  Homotopy analysis method (HAM) was proposed by Liao in 1992 in his PhD thesis
for non-linear problems and was applied it in many different problems of
mathematical-physics and engineering. In this note, a new development of
homotopy analysis method (ND-HAM) is demonstrated for non-linear
integro-differential equation (NIDEs) with initial conditions. Practical
investigations revealed that ND-HAM leads easy way how to find initial guess
and it approaches to the exact solution faster than the standard HAM, modified
HAM (MHAM), new modified of HAM (mHAM) and more general method of HAM (q-HAM).
Two examples are illustrated to show the accuracy and validity of the proposed
method. Five methods are compared in each example
