Journal of Mathematical Chemistry最新文献

Applying a method with vanished phase–lag might potentially eliminate the phase–lag and its first, second, and third derivatives. Improving algebraic order (AOR) and decreasing function evaluations (FEvs) are the goals of the new strategy called the cost–efficient approach. Equation PF3DPHFITN142SPS demonstrates the unique method. The suggested approach is P–Stable, meaning it is indefinitely periodic. The suggested approach is applicable to a wide variety of periodic and/or oscillatory issues. The challenging problem of Schrödinger-type coupled differential equations was solved in quantum chemistry by using this novel approach. Since the new method only needs 5FEvs to run each stage, it may be considered a cost–efficient approach. With an AOR of 14, we can significantly improve our present predicament.

In this work, two numerical techniques are compared for solving one and two-dimensional convection-diffusion equations. First technique is referred as “MCUAT tension B-spline," and the second technique is labeled as “MCUAH tension B-spline." Various aspects are examined to validate the compatibility of results, including comparisons between numerical and exact solutions and evaluation of different error norms. Present errors are compared with existing literature, presenting a remarkable improvisation. Statistical validation of work is tested via a correlation matrix heatmap generated in Python. The order of convergence of the proposed work is also included. Via an observation of comparison of results, it is claimed that UAT results are slightly better than UAH results. Different aspects of correlation, such as strongly negative correlation and perfect positive correlation, are notified. The present work will introduce new dimensions to the field of numerical techniques.

The optimized derivative fast Fourier transform (dFFT) simultaneously increases resolution and reduces noise in spectra reconstructed from encoded time signals. The pertinent applications have recently been published for time signals encoded with and without water suppression by in vitro and in vivo magnetic resonance spectroscopy (MRS). Even with the employed lower derivative orders, genuine resonances were narrowed, their intensities enhanced and the background baselines flattened. This unequivocally separated many overlapped peaks that are the thorniest problem in data analysis by signal processing. However, it has been common knowledge that higher-order derivative spectra quickly deteriorate with the increased derivative order. The optimized dFFT can challenge such findings. An unprecedented resilience of this processor to derivative-induced distortions is presently demonstrated for high derivative orders (up to 20). The salient illustrations are given for the water residual, lactate quartet and lactate doublet alongside their close surroundings. These applications of diagnostic relevance for patients with cancer are reported for time signals encoded with water suppression by in vitro proton MRS of human ovary.

We overview the main equations of the Rayleigh–Ritz variational method and discuss their connection with the problem of simultaneous diagonalization of two Hermitian matrices.

This article aims at developing a computational scheme for solving the time fractional reaction-subdiffusion (TFRSD) equation in two space dimensions. The Caputo fractional derivative is used to describe the time-fractional derivative appearing in the problem and it is approximated by using the L1 scheme. A compact difference scheme of order four is utilized for discretization of the spatial derivatives. Some test problems are solved to investigate the accuracy of the scheme. The computed results confirm that the scheme has convergence of order four in space and an order of ({min {{2-alpha ,1+alpha }}}) in the time direction, where (alpha in (0,1)) is the order of fractional derivative. Moreover, the computed results are compared with those obtained by other methods in order to justify the advantage of proposed algorithm.

In this paper, we use weight function approach to construct a new King-like family of methods to solve nonlinear equations with multiple roots. Here the weight functions are chosen appropriately to reach the maximum convergence order eight and the family is optimal in the sense of Kung–Traub conjecture. Moreover, local convergence of a fourth order modified King’s family for multiple roots is also studied. Radii of convergence balls of fourth order schemes are computed and compared with an existing method. Numerical examples have been presented based on applications of some real life problems and the results obtained show the superiority of our eighth order schemes over the existing ones. To study the dynamical behaviour of the proposed schemes, basins of attraction have also been presented which verifies that proposed eighth order schemes have more convergent points and requires less number of iterations in comparison to the existing methods.

