Eigen Functions Research Articles

Dispersion in Presence of Slip and Chemical Reactions in Porous-Wall Tube Flow

This paper deals with the problem of studying the effect of tangential velocity slip on the concentration profiles of a solute which is convectively diffusing and simultaneously undergoing irreversible, first order homogeneous and heterogeneous chemical reactions in a porous wall tube, under isothermal laminar flow conditions. It is shown that the presence of velocity slip at the porous wall induces an increase in the concentration of solute across the entirecross-section of the tube while an increase in the rates of chemical reactions is found to reduce the concentration under both slip and no-slip boundary conditions. Accurate eigen values, eigen functions and the concentration profiles are determined for the cases when slip or no-slip boundary condition holds at the wall.

多光束干渉図形の解析による高分解能分光法

Using the theory of Hilbert and Schmidt, spectrum can be reconstructed from the multiplebeam interferogram obtained with the Fabry-Perot interferometer . The resolution of this spectroscopic method is studied by the computer simulation. The effect of noise involved in the interferogram can be reduced by discarding the high frequency terms in eigenf unction expansion. The resolving power of this method turned out to be several millions and is about ten times larger than that obtained with conventional method . Results obtained by the method of Hilbert and Schmidt are compared with those obtained with the direct method. In the direct method, reconstructed spectrum using the interferogram involving noise is usually distorted. But it can be improved to some extent by applying the Fourier analysis.

Electromagnetic Particle Simulation Code PS2M for Bounded Plasmas with Conducting Walls

A computational method for solving boundary-value problems for the Maxwell equation in the particle simulation is developed and is emlcodied in a code PS2M, which is a new Particle Simulation code in 2.5 dimensions (2 space-dimensions and 3 velocity-dimensions) for electroMagnetic plasmas. This method is formulated by introducing a double representation of the δ functions which are expanded by two independent sets of eigen functions in a system. The validity of solutions of the boundary-value problems is proved by invoking the mathematical law of Green's functions. The high-order spline spatial interpolation was confirmed to be very useful in the simulation for electromagnetic plasmas, because its use removes the limit on the grid spacing, which has been severely limited of order of the Debye length in the conventional methods. This code, PS2M, has been used to study RF stabilization of the flute mode and ECRH.

Solutions to Three-dimensional Second-order Diffraction Problems by Means of Simple-source Integral-equation Method

Second-order diffraction problems for a three-dimensional body are solved by the use of simple sources and eigen functions. Numerical results of second-order velocity potentials are available as well as second-order wave forces, which are usually computed from first-order velocity potentials only, invoking Green's theorem.Numerical analyses are done for a vertical circular cylinder in deep water so as to verify the validity of the method. Agreement between the results by the present method and existing results are satisfactory. It is also confirmed that the second-order velocity potentials on the weather side penetrate the water much deeper than the first-order ones, i. e., the second-order velocity potentials surpass the first-order ones in the depth.

Effect of Free Surfaces and Interfaces on Dopant Distribution Profiles

ABSTRACTThe drift force exerted by a free surface, a coherent or a noncoherent interface on dopant atoms has been analyzed. We have considered a coherent or a noncoherent interface present in an epilayer of finite thickness. The difference in the shear modulus of the matrix and of the second phase is responsible for the configurational energy of a dopant atom in a coherent two-phase medium. On the other hand, the hydrostatic component of stress associated with a misfit dislocation in a noncoherent interface gives rise to a first-order size interaction. The drift forces are responsible for dopant diffusion towards a free surface or an interface. The diffusion equation including the drift term is solved using an eigen function expansion method with appropriate boundary conditions. The concentration profiles obtained from the analysis of the diffusion equation are in qualitative agreement with those obtained experimentally. The attractive drift force gives rise to a peak in the dopant concentration at the interface followed by a minimum near the interface. These calculations and the observed concentration profiles enable us to evaluate the interaction energy term of the dopants with free surfaces and interfaces and the dilatation associated with the point defects.

Stress singularities at the fiber end in short fiber reinforced composites.

The plane strain elastostatic problem for short fiber reinforced composite material is examined. Using the eigen function expansion method, stress singularities at the corner point of a fiber end and the tip of a fiber end crack are estimated. The effect of material combinations of fiber and matrix on the singularities are investigated. The singularity at the fiber end crack is numerically examined for carbon fiber reinforced epoxy by the boundary element technique. Based on the concept of linear fracture mechanics, the coefficient of the singular term for the fiber end crack problem is obtained by the numerical technique.

Microscopic Theory of the Proximity Effect in the Superconducting Transistor

Based on exact single-particle eigen functions of a superconductor-inversion layer model system, microscopic theory of the superconducting proximity effect is developed. The general integral kernel of Gorkov equation is obtained, which connects the clean and the dirty limit. Numerical solutions of the pair potential are obtained for this kernel. Based on this, the boundary condition of the pair potential, and the behaviour of the extrapolation length are clarified. The result connects naturally the dirty limit, where the phenomenological theory by de Gennes is valid, to the clean limit where the wave character of the Cooper pair prevails.

