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.

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.