Graph theory is used in many areas of chemical sciences, especially in molecular chemistry. It is particularly useful in the structural analysis of chemical compounds and in modeling chemical reactions. One of its applications concerns determining the structural formula of a chemical compound. This can be modeled as a variant of the well-known graph realization problem. In the classical version of the problem, a sequence of natural numbers is given, and the question is whether there exists a graph in which the vertices have degrees equal to the given numbers. In the variant considered in this paper, instead of a sequence of natural numbers, a sequence of sets of natural numbers is given, and the question is whether there exists a multigraph such that each of its vertices has a degree equal to a number from one of the sets. This variant of the graph realization problem matches the nature of the problem of determining the structural formula of a chemical compound better than other variants considered in the literature. We propose a polynomial time exact algorithm solving this variant of the problem.

The numerical solution of a class of second-order singularly perturbed three-point boundary value problems (BVPs) in 1D is achieved using a uniformly convergent, stable, and efficient difference method on a piecewise-uniform mesh. The presence of a boundary layer(s) on one (or both) of the interval’s endpoints is caused by the presence of the tiny parameter in the highest order derivative. As the perturbation parameter approaches 0, traditional numerical techniques on the uniform mesh become insufficient, resulting in poor accuracy and large blows without the use of an excessive number of points. Specially customised techniques, such as fitted operator methods or methods linked to adapted or fitted meshes that solve essential characteristics such as boundary and/or inner layers, are necessary to overcome this drawback. We developed a fitted-mesh technique in this paper that works for all perturbation parameter values. The monotone hybrid technique, which includes midway upwinding in the outer area and centre differencing in the layer region on a fitted-mesh condensing in the border layer region, is the basis for our difference scheme. In a discrete (L^infty ) norm, uniform error estimates are constructed, and the technique is demonstrated to be parameter-uniform convergent of order two (up to a logarithmic factor). To show the effectiveness of the recommended technique and to corroborate the theoretical findings, a numerical example is presented. In practise, the convergence obtained matches the theoretical expectations.

At higher altitudes near space shuttles moving at hypersonic speed the air is excited to high temperatures. Then not only mechanical collisions are affecting the gas flow, but also chemical reactions have an impact on such hypersonic flows. In this work we insert chemical reactions, in form of dissociations and associations, in a model for a mixture of mono- and polyatomic (non-reacting) species. More general chemical reactions, e.g., bimolecular ones, can be obtained by instant combinations of the considered reactions. Polyatomicity is here modelled by a continuous internal energy variable and the evolution of the gas is described by a Boltzmann equation. In the Chapman-Enskog process—and related half-space problems—the linearized Boltzmann collision operator plays a central role. Here we extend some important properties of the linearized operator to the considered model with chemical reactions. A compactness result, that the linearized operator can be decomposed into a sum of a positive multiplication operator—the collision frequency—and a compact integral operator, is obtained. The terms of the integral operator are shown to be (at least) uniform limits of Hilbert-Schmidt integral operators and, thereby, compact operators. Self-adjointness of the linearized operator follows as a direct consequence. Also, bounds on—including coercivity of—the collision frequency is obtained for hard sphere, as well as hard potentials with cutoff, like models. As consequence, Fredholmness as well as the domain of the linearized operator are obtained.

This article compared two high-order numerical schemes for convection-diffusion delay differential equation via two fractional operators with singular kernels. The objective is to present two effective schemes that give ((3-alpha )) and second order of accuracy in the time direction when (alpha in (0,1)) using Caputo and Modified Atangana-Baleanu Caputo derivatives, respectively. We also implemented a trigonometric spline technique in the space direction, giving second order of accuracy. Moreover, meticulous analysis shows these numerical schemes to be unconditionally stable and convergent. The efficiency and reliability of these schemes are illustrated by numerical experiments. The tabulated results obtained from test examples have also shown the comparison of these operators.