On the Square Eigenfunction Expansion for the Heisenberg Spin Chain

From the inverse scattering problem of the Heisenberg Spin chain, we have deduced the Square eigen function expansion of the variation of the non-linear Spin variables. The situation is different from the usual A.K.N.S. problem as the linear problem is not an eigenvalue problem in literal sense, and this also makes, the formulation of the equation for square eigen function difficult. So we have considered the corresponding equivalent matrix Schrödinger problem, and proved the orthogonality of square eigen-functions after having expressed the variation of the scattering data in terms of the field variables and square eigen-functions.

Discrete ordinate solution of a Fokker-Planck equation in laser physics

Abstract A newly developed discrete ordinate method of solution of differential equations is applied to the solution of the Fokker-Plank equation employed in the a nalysis is of optical bistability. The solution is written in terms of the eigenfunctions of the Fokker-PlancK operator and the time evolution of the probability density function is studied. The present analysis includes a study of the convergence of the eigenualues and eigen functions for different choices of the quadrature rule upon which the discrete ordinate method is based.

WAVE INTERACTION WITH MOORED SLOPING BREAKWATER

Interaction of periodic waves with a moored inclined floating breakwater has been studied theoretically and numerically. The floating breakwater is inclined at a well defined angle with the sea bottom; its seaward end in protruding above the water surface. In static equilibrium, without incoming waves, the body weight, the buoyance force, and the restoring forces from the mooring lines which are modeled using linear springs keep the breakwater at a fixed angle. The theoretical formulation is based on a suitable variational principle. For the numerical solution a combination of finite element approximation as well as eigen function expansion technique is used. The result is obtained in terms of wave transmission and reflection coefficient as well as the sway, heave and roll motion of the breakwater. The sensitivity of the solution on the parameters such as the bottom gap size, angle of inclination, and the mooring line stiffness are investigated over a range of incident wave transmission coefficient for dimensionless wave number hk > 0.60 ( k is wave number, h is the water depth). The results suggest that a certain degree of sheltering effect can be realized by employing this type of sloping breakwater.

The eigenvalue difference problem for the fourth order elliptic operator with mixed boundary conditions

The accuracy of the net method when the principal and natural boundary conditions are specified at the boundary is established. The interchange points of the conditions type are found at the corners of the rectangle. Estimates of the error of the solution are obtained under the assumption that the eigen functions belong to the Sobolev space W 2 4(Ω) .

Computation of the input impedances of a catheter for cardiac volumetry.

Electrical impedances between the electrodes of a catheter in a spheroidal model of the left ventricle are calculated analytically. The fields in the configuration are computed using eigen function expansions of the Laplace equation. For certain realistic combinations of specific conductances of the blood and the myocardium, numerical results are given. These results show the dependence of the impedances on the volume of the ventricle while keeping the length of the major axis of the ventricle constant. Also, the accuracy of the model and the dependence of the total conductance of the ventricle on its shape, and on the conductivity of the tissues surrounding it, are discussed. The theoretical results are compared against the experimental values of conductances, measured post mortem in a canine heart, for different volumes.

Direct intersubband optical absorption of semiconducting thin wire

The optical absorption due to direct intersubband transitions and including the broadening effects has been calculated by using an electric dipole approximation for a quasi-one-dimensional semiconducting thin wire. Effective mass approximation is adopted to determine the eigen functions and energies of the conduction electrons in Schrodinger equation by assuming a two-dimensional infinite potential well. When the electromagnetic wave is polarized in a ẑ direction which possesses a size-quantization effect in the thin wire, the selection rule requires that the sum of the quantum numbers of the initial state and finial state must be odd integers. ac and dc conductivities, and the absorption line shapes due to collisions are discussed.

Continuous Mode in the Theory of the Barotropic Instability of Zonal Flows

Evolution of the two-dimensional small disturbance superimposed on a viscous parallel shear flow is investigated under the beta-plane approximation of revolution. Formal solution is obtained for an arbitrarily localized initial distribution of vorticity. The eigenfunctions of the normal mode are proved to compose a complete set. In the region where the basic flow is uniform the eigenfunction of the continuous mode behaves as the Rossby wave. The interaction between the shear and the Rossby wave is found to be an inherent character of the eigen function of the continuous mode. The properties of the critical latitude of the Rossby wave are attributable to those of the friction layer in the eigenfunction.

