Search

We consider the behavior of buckling driven thin-film blisterings using von Karman’s plate theory. Our focus is on the setting where the blistered region is the unit square with clamped boundary conditions at the vertical sides and periodic ones along the horizontal sides. In this setting, we prove rigorous upper and lower bounds for the elastic energy which are of the same order as the film thickness. We also present a convincing argument for the necessity of branching of folds near the boundary as has been observed in experiments.

This paper is concerned with the stability and uniqueness of traveling waves of a delayed diffusive susceptible-infective-removed (SIR) epidemic model. We first prove the exponential stability of traveling waves by using the weighted energy method, where the traveling waves are allowed to be non-monotone. Then we establish the exact asymptotic behavior of traveling waves at −∞ by using Ikehara’s theorem. Finally, the uniqueness of traveling waves is proved by the stability result of traveling waves.

One of the main motivations for developing the theory of interpolation was to apply it to the theory of partial differential equations (PDEs). Nowadays interpolation theory has been developed in an almost unbelievable way {see the bibliography of Maligranda [Interpolation of Operators and Applications (1926-1990), 2nd ed. (Luleå University, Luleå, 1993), p. 154]}. In this article some model examples are presented which display how powerful this theory is when dealing with PDEs. One main aim is to point out when it suffices to use classical interpolation theory and also to give concrete examples of situations when nonlinear interpolation theory has to be applied. Some historical remarks are also included and the relations to similar results are pointed out Godkänd; 1994; 20070208 (kani)

The wave‐splitting Green’s function approach to one‐dimensional electromagnetic inverse problems of simultaneous reconstructions with different types of scattering data is analyzed. Theorems of existence, uniqueness, and stability are given for the following types of scattering data: reflection and transmission data for one round trip, two‐sided reflection data for one round trip, reflection data for two round trips.

A connection between dissipation anomaly in fluid dynamics and Colombeau’s theory of products of distributions is exemplified by considering Burgers equation with a passive scalar. Besides the well-known viscosity-independent dissipation of energy in the steadily propagating shock wave solution, the lesser known case of passive scalar subject to the shock wave is studied. An exact dependence of the dissipation rate ϵθϵθ of the passive scalar on the Prandtl number PrPr is given by a simple analysis: we show, in particular, ϵθ∝1/Pr−−√ϵθ∝1/Pr for large PrPr. The passive scalar profile is shown to have a form of a sum of tanh2n+1 xtanh2n+1 x with suitably scaled xx, thereby implying the necessity to distinguish HH from HnHn when PrPr is large, where HH is the Heaviside function and nn is a positive integer. An incorrect result of ϵθ∝1/Prϵθ∝1/Pr would otherwise be obtained. This is a typical example where Colombeau calculus for products of weak solutions is required for a correct interpretation. A Cole–Hopf-type transform is also given for the case of unit Prandtl number.

The continuum calculus, proposed earlier [J. Math. Phys. 17, 1988 (1976)], is here applied to the development of a functional version of Laplace transform and a method for the evaluation of path integrals containing a Gaussian‐like measure. Two methods in functional integration are proposed. First, the Gaussian integral for polynomial functionals and consequently functionals that can be expanded in Taylor series are examined. Formula (2.12) is derived. Next, we define the Laplace transform in the function space through the weak distribution formulation of Skorohod. Comparison of both approaches enables us to determine the expression of the functional integeral through a series of Laplace transformations. The second formula is given in Eq. (4.5). The latter formula is applicable to all Laplace transformable functionals. As an illustration of the utility of the formulas derived, we evaluate the integral of a cosine functional by methods 1 and 2, and obtain consistent results. Further applications to quantum mechanics are also presented. We examine the cases of a free particle,the quantum harmonic oscillator, the forced oscillator, and charged particle in a magnetic field. In all cases, we obtain correct results in comparison with known expressions. A numerical procedure is employed in the calculation of infinite products. The usefulness of the p‐integral method is stressed.

A stochastic Eulerian‐Lagrangian procedure is applied to steady‐state sound propagation from a small, collimated acoustic source to an omnidirectional point receiver imbedded in an infinite, continuous, statistically isotropic medium. An analytic procedure is developed for obtaining the Lagrangian measure function B(x, ξ | s) from its characteristic function φ(k, ξ | s) for stochastic‐Fermat media. The results include a coefficient of intensity variation V that evinces a frequency‐dependent, phase‐dominance region and a frequency‐independent, amplitude‐dominance region. The methods employed in this study are new to the problem of sound propagation through continuous stochastic media and avoid three common difficulties: (1) range limitations due to cumulative phase effects, (2) discrete scattering assumptions, and (3) restriction to an Eulerian path.

The problem of propagation over a wedge‐shaped overburden has been analyzed using a full‐wave‐solution approach. The relationship between this solution and an earlier one employing the compensation theorem is discussed in detail. The compensation theorem method makes use of the simplifying concept of of the surface impedance which is determined by a geometrical‐optical approach. The applicability and limitations of the latter approach are carefully studied for both dissipative and nondissipative over‐burdens.