This study examines the dynamics of a Listeria bacterium in a packaged food scenario using a mathematical framework that incorporates fractional derivatives in the Caputo sense. The model is solved using two primary methods: the homotopy analysis method (HAM), a semi-analytical approach that provides a series solution for fractional-order nonlinear ordinary differential equations (ODEs), and the Runge-Kutta 4th order (RK4) method, a well-known numerical technique for solving ODE systems. The HAM solution is derived by selecting a convergence control parameter, which produces a series of polynomials, whereas the RK4 method yields a fully numerical solution. The accuracy and dependability of both methods are evaluated by comparing their results to those of an ND Solver. The model's behavior under different parameter values is examined. All computations are carried out using Scilab and Mathematica software. The findings are presented in extensive graphs and tables, and formal theorems are used to demonstrate the convergence of the solution. This comparative method reveals the efficacy of fractional derivative modeling for analyzing bacterial activity in packaged food environments, as well as the effectiveness of HAM and RK4 in handling complex biological systems.
In this study, we first develop some helpful generating functions for the Jacobsthal and Jacobsthal-Lucas numbers. We have discussed generating functions for $\mathbf{J}_{2n}$ and $\mathbf{j}_{2n}$ with just even and odd terms and generating functions for $\mathbf{J}_{2n+1}$, $\mathbf{j}_{2n+1}$, $\mathbf{J}_n\mathbf{J}_{n+1}$, $\mathbf{j}_n\mathbf{j}_{n+1}$, $\mathbf{J}_n^2$ and $\mathbf{j}_n^2$, where $\mathbf{J}_n$ and $\mathbf{j}_n$ are $n^{th}$ Jacobsthal and Jacobsthal-Lucas numbers, respectively. Using these generating functions, we have found sum formulas for square Jacobsthal and Jacobsthal-Lucas numbers. Finally, we derive numerous useful convolution formulas including Jacobsthal numbers and similar expressions for Jacobsthal-Lucas numbers.
In this paper, we introduce non-homogeneous weighted Besov-type spaces and non-homogeneous weighted Triebel-Lizorkin-type spaces. We then establish their characterizations in terms of local means and prove some remarkable embeddings. Moreover, we also construct Rychkov's linear, bounded universal extension operator for these spaces.
In this work, we study the existence and uniqueness of solutions to a class of fractional differential equations of ABC type with impulse conditions. Sufficient conditions are established to prove the existence results by using fixed point theorems such as the Banach contraction principle and the Schauder fixed point theorem. In addition, the Ulam stability of the solution is considered. Finally, an example is provided to illustrate the main results.
This paper introduces a closed-form expression for generalized bivariate complex polynomials using a tridiagonal determinant in symbolic form. Subsequences with even and odd indices are also represented as tridiagonal determinants. Furthermore, we achieve factorizations of the tridiagonal determinant in terms of eigenvalues and demonstrate the versatility of the established closed-form formula through its application in deriving Chebyshev polynomials. This generalized sequence encompasses several well-known polynomial families in one or two variables, including the Fibonacci, Lucas, Pell, Pell–Lucas, and Chebyshev polynomial sequences in tridiagonal representation. Finally, we explore identities involving complex bivariate generalized polynomials, expressed as second-order determinants.
