the three-stage model is flexible enough to fit patterns that are more complex than the classic two-stage model can handle.     

Hydrodynamics of macromolecular complexes. II. Rotation

We have used the modified Oseen hydrodynamic interaction tensor along with iterative numerical solution of the coupled hydrodynamic interaction equations to calculate the rotational diffusion coefficients of macromolecular complexes composed of nonidentical spherical subunits. For the one structure, a prolate ellipsoid of revolution, for which exact solutions are available, a subunit model with the same length and volume gives asymptotic agreement with the Perrin equations. Other structures considered include plane polygonal rings, lollipops, and dumbbells.