We present a method to solve initial and boundary value problems using arti ial neural networks. A trial solution of the di erential equation is written as a sum of two parts. The rst part satis es the initial/boundary onditions and ontains no adjustable parameters. The se ond part is onstru ted so as not to a e t the initial/boundary onditions. This part involves a feedforward neural network, ...

This work develops fast and adaptive algorithms for numerically solving nonlinear partial di erential equations of the form ut = Lu +N f(u) where L and N are linear di erential operators and f(u) is a nonlinear function. These equations are adaptively solved by projecting the solution u and the operators L and N into a wavelet basis. Vanishing moments of the basis functions permit a sparse repr...

The Shuttle's two-dimensional entry trajectory planning method is extended to threedimensions. Both angle of attack and angle of bank variations are used to control the entry trajectory. The trajectory planning is done with a third-order system of di erential equations using the drag and lateral accelerations as intermediate controls. The reducedorder planning problem is solved as an optimal co...

The in ltration process is generally described by a nonlinear di erential equation, which can be solved by iteration methods such as a Newton-Raphson method. In this paper we propose a Discrete Event System Speci cation(DEVS) model for Green-Ampt in ltration. We show that this model can be approximated using Genetic Algorithm optimization of a fuzzy system. The fuzzy approximation is shown to b...

We present a full implementation of the parareal algorithm—an integration technique to solve di erential equations in parallel— in the Julia programming language for a fully general, rst-order, initial-value problem. Our implementation accepts both coarse and ne integrators as functional arguments. We use Euler’s method and another Runge-Kutta integration technique as the integrators in our exp...

This paper presents a family of generalized multistep methods that evolves the numerical solution of ordinary di erential equations on con guration spaces formulated as homogeneous manifolds. Any classical multistep method may be employed as an invariant method, and the order of the invariant method is as high as in the classical setting. We present numerical results that re ect some of the pro...

For the approximate representation of large data sets over a parameter domain in IR2, many geological and other applications require the construction of surfaces which have minimal energy, i.e., minimal curvature. One way to achieve this is by solving a fourth order elliptic partial di erential equation. Its discretization by a di erence scheme makes it particularly easy to incorporate (appropr...

Observers are usually formulated as explicit systems of di erential equations and implemented using standard ODE solvers. In this paper, we show that there can be advantages in formulating the observer as a DAE (Di erential-Algebraic Equation). We review the general idea of DAE observer design of [10]. We give two special normal forms for which DAE observer design yields an observer with linear...

This paper concerns a new type of approximant for series analysis. By incorporating linear recurrence techniques, such as are involved with second order di erential approximants, algebraic approximants can be extended to nonlinear di erential approximants. The discussion of nonlinearities is constrained by discussing approximants just one step removed from second order algebraic approximants. T...