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Various numerical methods for solving differential equations were analyzed and refined in an effort to develop a method which was adaptable to a large class of problems. The prime capabilities of the method included accuracy, numerical stability, and economic use of computer time. In multistep processes the corrector was changed at each step.
Optimal control of linear functional differential equations
Linear functional differential equations control with quadratic cost
Critical cases for neutral functional differential equations
Investigating asymptotic solutions of ordinary differential equations
Hybrid computers solution of linear differential equations, analyzing Taylor series /derivatives/ use for compensation
If a system of ordinary differential equations represents a property conserving system that can be expressed linearly (e.g., conservation of mass), it is then desirable that the numerical integration method used conserve the same quantity. It is shown that both linear multistep methods and Runge-Kutta methods are 'conservative' and that Newton-type methods used to solve the implicit equations preserve the inherent conservation of the numerical method. It is further shown that a method used by several authors is not conservative.
The hyperbolic system of partial differential equations with a real constant square coefficient matrix A is considered. The problem of finding an energy conserving norm for the solution of the system is reduced to the problem of characterizing those matrices appearing in the boundary conditions which satisfy two specific matrix equations. Necessary and sufficient conditions on the coefficient matrix A and the matrices appearing in boundary conditions are derived for an energy conserving norm. The conditions serve as criteria on a given system which determine whether or not the solution will have its energy conserved in some norm. Examples of specific systems and boundary conditions are also provided.
Classical asymptotic analysis of ordinary differential equations derives approximate solutions that are numerically stable. However, the analysis also leads to tedious expansions in powers of the relevant parameter for a particular problem. The expansions are replaced with integrals that can be evaluated by numerical integration. The resulting numerical solutions retain the linear independence that is the main advantage of asymptotic solutions. Examples, including the Falkner-Skan equation from laminar boundary layer theory, illustrate the method of asymptotic analysis with numerical integration.
The differential equation description of exospheric lateral transport of Hodges and Johnson is reformulated to extend its utility to light gases. Accuracy of the revised equation is established by applying it to terrestrial hydrogen. The resulting global distributions for several static exobase models are shown to be essentially the same as those that have been computed by Quessette using an integral equation approach. The present theory is subsequently used to elucidate the effects of nonzero lateral flow, exobase rotation, and diurnal tidal winds on the hydrogen distribution. Finally it is shown that the differential equation of exospheric transport is analogous to a diffusion equation. Hence it is practical to consider exospheric transport as a continuation of thermospheric diffusion, a concept that alleviates the need for an artificial exobase dividing thermosphere and exosphere.
Finite difference method for solving symmetric positive linear differential equations
Stability of processes defined by stochastic difference and differential equations
Formulation and study of the initial value problem for neutral functional differential equations. The existence, uniqueness, and continuation of solutions to this problem are investigated, and an analysis is made of the dependence of the solutions on the initial conditions and parameters, resulting in the derivation of a continuous dependence theorem in which the fundamental mathematical principles underlying the continuous dependence problem for a very general system of nonlinear neutral functional differential equations are separated out.
Second order differential equation for undamped forced nonlinear oscillations
Dimensional analysis and group theory methods of solving ordinary and partial differential equations
The propagation of errors in the solutions of the differential equations for the orbital elements of perturbed two-body motion is investigated. It is shown that the error in the time-element grows linearly for differential equations for orbital elements when only perturbations are present on the right-hand side, cubically for formulations which have a two-body term on the right-hand side, and linearly for formulations based upon extended phase space Hamiltonians.
Solution of general inhomogenous linear differential equations of second order using Lie groups
Techniques are presented for obtaining generalized finite-difference solutions to partial differential equations of the parabolic type. It is shown that the advantages of similarity in the solution of similar problems are generally not lost if the solution to the original partial differential equations is effected in the physical plane by finite-difference methods. The analysis results in a considerable saving in computational effort in the solution of both similar and nonsimilar problems. Several examples, including both the heat-conduction equation and the boundary-layer equations, are given. The analysis also provides a practical means of estimating the accuracy of finite-difference solutions to parabolic equations.