An approximate method for computing the temperature distribution over material thickness during hot flat rolling

An analytic solution for computing the temperature distribution over material thickness during hot flat rolling is so complex that finding the exact answer takes too much time. We approximate the analytic solution to cut significantly the run time required in calculating the temperature distribution. The proposed approach finds a finite number of eigenvalues instead of an infinite number of eigenvalues via minimizing the natural metric of definite real-valued functions. The initial condition of the cooling conditions altering sequentially is approximated using a sixth-order polynomial function so that we can perform the integral inherent in the analytic solution without numerical analysis.