A problem of diffraction by an elongated body of revolution is studied. The incident wave falls along the axis. The wavelength is small comparatively to the dimensions of the body. The parabolic equation of the diffraction theory is used to describe the diffraction process. A boundary integral equation is derived. The integral equation is solved analytically and by iterations for diffraction by a cone.
We study the flow of an incompressible liquid film down a wavy incline. Applying a Galerkin method with only one ansatz function to the Navier-Stokes equations we derive a second order weighted residual integral boundary layer equation, which in particular may be used to describe eddies in the troughs of the wavy bottom. We present numerical results which show that our model is qualitatively and quantitatively accurate in wide ranges of parameters, and we use the model to study some new phenomena, for instance the occurrence of a short wave instability for laminar flows which does not exist over flat bottom.
We study a certain one dimensional, degenerate parabolic partial differential equation with a boundary condition which arises in pricing of Asian options. Due to degeneracy of the partial differential operator and the non-smooth boundary condition, regularity of the generalized solution of such a problem remained unclear. We prove that the generalized solution of the problem is indeed a classical solution.
The current paper is the second part of a series of two papers dedicated to 2D problem of diffraction of acoustic waves by a segment bearing impedance boundary conditions. In the first part some preliminary steps were made, namely, the problem was reduced to two matrix Riemann-Hilbert problem. Here the Riemann-Hilbert problems are solved with the help of a novel method of OE-equations. Each Riemann-Hilbert problem is embedded into a family of similar problems with the same coefficient and growth condition, but with some other cuts. The family is indexed by an artificial parameter. It is proven that the dependence of the solution on this parameter can be described by a simple ordinary differential equation (ODE1). The boundary conditions for this equation are known and the inverse problem of reconstruction of the coefficient of ODE1 from the boundary conditions is formulated. This problem is called the OE-equation. The OE-equation is solved by a simple numerical algorithm.
In this paper we formulate new curvature functions on $mathbb{S}^n$ via integral operators. For certain even orders, these curvature functions are equivalent to the classic curvature functions defined via differential operators, but not for all even orders. Existence result for antipodally symmetric prescribed curvature functions on $mathbb{S}^n$ is obtained. As a corollary, the existence of a conformal metric for an antipodally symmetric prescribed $Q-$curvature functions on $mathbb{S}^3$ is proved. Curvature function on general compact manifold as well as the conformal covariance property for the corresponding integral operator are also addressed, and a general Yamabe type problem is proposed.
In this paper we analyze a nonlinear parabolic equation characterized by a singular diffusion term describing very fast diffusion effects. The equation is settled in a smooth bounded three-dimensional domain and complemented with a general boundary condition of dynamic type. This type of condition prescribes some kind of mass conservation; hence extinction effects are not expected for solutions that emanate from strictly positive initial data. Our main results regard existence of weak solutions, instantaneous regularization properties, long-time behavior, and, under special conditions, uniqueness.
A.V. Shanin
,A.I. Korolkov
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(2017)
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"Diffraction by an elongated body of revolution. A boundary integral equation based on the parabolic equation"
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Andrey Shanin V
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