Classical Separation Of Variables Research Articles

Application of generalized separation of variables to solving mixed problems with irregular boundary conditions

One of the methods for solving mixed problems is the classical separation of variables (the Fourier method). If the boundary conditions of the mixed problem are irregular, this method, generally speaking, is not applicable. In the present paper, a generalized separation of variables and a way of application of this method to solving some mixed problems with irregular boundary conditions are proposed. Analytical representation of the solution to this irregular mixed problem is obtained.

Motion on then-dimensional ellipsoid under the influence of a harmonic force revisited

The $n$ integrals in involution for the motion on the $n$-dimensional ellipsoid under the influence of a harmonic force are explicitly found. The classical separation of variables is given by the inverse momentum map. In the quantum case the Schr\odinger equation separates into one-dimensional equations that coincide with those obtained from the classical separation of variables. We show that there is a more general orthogonal parametrisation of Jacobi type that depends on two arbitrary real parameters. Also if there is a certain relation between the spring constants and the ellipsoid semiaxes the motion under the influence of such a harmonic potential is equivalent to the free motion on the ellipsoid.

Small time approximations of Green's functions for one‐dimensional heat conduction problems with convective boundaries

AbstractSmall time approximations are obtained for Green's functions in one‐dimensional heat conduction problems with convective boundaries. The method of images similar to those for cases with Dirichlet or Neumann boundaries is used. Multiple reflections in this method generate infinite series representations for the Green's functions whose general terms involve convolution integrals with integrable singularities. We truncate the series after single and double reflections (n = 1, 2) and transform the convolution integrals to known closed‐form expressions. Numerical results are presented for n = 1, 2 and compared with exact results obtained using the classical separation of variables. It is seen that results with n = 1, 2 provide accurate approximations for dimensionless time α t / L2 up to 0.3 and complementary to those obtained by separation of variables.

Harmonics on the quantum Euclidean space

We study harmonic polynomials on the quantum Euclidean space ENq generated by quantum coordinates xi, i = 1, 2, ..., N, on which the quantum group SOq(N) acts. They are defined as solutions of the equation Δqp = 0, where Δq is the q-Laplace operator on ENq. We construct a q-analogue of the classical zonal polynomials and associated spherical polynomials with respect to the quantum subgroup SOq(N − 2). The associated spherical polynomials constitute an orthogonal basis of the spaces of homogeneous harmonic polynomials. They are represented as products of polynomials depending on q-radii and xj, xj', j' = N − j + 1. This representation is, in fact, a q-analogue of the classical separation of variables.

Neoclassical Solution of Transient Interaction of Plane Acoustic Waves with a Spherical Elastic Shell

A detailed solution to the transient interaction of plane acoustic waves with a spherical elastic shell was obtained more than a quarter of a century ago based on the classical separation of variables, series expansion, and Laplace transform techniques. An eight­ term summation of the time history series was sufficient for the convergence of the shell deflection and strain, and to a lesser degree, the shell velocity. Since then, the results have been used routinely for validation of solution techniques and computer methods for the evaluation of underwater explosion response of submerged structures. By utilizing modern algorithms and exploiting recent advances of computer capacities and floating point mathematics, sufficient terms of the inverse Laplace transform series solution can now be accurately computed. Together with the application of the Cesaro summation using up to 70 terms of the series, two primary deficiencies of the previous solution are now remedied: meaningful time histories of higher time derivative data such as acceleration and pressure are now generated using a sufficient number of terms in the series; and uniform convergence around the discontinuous step wave front is now obtained, completely eradicating spurious oscillations due to the Gibbs' phenomenon. New results of time histories of response items of interest are pre­ sented. © 1996 John Wiley & Sons, Inc.

Neoclassical Solution of Transient Interaction of Plane Acoustic Waves with a Spherical Elastic Shell

A detailed solution to the transient interaction of plane acoustic waves with a spherical elastic shell was obtained more than a quarter of a century ago based on the classical separation of variables, series expansion, and Laplace transform techniques. An eight-term summation of the time history series was sufficient for the convergence of the shell deflection and strain, and to a lesser degree, the shell velocity. Since then, the results have been used routinely for validation of solution techniques and computer methods for the evaluation of underwater explosion response of submerged structures. By utilizing modern algorithms and exploiting recent advances of computer capacities and floating point mathematics, sufficient terms of the inverse Laplace transform series solution can now be accurately computed. Together with the application of the Cesaro summation using up to 70 terms of the series, two primary deficiencies of the previous solution are now remedied: meaningful time histories of higher time derivative data such as acceleration and pressure are now generated using a sufficient number of terms in the series; and uniform convergence around the discontinuous step wave front is now obtained, completely eradicating spurious oscillations due to the Gibbs' phenomenon. New results of time histories of response items of interest are presented.

