In this paper, we proposed some new integral inequalities for subadditive functions and the product of subadditive functions. Additionally, a novel integral identity was established and a number of inequalities of the Hermite-Hadamard type for subadditive functions pertinent to tempered fractional integrals were proved. Finally, to support the major results, we provided several examples of subadditive functions and corresponding graphs for the newly proposed inequalities.

In this paper, we suggest the Rishi transform, which may be used to find the analytic (exact) solution to multi-high-order linear fractional differential equations, where the Riemann-Liouville and Caputo fractional derivatives are used. We first developed the Rishi transform of foundational mathematical functions for this purpose and then described the important characteristics of the Rishi transform, which may be applied to solve ordinary differential equations and fractional differential equations. Following that, we found an exact solution to a particular example of fractional differential equations. We looked at four numerical problems and solved them all step by step to demonstrate the value of the Rishi transform. The results show that the suggested novel transform, "Rishi Transform, " yields exact solutions to multi-higher-order fractional differential equations without doing complicated calculation work.

We show that a class of fractional differences with Mittag–Leffler kernel can be negative and yet monotonicity information can still be deduced. Our results are complemented by numerical approximations. This adds to a growing body of literature illustrating that the sign of a fractional difference has a very complicated and subtle relationship to the underlying behavior of the function on which the fractional difference acts, regardless of the particular kernel used.

In this study, we consider the analysis of monotonicity for the Riemann-Liouville fractional differences of sequential type. The results are defined on the subsets of $(0,1)\times (0,1)$ with a certain restriction. By analysing the difference operator in the point-wise form into a delta form, we use the standard sequential formulas as stated in Theorems 2.1 and 2.2 to establish the positivity of the delta difference operator of the proposed the discrete sequential operators. Finally, some numerical experiments are conducted which confirm our theoretical monotonicity results.

In this study, we consider positivity and other related concepts such as $\alpha -$convexity and $\alpha -$monotonicity for discrete fractional operators with exponential kernel. Namely, we consider discrete $\mathrm{\Delta }$ fractional operators in the Caputo sense and we apply efficient initial conditions to obtain our conclusions. Note positivity results are an important factor for obtaining the composite of double discrete fractional operators having different orders.

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.

We established positivity of $\mathrm{\nabla }f$ obtained from a systematic computation of a composition of sequential fractional differences of the function $f$ that satisfy certain conditions in a negative lower bound setup. First, we considered the different order sequential fractional differences in which we need a complicated condition. Next, we equalled the order of fractional differences and we saw that a simpler condition will be needed. We illustrated our positivity results for an increasing function of the rising type.