Isogeometric analysis (IGA), introduced by Hughes, et al. [2,3], is a mathematical approach that combines Finite Element analysis (FEA) in conjunction with engineering design tools, such as CAD, which allows analysis, testing and redesign of structural elements via the same data set. Prior to implementing a new material into a manufacturing process, it is necessary to design the shape of the object and then analyze the durability of the design. Generally NURBS basis functions are used to design complex structures. Isogeometric analysis is effective in the design-analysis-manufacture loop.Babuska and Oh [10] introduced mapping techniques called the Method of Auxiliary Mapping (MAM) to handle singularities that occur in partial differential equations (PDEs). However, this method is unable to follow the framework of IGA. Thus, we are looking for another way to handle singularity in IGA using the Collocation method. In order to develop methodology for solving PDEs containing singularities, the B-spline basis functions are first modified using partition unity functions. By using these modified basis functions the neighborhood of singularity will be enriched so that they can capture the singular behavior of the true solution. In this dissertation, this method is tested to one-dimensional and two-dimensional problems. Also, this method is more effective and economical than other existing methods in handling problems containing singularities because the Collocation method requires less computation than the Galerkin method or any other existing methods. Schwarz alternating method in the framework of IGA-Collocation is also introduced in this dissertation. In this method, a domain is decomposed into two subdomains and then the problem is solved by solving subproblems in each subdomain. The iterative process starts with an initial guess and iterates until it arrives at a solution of desired accuracy. This technique has been applied to one- and two-dimensional problems for overlapping as well as non-overlapping subdomains. Elasticity problems containing singularities are also solved using this method. Numerical results are presented and compared with the results obtained by the IGA-Galerkin method.