  In the present work a new set of differential equations for the Hopfield
Neural Network (HNN) method were established by means of the Linear Extended
Gateaux Derivative (LEGD). This new approach will be referred to as
G\^ateaux-Hopfiel Neural Network (GHNN). A first order Fredholm integral
problem was used to test this new method and it was found to converge 22 times
faster to the exact solutions for {\alpha} > 1 if compared with the HNN integer
order differential equations. Also a limit to the learning time is observed by
analysing the results for different values of {\alpha}. The robustness and
advantages of this new method will be pointed out.
