Derivatives of the Evans function and (modified) Fredholm determinants for first order systems

2011 ◽  
Vol 284 (13) ◽  
pp. 1592-1638 ◽  
Author(s):  
Mita Das ◽  
Yuri Latushkin
Keyword(s):  
Evans Function ◽  
Fredholm Determinants ◽  
First Order ◽  
Derivatives Of

Complete monotonicity of entropy production in chemical reaction

1981 ◽  
Vol 46 (2) ◽  
pp. 452-456
Author(s):  
Milan Šolc
Keyword(s):  
Entropy Production ◽  
Equilibrium Constant ◽  
Chemical Reaction ◽  
Relative Entropy ◽  
Order Reaction ◽  
Complete Monotonicity ◽  
First Order Reaction ◽  
First Order ◽  
Completely Monotonic ◽  
Derivatives Of

The successive time derivatives of relative entropy and entropy production for a system with a reversible first-order reaction alternate in sign. It is proved that the relative entropy for reactions with an equilibrium constant smaller than or equal to one is completely monotonic in the whole definition interval, and for reactions with an equilibrium constant larger than one this function is completely monotonic at the beginning of the reaction and near to equilibrium.

Hodograph Method for Solving the Problem of Shallow Water Under a Solid Lid

2021 ◽  
pp. 15-24
Author(s):  
Tatiana F. Dolgikh
Keyword(s):  
Cauchy Problem ◽  
Green Function ◽  
Differential Equations ◽  
Shallow Water ◽  
Ordinary Differential Equations ◽  
Riemann Invariants ◽  
Hodograph Method ◽  
First Order ◽  
The Cauchy Problem ◽  
Derivatives Of

One of the mathematical models describing the behavior of two horizontally infinite adjoining layers of an ideal incompressible liquid under a solid cover moving at different speeds is investigated. At a large difference in the layer velocities, the Kelvin-Helmholtz instability occurs, which leads to a distortion of the interface. At the initial point in time, the interface is not necessarily flat. From a mathematical point of view, the behavior of the liquid layers is described by a system of four quasilinear equations, either hyperbolic or elliptic, in partial derivatives of the first order. Some type shallow water equations are used to construct the model. In the simple version of the model considered in this paper, in the spatially one-dimensional case, the unknowns are the boundary between the liquid layers h(x,t) and the difference in their velocities γ(x,t). The main attention is paid to the case of elliptic equations when |h|<1 and γ>1. An evolutionary Cauchy problem with arbitrary sufficiently smooth initial data is set for the system of equations. The explicit dependence of the Riemann invariants on the initial variables of the problem is indicated. To solve the Cauchy problem formulated in terms of Riemann invariants, a variant of the hodograph method based on a certain conservation law is used. This method allows us to convert a system of two quasilinear partial differential equations of the first order to a single linear partial differential equation of the second order with variable coefficients. For a linear equation, the Riemann-Green function is specified, which is used to construct a two-parameter implicit solution to the original problem. The explicit solution of the problem is constructed on the level lines (isochrons) of the implicit solution by solving a certain Cauchy problem for a system of ordinary differential equations. As a result, the original Cauchy problem in partial derivatives of the first order is transformed to the Cauchy problem for a system of ordinary differential equations, which is solved by numerical methods. Due to the bulkiness of the expression for the Riemann-Green function, some asymptotic approximation of the problem is considered, and the results of calculations, and their analysis are presented.

scholarly journals Hexagonal Grid Computation of the Derivatives of the Solution to the Heat Equation by Using Fourth-Order Accurate Two-Stage Implicit Methods

2021 ◽  
Vol 5 (4) ◽  
pp. 203
Author(s):  
Suzan Cival Buranay ◽  
Nouman Arshad ◽  
Ahmed Hersi Matan
Keyword(s):  
Heat Equation ◽  
Fourth Order ◽  
Boundary Value ◽  
Time Variable ◽  
Implicit Methods ◽  
First Order ◽  
Mixed Derivatives ◽  
Hexagonal Grids ◽  
Spatial Derivatives ◽  
Derivatives Of

We give fourth-order accurate implicit methods for the computation of the first-order spatial derivatives and second-order mixed derivatives involving the time derivative of the solution of first type boundary value problem of two dimensional heat equation. The methods are constructed based on two stages: At the first stage of the methods, the solution and its derivative with respect to time variable are approximated by using the implicit scheme in Buranay and Arshad in 2020. Therefore, Oh4+τ of convergence on constructed hexagonal grids is obtained that the step sizes in the space variables x1, x2 and in time variable are indicated by h, 32h and τ, respectively. Special difference boundary value problems on hexagonal grids are constructed at the second stages to approximate the first order spatial derivatives and the second order mixed derivatives of the solution. Further, Oh4+τ order of uniform convergence of these schemes are shown for r=ωτh2≥116,ω>0. Additionally, the methods are applied on two sample problems.

