A piecewise homotopy Padé technique to approximate an arbitrary function

The Padé approximation and its enhancements provide a more accurate approximation of functions than the Taylor series truncation. A new technique for approximating functions into rational functions is proposed in this paper. This technique is based on the homotopy Padé technique and introduces new parameters known as merging parameters. These parameters are added to the Tayler series before the Padé process is computed. To control error, the merging parameters and dividing the interval into subintervals are used. Two illustrative examples are used to demonstrate the validity and reliability of the proposed novel approximation. The robustness and efficiency of the proposed approximation were demonstrated by computing the absolute error and comparing the results to those of the standard Padé technique and the generalized restrictive Padé technique. Also, Hard-core scattering problem and Debye-Hukel function are tested by the proposed technique. The piecewise homotopy Padé method is an excellent path to approximate any function. The proposed new approximation's efficacy and accuracy have been validated using Mathematica 12.