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.

This article examines the space-time fractional phi-four (PF) model which is an improvement of the Klein-Fock-Gordon (KFG) model that is relevant in quantum mechanics, transmission of ultra-short pulses in optical fibers, de Broglie wave duality, and compound particles in spinless relativistic systems. New solitary wave solutions of the fractional phi-four equation were found with the help of the Riccati-Bernoulli sub-ODE method together with the Bäcklund transformation. The conformable fractional derivative played the role of improving the modeling of the complex dynamical systems. The obtained solutions, which were illustrated by giving 2D, 3D MATLAB plots and contour illustrations, were s eful in nuclear, particle physics and fluid mechanics. Thus, the current work demonstrates the applicability of the proposed approach in handling the fractional order deviations and contributes to the investigation of the dynamic behavior of the PF model.

This study delved into the analytical investigation of two significant nonlinear partial differential equations, namely the fractional Kawahara equation and fifth-order Korteweg-De Vries (KdV) equations, utilizing advanced analytical techniques: the Aboodh residual power series method and the Aboodh transform iterative method. Both equations were paramount in various fields of applied mathematics and physics due to their ability to describe diverse nonlinear wave phenomena. Here, we explored using the Aboodh methods to efficiently solve these equations under the framework of the Caputo operator. Through rigorous analysis and computational simulations, we demonstrated the efficacy of the proposed methods in providing accurate and insightful solutions to the time fractional Kawahara equation and fifth-order KdV equations. Our study advanced the understanding of nonlinear wave dynamics governed by fractional calculus, offering valuable insights and analytical tools for tackling complex mathematical models in diverse scientific and engineering applications.

The objective of this work is to provide the method of getting the closed-form solitary wave solution of the fractional $(3+1)$-generalized nonlinear wave equation that characterizes the behavior of liquids with gas bubbles. The same phenomena are evident in science, engineering, and even in the field of physics. This is done by employing the Riccati-Bernoulli sub-ode in a systematic manner as applied to the Bäcklund transformation in the study of this model. New soliton solutions, in the forms of soliton, are derived in the hyperbolic and trigonometric functions. The used software is the computational software Maple, which makes it possible to perform all the necessary calculations and the check of given solutions. The result of such calculations is graphical illustrations of the steady-state characteristics of the system and its dynamics concerning waves and the inter-relationships between the parameters. Moreover, the contour plots and the three-dimensional figures describe the essential features, helping readers understand the physical nature of the model introduced in this work.