Periodic solutions of integro-differential equations which arise in population dynamics

The problem of the existence and stability of periodic solutions of infinite-lag integra-differential equations is considered. Specifically, the integrals involved are of the convolution type with the dependent variable being integrated over the range (- ∞,t), as occur in models of population growth. It is shown that Hopf bifurcation of periodic solutions from a steady state can occur, when a pair of eigenvalues crosses the imaginary axis. Also considered is the existence of traveling wave solutions of a model population equation allowing spatial diffusion in addition to the usual temporal variation. Lastly, the stability of the periodic solutions resulting from Hopf bifurcation is determined with aid of a Floquet theory.

The first chapter is devoted to linear integro-differential equations with constant coefficients utilizing the method of semi-groups of operators. The second chapter analyzes the Hopf bifurcation providing an existence theorem. Also, the two-timing perturbation procedure is applied to construct the periodic solutions. The third chapter uses two-timing to obtain traveling wave solutions of the diffusive model, as well as providing an existence theorem. The fourth chapter develops a Floquet theory for linear integro-differential equations with periodic coefficients again using the semi-group approach. The fifth chapter gives sufficient conditions for the stability or instability of a periodic solution in terms of the linearization of the equations. These results are then applied to the Hopf bifurcation problem and to a certain population equation modeling periodically fluctuating environments to deduce the stability of the corresponding periodic solutions.

Existence, uniqueness, and stability of solutions of a class of nonlinear partial differential equations

In this study we investigate the existence, uniqueness and asymptotic stability of solutions of a class of nonlinear integral equations which are representations for some time dependent non- linear partial differential equations. Sufficient conditions are established which allow one to infer the stability of the nonlinear equations from the stability of the linearized equations. Improved estimates of the domain of stability are obtained using a Liapunov Functional approach. These results are applied to some nonlinear partial differential equations governing the behavior of nonlinear continuous dynamical systems.

The stability of traveling wave solutions of parabolic equations

A method for determining by inspection the stability or instability of any solution u(t,x) = ɸ(x-ct) of any smooth equation of the form u_t = f(u_(xx),u_x,u where ∂/∂a f(a,b,c) > 0 for all arguments a,b,c, is developed. The connection between the mean wavespeed of solutions u(t,x) and their initial conditions u(0,x) is also explored. The mean wavespeed results and some of the stability results are then extended to include equations which contain integrals and also to include some special systems of equations. The results are applied to several physical examples.

New variational principles for systems of partial differential equations

The question of finding variational principles for coupled systems of first order partial differential equations is considered. Using a potential representation for solutions of the first order system a higher order system is obtained. Existence of a variational principle follows if the original system can be transformed to a self-adjoint higher order system. Existence of variational principles for all linear wave equations with constant coefficients having real dispersion relations is established. The method of adjoining some of the equations of the original system to a suitable Lagrangian function by the method of Lagrange multipliers is used to construct new variational principles for a class of linear systems. The equations used as side conditions must satisfy highly-restrictive integrability conditions. In the more difficult nonlinear case the system of two equations in two independent variables can be analyzed completely. For systems determined by two conservation laws the side condition must be a conservation law in addition to satisfying the integrability conditions.

Some approximate solutions of dynamic problems in the linear theory of thin elastic shells

Some aspects of wave propagation in thin elastic shells are considered. The governing equations are derived by a method which makes their relationship to the exact equations of linear elasticity quite clear. Finite wave propagation speeds are ensured by the inclusion of the appropriate physical effects.

The problem of a constant pressure front moving with constant velocity along a semi-infinite circular cylindrical shell is studied. The behavior of the solution immediately under the leading wave is found, as well as the short time solution behind the characteristic wavefronts. The main long time disturbance is found to travel with the velocity of very long longitudinal waves in a bar and an expression for this part of the solution is given.

When a constant moment is applied to the lip of an open spherical shell, there is an interesting effect due to the focusing of the waves. This phenomenon is studied and an expression is derived for the wavefront behavior for the first passage of the leading wave and its first reflection.

For the two problems mentioned, the method used involves reducing the governing partial differential equations to ordinary differential equations by means of a Laplace transform in time. The information sought is then extracted by doing the appropriate asymptotic expansion with the Laplace variable as parameter.