Reliability Analysis of Lifeline Systems

Considering that most of the lifeline systems are composed of a few kinds of many structures, the reliability of the system is represented by the use of two functions, one pertaining to the reliability of the structures, and the other pertaining to the system configuration and redundancy. The former function is calculated by the use of the reliability of each kind of structure, which is effected by the external force and resisting capacity of structures. The latter function has nothing to do with the external force nor the resisting capacity of the structure and is an eigen function that depends on the configuration of the system and also depends on the safety criterion of the system. By the use of these two functions, an economical method to evaluate the reliability of the system is derived. Then, for the case that these two functions can be assumed to be the Gaussian distributions, a simple method to compute the reliability of the system is developed. For the system composed of only one kind of structures, a safety index is proposed.

Global shearing modes of galactic disks

A new scheme has been devised to calculate discrete unstable global shearing spiral modes of gaseous disk models of galaxies. The scheme makes use of the Legendre expansion of eigen functions and the problem of stability analysis is reduced to an eigenvalue problem of an infinite matrix. The spiral patterns of these shearing wave solutions of linearized equations change their form in the course of time due to the differential rotation of the equilibrium disk. These shearing wave solutions are presumed to have intermediate characteristics between so-called density-waves and material arms. Comparison between these shearing modes and the non-shearing normal modes for a series of disk models is presented.

Coupling instability of the shear Alfvén wave in the magnetosphere with the ionospheric ion drift wave, 2. Numerical analysis

In the present paper we describe in detail results of numerical calculations for eigen oscillations of the shear Alfvén wave coupled with the ionospheric ion drift wave in the presence of the steady electric field. The instability mechanism has been discussed in a companion paper (Tamao and Miura, this issue). For a model magnetosphere with a nonuniform density distribution along dipole magnetic field lines, together with ionospheric boundary conditions through which the coupling with the ion drift wave is taking place, we have solved numerically a system of differential equations describing the localized torsional oscillations of field lines. Dependences of the eigenfrequency and the growth (or damping) rate of oscillations on the height‐integrated ionospheric conductivity are examined, and it is shown that the nighttime oscillations exhibit a stronger instability than those for daytime. There is a minimum threshold intensity of the steady electric field and a maximum scale length in the north‐south direction of the localized oscillation in the ionosphere for occurrence of the instability. These are ten to a few tens of mV/m and a few tens of kilometers, respectively. The maximum scale length decreases with decreasing invariant latitude of the field line, while the peak value of the growth rate increases. Eigen function distributions along the field line are also determined, which makes it possible to infer dependences of perturbations at the level just above the ionosphere on the Pedersen conductivity in the ionosphere.

Free and forced shallow-water oscillations in a rotating channel of parabolic cross-section

Free and forced oscillations of shallow water in an infinitely long rotating channel of parabolic cross-section are analyzed. The pure cross-channel oscillations ofChrystal (1905) and solutions for zero rotation first discussed byProudman (1925) andHidaka (1932) are special asymptotic solutions for the free modes of this model. However, for increasingly large, along-shore wave number, our solutions donot uniformly approach those ofReid (1958) andBall (1967) for a single shore-line and semi-infinite ocean. A method of computing eigen frequencies and eigen functions for the general problem is described, and a sufficient number of these are exhibited graphically to permit visualization of the transitions between the asymptotic regions.

Structure of tail states of Landau subbands in a two dimensional electron system under strong magnetic fields

Eigenvalue problem for an effective Hamiltonian to describe the dynamics of the center coordinates of the cyclotron motion, derived by Tsukada, is explicitly solved in the case of a single Gaussian type potential. The eigen function does not depend on the range of the potential while the eigenergy does. Assuming random distributions of the strength and range of the potentials and of the positions of potential centers, the frequency dependence of the dynamical conductivity due to the tail states is expected to be given by D(E F ) 2 ω 2 ℓnω in the low frequency and low temperature limit where D(E F ) is the density of states at the Fermi energy E F .

Eigenfunction Expansions for a Singular Eigenvalue Problem with Eigenparameter in the Boundary Condition

We study the singular boundary value problem $\begin{gathered} \tau (u) = \frac{1}{r}\left\{ { - \left( {pu'} \right)^\prime + qu} \right\} = \lambda u,\quad a \leqq x < b < \infty \hfill \\ - \beta _1 u(a) + \beta _2 \left( {pu'} \right)(a) = \lambda \left[ {\beta '_1 u(a) - \beta '_2 \left( {pu'} \right)(a)} \right] \hfill \\ \end{gathered} $ Conditions are placed on the coefficients which ensure the spectrum is bounded below and the essential spectrum is empty. An eigenf unction expansion theory is developed for a class $\mathcal{D}_1 $ of functions. For this class the convergence is uniform and absolute on compact intervals. When the eigenvalues are all nonnegative, the class $\mathcal{D}_1 $ is shown to be the domain of $A^{{1 / 2}} $, where A is the self -adjoint operator associated with the boundary value problem.