Solution of a cauchy problem for a diffusion equation in a Hilbert space by a Feynman formula

The Cauchy problem for a class of diffusion equations in a Hilbert space is studied. It is proved that the Cauchy problem in well posed in the class of uniform limits of infinitely smooth bounded cylindrical functions on the Hilbert space, and the solution is presented in the form of the so-called Feynman formula, i.e., a limit of multiple integrals against a gaussian measure as the multiplicity tends to infinity. It is also proved that the solution of the Cauchy problem depends continuously on the diffusion coefficient. A process reducing an approximate solution of an infinite-dimensional diffusion equation to finding a multiple integral of a real function of finitely many real variables is indicated.     

Diffusion over a saddle with a langevin equation

The diffusion problem over a saddle is studied using a multidimensional Langevin equation. An analytical solution is derived for a quadratic potential and the probability to pass over the barrier deduced. A very simple solution is given for the one-dimensional problem and a general scheme is shown for higher dimensions.     

Wave propagation in a transient plasma-development of a Green'sfunction

The effect of a transient plasma on a source wave, when the rise time of the electron density profile is comparable to the source wave period, is considered. The problem is not amenable to a sudden-switching approximation or an adiabatic approximation. It is investigated by developing a causal Green's function for temporally unlike media. A perturbation series based on the Green's function gives the solution to the desired accuracy. As compared to the numerical methods that give the total fields, this method has the intrinsic advantage of processing the forward propagating and backward propagating waves explicitly. The validity of the method is illustrated by comparing the perturbation solution to the exact solution of a linear profile. The scattering coefficients for a “hump profile,” which has no obvious exact solution, are obtained to illustrate the general applicability of the new approach     

Motion of a particle in a static magnetic field and the field of a monochromatic electromagnetic plane wave

A solution of the equation of motion of a charged particle in an external electromagnetic field comprising a superposition of a uniform static magnetic field and the field of a monochromatic, elliptically polarized electromagnetic plane wave is obtained as the solution of a Cauchy problem. The resonance case is investigated. An analysis of the resulting solution is given. Zh. Tekh. Fiz. 69, 106–110 (May 1999)     

The non-stationary freefield response for a moving load with a random amplitude

This paper presents a stochastic solution procedure for the calculation of the non-stationary freefield response due to a moving load with a random amplitude. In this case, a non-stationary autocorrelation function and a time-dependent spectral density are required to characterize the response at a fixed point in the freefield. The non-stationary solution is derived from the solution in the case of a moving load with a deterministic amplitude. It is shown how the deterministic solution can be calculated in an efficient way by means of integral transformation methods if the problem geometry exhibits a translational invariance in the direction of the moving load. A key ingredient is the transfer function between the source and the receiver that represents the fundamental response in the freefield due to an impulse load at a fixed location. The solution in the case of a moving load with a random amplitude is formulated in terms of the double forward Fourier transform of the non-stationary autocorrelation function. The solution procedure is illustrated with an example where the non-stationary autocorrelation function and the time-dependent standard deviation of the freefield response are computed for a moving harmonic load with a random phase shift. The results are compared with the response in the deterministic case.     

On MHD transient flow of a Maxwell fluid in a porous medium and rotating frame

Analytic solution for unsteady magnetohydrodynamic (MHD) flow is constructed in a rotating non-Newtonian fluid through a porous medium. Constitutive equations for a Maxwell fluid have been taken into consideration. The hydromagnetic flow in the uniformly rotating fluid is generated by a suddenly moved infinite plate in its own plane. Analytic solution of the governing flow problem is obtained by means of the Fourier sine transform. It is shown that the obtained solution satisfies both the associate partial differential equation and the initial and boundary conditions. The solution for a Navier-Stokes fluid is recovered if λ→0. The steady state solution is also obtained for t→∞.     

Diffraction of a plane electromagnetic wave by a slot in a conducting screen with a transverse dielectric layer

The problem of diffraction of a plane electromagnetic wave by a slot in a planar perfectly conducting arbitrary thick screen with an infinite planar dielectric layer passing through the slot transversely to the screen is solved rigorously. In each of the field existence domains (two domains on either side of the screen and the interior of the slot), the solution is represented as an expansion in piecewise harmonic or exponential modes that allow for reflection and refraction at the boundaries of the dielectric layer. It is found that a set of functions describing such modes is complete enough to construct a solution satisfying all boundary conditions of the diffraction problem. The procedures of solution construction for the case at hand and for the same diffraction structure without the dielectric layer are compared.     

Wrinkling of a bilayer resting on a soft substrate under in-plane compression

The wrinkling problem of a bilayer film resting on a soft substrate under in-plane compression is investigated via theoretical analysis and numerical simulations. Such a system may undergo two different wrinkling modes, namely surface wrinkling of the top single layer on a composite substrate or bilayer wrinkling on a homogeneous substrate, depending on the geometric and material parameters. In the present paper, a theoretical solution is first derived for the former case, while a solution available in the literature is applied for the latter. The solution derived here can degenerate both to the classical solution for a stiff layer resting on a semi-infinite soft substrate and that for a stiff layer lying on a compliant substrate of finite thickness. A series of finite element simulations are performed to demonstrate the accuracy of the theoretical solution. In addition, it is found that, at a given compressive strain, a certain change in the elastic modulus of the intermediate layer may lead to a dramatic surface pattern transformation between the two wrinkling modes, indicating a novel approach to realise surface pattern switching.     

