In this thesis , we focused our attention to two problems in inhomogenities in superconductor systems . The first one is the calculation of Andreev Bound States in the junctions of superconductor and normal metal . Solving the Bogoliubov – deGennes mean field equation for nonhomogeneous superconductor systems gives the bound and quantized states with energy below the superconductor energy gap . For more complex geometries , numerical methods are inevitable . In this regard , a new numerical method , Kernel Polynomial Method (KPM) , has been adopted . This method is based on the expansion of the physical quantities - specially electronic Density of States (DOS) - in terms of the Chebishev polynomial . In chapter , we took advanatage of KPM for solving the Bogoliubov-deGennes equation for superconductor – normal metal junctions . We have used the local density of states to distinguish the bound states from propagating ones; i.e . local DOS for bound states has a finite value in the normal region while it vanishes rapidly in the superconductor region. The second topic presented in this thesis is calculation of the supercurrent in the Graphene Josephson Junction .Recently , some experiments have been conducted to see the Josephson effect in the Junctions based on the Graphene and measure the supercurrent in these systems . Consequently , this quantity has been calculated using various methods . Temprature and lenghth dependency of the supercurrent have been determined . In a very recently experiment , current-phase relation has been measured . In this work , we introduced a new method for performing this task , that is the perturbative Green’s Function method in framework of the path integral , using the tunneling Hamiltonian between superconductor region and graphene normal region . The main feature of our work is that we assumed that , superconductivity will be induced on the honycomb lattice of graphene , while the pairing occures between electrons on different valleys . It will be shown that the fourth order is the lowest order of perturbation that contributes in the Josephson supercurrent . Tunneling Hamiltonian involves the two kinds of tunnling , interavalley and intervalley . Considering the intervalley tunnlling , we achieved results that have not been seen in previous works . We have seen sharp oscillations in the diagrams which show the dependency of the supercurrent on the distance between the superconductors electrods . But if we solely include intravalley tunnlling , our result will match exactly to the others' works . Keywords: Bogoliubov-de Gennes equation , KPM , Andreev bound states , Graphene , Josephson supercurrent.