Quantum calculus, similar to calculus without limits, is the same as ordinary “infinitesimal calculus”. Quantum Hermite–Hadamard-type inequalities, according to quantum calculus, recently discovered improvements within quantum Hermite–Hadamard-type inequalities. New results about the derivatives and integrals identities related to both qm1-derivatives and q^m2-integrals will be obtained. This research work is motivated by this fact, so using properties of “generalized higher-order strongly preinvex” and “quasi-preinvex” functions, we drive innovative Quantum Hermite–Hadamard-type inequalities. As applications, new Hermite–Hadamard-type inequalities Hermite–Hadamard qm1-integral and q^m2-integral-type inequalities will be obtained. These types of identities are applied to “preinvex functions”. The newly obtained important outcomes are present. The results of these new generalizations are used to assess a variety of mathematical difficulties. These new findings have a huge impact on integrated symmetrical functions and approximations, functions with a symmetric degree. These visions are encouraging new and significant achievements in a wide range of mathematics and engineering disciplines. The generalized strongly “preinvex functions” are the “quasi-preinvex function” studied using “elementary Quantum” Calculus methods.

Gas Dynamics Equations (GDEs) play a fundamental role in modeling fluid flows phenomena across a range of applications, from environmental systems to aerospace engineering. These equations mathematically represent the fundamental laws of mass, momentum and energy conservation. Modeling complex gas flows often requires advanced mathematical tools capable of capturing nonlocal and memory-dependent behavior. Therefore, this study explores the implementation of the Caputo–Fabrizio Fractional Derivative (CFFD) in the analysis of GDEs, highlighting its potential to accurately capture the behavior of complex fluid dynamics systems. A two-dimensional (2D) wavelet-based approach combined with suitable collocation grids is employed to approximate the solutions for spacetime Fractional Gas Dynamics Equations (FGDEs). The concept of the CFFD, with its nonsingular kernel, is integrated into the framework of GDEs, offering a more precise analytical perspective. By reformulating the FGDEs into a system of algebraic equations, the proposed approach enables efficient computation via iterative technique. The error analysis of the numerical results is presented through graphs and tables for three illustrative examples with varying fractional values, demonstrating a strong correlation between the analytical and approximated solutions. The performance of the scheme is evaluated using multiple error metrics, including minimum absolute error, L∞ error, L2 error, and LRMS errors. The absolute errors demonstrate the improvement in results of FGDEs as the wavelet basis increases. The results validate the reliability and ease of implementation of the suggested approach for solving the FGDEs. The study demonstrates the method’s reliability, and potential for solving a wide class of nonlinear fractional models governed by nonlocal dynamics.