The Research Dialogue

Abstract

Fractional differential equations (FDEs) have gained considerable prominence in modeling scientific and engineering problems involving memory and hereditary effects. In such models, Caputo fractional derivatives of order between 0 and 1 are commonly employed to represent memory-dependent processes while preserving the applicability of classical initial conditions. The resulting dynamic models are generally nonlinear and, in most practical situations, do not admit exact analytical solutions. Because fractional derivatives provide a more realistic description of the internal dynamics of materials and complex systems, FDEs have become important for understanding biological population models, diffusion phenomena, chemical reactions, conductive heat transfer, engineering applications, and other complex processes. Consequently, the development of effective techniques for obtaining approximate solutions has become an important area of research. Fractionalization of mathematical models can be achieved through several approaches, among which the Caputo-type fractionalization has proved to be particularly effective for constructing mathematically consistent models capable of describing temporal dynamical behavior. The present work focuses on nonlinear Caputo-type FDEs and employs the Homotopy Analysis Method (HAM), a powerful analytical technique that has attracted significant attention due to its flexibility and effectiveness in handling nonlinear problems. The convergence of the fractional HAM is investigated for nonlinear Caputo fractional-order differential equations, and the existence and convergence of the obtained solutions are established within the framework of the convergence-controlled homotopy analysis method. The associated deformation equations are derived, and sufficient conditions for controlling the truncation error are obtained through the auxiliary operator, thereby improving the accuracy and reliability of the proposed approach.

Keywords: Fractional differential equations; Caputo fractional derivative; Homotopy Analysis Method (HAM); Convergence-controlled homotopy analysis; Nonlinear fractional-order systems.

Pandya^, Harshil U., Gohil, Dr. V. P. (2026). A Convergence-Controlled Homotopy Analysis Framework for Nonlinear Fractional Differential Equations. The Research Dialogue, 5(2), 440–451, Volume-05, Issue-02, July-2026.

DOI: https://doi.org/10.64880/theresearchdialogue.v5i2.47