Reference to formal approach to homotopy analysis method

Question

I'm currently reading a book about the Homotopy Analysis Method (HAM), but it isn't very rigorous (it explains most things with a single example), which is bothering me.

I'm searching for papers where HAM is more rigorously approached (I'm mostly interested in theorems regarding the analyticity of the considered homotopies')

I'm assuming that Shijun Liao's original paper ("On the proposed homotopy analysis technique for nonlinear problems and its applications") (PHD Thesis, 1992) has rigorous results, but I can't find a PDF or a place where I could purchase it anywhere.

Reference to any other papers that have rigorous results would also be greatly appreciated!

Answer 1

The paper by Shijun Liao "Notes on the homotopy analysis method: Some definitions and theorems" Communications in Nonlinear Science and Numerical Simulation, 14(4) 2009 pp. 983-997 may be what you're after.

The author notes in the first section of the paper:

At the current stage of the HAM, it is urgently necessary to redescribe this method in a more rigorous way. So, in this paper, definitions of some new concepts such as the homotopy-derivative, the convergence-control parameter, the convergence-control vector, and so on, are given so as to redescribe the method more rigorously.

Edit: see M.Turkyilmazoglu, "A note on the homotopy analysis method" Applied Math. Letters, 23 (2010) pp. 1226-1230 for an extension of some of the theorems in Liao's paper.