Multivariate linear recurrences appear in such diverse elds of mathematics as combinatorics, probability theory, and numerical resolution of partial diierential equations. Whereas in the univariate case the solution of a constant-coeecient recurrence always has a rational generating function, this is no longer true in the multivariate case where this generating function can be non-rational, non...

We study the complexity of randomized solution of initial value problems for systems of ordinary differential equations (ODE). The input data are assumed to be γ-smooth (γ = r+% : the r-th derivatives satisfy a %-Hölder condition). Recently, the following almost sharp estimate of the order of the n-th minimal error was given by Kacewicz (Almost optimal solution of initial-value problems by rand...

in Hilbert space and their realizations in function spaces as initial-boundary value problems for partial differential equations which may contain degenerate or singular coefficients. The Cauchy problem consists of solving (1.1) subject to the initial condition Jdu(0) = h. We are concerned with the case where the solution is given by an analytic semigroup; it is this sense in which the Canchy p...

This paper is devoted to the study of discrete fractional calculus; the particular goal is to define and solve well-defined discrete fractional difference equations. For this purpose we first carefully develop the commutativity properties of the fractional sum and the fractional difference operators. Then a ν-th (0 < ν ≤ 1) order fractional difference equation is defined. A nonlinear problem wi...

The acceleration technique known as Iterated Defect Correction (IDeC) for the numerical solution of singular initial value problems is investigated. IDeC based on the implicit Euler method performs satisfactorily and can thus be used for the eÆcient solution of singular boundary value problems with the shooting method. Higher order one-step methods like the box scheme or the trapezoidal rule ca...

Finding the exact solution of initial value problems is the main topic of this thesis. Because differential equations in practice can be hard to solve, we require algorithms to solve them on a computer. Since the solution of the equation needs to be stored on a computer, we first discuss the representation of real functions. We show that approximating a function by polynomials gives a suitable ...

This paper reviews various aspects of stiffness in the numerical solution of initial-value problems for systems of ordinary differential equations. In the literature on numerical methods for solving initial value problems the term "stiff" has been used by various authors with quite different meanings, which often causes confusion. This paper attempts to clear up this confusion by reviewing some...

Introduction Partial differential equations on bounded domains of R n have traditionally been equipped with homogeneous boundary conditions (usually Dirichlet, Neumann, or Robin). However, other kinds of boundary conditions can also be considered, and for a number of concrete application it seems that dynamic (i.e., time-dependent) boundary conditions are the right ones. Motivated by physical p...