A convergence condition of the homotopy analysis method

In this chapter, we present a condition enabling the homotopy analysis method (HAM) to converge to the exact solution of the sought solution of algebraic, highly nonlinear differential-difference, integro-differential, fractional differential and ordinary or partial differential equations or systems. The previous notions of convergence control parameter are carefully reviewed and a novel description is proposed to find out an optimal value for the convergence control parameter, which, although it is completely different from the classical definition by means of the squared residual error as often used in the literature, yields nearly the same interval of convergence and optimal convergence parameters as those found from the squared residual error. When an unknown parameter is embedded into the governing equations, the convergence of the HAM is better pursued by the ratio relevant to this parameter rather than the ratio of other functions involving much harder integrations. An error estimate for the HAM is also provided. Physical and mechanical examples, including the Volterra differential-difference equation, the Fredholm integro-differential equation for the static beam, the Airy equation, the undamped and dumped Duffing oscillators, the Thomas–Fermi equation, the Gelfand problem, the fractional differential equation, the rotating sphere, and more, clearly illustrate the validity of the new approach and further provide knowledge on why the corresponding homotopy series generated by the HAM should converge to the exact solution in the domain of interest.