Constrained q-fractal rational spline through q-iterated function system with applications in differential equations

By interweaving q-calculus and theory of iterated function system, a novel spline, namely, q-fractal rational spline is cultivated in this article. The proposed q-fractal rational spline is the fixed point of the Read-Bajraktarević operator defined on a suitable function space. The proposed q-fractal rational spline consists of a q-vector, a scaling vector and a shape vector. Using the sign-preserving property of a continuous function at a point, we establish the existence constrained q-fractal rational spline. More precisely, we show that there exists a q-fractal rational spline which lies (i) above a prescribed piecewise straight line, (ii) below a prescribed piecewise straight line, (iii) between the prescribed piecewise straight lines, or (iv) within a prescribed rectangle. We obtain an upper bound for error in approximation of the original function by the proposed q-fractal rational spline. Utilizing q-fractal rational spline, we develop a methodology for solving first order ordinary differential equation having continuous and nowhere differentiable force function (for instance, Weierstrass function). The proposed theoretical results are demonstrated through numerical examples.