We i nvestigate the optical phenomenon responsible for the colored shine that sometimes a ppears at the surface of ink layers i n t he s pecular direction, o ften ca lled bronzing or gloss di fferential. I t seems to co me from the w avelengthdependent refractive index of t he ink, which induces a wavelength-dependent reflectance of t he i nk-air interface. Our experiments o n c yan and magenta...

Consider the Cauchy problem for an ordinary diierential equation _ x = g(t; x); x(0) = x; t 2 0; T]: (1:1) When g is continuous, the local existence of solutions is provided by Peano's theorem. Several existence and uniqueness results are known also in the case of a discontinuous right hand side 7]. We recall here the classical theorem of Carath eodory 8]: Theorem A. Let g : 0; T] IR n 7 ! IR n...

Here we will provide an overview of the method of nding symmetries of partial di erential equations. First we will discuss some basic facts regarding local groups of transformations, symmetries of algebraic equations and group actions on functions. We will then proceed to a discussion of how to interpret the task of solving a PDE in terms of jet spaces, and the meaning of group actions in this ...

On q-Di®erence Equations Moemen H. Abu-Risha Department of Mathematics Cairo University, Eygpt E-mail: [email protected] We give an investigation of some properties of solutions of linear q¡di®erence equations de ̄ned on an interval [a;1): The cases a = 0 and a > 0 are essentially di®erent. The study of the latter case is necessary for the derivation of asymptotic formulae and oscil...

we have developed a three level implicit method for solution of the helmholtz equation. using the cubic spline in space and nite dierence in time directions. the approach has been modied to drive numerov type nite dierence method. the method yield the tri- diagonal linear system of algebraic equations which can be solved by using a tri-diagonal solver. stability and error estimation of the...

One of the most important theorems in Ordinary Di↵erential Equations is Picard’s Existence and Uniqueness Theorem for first-order ordinary di↵erential equations. Why is Picard’s Theorem so important? One reason is it can be generalized to establish existence and uniqueness results for higher-order ordinary di↵erential equations and for systems of di↵erential equations. Another is that it is a g...

We propose a multilevel preconditioning strategy for the iterative solution of large sparse linear systems arising from a nite element discretization of the radiation di usion equations. In particular, these equations are solved using a mixed nite element scheme in order to make the discretization discontinuous, which is imposed by the application in which the di usion equation will be embedded...

Some oscillation criteria for the oscillatory behavior of fourth order superlinear dynamic equations on time scales are established. Criteria are proved that ensure that all solutions of superlinear and linear equations are oscillatory. Many of our results are new for corresponding fourth order superlinear di¤erential equations and fourth order superlinear di¤erence equations.

We describe some new algorithm and heuristics for computing the polynomial and rational solutions of bounded degree of a class of ordinary di erential equations, which includes generalized Riccati equations. As a consequence, our methods can be used for factoring linear ordinary di erential equations. Since they generate systems of algebraic equations in at most n unknowns, where n is the order...

The method of modi®ed equations is studied as a technique for the analysis of ®nite di€erence equations. The non-uniqueness of the modi®ed equation of a di€erence method is stressed and three kinds of modi®ed equations are introduced. The ®rst modi®ed or equivalent equation is the natural pseudo-di€erential operator associated to the original numerical method. Linear and nonlinear combinations ...