A new contribution to the Newton-Leibnitz differentiation of polynomial approximations in thermal processes

This paper presents the determining of the critical cooling time and preheating temperature for welding. The primary equation which represents the cooling temperature is transcendent and, as such, it is not suitable for explicit solving. Hence, the author has investigated the appropriate interpolation polynomial which should align the polynomial level with the possibility of providing good accuracy of the approximation. Since the application and design of the welding included calculation of the derivatives of the polynomial, it is concluded that the accuracy of the derivative is not precise, nor does it follow the accuracy of the polynomial approximation. This research showed that the current theory for differentiation of the polynomial approximation does not provide precise results, although all the requirements of differentiability are provided. There are certain shortcomings in these actions, which imply that the currently known method of differentiation cannot be used. The new method of differentiating includes relatively small differentiating error regarding the exact value of the derivative, and the error for the polynomial approximation relating to the exact formula is not surpassed. The attempt to solve the design problem in thermal processes also makes a contribution in differentiation.