Explicit and Implicit Methods In Solving Differential Equations

Differential equations are equations that involve an unknown function and derivatives. Euler's method are efficient methods to yield fairly accurate approximations of the actual solutions. By manipulating such methods, one can find ways to provide good approximations compared to the exact solution of parabolic partial differential equations and nonlinear parabolic differential equations.

Differential invariants of second-order ordinary differential equations

The notion of a differential invariant for systems of second-order differential equations on a manifold M with respect to the group of vertical automorphisms of the projection is de?ned and the Chern connection attached to a SODE allows one to determine a basis for second-order differential invariants of a SODE.

Kinematic geometry of mass-triangles and reduction of Schrödinger’s equation of three-body systems to partial differential equations solely defined on triangular parameters

Schrödinger’s equation of a three-body system is a linear partial differential equation (PDE) defined on the 9-dimensional configuration space, ℝ9, naturally equipped with Jacobi’s kinematic metric and with translational and rotational symmetries. The natural invariance of Schrödinger’s equation with respect to the translational symmetry enables us to reduce the configuration space to that of a 6-dimensional one, while that of the rotational symmetry provides the quantum mechanical version of angular momentum conservation. However, the problem of maximizing the use of rotational invariance so as to enable us to reduce Schrödinger’s equation to corresponding PDEs solely defined on triangular parameters—i.e., at the level of ℝ6/SO(3)—has never been adequately treated. This article describes the results on the orbital geometry and the harmonic analysis of (SO(3),ℝ6) which enable us to obtain such a reduction of Schrödinger’s equation of three-body systems to PDEs solely defined on triangular parameters.

Efficient algorithms for solving systems of ordinary differential equations for ecosystems modeling /

Mode of access: Internet.

Partial differential equations.

Mode of access: Internet.

Existence theorems in partial differential equations.

"Only the material on elliptic equations will appear in these notes."

Differential equations; geometric theory.

Mode of access: Internet.

Partial differential equations,

Mimeographed.

Plane waves and spherical means applied to partial differential equations.

Bibliography: p. 165-170.

Numerical integrators for stiff and highly oscillatory differential equations /

Bibliography: p. 27-28.

An automatic integrator for ordinary differential equations for ILLIAC IV /

"June 1, 1969."

Numerical solution of stiff ordinary differential equations using collocation methods /

Bibliography: p. 79-80.

A computer graphic technique for finding numerical methods for ordinary differential equations.

On cover: COO-1469-0106.

High order stiffly stable methods for ordinary differential equations

Supported in part by contract U.S. AEC AT(11-1) 1469 and in part by National Science Foundation grant NSF-GJ-217.

Multi-derivative numerical methods for the solution of stiff ordinary differential equations /

"Under contracts US AEC AT(11-1)2383 and US AEC AT(11-1)1469."