Applications including generating functions of spline type curves and their mathematical models

In this thesis, definitions, some properties and spline type curves families are given. Linear (first-order, linear), quadratic (second-order) and k. spline, B-spline, cardinal spline, exponential spline, and exponential Euler spline, known as the most important spline curve families, are given. Additionally, definitions and some properties of some well-known special numbers and polynomials are also given. Their relationships with spline curves families are examined. By applying the derivative operator to the generating functions for higher order Apostol-Euler numbers and polynomials, some calculation formulas, derivative formulas, and relations are obtained. Using the mathematical induction method and functional equations of generating functions, higher order partial differential equations containing the complex number order Apostol-Euler numbers and polynomials are obtained. With the help of these partial derivatives, computational formulas containing these numbers and polynomials are derived. With the help of these new formulas, a newly modified complex number order exponential Euler type spline curves family with degree v are defined. In addition, relations among the spline curves related to Beta type rational functions, special numbers and polynomials are found. Some certain finite sums involving these spline curves and these numbers and polynomials are given. In addition, new families of spline type curves are also defined by using the relationship between complex numbers order Apostol-Euler numbers and polynomials of Apostol-Genocchi numbers and polynomials. Moreover, after giving some definitions and properties of mathematical modeling, some graph examples and Mathematica codes related to mathematical models covering spline curves are also given with the help of the Wolfram Mathematica package program.