The Wave Propagation Method for the Analysis of Boundary Stabilization in Vibrating Structures

In the modern vibration control of flexible space structures and flexible robots, various boundary feedback schemes have been employed to cause energy dissipation and damping, thereby achieving stabilization. The mathematical analysis of eigenspectrum of vibration is usually carried out by classical separation of variables and by solving the transcendental equations. This involves rather lengthy and tedious work due to the complexity and the numerous boundary conditions. A different approach, developed by Keller and Rubinow, uses ideas from wave propagation to obtain asymptotic estimates of eigenvalues for multidimensional scattering problems. This approach is powerful and yields accurate eigenvalue estimates even at a relatively low frequency range [Ann. Physics, 9 (1960), pp. 24–75]. In this paper, we take advantage of this wave approach to study one-dimensional vibration problems with boundary damping. We decompose vibration waves into incident, reflected (including transmitted) and evanescent waves. Based on their amplitude and reflection coefficients we are able to derive eigenfrequency estimates and compute numerical values. This wave propagation method greatly simplifies the asymptotic estimation procedures and reproduces earlier results in [SIAM J. Appl. Math., 47 (1987), pp. 751–780], [Lecture Notes in Pure and Applied Mathematics, Vol. 108, Marcel Dekker, New York, 1987], [ SIAM J. Appl., Math., 49 (1989), pp. 1665–1693]. It also yields new insights into various structural control problems such as feedback with tipped mass [12], nonrobustness of feedback delays [SIAM J. Control Optim., 24 (1986), pp. 152–156], [SIAM J. Control Optim., 26 (1988), pp. 697–713], noncollocated sensors and actuators, and special medium-low frequency structural damping patterns.

Thermal conductivity of coated filler composites

The temperature field and thermal conductivity of a coated-fiber composite were studied analytically. The analytical model is based on the classical separation of variables and equivalent-inclusion method. Closed-form solutions are given for the thermal conductivity of a coated-fiber composite. The present results on the temperature gradient in a thin coating were compared with those based on the thin-coating model by Walpole, resulting in a complete agreement when the thickness of the coating approaches zero. Then, a parametric study was conducted to study the effect of coating properties and other parameters on the thermal conductivity of the composite. It was found in this parametric study that the use of highly conductive (or resistive) coating, even if its thickness is small, is quite effective in enhancing the overall thermal conductivity (or resistivity) of the composite.

Lie Group Representations and Harmonic Polynomials of a Matrix Variable

The first part of this paper deals with problems concerning the symmetric algebra of complex-valued polynomial functions on the complex vector space of n by k matrices. In this context, a generalization of the so-called “classical separation of variables theorem” for the symmetric algebra is obtained. The second part is devoted to the study of certain linear representations, on the above linear space (the symmetric algebra) and its subspaces, of the complex general linear group of order k and of its subgroups, namely, the unitary group, and the real and complex special orthogonal groups. The results of the first part lead to generalizations of several well-known theorems in the theory of group representations. The above representation, of the real special orthogonal group, which arises from the right action of this group on the underlying vector space (of the symmetric algebra) of matrices, possesses interesting properties when restricted to the Stiefel manifold. The latter is defined as the orbit (under the action of the real special orthogonal group) of the n by k matrix formed by the first n row vectors of the canonical basis of the k-dimensional real Euclidean space. Thus the last part of this paper is involved with questions in harmonic analysis on this Stiefel manifold. In particular, an interesting orthogonal decomposition of the complex Hilbert space consisting of all square-integrable functions on the Stiefel manifold is also obtained.

Pivoted Plane Pad Bearings: A Variational Solution

It is known that the solution of Reynolds’ equation for the plane pad bearing presents no theoretical, but much computational, difficulty. In this paper, after a classical separation of variables is effected, the eigenvalue problem that arises is replaced by a Rayleigh-Ritz matrix. Then it is shown that the design variables of the bearing—all of which depend on integrals of the pressure—can be calculated quite accurately from inner products involving two vectors and a function of the Rayleigh-Ritz matrix, without solving any eigenvalue problem.