SPECTRAL DECOMPOSITION OF FRACTIONAL SWIFT–HOHENBERG EQUATIONS USING THE FRACTIONAL FOURIER TRANSFORM

Abstract

This paper introduces a spectral decomposition framework for addressing time–fractional Swift–Hohenberg (S–H) equations through the use of the Fractional Fourier Transform (FrFT). In contrast to classical transforms such as Laplace or Elzaki, the FrFT offers a more versatile spectral mechanism that bridges time and frequency domains, enabling an effective diagonalization of differential operators. Building on this idea, we propose the Fractional Fourier Transform Decomposition Method (FrFT–DM), which integrates the spectral features of FrFT with decomposition strategies to derive convergent series solutions for both linear and nonlinear fractional Swift–Hohenberg models.

The theoretical foundation of the method is supported by rigorous results on existence, uniqueness, and convergence, along with a precise truncation error estimate. To ensure stability and boundedness, several new lemmas and corollaries are established, providing spectral justification for the approach. The effectiveness of the method is illustrated through six representative examples, encompassing both linear and nonlinear cases of the Swift–Hohenberg equation. Numerical simulations demonstrate that the approach accurately captures the solution behavior across various fractional orders and reduces to the classical integer–order solutions as .