We consider the existence and uniqueness of the solution of the Wigner equation in the presence of boundary conditions. The equation, describing electron transport in nanostructures, is analyzed in terms of the Neumann series expansion of the corresponding integral form, obtained with the help of classical particle trajectories. It is shown that the mathematical aspects of the solution can not ...

In this paper a finite element method involving Petrov-Galerkin method with quintic B-splines as basis functions and septic B-splines as weight functions has been developed to solve a general seventh order boundary value problem with a particular case of boundary conditions. The basis functions are redefined into a new set of basis functions which vanish on the boundary where the Dirichlet and ...

In this article, we propose a new technique based on 2-D shifted Legendre poly?nomials through the operational matrix integration method to find numeri?cal solution of stochastic heat equation with Neumann boundary conditions. For proposed technique, convergence criteria and error estima?tion are also discussed in detail. This is tested two exam?ples, it observed that very easy handle such prob...