In this article, the Montgomery identity and Ostrowski inequality are established for univariate first-order diamond-alpha differentiable functions. We also investigate the generalization of Ostrowski-type inequalities for bivariate functions with bounded second-order diamond-alpha derivatives by applying integration by parts for ${\mathrm{♢}}_{\alpha }$-integrals. Moreover, some extensions of dynamic trapezoid- and Grüss-type inequalities are also obtained by using the Montgomery identity.

This paper introduces a novel numerical scheme, the conformable finite difference method (CFDM), for solving time-fractional gas dynamics equations. The method was developed by integrating the finite difference method with conformable derivatives, offering a unique approach to tackle the challenges posed by time-fractional gas dynamics models. The study explores the significance of such equations in capturing physical phenomena like explosions, detonation, condensation in a moving flow, and combustion. The numerical stability of the proposed scheme is rigorously investigated, revealing its conditional stability under certain constraints. A comparative analysis is conducted by benchmarking the CFDM against existing methodologies, including the quadratic B-spline Galerkin and the trigonometric B-spline functions methods. The comparisons are performed using $L_2$ and $L_{\mathrm{\infty }}$ norms to assess the accuracy and efficiency of the proposed method. To demonstrate the effectiveness of the CFDM, several illustrative examples are solved, and the results are presented graphically. Through these examples, the paper showcases the capability of the proposed methodology to accurately capture the behavior of time-fractional gas dynamics equations. The findings underscore the versatility and computational efficiency of the CFDM in addressing complex phenomena. In conclusion, the study affirms that the conformable finite difference method is well-suited for solving differential equations with time-fractional derivatives arising in the physical model.