Shiksha Samvad

Abstract

Fractional differential equations (FDEs) provide an effective mathematical framework for describing phenomena that evolve in both time and space and exhibit memory and hereditary effects. Their applications span a wide range of fields, including control theory, polymer mechanics, bioengineering, and other areas of applied science and engineering. The Homotopy Analysis Method (HAM) is a powerful analytical technique that constructs the solution in the form of an infinite series without requiring linearization, perturbation assumptions, or small parameters. HAM can be extended to fractional-order models involving Riemann–Liouville or Caputo fractional derivatives, leading to a comprehensive formulation of the method and providing valuable insight into the qualitative behaviour of the solutions. The present study documents the detailed application of HAM to nonlinear FDEs, with the objectives of demonstrating its analytical capability, establishing theoretical results concerning the existence and uniqueness of solutions, and investigating convergence behaviour together with error estimation. The analysis is restricted to nonlinear time-fractional differential equations involving fractional derivatives with respect to time only. A key feature of HAM is the freedom to choose the auxiliary linear operator. Although a linear counterpart is often selected to simplify the governing system, formulations based on Riemann–Liouville derivatives may become mathematically cumbersome. To avoid this difficulty, the proposed approach chooses the linear operator to be zero, which significantly simplifies the implementation while preserving the generality of the method. The nonlinear FDE is retained in its general form to highlight the flexibility of HAM and its capability to handle highly nonlinear problems, and appropriate boundary conditions are explicitly imposed to ensure a well-defined solution procedure. The formulation assumes Caputo fractional derivatives in the nonlinear term, consistent with the time-fractional integration operator appearing on the left-hand side of the equation (Mohammed & Khlaif, 2014).

Keywords: Fractional differential equations; Homotopy Analysis Method (HAM); Caputo fractional derivative; Nonlinear time-fractional equations; Convergence and error analysis.