The aim of this article is to solve the Volterra-Fredholm integro-differential equations of fractional order numerically by using the shifted Jacobi polynomial collocation method. The Jacobi polynomial and collocation method properties are presented. This technique is used to convert the problem into the solution of linear algebraic equations. The fractional derivatives are considered in the Caputo sense. Numerical examples are given to show the accuracy and reliability of the proposed technique.

Each sport has its own energy requirements that differ from the energy requirements of other sports, and a different method is used in each of them, so the trainer must first rely on the principle of privacy in training first, that is, privacy according to the working energy system, that is, he defines the controlling energy system In that event, and how the muscles use the available energy to perform according to the energy production systems. As we find the serving skill is the first volleyball skill with which the team starts the match in order to be able to gain points directly, through knowledge it turns out that there is a weakness in the skill performance, especially the skill of serving and being The key to victory for volle

The heat exchanger is a device used to transfer heat energy between two fluids, hot and cold. In this work, an output feedback adaptive sliding mode controller is designed to control the temperature of the outlet cold water for plate heat exchanger. The measurement of the outlet cold temperature is the only information required. Hence, a sliding mode differentiator was designed to estimate the time derivative of outlet hot water temperature, which it is needed for constructing a sliding variable. The discontinuous gain value of the sliding mode controller is adapted according to a certain adaptation law. Two constraints which imposed on the volumetric flow rate of outlet cold (control input) were considered within the rules of the proposed