Rationalized Haar wavelet bases to approximate the solution of the first Painlev'e equations

In this article, using the properties of the rationalized Haar (RH) wavelets and the matrix operator, a method is presented for calculating the numerical approximation of the first  Painlev\'e equations solution. Also, an upper bound of the error is given and by applying the Banach fixed point theorem  the convergence analysis of the method is stated. Furthermore, an algorithm to solve the first Painlev\'e equation is proposed. Finally, the reported results are compared with some other methods to show the effectiveness of the proposed approach.

Keywords:

Wave equation , first Painlev'e equation , Volterra integral equation , RH wavelet