We prove the existence and uniqueness of fundamental solution for Kolmogorov operators associated to some stochastic processes, that arise in Black & Scholes setting pricing problem relevant path dependent options. improve previous results we provide a closed form expression Cauchy under weak regularity assumptions on coefficients differential operator. Our method is based limiting procedure, w...

This chapter describes an application of the recently proposed Modified Method of Fundamental Solutions (MMFS) to the potential flow problems. The solution in two dimensional Cartesian coordinates is represented in terms of the fundamental solution of the Laplace equation together with the first order polynomial augmentation. The collocation is used for determination of the expansion coefficien...

In this paper, a new collocation method, which is based on Boubaker polynomials, is introduced for the approximate solutions of a class of two-dimensional linear Fredholm integral equationsof the second kind. The properties of two-dimensional Boubaker functions are presented. The fundamental matrices of integration with the collocation points are utilized to reduce the solution of the integral ...

A new max-plus fundamental solution semigroup is presented for a class of lossless wave equations. This new semigroup is developed by employing the action principle to encapsulate the propagation of all possible solutions of a given wave equation in the evolution of the value function of an associated optimal control problem. The max-plus fundamental solution semigroup for this optimal control ...

Based on the fact that the Neumann-Green function can be constructed as a perturbation of the fundamental solution by a single-layer potential, we establish a gaussian lower bound and a gaussian type upper bound for the Neumann-Green function for a general parabolic operator. We build our analysis on old tools coming from the construction of a fundamental solution of a general parabolic operato...

and Applied Analysis 3 equation. Therefore, we will choose fundamental solution of the principal part of equation 2.1 , that is, for two-dimensional Laplace equation as follows: U x − ξ 1 2π ln|x − ξ|. 3.1 4. Basic Relations At first, we construct Green’s second formula connected with 2.1 and fundamental solution 3.1 , that is, we multiply 2.1 by fundamental solution 3.1 and integrate it with r...

in this paper, we present the numerical solution of ordinary dierential equations (or sdes), from each order especially second-order with time-varying and gaussian random coecients. we indicate a complete analysis for second-order equations in special case of scalar linear second-order equations (damped harmonic oscillators with additive or multi- plicative noises). making stochastic dierent...

Based on the radial basis function (RBF), nonsingular general solution, and dual reciprocity method (DRM), this paper presents an inherently meshless, integration-free, boundaryonly RBF collocation technique for numerical solution of various partial differential equation systems. The basic ideas behind this methodology are very mathematically simple. In this study, the RBFs are employed to appr...