In this paper, we present the monotonicity analysis for the nabla fractional differences with discrete generalized Mittag-Leffler kernels $( {}^{ABR}_{a-1}{\nabla }^{\delta ,\gamma }y )(\eta )$ ( a − 1 A B R ∇ δ , γ y ) ( η ) of order $0<\delta <0.5$ 0 < δ < 0.5 , $\beta =1$ β = 1 , $0<\gamma \leq 1$ 0 < γ ≤ 1 starting at $a-1$ a − 1 . If $({}^{ABR}_{a-1}{\nabla }^{\delta ,\gamma }y ) ( \eta )\geq 0$ ( a − 1 A B R ∇ δ , γ y ) ( η ) ≥ 0 , then we deduce that $y(\eta )$ y ( η ) is $\delta ^{2}\gamma $ δ 2 γ -increasing. That is, $y(\eta +1)\geq \delta ^{2} \gamma y(\eta )$ y ( η + 1 ) ≥ δ 2 γ y ( η ) for each $\eta \in \mathcal{N}_{a}:=\{a,a+1,\ldots\}$ η ∈ N a : = { a , a + 1 , … } . Conversely, if $y(\eta )$ y ( η ) is increasing with $y(a)\geq 0$ y ( a ) ≥ 0 , then we deduce that $({}^{ABR}_{a-1}{\nabla }^{\delta ,\gamma }y )(\eta ) \geq 0$ ( a − 1 A B R ∇ δ , γ y ) ( η ) ≥ 0 . Furthermore, the monotonicity properties of the Caputo and right fractional differences are concluded to. Finally, we find a fractional difference version of the mean value theorem as an application of our results. One can see that our results cover some existing results in the literature.
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.
Monotonicity analysis of delta fractional sums and differences of order υ∈(0,1] on the time scale hZ 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(z) is υ-increasing on Ma+υh,h, where the delta Riemann–Liouville fractional h-difference of order υ of a function y(z) starting at a+υh is greater or equal to zero, and then, we can show that y(z) is υ-increasing on Ma+υh,h, where the delta Caputo fractional h-difference of order υ of a function y(z) starting at a+υh is greater or equal to −1Γ(1−υ)(z−(a+υh))h(−υ)y(a+υh) for each z∈Ma+h,h. Conversely, if y(a+υh) is greater or equal to zero and y(z) is increasing on Ma+υh,h, we show that the delta Riemann–Liouville fractional h-difference of order υ of a function y(z) starting at a+υh is greater or equal to zero, and consequently, we can show that the delta Caputo fractional h-difference of order υ of a function y(z) starting at a+υh is greater or equal to −1Γ(1−υ)(z−(a+υh))h(−υ)y(a+υh) on Ma,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 hZ utilizing the monotonicity results.
<abstract><p>In present study, the Boussinesq equation is obtained by means of the Sardar Sub-Equation Technique (SSET) to create unique soliton solutions containing parameters. Using this technique, different solutions are obtained, such as the singular soliton, the dark-bright soliton, the bright soliton and the periodic soliton. The graphs of these solutions are plotted for a batter understanding of the model. The results show that the technique is very effective in solving nonlinear partial differential equations (PDEs) arising in mathematical physics.</p></abstract>
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