Approximation of 3D convection diffusion equation using DQM based on modified cubic trigonometric B-splines

2020 ◽  
pp. 1-10
Author(s):  
Mohammad Tamsir ◽  
Neeraj Dhiman ◽  
F.S. Gill ◽  
Robin
Keyword(s):  
Diffusion Equation ◽  
Numerical Solutions ◽  
Basis Functions ◽  
B Spline ◽  
Convection Diffusion ◽  
B Splines ◽  
Convection Diffusion Equation ◽  
First Order ◽  
The Stability ◽  
Derivatives Of

This paper presents an approximate solution of 3D convection diffusion equation (CDE) using DQM based on modified cubic trigonometric B-spline (CTB) basis functions. The DQM based on CTB basis functions are used to integrate the derivatives of space variables which transformed the CDE into the system of first order ODEs. The resultant system of ODEs is solved using SSPRK (5,4) method. The solutions are approximated numerically and also presented graphically. The accuracy and efficiency of the method is validated by comparing the solutions with existing numerical solutions. The stability analysis of the method is also carried out.

scholarly journals Polycyclic Aromatic Hydrocarbon Metabolism by White Rot Fungi and Oxidation by Coriolopsis gallica UAMH 8260 Laccase

1999 ◽  
Vol 65 (9) ◽  
pp. 3805-3809 ◽  
Author(s):  
Michael A. Pickard ◽  
Rosa Roman ◽  
Raunel Tinoco ◽  
Rafael Vazquez-Duhalt
Keyword(s):  
Oxidation Rate ◽  
Mass Spectra ◽  
White Rot Fungi ◽  
Order Rate Constant ◽  
Maximum Activity ◽  
White Rot ◽  
First Order ◽  
Polycyclic Aromatic ◽  
Derivatives Of

ABSTRACT We studied the metabolism of polycyclic aromatic hydrocarbons (PAHs) by using white rot fungi previously identified as organisms that metabolize polychlorinated biphenyls. Bran flakes medium, which has been shown to support production of high levels of laccase and manganese peroxidase, was used as the growth medium. Ten fungi grown for 5 days in this medium in the presence of anthracene, pyrene, or phenanthrene, each at a concentration of 5 μg/ml could metabolize these PAHs. We studied the oxidation of 10 PAHs by using laccase purified from Coriolopsis gallica. The reaction mixtures contained 20 μM PAH, 15% acetonitrile in 60 mM phosphate buffer (pH 6), 1 mM 2,2′-azinobis-(3-ethylbenzthiazoline-6-sulfonate) (ABTS), and 5 U of laccase. Laccase exhibited 91% of its maximum activity in the absence of acetonitrile. The following seven PAHs were oxidized by laccase: benzo[a]pyrene, 9-methylanthracene, 2-methylanthracene, anthracene, biphenylene, acenaphthene, and phenanthrene. There was no clear relationship between the ionization potential of the substrate and the first-order rate constant (k) for substrate loss in vitro in the presence of ABTS. The effects of mediating substrates were examined further by using anthracene as the substrate. Hydroxybenzotriazole (HBT) (1 mM) supported approximately one-half the anthracene oxidation rate (k = 2.4 h−1) that ABTS (1 mM) supported (k = 5.2 h−1), but 1 mM HBT plus 1 mM ABTS increased the oxidation rate ninefold compared with the oxidation rate in the presence of ABTS, to 45 h−1. Laccase purified from Pleurotus ostreatus had an activity similar to that ofC. gallica laccase with HBT alone, with ABTS alone, and with 1 mM HBT plus 1 mM ABTS. Mass spectra of products obtained from oxidation of anthracene and acenaphthene revealed that the dione derivatives of these compounds were present.

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