A normalized Caputo–Fabrizio fractional diffusion equation

We propose a normalized Caputo–Fabrizio (CF) fractional diffusion equation. The CF fractional derivative replaces the power-law kernel in the Caputo derivative with an exponential kernel, which avoids singularities. Compared to the Caputo derivative, the CF derivative is better suited for systems where memory effects decay smoothly rather than following a power law. However, the kernel is not normalized in the sense that its weighting function does not integrate to unity. To resolve this limitation, we develop a modified formulation that ensures proper normalization. To investigate the fractional order’s effect on evolution dynamics, we perform computational tests that highlight memory effects.