An Efficient Hermite Polynomial Solution to the Fractional Differential Equations

Abstract

Fractional differential equations (FDEs) play a fundamental role in modeling complex physical, biological, and engineering phenomena characterized by memory effects and nonlocal dynamics. This paper presents an efficient numerical framework based on a fractional Hermite interpolation formula for solving FDEs. The proposed method extends the classical Hermite interpolation scheme to fractional calculus by embedding fractional derivatives within the interpolation structure, thereby improving approximation precision and convergence behavior. To enhance computational performance, optimized interpolation nodes and refined fractional derivative approximations are introduced, effectively reducing truncation errors and improving numerical stability. The method is systematically formulated and implemented in a computational environment, with numerical experiments verifying its robustness and accuracy. Results confirm that the proposed scheme achieves superior stability and precision compared with conventional numerical techniques, demonstrating its potential for broad application in the solution of fractional-order models across scientific and engineering domains.