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Initial value problem for singularly perturbed differential-difference equation which is scalar, linear and has constant coefficients
Asymptotic rate of convergence of Gauss-Seidel type iterative processes for nonlinear difference equations
LaSalle-like stability theorems using invariance properties of ordinary difference equations solutions, deriving convergence region
Asymptotic solutions of linear difference equations, discussing applications to eigenvalue problems of free oscillations of model galaxies
Discrete approximation schemes for the solution of nonlinear hereditary control problems are constructed. The methods involve approximation by a sequence of optimal control problems in which the original infinite dimensional state equation has been approximated by a finite dimensional discrete difference equation. Convergence of the state approximations is argued using linear semigroup theory and is then used to demonstrate that solutions to the approximating optimal control problems in some sense approximate solutions to the original control problem. Two schemes, one based upon piecewise constant approximation, and the other involving spline functions are discussed. Numerical results are presented, analyzed and used to compare the schemes to other available approximation methods for the solution of hereditary control problems.
Discrete approximation schemes for the solution of nonlinear hereditary control problems are constructed. The methods involve approximation by a sequence of optimal control problems in which the original infinite dimensional state equation has been approximated by a finite dimensional discrete difference equation. Convergence of the state approximations is argued using linear semigroup theory and is then used to demonstrate that solutions to the approximating optimal control problems in some sense approximate solutions to the original control problem. Two schemes, one based upon piecewise constant approximation, and the other involving spline functions are discussed. Numerical results are presented, analyzed and used to compare the schemes to other available approximation methods for the solution of hereditary control problems. Previously announced in STAR as N83-33589
The propagation and dissipation of wavelike solutions to finite difference equations is analyzed on the basis of an asymptotic approach in which a wave solution is expressed as a product of a complex amplitude and an oscillatory phase function whose frequency and wavenumber may also be complex. An asymptotic expansion leads to a local dispersion relation for wavenumber and frequency; the first-order terms produce an equation for the amplitude in which the local group velocity appears as the convection velocity of the amplitude. Equations for the motion of wavepackets and their interaction at boundaries are derived, and a global stability analysis is carried out.
A semi-implicit iterative technique is given for the solution of implicit matrix difference equations which arise in variable-step direct time integration of dynamic systems equations. For all practical purposes, the technique avoids repeated refactoring of the stepsize-dependent solution matrix and achieves the solution convergence for engineering accuracy requirements within a permissible number of iterations. Extension of the present method to nonlinear problems treated by the pseudo-force procedure is straightforward. For nonlinear solution procedures based on the tangent modulus formulation, the technique requires modification to make use of the tangent stiffness matrix. It is noted that the question of whether to include the nonlinear terms within the iteration loop has not yet been resolved.
Linear static deformation and buckling of shallow spherical shells under asymmetric load by solving two dimensional finite difference equations
The classical method of equivalent linearization is extended to a particular class of nonlinear difference equations. It is shown that the method can be used to obtain an approximation of the periodic solutions of these equations. In particular, the parameters of the limit cycle and the limit points can be determined. Three examples illustrating the method are presented.
A technique to construct a uniformly valid perturbation series solution to a particular class of nonlinear difference equations is shown. The method allows the determination of approximations to the periodic solutions to these equations. An example illustrating the technique is presented.
A fast method for generating three-dimensional grids for fuselage-wing transonic flow calculations using parabolic difference equations is described. No iterative scheme is used in the three-dimensional sense; grids are generated from one grid surface to the next starting from the fuselage surface. The computational procedure is similar to the iterative solution of the two-dimensional heat conduction equation. The proposed method is at least 10 times faster than the elliptic grid generation method and has much smaller memory requirements. Results are presented for a fuselage and wing of NACA-0012 section and thickness ratio of 10 percent. Although only H-grids are demonstrated, the present technique should be applicable to C-grids and O-grids in three dimensions.
In many biological and physical systems, feedback mechanisms depend on a set of thresholds associated with the state variables. Each feedback has a characteristic time scale. It is suggested that delay-difference equations for Boolean-valued variables are an appropriate mathematical framework for such situations: the feedback thresholds result in the discrete, on-off character of the variables, and the interaction time scales of the feedbacks are expressed as delays. The initial-value problem for Boolean delay equations (B-Delta-Es) is formulated, and shown to have unique solutions for all times. Examples of periodic and aperiodic solutions are given. Aperiodic solutions have increasing complexity which depends on time t roughly as t to the l-1 power, l being the number of delays. Stability of solutions is defined, and some examples of stability analysis are given; additional stability questions are raised. The present formulation of (B-Delta-Es) is compared with related work and generalizations are suggested.
It is shown that a discrete multi-time method can be constructed to obtain approximations to the periodic solutions of a special class of second-order nonlinear difference equations containing a small parameter. Three examples illustrating the method are presented.
Future space vehicles will be relatively large and flexible, and active control will be necessary to maintain geometrical configuration. While the stresses and strains in these space vehicles are not expected to be excessively large, their cumulative effects will cause significant geometrical nonlinearities to appear in the equations of motion, in addition to the nonlinearities caused by material properties. Since the only effective tool for the analysis of such large complex structures is the digital computer, it will be necessary to gain a better understanding of the nonlinear ordinary difference equations which result from the time discretization of the semidiscrete equations of motion for such structures.
The evolution of a completely excited system of N two-level atoms, distributed over a large region and interacting with all modes of radiation field, is studied. The distinction between r-conserving (RC) and r-nonconserving (RNC) processes is emphasized. Considering the number of photons emitted as the discrete independent variable, the evolution is described by a partial difference equation. Numerical solution of this equation shows the transition from RNC dominance at the beginning to RC dominance later. This is also a transition from incoherent to coherent emission of radiation.
The dependence of the brightness of chromospheric network elements on latitude was investigated.
The theory of Massera and Schaeffer relating the existence of unique almost periodic solutions of an inhomogeneous linear equation to an exponential dichotomy for the homogeneous equation was completely extended to discretizations by a strongly stable difference scheme. In addition it is shown that the almost periodic sequence solution will converge to the differential equation solution. The preceding theory was applied to a class of exponentially stable partial differential equations to which one can apply the Hille-Yoshida theorem. It is possible to prove the existence of unique almost periodic solutions of the inhomogeneous equation (which can be approximated by almost periodic sequences) which are the solutions to appropriate discretizations. Two methods of discretizations are discussed: the strongly stable scheme and the Lax-Wendroff scheme.