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By
Glowinski, Roland; Li, ChinHsien; Lions, JacquesLouis
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63 Citations
In this paper we discuss the numerical implementation of a systematic method for the exact boundary controllability of the wave equation, concentrating on the particular case of Dirichlet controls. The numerical methods described here consist in a combination of: finite element approximations for the space discretization; explicit finite difference schemes for the time discretization; a preconditioned conjugate gradient algorithm for the solution of the discrete problems; a pre/post processing technique based on a biharmonic Tychonoff regularization. The efficiency of the computational methodology is illustrated by the results of numerical experiments.
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By
Arai, Masaharu; Okamoto, Kazuo; Kametaka, Yoshinori
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2 Citations
The third author discovered numerically an interesting phenomenon that the Aitken acceleration to the ratiop_{n}=f_{n−1}/f_{n} of the successive Fibonacci numbersf_{n−1} andp_{n} in exactlyp_{2n}. This fact has a natural extension to the case that Aitken acceleration is replaced by εalgorithm and Fibonacci numbers can be replaced by solutions of second order difference equations. As a byproduct we will show annplication formula of cotz.
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By
Kuroda, S. T.; Suzuki, Toshio
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6 Citations
A method is proposed to compute an eigenfunction and the associated eigenvalue of the Schrödinger operatorH=−Δ+V(x), the eigenvalue being required to be closest to a given numberE. The idea is to use a timedependent Schrödinger equation with the inhomogeneous term exp(−irE)h to excite the target mode. A device is introduced to avoid exciting remote resonant eigenvalues. Some analysis of the schemes is given and it is shown that the use of extrapolations is effective. Some numerical examples are presented. They indicate that the proposed method produces rather accurate results.
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By
Ikehata, Masaru; Nakamura, Gen
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2 Citations
We consider the decaying mode of the local energy of the solution of an initial boundary value problem in the exterior region outside a spherical obstacle for the equation of elasticity as time tends to infinity. It is shown that Rayleigh’s surface wave prevents the exponential decay of the local energy in the case of the Neumann boundary condition and that the local energy blows up at the rate of the square of the time variable in the case of the Robin boundary condition.
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By
Ukai, Seiji
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31 Citations
The nonlinear Bolzmann equation is discussed without cutoff approximations on potentials of infinite range. The Cauchy problem is solved locally in time, for both the spatially homogeneous and inhomogeneous cases. For the former case, this is done in function spaces of Gevrey classes in the velocity variables, and for the latter, in spaces of functions which are analytic in the space variables and of Gevrey classes in the velocity variables. The obtained existence theorem is of CauchyKowalewski type. Also, the convergence of Grad’s angular cutoff approximations is established.
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By
Yamamoto, Norio
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Period doubling is considered when two or more different branches intersect at a bifurcation point and one of them consists of πperiodic solutions and the others consist of 2πperiodic solutions, and a method for computing such a bifurcation point with high accuracy is proposed.
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By
Shiota, Yasunobu
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2 Citations
We will generalize Hata’s theorems on the selfsimilarity and show that the graph of the generalized Takagi function is a typical example of our result. We will also investigate the relation between selfaffine functions and selfsimilar sets.
By
Li, TienYien; Sauer, Tim
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5 Citations
Most systems of polynomials which arise in applications have fewer than the expected number of solutions. A simple homotopy is presented for finding all solutions of such a “deficient” system. Different from current homotopies used for such systems, only one parameter is needed to regularize the problem. Within some limits an arbitrary starting problem can be chosen, as long as its solution set is known.
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By
Kabaya, Keiko; Iri, Masao
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7 Citations
It is shown that the probability density functionp_{n}(x) for the random variableX_{n} = α[U_{n} + (1 − α)U_{n − 1} + ... + (1 − α)^{n − 2}U_{2} + (1 − α)^{n − 1}U_{1}] (where 0<α<1 andU_{i}^{′}
s are independent random variables subject to the uniform distribution on the interval [−1, 1]) uniformly converges to an infinitely differentiable functionp_{∞}(x) asn→∞, and thatp_{∞}(x) is nonanalytic at infinitely many points. In particular, for α=1/2,p_{∞}(x) is nowhere analytic on [−1, 1].
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By
Hamdache, Kamel
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26 Citations
This work is devoted to the global existence and asymptotic behaviour of the solutions of several models for the Boltzmann equations with angular cutoff. Those models contain the so called soft and hard interaction potentials (7/3<s≤+∞).
