Monotonicity analysis of delta fractional sums and differences of order $\upsilon \in (0,1]$ on the time scale $h\mathrm{Z}$ are presented in this study. For this analysis, two models of discrete fractional calculus, Riemann–Liouville and Caputo, are considered. There is a relationship between the delta Riemann–Liouville fractional h-difference and delta Caputo fractional h-differences, which we find in this study. Therefore, after we solve one, we can apply the same method to the other one due to their correlation. We show that $y\left(z\right)$ is $\upsilon$-increasing on ${\mathrm{M}}_{a+\upsilon h,h}$, where the delta Riemann–Liouville fractional h-difference of order $\upsilon$ of a function $y\left(z\right)$ starting at $a+\upsilon h$ is greater or equal to zero, and then, we can show that $y\left(z\right)$ is $\upsilon$-increasing on ${\mathrm{M}}_{a+\upsilon h,h}$, where the delta Caputo fractional h-difference of order $\upsilon$ of a function $y\left(z\right)$ starting at $a+\upsilon h$ is greater or equal to $-\frac{1}{\Gamma (1-\upsilon )}{(z-(a+\upsilon h))}_h^{(-\upsilon )}y(a+\upsilon h)$ for each $z\in {\mathrm{M}}_{a+h,h}$. Conversely, if $y(a+\upsilon h)$ is greater or equal to zero and $y\left(z\right)$ is increasing on ${\mathrm{M}}_{a+\upsilon h,h}$, we show that the delta Riemann–Liouville fractional h-difference of order $\upsilon$ of a function $y\left(z\right)$ starting at $a+\upsilon h$ is greater or equal to zero, and consequently, we can show that the delta Caputo fractional h-difference of order $\upsilon$ of a function $y\left(z\right)$ starting at $a+\upsilon h$ is greater or equal to $-\frac{1}{\Gamma (1-\upsilon )}{(z-(a+\upsilon h))}_h^{(-\upsilon )}y(a+\upsilon h)$ on ${\mathrm{M}}_{a,h}$. Furthermore, we consider some related results for strictly increasing, decreasing, and strictly decreasing cases. Finally, the fractional forward difference initial value problems and their solutions are investigated to test the mean value theorem on the time scale $h\mathrm{Z}$ utilizing the monotonicity results.

In this work, we established new inequalities of Hermite–Hadamard type for convex functions via conformable fractional integrals. Through the conformable fractional integral inequalities, we found some new inequalities of Hermite–Hadamard type for convex functions in the form of classical integrals.

The discrete delta Caputo-Fabrizio fractional differences and sums are proposed to distinguish their monotonicity analysis from the sense of Riemann and Caputo operators on the time scale $\mathcal{Z}$. Moreover, the action of $\mathcal{Q}-$ operator and discrete delta Laplace transform method are also reported. Furthermore, a relationship between the discrete delta Caputo-Fabrizio-Caputo and Caputo-Fabrizio-Riemann fractional differences is also studied in detail. To better understand the dynamic behavior of the obtained monotonicity results, the fractional difference mean value theorem is derived. The idea used in this article is readily applicable to obtain monotonicity analysis of other discrete fractional operators in discrete fractional calculus.

This research presents a novel conformable-Caputo fractional non-polynomial spline method for solving the time-fractional Korteweg-de Vries (KdV) equation. Emphasizing numerical analysis and algorithm development, the method offers enhanced precision and modeling capabilities. Evaluation via the Von Neumann method demonstrates unconditional stability within defined parameters. Comparative analysis, supported by contour and 2D/3D graphs, validates the method’s accuracy and efficiency against existing approaches. Quantitative assessment using L₂ and L_∞ error norms confirms its superiority. In conclusion, the study proposes a robust solution for the time-fractional KdV equation.