(QUANTUM, DEFORMED)-FRACTIONAL COMPLEX STEPS: THEORY, ENTROPY ANALYSIS, AND IMAGING APPLICATIONS

3D complex surface plots of $ |\Gamma_{q, \tau}(x+iy)| $ over the complex domain $ -1 \le \Re(z) \le 2 $, $ -1.5 \le \Im(z) \le 1.5 $ for different parameter pairs $ (q, \tau) = (0.5, 1), (0.5, 1.5), (0.7, 1) $, and $ (0.7, 1.5) $. Increasing $ \tau $ amplifies deformation scaling, while increasing $ q $ modifies the decay characteristics, illustrating the combined influence of quantum deformation and memory.

Comparison of basic and symmetric $ (q, \tau) $ -Fractional Complex Step Method (FCSM) for approximating the Caputo $ (q, \tau) $ -fractional derivative of $ f(x)=x^{\beta} $ with $ \alpha=0.8 $, $ \beta=2.5 $, $ q=0.5 $, $ \tau=1.2 $, and $ x_0=1.3 $. The symmetric scheme exhibits reduced error across multiple scales of the step size $ h $, confirming its superior accuracy in practical computation.

Synthetic verification of the entropy identity for $ \alpha=0.8 $, $ D=0.1 $, $ \varepsilon=0.2 $. The left-hand side $ \, {}^{C}D_t^{\alpha} S_1(t)\, $ is computed by the symmetric complex-step estimator on a local polynomial fit of $ S_1(t) $; the right-hand side $ D\!\int |\nabla p|^2/p $ is evaluated by quadrature. The absolute difference is small across all test times, supporting the identity and the numerical consistency of the FCSM-based estimator for entropy rates.

Log–log error of the calibrated symmetric $ (q, \tau) $-FCSM at $ (x_0, t_0)=(0.5, 1.0) $. The slope approaches $ 2\alpha=1.6 $–$ 2.0 $, consistent with $ \mathcal{O}(h^{2\alpha}) $.

Log–log pointwise PDE residual $ |\mathcal{R}(h)| $ at $ (x_0, t_0) $. Residual decay mirrors the calibrated derivative error.

Synthetic pollution images interpreted under the $ (q, \tau) $ -fractional framework for different memory parameters $ \tau=1, 1.5, 1.75, 2 $ ($ q=0.5 $, $ \alpha=0.8 $). Increasing $ \tau $ corresponds to stronger memory effects, leading to reduced spatial diffusion and more persistent pollution patterns. The entropy values reported in each panel confirm slower entropy production for larger $ \tau $, consistent with the fractional $ H $-theorem.

Tsallis entropy density maps for a synthetic pollution image with entropic index $ r=1.5 $ under different memory parameters $ \tau=1, 1.5, 1.75, 2 $ ($ q=0.5 $). Increasing $ \tau $ corresponds to stronger memory effects, leading to more localized entropy structures and reduced spatial smoothing. This behavior reflects slower entropy production predicted by the $ (q, \tau) $-fractional diffusion framework.