Fractal cubic multiquadric quasi-interpolation

In this article, we propose a novel class of fractal cubic multiquadric functions that generalize the classical cubic multiquadric functions. By employing these fractal multiquadric functions, we develop a fractal quasi-interpolation operator, denoted by $L_d^{\alpha }$. We examine various properties of these fractal cubic multiquadric approximants, such as shape-preserving characteristics and the ability to reproduce quadratic polynomials. Error estimates for these approximants are also derived, and estimates for the box-dimension of the graphs of fractal cubic multiquadric approximants are given. Numerical examples are presented to validate these theoretical findings and highlight the benefits of the fractal quasi-interpolant $L_d^{\alpha }f$. Additionally, we apply the proposed fractal quasi-interpolants to solve an integral equation with a non-smooth degenerate kernel. This approach shows a high rate of convergence to the exact solution. Box-dimension results for the numerical solution of this integral equation are also established.