The Talbot effect is usually modeled using the Helmholtz equation, but its main experimental features are captured by solution to free Schr\"odinger equation with Dirac comb as initial datum. This simplified description a consequence of paraxial approximation in geometric optics. However, it heuristic that not mathematically well justified, so K. I. Oskolkov raised problem "mathematizing" it. W...

In this paper, we intend to solve special kind of ordinary differential equations which is called Heun equations, by converting to a corresponding stochastic differential equation(S.D.E.). So, we construct a stochastic linear equation system from this equation which its solution is based on computing fundamental matrix of this system and then, this S.D.E. is solved by numerically methods. Moreo...

In this paper, two inverse problems of Stephen kind with local (Dirichlet) boundary conditions are investigated. In the first problem only a part of boundary is unknown and in the second problem, the whole of boundary is unknown. For the both of problems, at first, analytic expressions for unknown boundary are presented, then by using these analytic expressions for unknown boundaries and bounda...

This paper proposes a hybrid method to find cumulative distribution function (CDF) of completion time of GERT-type networks (GTN) which have no loop and have only exclusive-or nodes. Proposed method is cre-ated by combining an analytical transformation with Gaussian quadrature formula. Also the combined crude Monte Carlo simulation and combined conditional Monte Carlo simulation are developed a...

The matrix differential Riccati equation (DRE) is ubiquitous in control and systems theory. The presence of the quadratic term implies that a simple linear-systems fundamental solution does not exist. Of course it is well-known that the Bernoulli substitution may be applied to obtain a linear system of doubled size. Here however, tools from max-plus analysis and semiconvex duality are brought t...

In this paper, we investigate infinite product representation of the solution of a Sturm- Liouville equation with an indefinite weight function which has two zeros and/or singularities in a finite interval. First, by using of the asymptotic estimates provided in [W. Eberhard, G. Freiling, K. Wilcken-Stoeber, Indefinite eigenvalue problems with several singular points and turning points, Math. N...