Some classical and fractal splines for constrained and shape preserving interpolation and applications in solving differential equations

In this paper, we develop constrained and shape preserving classical splines in one and two variables and fractal splines in two variables. First, we obtain a classical rational spline lying above or below the prescribed piecewise line or contained in a prescribed rectangle. Using a blending technique, we construct a classical bivariate rational spline and study its constrained and shape preserving aspects. The proposed constrained and shape preserving splines are exploited for solving constraint differential equations. We construct a bivariate fractal spline [which is the fixed point of the Read-Bajraktarević (RB) operator] as a perturbation of our classical bivariate rational spline proposed in this paper. Further, we investigate suitable conditions on the shape parameters and scaling factors so that the proposed fractal surface lies above or below the given plane, or within a prescribed cuboid. Also, we study the positivity and monotonicity preserving aspects of the proposed bivariate fractal spline. It is observed that our bivariate fractal spline converges to the original function as the mesh norm goes to zero. The proposed theoretical results are verified through numerical examples.