A finite difference numerical method is investigated for fractional order diffusion problems in one space dimension. The basis of the mathematical model and the numerical approximation is an appropriate extension of the initial values, which incorporates homogeneous Dirichlet or Neumann type boundary conditions. The wellposedness of the obtained initial value problem is proved and it is pointed out that each extension is compatible with the original boundary conditions. Accordingly, a finite difference scheme is constructed for the Neumann problem using the shifted Grünwald-Letnikov approximation of the fractional order derivatives, which is based on infinite many basis points. The corresponding matrix is expressed in a closed form and the convergence of an appropriate implicit Euler scheme is proved.

Space-fractional diffusion problems are investigated from the modeling point of view. It is pointed out that the elementwise power of the Laplacian operator in R n is an inadequate model of fractional diffusion. Also, the approach with fractional calculus using zero extension is not a proper model of homogeneous Dirichlet boundary conditions. At the time, the spectral definition of the fractional Dirichlet Laplacian seems to be in many aspects a proper model of fractional diffusion.

In the present work we introduce a computational approach to the absolute rovibrational quantum partition function using the path-integral formalism of quantum mechanics in combination with the nested sampling technique. The numerical applicability of path-integral nested sampling is demonstrated for small molecules of spectroscopic interest. The computational cost of the method is determined by the evaluation time of a point on the potential energy surface (PES). For efficient PES implementations, the path-integral nested sampling method can be a viable alternative to the direct-Boltzmann-summation technique of variationally computed rovibrational energies, especially for medium-sized molecules and at elevated temperatures.

A finite element numerical method is investigated for fractional order elliptic boundary value problems with homogeneous Dirichlet type boundary conditions. It is pointed out that an appropriate stiffness matrix can be obtained by taking the prescribed fractional power of the stiffness matrix corresponding to the non-fractional elliptic operators.It is proved that this approach, which is also called the matrix transformation or matrix transfer method, delivers optimal rate of convergence in the L 2 -norm.

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