Exploiting delay differential equations solved by Eta functions as suitable mathematical tools for the investigation of thickness controlling in rolling mill

This work is dedicated to introducing the properties and application of Eta functions. We derive the properties of the Eta function, such as the generating function, integral representation, and the Laplace transform. Also, some properties of the Eta-based functions are introduced. To show the advantages of the Eta-based functions in the computational method, we develop a new numerical method to solve the state-dependent and time-dependent neutral delay differential equation based on the Eta-based function. We introduce the operational matrix of derivative for the Eta-base functions to develop the new numerical method. This method uses the operational matrix of derivative and collocation method to convert the delay differential equation to a system of nonlinear algebraic equations. We derive the technique’s error bound and establish the method’s accuracy by solving some examples, which are state-dependent and time-dependent delay differential equations. In the end, we study the model of the metal forming process by rolling the mill using the new numerical method to show the advantages of using the Eta-based function for solving a more practical problem. •We study the model of the metal forming process by rolling mill.•We study the behavior of the rolling mill thickness control system using Eta functions.•The method is based on using Eta functions as a set of base functions in the best approximation.•We use Eta functions to introduce a new numerical method for solving the state-dependent and time-dependent delay differential equations.•We derive the error bounds of the numerical method and demonstrate the high precision of the new numerical technique.