Direct calculation of the reactive transition matrix by L2 quantum mechanical variational methods with complex boundary conditions

1989 ◽  
Vol 91 (3) ◽  
pp. 1643-1657 ◽  
Author(s):  
Yan Sun ◽  
Chin‐hui Yu ◽  
Donald J. Kouri ◽  
David W. Schwenke ◽  
Philippe Halvick ◽  
...  
Keyword(s):  
Boundary Conditions ◽  
Variational Methods ◽  
Transition Matrix ◽  
Direct Calculation ◽  
Quantum Mechanical ◽  
Complex Boundary

Quantum mechanical algebraic variational methods for inelastic and reactive molecular collisions

1988 ◽  
Vol 92 (11) ◽  
pp. 3202-3216 ◽  
Author(s):  
David W. Schwenke ◽  
Kenneth Haug ◽  
Meishan Zhao ◽  
Donald G. Truhlar ◽  
Yan Sun ◽  
...  
Keyword(s):  
Variational Methods ◽  
Quantum Mechanical ◽  
Molecular Collisions

Free Vibration of Orthotropic Skew Plates

1999 ◽  
Vol 122 (3) ◽  
pp. 313-317 ◽  
Author(s):  
A. M. Farag ◽  
A. S. Ashour
Keyword(s):  
Boundary Conditions ◽  
Transition Matrix ◽  
Longitudinal Direction ◽  
Displacement Function ◽  
Characteristic Matrix ◽  
Skew Angle ◽  
Kantorovich Method ◽  
Vibration Effect ◽  
Finite Strip Method ◽  
Semianalytical Method

The main purpose of this paper is to develop a fast converging semianalytical method for assessing the vibration effect on thin orthotropic skew (or parallelogram/oblique) plates. Since the geometry of the skew plate is not helpful in the mathematical treatments, the analysis is often performed by more complicated and laborious methods. A successive conjunction of the Kantorovich method and the transition matrix is exploited herein to develop a new modification of the finite strip method to reduce the complexity of the problem. The displacement function is expressed as the product of a basic trigonometric series function in the longitudinal direction and an unknown function that has to be determined in the other direction. Using the new transition matrix, after necessary simplification and the satisfaction of the boundary conditions, yields a set of simultaneous equations that leads to the characteristic matrix of vibration. The influence of the skew angle, the aspect ratio, the properties of orthotropy, and the prescribed boundary conditions are investigated. Convergence of the solution is investigated and the accuracy of the results is compared with that available from other numerical methods. The numerical results show that the convergence is rapidly deduced and the comparisons agree very well with known results. [S0739-3717(00)00202-6]

scholarly journals Antiperiodic Solutions forp-Laplacian Systems via Variational Approach

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Lifang Niu ◽  
Kaimin Teng
Keyword(s):  
Boundary Conditions ◽  
Variational Methods ◽  
Variational Approach ◽  
Existence Of Solutions

We establish the existence of solutions forp-Laplacian systems with antiperiodic boundary conditions through using variational methods.

scholarly journals Spectrum of discrete 2n-th order difference operator with periodic boundary conditions and its applications

2021 ◽  
Vol 41 (4) ◽  
pp. 489-507
Author(s):  
Abdelrachid El Amrouss ◽  
Omar Hammouti
Keyword(s):  
Boundary Conditions ◽  
Variational Methods ◽  
Critical Point Theory ◽  
Periodic Boundary ◽  
Eigenvalue Problems ◽  
Difference Operator ◽  
Periodic Boundary Conditions ◽  
Point Theory ◽  
Existence Of A Solution ◽  
Order Difference

Let \(n\in\mathbb{N}^{*}\), and \(N\geq n\) be an integer. We study the spectrum of discrete linear \(2n\)-th order eigenvalue problems \[\begin{cases}\sum_{k=0}^{n}(-1)^{k}\Delta^{2k}u(t-k) = \lambda u(t) ,\quad & t\in[1, N]_{\mathbb{Z}}, \\ \Delta^{i}u(-(n-1))=\Delta^{i}u(N-(n-1)),\quad & i\in[0, 2n-1]_{\mathbb{Z}},\end{cases}\] where \(\lambda\) is a parameter. As an application of this spectrum result, we show the existence of a solution of discrete nonlinear \(2n\)-th order problems by applying the variational methods and critical point theory.

