Quantum calculus-based nonlinear analysis and neural approximation of fuzzy partial differential models in fluid dynamics
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Keywords:
Quantum calculus; Fuzzy partial differential equations; Banach fixed-point theory; Lyapunov stability analysis; Physics-informed neural networks.Abstract
This paper constructs a firm quantum calculus based nonlinear analytical framework of fuzzy partial differentiable models that are encountered in fluid dynamics. The fuzzy nonlinear equation governing is expressed in terms of a q-time derivative operator and reduced to a corresponding operator equation in a Banach space. The locality of solutions and existence are proved through contraction mapping principles on the appropriate Lipschitz and coerciveness conditions. The exponential stability is obtained using a Lyapunov functional method and provides explicit decay rates which are proportional to viscosity and nonlinear growth parameters. In an attempt to improve computationally the tractability of computations and maintain analytical properties, a residual-based scheme of
neural approximations is proposed. A nonlinear residual functional is then formulated so that it would be consistent with the governing fuzzy quantum PDE and convergence estimates are derived in the correct Sobolev norms. The contribution of the quantum parameter is calculated in a systematic manner and its impact on the stability rates, minimization of the residual and damping behaviour
have been identified.
The theoretical findings are confirmed by the numerical experiments that show that neural approximation error decays polynomially, fuzzy uncertainty propagation is bounded by 8 levels of the 8-level, and the optimization dynamics is stable on a variety of quantum parameter space. The suggested framework combines the quantum operator theory, nonlinear functional analysis, and neural approximation into a consistent approach to uncertain modelling of nonlinear fuzzy fluid flow systems.
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