Hybrid Caputo-Type Fractional Parallel Schemes for Nonlinear Elliptic PDEs with Chaos- and Bifurcation-Based Acceleration

In this work, we propose a fractional Jacobian–based parallel two-stage iterative framework for the numerical solution of nonlinear systems arising from elliptic PDE discretizations. The core of the approach is a high-order fractional two-step scheme (S1), which combines a linear Newton-type correction with a quadratic fractional correction and incorporates a structured parallel interaction mechanism inspired by Weierstrass-type schemes. Under standard regularity assumptions, a rigorous local convergence analysis shows that the S1 scheme provides a high-order local correction mechanism, yielding a convergence order of 2μ+3 under suitable local accuracy conditions. To enhance robustness with respect to the choice of initial guesses, a safeguarded realization of the method, denoted by SBVM*, is introduced. Since the safeguard mechanism may modify the local iteration map, convergence of SBVM* is ensured under appropriate acceptance conditions, while its asymptotic behavior coincides with that of the S1 scheme once the safeguard becomes inactive. The dynamical behavior of the resulting iterative map is further investigated through bifurcation diagrams and Lyapunov exponent analysis, providing practical guidelines for parameter selection and enabling the identification of stable operating regimes while avoiding chaotic behavior. Extensive numerical experiments involving linear and nonlinear elliptic benchmark problems from engineering and biomedical applications demonstrate that SBVM* achieves improved convergence behavior, enhanced numerical stability, and reduced computational cost relative to existing parallel solvers such as ELVM* and ACVM*. The proposed framework therefore provides an effective and scalable numerical approach for the solution of nonlinear elliptic models arising in biomedical and engineering contexts.