Journal of Mathematical Chemistry最新文献

In this paper, we first introduce a Fourier–Legendre spectral collocation method to solve the two-dimensional static Cauchy–Navier equations with variable coefficients in irregular annular domains. We then present a space-time Fourier–Legendre spectral collocation method for time-dependent Cauchy–Navier equations in such domains. The process begins by applying a polar coordinate transformation to map the irregular annular domain onto a regular one, followed by a linear transformation to map this domain onto the reference element. Classical spectral collocation methods are then employed for numerical simulation on the reference element. The numerical results demonstrate that the proposed method achieves high accuracy.

In this work, we develop multi-step vectorial iterative schemes for solving nonlinear systems, achieving fourth and sixth-order convergence. The proposed methods are designed to minimize computational costs by employing a single inverse operator and reducing the number of functional evaluations per iteration. Furthermore, we generalize the sixth-order three-step scheme into a ((q+1))-step family, increasing the convergence order to (2q+2). While standard local convergence analysis based on Taylor series expansion is common, it limits applicability as it requires the use of higher-order derivatives. To overcome this limitation, our theoretical analysis is conducted in a Banach space setting and relies solely on first-order derivatives. The existence of a unique solution is guaranteed within a specific domain, whose radius of convergence is formally obtained using Lipschitz constants. A detailed computational complexity analysis confirms the superior efficiency of our methods compared to existing approaches. Numerical experiments on different problems demonstrate significantly improved performance, while stability is validated through basins of attraction in the complex plane.

The definition of the recently introduced hitting time index suggests its close relation with the Kirchhoff index. Here, this relation is computationally investigated for trees and molecular trees. Additionally, the usability of these molecular descriptors as tools for modeling physicochemical properties of alkanes is compared. The second part of the paper is reserved for closed formulas of the hitting time index for bi-stars and the broom graphs. Finally, the upper and lower bounds, in terms of the maximum degree, the hyper-Wiener index, the Wiener index, and the Harary index for the hitting time index of trees are derived.

Using the transfer matrix method, we obtain the calculation method for the numbers of perfect matchings of m-layer hexagonal chains. Especially, we give out the linear recurrences, generating functions and general terms for the numbers of the perfect matchings in two types of special m-layer hexagonal chains–alternating and parallelogram hexagonal chains.

This work presents a numerical method for solving a three-dimensional time-fractional reaction-diffusion equation (TFRDE). The solution to this problem has a weak singularity near the initial time. The fractional time derivative is discretized using the L1 method on a nonuniform time grid, while the spatial derivatives are approximated by a fourth-order compact finite difference (CFD) scheme. The resulting fully discrete formulation is computationally expensive, therefore, an alternating direction implicit (ADI) technique is introduced to improve efficiency. The stability and convergence of the proposed scheme are rigorously analyzed. Two numerical experiments are conducted to verify the accuracy and computational efficiency of the proposed method. Theoretical analysis demonstrates that the proposed scheme attains a temporal convergence rate of (min { 2 - gamma ,, rgamma ,, 1 + gamma }) and fourth-order spatial accuracy. Numerical findings validate the theoretical convergence rates. To demonstrate the advantage of the proposed method, the numerical results obtained by the proposed method are compared with the result reported in Xiao et al., (Commun. Anal. Mech. 16(1):53–70, 2024).

In the current paper, we consider the numerical solution of a block circulant tridiagonal linear system which commonly originates from convolution equations under periodic boundary conditions. By leveraging the block-Toeplitz structure, we propose a novel structure-preserving factorization of the coefficient matrix. Based on the structure-preserving matrix factorization and Sherman-Morrison-Woodbury formula, we then develop an efficient numerical algorithm with linear time complexity for solving block circulant tridiagonal linear systems. Additionally, a theoretical error analysis is provided to ensure numerical stability, and a numerical formula for the determinant of the block circulant tridiagonal matrix is also presented. Numerical results with simulations in MATLAB implementation are provided to demonstrate the accuracy and efficiency of our proposed algorithm, and its competitiveness with the block (LU) decomposition method.

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.