Convergence of the homotopy analysis method

The homotopy analysis method is studied in the present paper. The question of convergence of the homotopy analysis method is resolved. It is proven that under a special constraint the homotopy analysis method does converge to the exact solution of the sought solution of nonlinear ordinary or partial differential equations. An optimal value of the convergence control parameter is given through the square residual error. An error estimate is also provided. Examples, including the Blasius flow, clearly demonstrate why and on what interval the corresponding homotopy series generated by the homotopy analysis method will converge to the exact solution.

💡 Research Summary

The paper provides a rigorous treatment of the convergence properties of the Homotopy Analysis Method (HAM), a semi‑analytical technique for solving nonlinear ordinary and partial differential equations. After a brief introduction that contrasts HAM with traditional perturbation and variational approaches, the authors lay out two essential assumptions: (1) the linear operator (L) used in the homotopy construction is invertible and bounded on the whole domain, and (2) the nonlinear operator (N) is sufficiently smooth and Lipschitz continuous. Under these hypotheses the homotopy series can be written as
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