Cubic spline fractal solutions of two-point boundary value problems with a non-homogeneous nowhere differentiable term

Abstract

Fractal interpolation functions (FIFs) supplement and subsume all classical interpolants. The major advantage by the use of fractal functions is that they can capture either the irregularity or the smoothness associated with a function. This work proposes the use of cubic spline FIFs through moments for the solutions of a two-point boundary value problem (BVP) involving a complicated non-smooth function in the non-homogeneous second order differential equation. In particular, we have taken a second order linear BVP: y′′(x)+Q(x)y′(x)+P(x)y(x)=R(x) with the Dirichlet’s boundary conditions, where P(x) and Q(x) are smooth, but R(x) may be a continuous nowhere differentiable function. Using the discretized version of the differential equation, the moments are computed through a tridiagonal system obtained from the continuity conditions at the internal grids and endpoint conditions by the derivative function. These moments are then used to construct the cubic fractal spline solution of the BVP, where the non-smooth nature of y′′ can be captured by fractal methodology. When the scaling factors associated with the fractal spline are taken as zero, the fractal solution reduces to the classical cubic spline solution of the BVP. We prove that the proposed method is convergent based on its truncation error analysis at grid points. Numerical examples are given to support the advantage of the fractal methodology.