In the HPM, You can study 

(He, JH., “Homotopy perturbation technique”, Comp. Meth. Appl. Mech. Eng., 178 ,

257-262, 1999 ) and 

(He, J.H., “Application of homotopy perturbation method to non-linear wave

Equation”, Chaos Soliton. Fract., 26, 695-700, 2005)   article

All semi numerical method like HPM, MAM, VIM and ADM solve only ordinary differential equations only!!. Many authors used this method to solve partial differential equations with initial conditions or boundary conditions or mixed conditions. In fact, these authors used an initial solution say u0 (t, x) contain the exact solution in x and the iterative process will generate the solution in term to see all examples in wazwaz book

The Adomian decomposition method applied to nonlinear evolution equations in solution theory, In: Encyclopedia of Complexity and Systems Science, Ed.: R. A. Meyers, Springer

A homotopy  construct with an embedding parameter p∈[0,1], which is considered as a “small parameter”.

You can see:

J.-H. He, “Comparison of homotopy perturbation method and homotopy analysis method,” Applied Mathematics and Computation, vol. 156, no. 2, pp. 527–539, 2004. 

Using homotopy Analysis Method (HAM) gives you more freedom to choose arbitrary parameters and functions. But, actually, which method to use depends on the difficulty of your application. You may also look for traditional perturbation methods if it gives reasonable accuracy as approximate methods.

For HAM I would suggest the following book:

1- Shijun Liao, Advances in the Homotopy Analysis Method, World Scientific Publishing Company 2014.

2- Shijun Liao, Beyond perturbation: introduction to the homotopy analysis method, Chapman and Hall/CRC, 2003.

thanks a lot for replay but I have question about initial approximation in homotopy perturbation method can I choose any( point or function) or I should choose the initial approximation depend on the assumption or satisfy the given equation and I would illustrated about why choose the initial approximation the source term or part of them in many problems of integral equations ?

You choose the initial boundary conditions depend on the problems

To my understanding, all these homotopy methods are almost the same or repeating some existing methods