The design basis functions in IGA are refined and enhanced by extra enrichment functions and various local refinements with the use of partition of unity (PU) functions with flat-top. These reconditioned and modified basis functions are pushed forward to the physical domain by the original design mapping for analysis. With this method (PU-IGA), the corresponding stiffness matrix has a smaller bandwidth, and local refinements become simpler. We apply PU-IGA to various singularly perturbed problems incorporating boundary layer enrichment functions developed by boundary layer analysis. Here, we construct the PU functions on the reference domain and push-forward them to a physical domain through geometric mapping for the construction of enriched global basis functions on a physical domain. Therefore, we have advantages in calculating stiffness matrices and load vectors with integrals over rectangular areas. Next, we apply PU-IGA, which is enriched with singular functions that resemble singularities, to fourth order differential equations containing singularities. This direct enrichment method yields an accurate numerical solution; however, it yields large matrix condition numbers and integrals of singular functions. To alleviate these limitations, we propose a mapping method by constructing a singular mapping from the reference domain onto the singular zone of the physical domain. This singular mapping transforms polynomials on the reference domain to singular basis functions on the physical domain. This mapping method has the same effect as the directly enriched PU-IGA but yields small condition numbers and no singular integrals.