Solutions of differential equations using fractal multiquadric RBF networks

Abstract In this article, we introduce a novel class of self-referential fractal multiquadric (MQ) functions that exhibit symmetry about the origin. To ensure the differentiability of the original classical multiquadric function, we carefully limit the scaling factors in the construction of the fractal MQ function. Utilizing the translates of a fractal MQ function at a given collocation point, we develop two numerical methods: (i) Approximation of a function with an irregular derivative (ii) the solution of differential equations involving irregular functions or irregular fractal functions. We apply the proposed methods to the solution of various differential equations, including a delay differential equation and demonstrate the advantages of fractal MQ functions over classical MQ functions by means of several examples.