In this work, new cubic, quartic and quintic B-spline approximations are proposed for approximate solution of initial and boundary value problems. The error analysis of these new approximations has been executed, which affirms their uniform convergence in entire spatial domain. The numerical algorithms based on B-spline functions assembled with new approximations are developed for numerical solution of second, third and fourth singular initial and boundary value problems. The L''Hospital''s Rule has been utilized for removal of singularities. The non-linear equation is first changed into a system of linear equations with the help of quasi-linearization technique and then solved by a modified form of well known Thomas algorithm. Moreover, numerical schemes based on usual finite difference formulation and new B-spline approximations are also presented for numerical solution of three non-linear mathematical models named as Klein-Gordon equation, Korteweg-de Vries equation and Kuramoto-Sivashinsky equation. The finite difference formulation is used for temporal discretization, whereas, new B--spline approximations based on Crank--Nicolson scheme have been used to interpolate the solution in space direction. The proposed numerical schemes are proved to be unconditionally stable subject to the Von-Neuman stability analysis. In order to corroborate this work, several test examples have been considered and the computational outcomes are compared with existing numerical techniques. It is revealed that by virtu of simple and straightforward implementation, our new B-spline approximations produce more accurate outcomes as compared to other variants on the topic.