Scattering in a Pekeris waveguide from a rough bottom using a two-way coupled mode approach

Scattering from a rough ocean bottom is described numerically with a two-way coupled-mode formalism that contains scattering effects to all orders and provides an exact solution to the wave equation. Both scattered field and direct blast components are computed within the formalism framework. A comparison of the scattered component solution from the coupled mode with the Born approximation (BA) solution for scattering from a rough bottom Pekeris waveguide shows that the BA predicts correctly the scattered field levels but not detailed structure. The transition from direct blast to scattered field dominance is identified in the total field time series.     

The gravitational field of a homogeneous plate with a non-zero cosmological constant

A static interior solution of Einstein gravitational equations with λ ≠ 0 generated by a plane plate of a constant mass density is found. It is shown how to join this interior solution with the exterior metric as well     

The spacetime of a Dirac fermion

We present an approximate solution to the minimally coupled Einstein-Dirac equations. We interpret the solution as describing a massive fermion coexisting with its own gravitational field. The solution is axisymmetric but is time dependent. The metric approaches that of a flat spacetime at the spatial infinity. We have calculated a variety of conserved quantities in the system.     

Capacitance of a tube

The electrostatic problem of a hollow, conducting tube of finite length held at a fixed potential is solved using two methods. A two-term Galerkin solution is constructed for the surface distribution of induced charge. The sum of a uniform component and a simple edge-condition term provides a variational solution to the dual integral equations that are the equations-of-motion for the mixed boundary value problem. Comparisons are made with the numerical results of an independent boundary element or moment method. The numerical solution uses collocation or point matching and a piecewise constant basis for the charge density.     

On the gravitational field of a mass possessing a magnetic dipole moment

A solution of the Einstein-Maxwell magnetostatic equations describing the gravitational field of a mass endowed with a magnetic dipole moment is presented. It reduces to the Schwarzschild solution if the magnetic field vanishes.     

小虫爬行引起的绕竖直轴转动的圆盘转动问题的严格解

给出了小虫在可绕竖直轴无摩擦转动圆盘上爬行时引起圆盘转动问题的严格解,并就一个特例检验了结果的正确性。     

Colorimetric detection of ssDNA in a solution

A solution containing ssDNA was detected by using colloidal gold nanoparticles upon a color change of solution. First of all, we prepared colloidal gold nanoparticles with a size of less than 20 nm by means of citrate reduction of HAuCl₄. Since colloidal gold nanoparticles modified with citrate anions have a negative charge, they are very well dispersed in the solution due to the negative charge repulsion, showing red color. If the electrostatic repulsion is screened by additives such as NaCl, the nanoparticles start to aggregate, leading to a color alteration. We found that the color alteration is retarded when the solution contains ssDNA, which plays a role in preventing the nanoparticles from aggregation. This allows the determination of whether or not ssDNA is present in a solution. However, the color alteration is not retarded when the solution contains dsDNA.     

On the theory of a boundary layer associated with wave motion on the free surface of a liquid

A modified theory of a boundary layer associated with a periodic capillary-gravitational motion on the free surface of an infinitely deep viscous liquid is proposed. The flow in the boundary layer is described in terms of a simplified (compared with the complete statement) model problem a solution to which correctly reflects the main features of an exact asymptotic solution: the rapid decay of the flow eddy part with depth of the liquid and insignificance of some terms appearing in the complete statement. The boundary layer thickness at which the discrepancy between the exact asymptotic solution and model solution is within a given margin is estimated.     

Numerical analysis of a shock-wave solution of the Enskog equation obtained via a Monte Carlo method

In this paper a planar stationary shock-wave-like solution of the Enskog equation obtained via a Monte Carlo technique is studied; both the algorithm used to obtain the solution and the qualitative behavior of the macroscopic quantities are discussed in comparison with the corresponding solution of the Boltzmann equation.     

On the stability of a self-similar spherical bubble of a scalar Higgs field in de Sitter space

An exact generalized discontinuous solution of the spherical-bubble type is obtained for a scalar Higgs field in de Sitter space. It is shown that the radius of such a generalized bubble evolves in accordance with one of the exact solutions to a dynamical problem considered previously for the bubble radius in the thin-wall approximation, where the bubble-wall thickness is negligible in relation to the bubble radius. Both the generalized solution and the self-similar bubble-type solution that was obtained earlier for the Higgs field in de Sitter space are studied for stability: it is shown that the former is stable, while the latter is unstable in this space. A physical interpretation of the reasons for the instability of the self-similar solution is given.     

Microscale heat transfer in a free jet against a plane surface

A new two-layer model has been proposed to study microscale heat transfer associated with a developing flow boundary layer. As an example, a cold, microscale film of liquid impinging on an isothermal hot, horizontal surface has been investigated. The boundary layer is divided into two regions: a micro layer at microscale away from the surface and a macro layer at macroscale away from the surface. An approximate solution for the velocity and temperature distributions in the flow along the horizontal surface is developed, which exploits the hydrodynamic similarity solution for microscale film flow. The approximate solution may provide a valuable basis for assessing microscale flow and heat transfer in more complex settings.