1 2 The paper deals with the basic integral equation of random field estimation theory by the criterion of minimum of variance of the error estimate. This integral equation is of the first kind. The corresponding integral operator over a bounded domain Ω in Rn is weakly singular. This operator is an isomorphism between appropriate Sobolev spaces. This is proved by a reduction of the integral eq...

The boundary value problem for the Laplace equation outside several cracks in a plane is studied. The jump of the solution of the Laplace equation and the boundary condition containing the jump of its normal derivative are specified on the cracks. The problem has unique classical solution under certain conditions. The new integral representation for the unique solution of this problem is obtain...

The payoffs of the barrier options depend on the time path of the underlying price as opposed to just the price at expiry. It implies that both the boundary conditions and the initial condition are imposed on the Black-Scholes partial differential equation. Therefore, the valuation of the barrier options is a boundary value problem. Since the Black-Scholes equation can be converted into a homog...

A hierarchical multilevel preconditioner is constructed for an efficient solution of a first kind boundary integral equation with the single layer potential operator discretized by a boundary element method. This technique is based on a hierarchical clustering of all boundary elements as used in fast boundary element methods. This hierarchy is applied to define a sequence of nested boundary ele...

It is well known that the parabolic partial differential equations in two or more space dimensions with overspecified boundary data, feature in the mathematical modeling of many phenomena. In this article, an inverse problem of determining an unknown time-dependent source term of a parabolic equation in general dimensions is considered. Employing some transformations, we change the inverse prob...