Journal of Mathematical Chemistry最新文献

We present closed-form expressions for the exchange integral between general hydrogenic orbitals and its derivatives with respect to effective decay parameters. This work is a sequel to our earlier Coulomb integral study, but here the structural difficulty is inverted: the Legendre expansion that simplified the Coulomb case becomes cumbersome due to surviving phase couplings, while the Laplace route is comparatively more tractable. The results enable fully analytic screening optimization incorporating both Coulomb and exchange contributions.

The Lengyel–Epstein mathematical model for the CIMA chemical reaction is studied. The concentrations depend on time and a single spatial coordinate, so that one-dimensional patterns in space are possible. A linearized solution for the spatial patterns is presented, and the question of pattern selection is addressed. Nonlinear patterns are discussed and compared against the predictions of linearized theory. It is found that spatially-homogeneous time-dependent oscillations exist, born from Hopf bifurcations. In addition, Turing bifurcations also occur, and give rise to steady-state patterns. Furthermore, these steady patterns can undergo further bifurcation at large amplitude. These one-dimensional stationary patterns are quasi-stable, in the sense that they may persist for some time, but ultimately, they collapse onto the spatially-homogeneous limit-cycle solutions.

This study presents a unified numerical strategy that eliminates higher-order partial derivatives by employing Genocchi wavelets, their operational matrix of integration, and the collocation method for derivative terms. This approach serves as an alternative to traditional iterative methods, which often struggle to handle highly nonlinear problems effectively. The analysis and numerical solution of elliptic partial differential equations are discussed within the framework of the Genocchi Wavelet Collocation Method (GWCM). In this study, we examine the convergence, error estimation, and rapid applicability of the proposed method to a diverse range of problems. The effectiveness of the approach is demonstrated through detailed numerical experiments, with results presented in both tabular and graphical formats for clear comparison. The findings confirm the superior performance of GWCM over traditional methods, particularly under various parameter variations. One of the key advantages of this method is its ease of implementation and computational efficiency. The obtained solutions closely match the exact solutions, and an interesting observation is that for elliptic differential equations with polynomial solutions of finite degree, the method produces zero error. All computations are carried out using the latest version of MATLAB, ensuring accuracy and reliability.

The authors of Roul et al. (J Math Chem 61:2146–2175, 2023) developed a numerical method for the time-fractional diffusion equation. In this method, the L1 scheme is employed on a uniform mesh for time discretization and a compact finite difference scheme for spatial discretization. They have ignored the initial weak singularity at (t=0). The present study applies the (L2text {-}1_{sigma }) scheme on a graded temporal mesh, providing an improvement over the L1 scheme by accurately approximating the Caputo time-fractional derivative and capturing the initial-time singularity. Spatial derivatives are approximated using a high-order compact finite difference scheme. The stability and convergence of the proposed scheme are rigorously proven using the energy method, in contrast to the Von-Neumann analysis used in Roul et al. (J Math Chem 61:2146–2175, 2023), which is limited to periodic and homogeneous boundary conditions. The proposed scheme achieves a temporal accuracy of (min {ralpha ,,2}), with (alpha in (0,1)), and fourth-order spatial accuracy. Numerical experiments validate the theoretical findings, and comparisons with Roul et al. (J Math Chem 61:2146–2175, 2023) and Roul (J Comput Appl Math 451:116033,2024) demonstrate the superior accuracy of the proposed approach.

In this work, we reconsider the classical, non-linear set of ordinary differential equations for Michaelis-Menten kinetics of enzyme reactions. As the first contribution, we prove non-negativity, boundedness by two conservation properties, existence and uniqueness globally in time of solutions to this time-continuous model. As the second contribution, we show that the unique equilibrium state is globally asymptotically stable by application of LaSalle’s invariance principle through a suitable Lyapunov function. As the third and main contribution, we introduce a non-standard finite-difference-method based on the implicit Eulerian time-stepping method for the discretization of the time-continuous dynamical system. We reformulate its numerical solution algorithm by an explicit scheme and demonstrate that all desirable properties of the time-continuous model transfer to the proposed time-discrete variant. Finally, we highlight the theoretical findings by numerical experiments.

Topological indices play a crucial role in the prediction of physicochemical and biological properties of molecules. Recently, Ivan Gutman introduced the Sombor index, a new vertex-degree-based topological index that possesses significant geometric meaning. Since its introduction, this index has garnered substantial attention in mathematical chemistry and graph theory, leading to a surge in research exploring its properties, applications, and extensions. Motivated by its geometric foundation, we investigate in this paper various generalizations of the integral Sombor indices, analyzing their structural characteristics and mathematical implications. In addition, we study the applications of these indices in modeling the entropy of octane isomers.

Distributed delays play an important role in biological modeling, and are often inevitable to obtain a sufficiently precise quantitative description of certain important processes. It is known from previous studies that mass-action type complex balanced chemical reaction networks (CRNs) containing distributed delays are stable. In this paper, we consider complex balanced biochemical reaction networks with distributed delays, containing a general class of reaction rates including Michaelis–Menten and Hill-type kinetics. We show that for integrable delay distributions defined on a finite time interval, there exists precisely one equilibrium in each stoichiometric compatibility class. The local stability of the equilibria is shown using a logarithmic Lyapunov–Krasovski functional. The theoretical results are demonstrated through two illustrative examples.

One of the most important pieces of information related to molecular graphs is the determination (when possible) of upper and lower bounds for their corresponding topological indices. Such bounds make it possible to establish the approximate range of the topological indices in terms of molecular structural parameters. The purpose of this paper is to provide new inequalities for the Gutman-Milovanović index, which generalizes important indices such as the general second Zagreb index and the general sum-connectivity index. Moreover, the characterization of extremal graphs with respect to many of these inequalities is obtained. It is a well-known fact that the main application of topological indices is focused on understanding the physicochemical properties of chemical compounds; nevertheless, some additional applications are given to the study of the physicochemical properties of octane isomers. These are compounds that are especially relevant in the chemical and petroleum industries, but are also beginning to be used in the pharmaceutical industry.

In this paper, the elliptic Sombor index is studied and new upper and lower bounds for it are established. These bounds involve terms related to the (alpha )-Sombor index, the symmetric division deg index, the Gutman–Milovanović index, the general sum-connectivity index, and the hyperbolicity constant. Also, physicochemical properties of polyaromatic hydrocarbons using the elliptic Sombor index are modeled.