Journal of Mathematical Chemistry最新文献

Energy-preserving algorithms, as one of the core research areas in numerical ordinary differential equations, have achieved great success by many methods such as symplectic methods and discrete gradient methods. This paper considers the numerical integration of quasi-bi-Hamiltonian systems, which, as a generalization of bi-Hamiltonian systems, can be expressed in two distinct ways: ({dot{y}} = P _ { 1 } ( y ) nabla H _ { 2 }(y) = frac{1}{rho (y)}P _ { 2 } ( y ) nabla H _ { 1 }(y)). The quasi-bi-Hamiltonian systems have two Hamiltonians (H_1(y)) and (H_2(y)). Conventional discrete gradient methods can only preserve one Hamiltonian at a time. In this paper, based on discrete gradient and projection, new energy-preserving integrators that can preserve the two Hamiltonians simultaneously are proposed. They show better qualitative behaviours than traditional discrete gradient methods do. Numerical integrations of Hénon-Heiles type systems and the Korteweg-de Vries (KdV) equation are conducted to show the effectiveness of the new integrators in comparison with traditional discrete gradient methods.

Power law systems have been studied extensively due to their wide-ranging applications, particularly in chemistry. In this work, we focus on power law systems that can be decomposed into stoichiometrically independent subsystems. We show that for such systems where the ranks of the augmented matrices containing the kinetic order vectors of the underlying subnetworks sum up to the rank of the augmented matrix containing the kinetic order vectors of the entire network, then the existence of the positive steady states of each stoichiometrically independent subsystem is a necessary and sufficient condition for the existence of the positive steady states of the given power law system. We demonstrate the result through illustrative examples. One of which is a network of a carbon cycle model that satisfies the assumptions, while the other network fails to meet the assumptions. Finally, using the aforementioned result, we present a systematic method for deriving positive steady state parametrizations for the mentioned subclass of power law systems, which is a generalization of our recent method for mass action systems.

The phase-lag and all of its derivatives (first, second, third, fourth, fifth, and sixth) might be eliminated using a phase-fitting technique. The new approach, which is referred to as the economical method, targets maximizing algebraic order (AOR) and reducing function evaluations (FEvs). The one-of-a-kind approach is demonstrated by Equation PF6DPFN142SPS.The proposed method is infinitely periodic i.e. P-Stable. To many periodic and/or oscillatory problems, the suggested strategy can be applied. Using this innovative method, the difficult issue of Schrödinger-type coupled differential equations was tackled in quantum chemistry. Every step of the new approach only requires 5FEvs to execute, making it a economic algorithm. By accomplishing a AOR of 14, this allows us to greatly enhance our current situation.

We refine the enumeration of fusene chains of a given length with respect to the number of turns by constructing bijections between such chains and ternary words. Explicit formulas thus obtained are then used to compute the expected values for the whole class of bond-additive degree-based topological indices over all such chains of a given length. The results are also applicable to several other classes of chemically interesting polycyclic chains, such as, e.g., phenylene and spiro chains.

The research article presents a novel approach for the numerical solution of three-component time fractional order Brusselator reaction-diffusion system using the innovative Vieta–Fibonacci wavelet and collocation method. The proposed method involves the derivation of operational matrices for both integer and fractional order derivatives, enable the accurate and efficient computation of the system. The existence, uniqueness of solution and Ulam–Hyers stability of the model are rigorously discussed. Furthermore, a comprehensive convergence analysis of the Vieta–Fibonacci wavelet method is presented, which demonstrates its effectiveness in approximating the fractional derivative of the Brusselator system. The numerical experiments showcase the superior performance of the method in terms of accuracy and computational efficiency. The application of the Vieta–Fibonacci wavelet method to the three-component fractional order Brusselator reaction-diffusion system marks a significant advancement in the field of computational mathematics. The successful implementation of the Vieta–Fibonacci wavelet method signifies a significant advancement in solving fractional-order reaction-diffusion problems.

When considering the possibility of storing information in the sequence of monomer residues within an AB-type copolymer chain, it is constructive to model that sequence as a string of ones and zeros. The intramolecular environment around any given digit (say a “1”) can then be represented by another string of integers—a code—obtained by summing pairs of digits at equivalent positions, in both directions, from that digit. The code can include only integers 0, 1 and 2, and can represent a number in any base b higher than 2. In base b = 3 the resulting set of codes includes all numbers (because only digits 0, 1 and 2 occur in ternary expansions), but in any base b > 3 the codes define a limited set of numbers comprising a fractal we term a Smith–Cantor set. The ¹H NMR spectrum of a random, AB-type co(polyester-imide) shows, on complexation with pyrene, a pattern of complexation shifts approximating very closely to the Smith–Cantor set for which b = 4. Other co(polyimide) complexes show a ¹H NMR pattern corresponding to a specific sub-set of this fractal. The sub-set arises from a “stop-at-zero” limitation, whereby digits in the initial string are set to zero for code-generating purposes if they occur beyond a zero, as viewed from the central “1”. The limitation arises in copolymers where pyrene binds by intercalation between pairs of adjacent diimide residues. This numerical approach provides a complete, unifying theory to account for the emergence of fractal character in the ¹H NMR spectra of AB-type copolymer complexes.

In this paper, we introduce a novel series of second-order Backward Differentiation Formulas (BDFs) specifically designed to address phase-lag and its first derivative in the numerical resolution of Initial Value Problems (IVPs) with orbital solutions. Our methodology commences with an in-depth analysis of phase-lag phenomena associated with second-order BDFs. Following this, we construct a suite of equations by embedding algebraic functions into the operational framework of the 3-step second-order BDF (SOBDF) method. Additionally, we elaborate on equations that precisely describe the phase-lag and its derivatives, with a concentrated focus on the 3-step SOBDF method. The culmination of this work is the presentation of six distinct methods, each methodically crafted to negate both the real and imaginary elements of phase-lag and its derivatives in numerical computations. The study advances with a meticulous examination of the local truncation error and the stability regions pertinent to the six phase-fitted methods introduced. Furthermore, we scrutinize their computational performance by deploying these methods across a spectrum of initial value problems, offering valuable insights into their effectiveness in varying contexts.

The accurate calculation of reaction rates from experimental data is crucial for understanding and characterizing chemical processes. However, the presence of noise in experimental data can introduce errors in rate calculations. In this study, we introduced a novel approach that utilizes the Legendre series expansion method to directly derive smooth reaction rates from noisy experimental data, eliminating the need for numerical differentiation methods. This approach proves to be highly effective in handling noisy thermogravimetric analysis (TGA) data obtained from the thermal decomposition of specific polymers. We demonstrated the robustness and reliability of this method and provided Gnu Octave codes as a free alternative to MATLAB, making the implementation more accessible. Furthermore, the smooth reaction rates obtained were used to evaluate the activation energy using the Friedman isoconversional method. The results showed excellent agreement with those obtained using the Vyazovkin integral method. Additionally, the proposed method can be applied to obtain smooth derivative thermogravimetric (DTG) curves using noisy TGA data set.

The dynamics exhibited by reaction networks is often a manifestation of their steady states. We show that there exists endotactic and strongly endotactic dynamical systems that are not weakly reversible and possess a family of infinitely many positive steady states. In addition, we prove for some of these systems that there exist no weakly reversible mass-action systems that are dynamically equivalent to mass-action systems generated by these networks. This extends a result by Boros, Craciun and Yu [1], who proved the existence of weakly reversible dynamical systems with infinitely many steady states.