scholarly journals On a p x -Biharmonic Kirchhoff Problem with Navier Boundary Conditions

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Hafid Lebrimchi ◽  
Mohamed Talbi ◽  
Mohammed Massar ◽  
Najib Tsouli
Keyword(s):  
Boundary Conditions ◽  
Variational Methods ◽  
Nontrivial Solution ◽  
Existence Of Solutions ◽  
Type Problem ◽  
Navier Boundary Conditions ◽  
Kirchhoff Type ◽  
Kirchhoff Type Problem ◽  
Kirchhoff Problem

In this article, we study the existence of solutions for nonlocal p x -biharmonic Kirchhoff-type problem with Navier boundary conditions. By different variational methods, we determine intervals of parameters for which this problem admits at least one nontrivial solution.

Numerical Methods of Higher-Order Accuracy for Diffusion- Convection Equations

1968 ◽  
Vol 8 (03) ◽  
pp. 293-303 ◽  
Author(s):  
H.S. Price ◽  
J.C. Cavendish ◽  
R.S. Varga
Keyword(s):  
Finite Difference ◽  
Boundary Conditions ◽  
Variational Methods ◽  
Miscible Displacement ◽  
Two Dimensional ◽  
Order Accuracy ◽  
One Dimensional ◽  
Finite Difference Approximations ◽  
Difference Approximations ◽  
Diffusion Convection

Abstract A numerical formulation of high order accuracy, based on variational methods, is proposed for the solution of multidimensional diffusion-convection-type equations. Accurate solutions are obtained without the difficulties that standard finite difference approximations present. In addition, tests show that accurate solutions of a one-dimensional problem can be obtained in the neighborhood of a sharp front without the need for a large number of calculations for the entire region of interest. Results using these variational methods are compared with several standard finite difference approximations and with a technique based on the method of characteristics. The variational methods are shown to yield higher accuracies in less computer time. Finally, it is indicated how one can use these attractive features of the variational methods for solving miscible displacement problems in two dimensions. Introduction The problem of finding suitable numerical approximations for equations describing the transport of heat or mass by diffusion and convection simultaneously has been of interest for some time. Equations of this type, which will be called diffusion-convection equations, arise in describing many diverse physical processes. Of particular interest here is the equation describing the process by which one miscible liquid displaces another liquid in a one-dimensional porous medium. The behavior of such a system is described by the following parabolic partial differential equation: (1) where the diffusivity is taken to be unity and c(x, t) represents a normalized concentration, i.e., c(x, t) satisfied 0 less than c(x, t) less than 1. Typical boundary conditions are given by ....................(2) Our interest in this apparently simple problem arises because accurate numerical approximations to this equation with the boundary conditions of Eq. 2 are as theoretically difficult to obtain as are accurate solutions for the general equations describing the behavior of two-dimensional miscible displacement. This is because the numerical solution for this simplified problem exhibits the two most important numerical difficulties associated with the more general problem: oscillations and undue numerical dispersion. Therefore, any solution technique that successfully solves Eq. 1, with boundary conditions of Eq. 2, would be excellent for calculating two-dimensional miscible displacement. Many authors have presented numerical methods for solving the simple diffusion-convection problem described by Eqs. 1 and 2. Peaceman and Rachford applied standard finite difference methods developed for transient heat flow problems. They observed approximate concentrations that oscillated about unity and attempted to eliminate these oscillations by "transfer of overshoot". SPEJ P. 293ˆ

Variational approaches to systems of Sturm–Liouville boundary value problems

2020 ◽  
pp. 2150032
Author(s):  
Shapour Heidarkhani ◽  
Ghasem A. Afrouzi ◽  
Shahin Moradi
Keyword(s):  
Boundary Conditions ◽  
Variational Methods ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Technical Approach ◽  
Three Solutions ◽  
Value System ◽  
Sturm Liouville ◽  
Variational Approaches

In this paper, we consider the existence of one solution and three solutions for the boundary value system with Sturm–Liouville boundary conditions [Formula: see text] for [Formula: see text]. Our technical approach is based on variational methods. In addition, examples are provided to illustrate our results.

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