ABSTRACT: In this paper we brie y describe the interaction of high power laser beam with metal material and its application in the engineering industry. We involve the mathematical model using the nite element method to solve the corresponding partial di erential equation of heat transfer numerically. For application in a laser industry we present results of some examples of heat eld distribution.

For linear di erential algebraic equations of tractability index the notion of the adjoint equation is analysed in full detail Its solvability is shown at the lowest possible smoothness The fundamental matrices of both equations are de ned and their relationships are characterized

| Tsirelson's stochastic di erential equation is called \celebrated and mysterious" by Rogers and Williams bd16ce. This note aims at making it a little more celebrated and a little less mysterious. Using a deterministic time-change, we translate the study of Tsirelson's equation into the study of \eternal" Brownian motion on the circle. This allows us to show that the ltration generated by any ...

in this paper the solution of a second order linear dierential equations with intu-itionistic fuzzy boundary value is described. it is discussed for two dierent cases: coecientis positive crisp number and coecient is negative crisp number. here fuzzy numbers aretaken as generalized trapezoidal intutionistic fuzzy numbers (gtrifns). further a numericalexample is illustrated.

The problem of nding bounds on the H∞-norm of systems with a nite number of point delays and distributed delay is considered. Su cient conditions for the system to possess an H∞-norm which is less or equal to a prescribed bound are obtained in terms of Riccati partial di erential equations (RPDE’s). We show that the existence of a solution to the RPDE’s is equivalent to the existence of a stabl...

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...

In this paper, we are interested in the one-dimensional porous medium equation when the initial condition is the distribution function of a probability measure. We associate a nonlinear martingale problem with it. After proving uniqueness for the martingale problem, we show existence owing to a propagation of chaos result for a system of weakly interacting di usion processes. The particle syste...

In general, a comparison Lemma for the solutions of Forward-Backward Stochastic Di erential Equations (FBSDE) does not hold. Here we prove one for the backward component at the initial time, relying on certain monotonicity conditions on the coeÆcients of both components. Such a result is useful in applications. Indeed, one can use FBSDE's to de ne a utility functional able to capture the disapp...

We extend Ehrhard Regnier's di erential linear logic along the lines of Laurent's polarization. We provide a denotational semantics of this new system in the well-known relational model of linear logic, extending canonically the semantics of both di erential and polarized linear logics: this justi es our choice of cut elimination rules. Then we show this polarized di erential linear logic re ne...