Some new operators based on Touchard polynomials

This paper introduced a numerical method for solving generalized Abel integral (GAI) equations and generalized Abel integro differential (GAID) equations. This method is based upon Touchard polynomials (TPs) approximation. The Touchard polynomials were first presented and the resulting Touchard matrices were utilized to transform the generalized Abel integral and integro differential equations into a system of linear algebraic equations. The results of the presented method were obtained through some examples of the first and second types of equations under study. All results for this method have been compared with those of the presented methods in the literature.

The idea of this research is to find approximate solution of Linear Volterra Integro-Differential Equation (LVIDE) of the first order and second kind by using different degrees of the Touchard polynomials is presented. Because the simplicity definition applying this method will give high resolution results in low cost and short time, also can be easily differentiated, integrated and converge to an exact solution.The algorithm and example given are to illustrate the solution in this way and compare it with the exact solution.

We introduce a novel integral representation for the Touchard polynomials. The notable utility of such rational functions is prominent in combinatorics and computer science. This modest discovery may aid algebraic experiments involving such functions and potential pursuers of the infamous Nested Radical Constant.

Using the well-known least-squares weighted residual method (LSM) in coupling with various degrees of Touchard polynomials (TPs), found the numerical solutions of Volterra–Fredholm integro-differential equations (VFIDEs) and mixed Volterra–Fredholm integro-differential equations (MIDEs) of the second kind. There exist many approaches that have evaluated the approximate solution of the integro-differential equations (VFIDEs (MIDEs)) like the Adomian decomposition method and modified Adomian decomposition method, Homotopy analysis method, Taylor polynomials, power series expansion and cubic Legendre spline collocation method. In this work, we presented a method based on combining LSM with TPs is an essential component of the suggested approach. By implementing such a method, a system of algebraic equations can be generated that can be solved by employing well-known linear algebraic methods. Several VFIDEs (MIDEs) examples were solved with a comparatively minimal number of reiterations to show the accuracy and effectiveness of the presented approach when comparing the current method with other methods already accessible in the scientific literature, as well as from the approximate solutions of each of these situations, researchers found that there was an apparent agreement with the exact solutions for some examples. The applicability of the proposed method was proven and the convergence analysis was discussed.

ABSTRACTWe present a novel model, which is a two-parameter extension of the Poisson distribution. Its normalizing constant is related to the Touchard polynomials, hence the name of this model. It is a flexible distribution that can account for both under- or overdispersion and concentration of zeros that are frequently found in non-Poisson count data. In contrast to some other generalizations, the Hessian matrix for maximum likelihood estimation of the Touchard parameters has a simple form. We exemplify with three data sets, showing that our suggested model is a competitive candidate for fitting non-Poisson counts.

Touchard like polynomials and generalized Stirling numbers