non-linear integro-differential equation

Homotopy analysis method (HAM) was proposed by Liao in 1992 in his PhD thesis for non-linear problems and was applied 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 an easy way how to find initial guess and it approaches the exact solution faster than the standard HAM, modified HAM (MHAM), new modified of HAM (mHAM) and more general method of HAM (q-HAM).