We discuss a leading-edge model used in the computation of the run-out length of dry-flowing avalanches. The model has the form of a singular initial value problem for a scalar ordinary differential equation describing the avalanche dynamics. Existence, uniqueness and smoothness properties of the analytical solution are shown. We also prove the existence of a unique root of the solution. Moreov...

This lecture is a survey of the theory of stability and error bounds for numerical methods for stii initial value problems. We begin with problems in singular perturbation form, where the distinction between \fast" and \slow" variables is clearest. Next we explain Kreiss's characterization of stii problems that includes singular perturbation problems as a special case. The requirement of unifor...

We consider a class of nonlinear singular perturbation problems of the form [ ] β = α = ∈ = + ′ + ′ ′ ε ) ( , ) ( ]; , [ ), ( ) , ( )) ( ( ) ( b y a y b a x x r y x q x y p x y with a boundary layer at one end point. Using the theory of singular perturbations, the original problem is reduced to an asymptotically equivalent first order initial value problem. Then, a variable step size in itial v...

How complex could the solution be to an initial value problem given by a polynomial-time computable function? This question can be given a natural and precise sense in computational complexity theory. In this note, we put together several results stating that more conditions on the problem make the solution simpler. In particular, Lipschitz continuity alone may leave the solution polynomial-spa...

where the initial time, t0, is a given real number, the initial position, ~ ξ0 ∈ IR, is a given vector and ~ F : IR × IR → IR is a given function. We shall assume throughout these notes that ~ F is C. By definition, a solution to the initial value problem (1) on the interval I (which may be open, closed or half–open, but which, of course, contains t0) is a differentiable function ~x(t) which obeys