I. Numerical solution of exact pair equations. II. Numerical solution of first-order pair equations

Part I

Numerical solutions to the S-limit equations for the helium ground state and excited triplet state and the hydride ion ground state are obtained with the second and fourth difference approximations. The results for the ground states are superior to previously reported values. The coupled equations resulting from the partial wave expansion of the exact helium atom wavefunction were solved giving accurate S-, P-, D-, F-, and G-limits. The G-limit is -2.90351 a.u. compared to the exact value of the energy of -2.90372 a.u.

Part II

The pair functions which determine the exact first-order wavefunction for the ground state of the three-electron atom are found with the matrix finite difference method. The second- and third-order energies for the (1s1s)¹S, (1s2s)³S, and (1s2s)¹S states of the two-electron atom are presented along with contour and perspective plots of the pair functions. The total energy for the three-electron atom with a nuclear charge Z is found to be E(Z) = -1.125•Z² +1.022805•Z-0.408138-0.025515•(1/Z)+O(1/Z²)a.u.

Suspension mechanics: I. Inertial and non-Newtonian migration of neutrally buoyant rigid spheres in two-dimensional unidirectional flows; II. The effect of viscoelasticity on the creeping motion of a train of neutrally buoyant Newtonian drops through a circular tube

The lateral migration of neutrally buoyant rigid spheres in two-dimensional unidirectional flows was studied theoretically. The cases of both inertia-induced migration in a Newtonian fluid and normal stress-induced migration in a second-order fluid were considered. Analytical results for the lateral velocities were obtained, and the equilibrium positions and trajectories of the spheres compared favorably with the experimental data available in the literature. The effective viscosity was obtained for a dilute suspension of spheres which were simultaneously undergoing inertia-induced migration and translational Brownian motion in a plane Poiseuille flow. The migration of spheres suspended in a second-order fluid inside a screw extruder was also considered.

The creeping motion of neutrally buoyant concentrically located Newtonian drops through a circular tube was studied experimentally for drops which have an undeformed radius comparable to that of the tube. Both a Newtonian and a viscoelastic suspending fluid were used in order to determine the influence of viscoelasticity. The extra pressure drop due to the presence of the suspended drops, the shape and velocity of the drops, and the streamlines of the flow were obtained for various viscosity ratios, total flow rates, and drop sizes. The results were compared with existing theoretical and experimental data.

Representing measures on the Royden boundary for solutions of Δu=Pu on a Riemannian manifold

Consider the Royden compactification R* of a Riemannian n-manifold R, Γ = R*\R its Royden boundary, Δ its harmonic boundary and the elliptic differential equation Δu = Pu, P ≥ 0 on R. A regular Borel measure m^P can be constructed on Γ with support equal to the closure of Δ^P = {q ϵ Δ : q has a neighborhood U in R* with _U^ʃ_ᴖR^{P ˂ ∞} }. Every enegy-finite solution to u (i.e. E(u) = D(u) + ^ʃ_Ru²P ˂ ∞, where D(u) is the Dirichlet integral of u) can be represented by u(z) = ^ʃ_Γu(q)K(z,q)dm^P(q) where K(z,q) is a continuous function on ^Rx Γ . A _P^~_E-function is a nonnegative solution which is the infimum of a downward directed family of energy-finite solutions. A nonzero _P^~_E-function is called _P^~_E-minimal if it is a constant multiple of every nonzero _P^~_E-function dominated by it. THEOREM. There exists a _P^~_E-minimal function if and only if there exists a point in q ϵ Γ such that m^P(q) > 0. THEOREM. For q ϵ Δ^P , m^P(q) > 0 if and only if m⁰(q) > 0 .

The equivalent conductance of solutions of picric acid

Picric acid possesses the property, which is rare among strong electrolytes, of having a convenient distribution ratio between water and certain organic solvents such as benzene, chloroform, etc. Because of this property, picric acid offers peculiar advantages for studying the well known deviations of strong electrolytes from the law of mass action, for; by means of distribution experiments, the activities of picric acid in various aqueous solutions may be compared.

In order to interpret the results of such distribution experiments, it is necessary to know the degree of ionization of picric acid in aqueous solutions.

At least three series of determinations of the equivalent conductance of picric acid have been published, but the results are not concordant; and therefore, the degree of ionization cannot be calculated with any degree of certainty.

The object of the present investigation was to redetermine the conductance of picric acid solutions in order to obtain satisfactory data from which the degrees of ionization of its solutions might be calculated.