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Work on the construction of finite difference models of differential equations having zero truncation errors is summarized. Both linear and nonlinear unidirectional wave equations are discussed. Results regarding the construction of zero truncation error schemes for the full wave equation and Burger's equation are also briefly reported.
Partial differential equations for modeling the structural dynamics and control systems of flexible spacecraft are applied here in order to facilitate systems analysis and optimization of these spacecraft. Example applications are given, including the structural dynamics of SCOLE, the Solar Array Flight Experiment, the Mini-MAST truss, and the LACE satellite. The development of related software is briefly addressed.
New Lie series method for numerical integration of ordinary differential equations, and restricted three-body problem application
Fortran ii computer program for use with univac 80 solid-state computer for solving differential equations of simply-supported cylindrical shells
We discuss an Adams-type predictor-corrector method for the numerical solution of fractional differential equations. The method may be used both for linear and for nonlinear problems, and it may be extended to multi-term equations (involving more than one differential operator) too.
A three-step hybrid analysis technique, which successively uses the regular perturbation expansion method, the Pade expansion method, and then a Galerkin approximation, is presented and applied to some model boundary value problems. In the first step of the method, the regular perturbation method is used to construct an approximation to the solution in the form of a finite power series in a small parameter epsilon associated with the problem. In the second step of the method, the series approximation obtained in step one is used to construct a Pade approximation in the form of a rational function in the parameter epsilon. In the third step, the various powers of epsilon which appear in the Pade approximation are replaced by new (unknown) parameters (delta(sub j)). These new parameters are determined by requiring that the residual formed by substituting the new approximation into the governing differential equation is orthogonal to each of the perturbation coordinate functions used in step one. The technique is applied to model problems involving ordinary or partial differential equations. In general, the technique appears to provide good approximations to the solution even when the perturbation and Pade approximations fail to do so. The method is discussed and topics for future investigations are indicated.
Liapunov stability theory generalized and applied to class of partial differential equations
Periodic solution of central differential equation in relativity theory of gravitation
Second order elliptic differential equation system solution reflection behavior, noting constant coefficients and boundary conditions
Linear differential equations with quasi- periodical coefficients solved using method of accelerated convergence
Chebyshev series solution of nonlinear ordinary differential equations - initial value problems
Basic theory of existence, uniqueness, and continuation for neutral functional differential equations
Iterative method for systems of two-point boundary value differential equations based on contraction mapping principle
Existence and uniqueness, stability and asymptotic stability of nonlinear operator differential equation
Existence theorem for periodic solutions of nonlinear differential equations with deviating arguments
Lighthill method with Lagrange expansion for approximative solutions of partial differential equations
A spline-based approximation scheme for nonlinear nonautonomous delay differential equations is discussed. Convergence results (using dissipative type estimates on the underlying nonlinear operators) are given in the context of parameter estimation problems which include estimation of multiple delays and initial data as well as the usual coefficient-type parameters. A brief summary of some of the related numerical findings is also given.
In obtaining solutions to the first-order nonlinear partial differential equations (PDEs) for synthesizing offset dual-shaped reflectors, it is found that previously observed computational problems can be avoided if the integration of the PDEs is started from an inner projected perimeter and integrated outward rather than starting from an outer projected perimeter and integrating inward. This procedure, however, introduces a new parameter, the main reflector inner perimeter radius p(o), when given a subreflector inner angle 0(o). Furthermore, a desired outer projected perimeter (e.g., a circle) is no longer guaranteed. Stability of the integration is maintained if some of the initial parameters are determined first from an approximate solution to the PDEs. A one-, two-, or three-parameter optimization algorithm can then be used to obtain a best set of parameters yielding a close fit to the desired projected outer rim. Good low cross-polarization mapping functions are also obtained. These methods are illustrated by synthesis of a high-gain offset-shaped Cassegrainian antenna and a low-noise offset-shaped Gregorian antenna.