Piecewise homotopy analysis method and convergence analysis for formally well-posed initial value problems

In this paper, we propose piecewise homotopy analysis method (PHAM) to extend the convergence region of the homotopy analysis solution for well-posed initial value problems. A convergence theorem of the homotopy analysis solution in the expression of polynomials in the form of Cauchy-Kowalevskaya theorem is given for the nonlinear equation which is analytical near the initial point. We also discuss the influence of the convergence-control parameter h. Our convergence result supports Liao’s conjecture (1): the convergence region increases with the increasing of h(<0). PHAM solution with the symbol of arbitrary initial values has a general form on every convergent subinterval, since the formal series solution is determined by governing equation and initial values. So, we only should substitute the initial values to general form of PHAM solution to obtain the analytical expression on a target subintervals. We also present three typical examples to illustrate the effectiveness and validity of PHAM and convergence results.