Multiple solutions and periodic oscillations in nonlinear diffusion processes

We study the oscillatory stationary states in the temperature and concentration fields occurring in tubular chemical reactors. Singular perturbation and multitime scale procedures are combined formally to clearly and simply reveal the mechanism controlling these oscillatory states. Their stability is also studied, and when coupled with previously obtained results on multiple steady states, this information completes the response (bifurcation) diagram in one-parameter range of the tubular reactor. The results apply also to more general nonlinear parabolic problems of which the first order tubular reactor is a special case

The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts (Rutherford Aris)

[Book review

New Eigenfunction Expansions and Alternative Representations for the Reduced Wave Equation

It has long been known that certain integral transforms and Fourier-type series can be used methodically for the resolution of certain kinds of classical boundary and initial value problems in separable coordinate systems. More recently, it has been shown that these classical transforms and series are spectral representations associated with an ordinary differential system which results on applying separation of variables to the given boundary value problem. This has been the basis for recent work concerned with systematically generating the proper spectral representation needed to solve a given problem. See [1] and [2] for a list of references

Multiple Solutions of Singular Perturbation Problems

Under certain conditions on g(x, u) we establish the existence and asymptotic behavior for small ε > 0 of multiple asymptotic solutions of the nonlinear boundary value problem εu" + u’ - g(x,u) = 0, 0 < x < 1, u’(0) - au(0)= A ≥ 0, a > 0, u’(1) + bu(1) = B > 0, b > 0. Formal techniques of singular perturbation theory clearly reveal the mechanism which controls the appearance of multiple solutions. Their existence is then established rigorously by iteration schemes and the so-called "shooting method" for ordinary differential equations

An Integral Transform Associated with Boundary Conditions Containing an Eigenvalue Parameter

It has long been known that certain integral transforms and Fourier-type series can be used to solve many classical boundary and initial value problems in separable coordinate systems. More recently, it has been shown that these classical transforms and series are spectral representations associated with an ordinary differential system which results on applying separation of variables to the given boundary value problem. This has been the basis for recent work concerned with systematically generating the proper spectral representation needed to solve a given problem. See [1]-[4] for a list of references. We shall consider the problem of finding the associated spectral representation when the resulting ordinary differential system has the eigenvalue parameter occurring in both the equation and one boundary condition. Moreover, the differential equation is to be satisfied on a semi-infinite interval, thus leading to a singular problem which does not seem to have been studied before. In §2 by using a transformation due to B. Friedman (which we modify appropriately for our singular case), we first give a formal derivation of the spectral representation, and then we rigorously prove the result. In §3 our representation is applied to solve an initial-boundary value problem arising in the theory of diffusion and heat flow in one dimension. We should bear in mind that even in cases where solutions are already known, our method systematically yields alternative representations which are often more rapidly convergent and from which asymptotic expansions of solutions with respect to parameters can often be found. In the problem to be considered a representation of the solution can also be found easily by a straight-forward application of the Laplace transform in time t, while the new transform derived in §2 yields the solution when applied in space x. The new transform, however, can be applied when the coefficients in the boundary value problem are time dependent, a situation which, in general, precludes the use of the Laplace transform.. As a general rule [1] there should be a spectral representation associated with each ordinary differential system resulting from applying separation of variables to the original boundary value problem, and each spectral representation should lead to a different representation of the solution of the original problem

Instabilities in chemically reacting mixtures

We shall study two different types of instability which arise in the theory of chemical and biochemical reactions [1], [2] and in the study of heat and mass transfer in porous catalysts [3], [4]. Peculiar physical effects involving local regions of oscillation and local instability (in a sense to be explained below) are observed experimentally. For each of the two different types of phenomena we believe that we have identified one possible mechanism for such occurrences. In § 2 we shall show that the sudden transition to localized temporal oscillation is reflected in a special situation first observed by N. Levinson [5]. The underlying chemistry and mathematics is introduced via a very simple initial value problem for a model system of reaction equations. A singular perturbation analysis clearly reveals the structure of the solution and also the mechanism which governs the occurrence of the oscillatory instabilities. In § 3 we consider the phenomenon of localized steady spatial oscillation for general reaction-diffusion equations. By combining singular perturbation and generalized WKB type methods we present a general technique for studying this type of phenomenon

Changing Time History in Moving Boundary Problems

A class of diffusion-stress equations modeling transport of solvent in glassy polymers is considered. The problem is formulated as a one-phase Stefan problem. It is shown that the moving front changes like √t initially but quickly behaves like t as t increases. The behavior is typical of stress-dominated transport. The quasi-steady state approximation is used to analyze the time history of the moving front. This analysis is motivated by the small time solution

A delay logistic equation with variable growth rate

A logistic equation with distributed delay is considered in the case where the growth rate oscillates sinusoidally about a positive mean value. A delay kernel is chosen which admits bifurcation of the equilibrium state into a periodic solution when the growth rate is constant. It is shown that the fluctuations in growth rate modulate the bifurcation into a quasiperiodic solution. In certain circumstances, however, it is shown that frequency locking can occur but that this is a local phenomenon which does not persist outside the immediate vicinity of the bifurcation point

Free boundary problems in controlled release pharmaceuticals. I: diffusion in glassy polymers

This paper formulates and studies two different problems occurring in the formation and use of pharmaceuticals via controlled release methods. These problems involve a glassy polymer and a penetrant, and the central problem is to predict and control the diffusive behavior of the penetrant through the polymer. The mathematical theory yields free boundary problems which are studied in various asymptotic regimes