This paper introduces a general nabla operator of order two that includes coefficients of various trigonometric functions. We also introduce its inverse, which leads us to derive the second-order $\overline{\theta (\mathtt{t})}$-Fibonacci polynomial, sequence, and its summation. Here, we have obtained the derivative of the $\overline{\theta (\mathtt{t})}$-Fibonacci polynomial using a proportional derivative. Furthermore, this study presents derived theorems and intriguing findings on the summation of terms in the second-order Fibonacci sequence, and we have investigated the bifurcation analysis of the $\overline{\theta (\mathtt{t})}$-Fibonacci generating function. In addition, we have included appropriate examples to demonstrate our findings by using MATLAB.

This article aims to establish sufficient conditions for qualitative properties of the solutions for a new class of a pantograph implicit system in the framework of Atangana-Baleanu-Caputo ( $\mathcal{A}\mathcal{B}\mathcal{C}$) fractional derivatives with respect to another function under integral boundary conditions. The Schaefer and Banach fixed point theorems (FPTs) are utilized to investigate the existence and uniqueness results for this pantograph implicit system. Moreover, some stability types such as the Ulam-Hyers $(\mathbb{U}\mathbb{H})$, generalized $\mathbb{U}\mathbb{H}$, Ulam-Hyers-Rassias $(\mathbb{U}\mathbb{H}\mathbb{R})$ and generalized $\mathbb{U}\mathbb{H}\mathbb{R}$ are discussed. Finally, interpretation mathematical examples are given in order to guarantee the validity of the main findings. Moreover, the fractional operator used in this study is more generalized and supports our results to be more extensive and covers several new and existing problems in the literature.

In this paper, we consider a coupled snap system in a fractional $\mathbb{G}$-Caputo derivative sense with integral boundary conditions. Hyers-Ulam stability criterion is investigated, and a numerical simulation will be supplied to some applications. Some numerical simulations are presented to guarantee the theoretical results.

In this study, some new Hermite-Hadamard type inequalities for co-ordinated convex functions were obtained with the help of conformable fractional integrals. We have presented some remarks to give the relation between our results and earlier obtained results. Moreover, an identity for partial differentiable functions has been established. By using this equality and concept of co-ordinated convexity, we have proven a trapezoid type inequality for conformable fractional integrals.

The features of analytical functions were mostly studied using a fuzzy subset and a $\mathfrak{q}$-difference operator in this study, as we investigate many fuzzy differential subordinations related to the $\mathfrak{q}$-analogue multiplier-Noor integral operator. By applying fuzzy subordination to univalent functions whose range is symmetric with respect to the real axis, we create a few new subclasses of analytical functions. We define numerous classes related to the family of linear $\mathfrak{q}$ -operators and introduce them. Here, we focus on the inclusion results and other integral features.

Many years of research have gone into spline functions, and they are now used in countless computational tasks. Splines have a lot of useful properties that make them an excellent tool for numerical problem solving, which account for their never-ending applications. The piecewise continuous functions known as spline functions yield smooth outcomes. The numerical solution to the nonhomogeneous time-fractional Banjamin-Bona-Mahony-Burger problem was presented in this study. The objective of the study was to obtain accurate numerical results by applying the Atangana-Baleanu fractional derivative with the help of the forward difference scheme for integer-order time derivative while the $\theta$-weighted scheme with the collaboration of cubic B-spline functions was used for the spatial derivatives. The stability of the proposed scheme was analyzed and proved to be unconditionally stable. The convergence analysis was also studied, and it was of the second order $O(h^2+(\mathrm{\Delta }s{)}^2)$. The proposed scheme was applicable and accurate, as demonstrated by numerical examples and their conceivable outcomes. The proposed scheme provided accuracy compared to other numerical techniques because it yielded numerical solutions in $C^2$ continuous piecewise form at each knot in the domain.

This work used the modified extended direct algebraic expansion method to find exact soliton solutions for the (2+1)-dimensional nonlinear Zoomeron equation. The modified extended direct algebraic technique employs a wave transformation and, in order to determine solutions, it then performs an algebraic expansion, compares coefficients, and balances the equation. The results were an effective acquisition of a variety of solitons with unique wave characteristics including bright, kink, periodic, singular periodic, and dark solitons. A stability investigation has confirmed the structural integrity of these solutions under minor perturbations. In the form of 2D, contour, and 3D graphical representations, the stability and propagation of these solutions were further investigated. The findings illustrate how effectively this technique can solve higher-dimensional nonlinear equations and yield more soliton solutions. Beyond broadening our knowledge of nonlinear wave behavior, this research could be beneficial in nonlinear optics, fluid motion, and plasma systems.

The Rössler attractor model is an important model that provides valuable insights into the behavior of chaotic systems in real life and is applicable in understanding weather patterns, biological systems, and secure communications. So, this work aims to present the numerical performances of the nonlinear fractional Rössler attractor system under Caputo derivatives by designing the numerical framework based on Ultraspherical wavelets. The Caputo fractional Rössler attractor model is simulated into two categories, (i) Asymmetric and (ii) Symmetric. The Ultraspherical wavelets basis with suitable collocation grids is implemented for comprehensive error analysis in the solutions of the Caputo fractional Rössler attractor model, depicting each computation in graphs and tables to analyze how fractional order affects the model’s dynamics. Approximate solutions obtained through the proposed scheme for integer order are well comparable with the fourth-order Runge-Kutta method. Also, the stability analyses of the considered model are discussed for different equilibrium points. Various fractional orders are considered while performing numerical simulations for the Caputo fractional Rössler attractor model by using Mathematica. The suggested approach can solve another non-linear fractional model due to its straightforward implementation.

In this article, we employ the Laplace transform (LT) method to study fractional differential equations with the problem of displacement of motion of mass for free oscillations, damped oscillations, damped forced oscillations, and forced oscillations (without damping). These problems are solved by using the Caputo and Atangana-Baleanu (AB) fractional derivatives, which are useful fractional derivative operators consist of a non-singular kernel and are efficient in solving non-local problems. The mathematical modelling for the displacement of motion of mass is presented in fractional form. Moreover, some examples are solved.