Application of a Generalized Fractional Fourier Transform with Memory to Nonlocal Fractional Differential Equations

Application of a Fractional Fourier Transform with Memory to nonlocal fractional differential
equations is proposed in this work. The new transform is defined as
\[
\mathcal{F}_{\alpha,\beta}[u](\omega) :=
\int_{-\infty}^{+\infty}
u(t)\,
\exp\!\Bigg(
i\, I^{\beta}\, g(\omega)\,
\int_{0}^{t} h(s)\,ds
\Bigg)\, dt ,
\]
where $g(\omega)$ and $h(s)$ characterize memory effects, while $I^{\beta}$ provides an
additional fractional degree of freedom. This formulation allows us to capture nonlocal and
hereditary behaviors commonly observed in fractional-order systems, and reduces to the
classical Fourier transform as a special case when $g(\omega)$ is constant, $h(s)=1$, and
$\beta=0$.

Beyond its formal definition, the proposed framework offers a more flexible mathematical
structure that is particularly well-suited to problems where memory effects play a dominant
role. Many physical and engineering systems—such as anomalous diffusion, viscoelasticity, and
biological processes—cannot be accurately described without explicitly incorporating
hereditary dynamics. The memory-based kernel of the transform provides a natural mechanism
for integrating such effects, thereby extending the applicability of Fourier analysis to a
much broader class of problems.

In addition to its theoretical relevance, the transform establishes new operational relations
with Caputo-type derivatives and fractional integrals. These relations enable the development
of efficient analytical and semi-analytical methods for solving nonlocal fractional
differential equations. Numerical illustrations further confirm that the proposed transform
achieves better adaptability to varying memory characteristics compared with the classical
Fourier and even standard fractional Fourier frameworks.

Overall, this study highlights the potential of the Fractional Fourier Transform with Memory
as a versatile and powerful tool, bridging the gap between traditional Fourier analysis and
the growing need for models that faithfully represent memory-dependent, nonlocal, and
fractional-order dynamics.