integration interval
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In this paper, Runge Kutta method of order 4 is used to study the electrical circuits designs through past, intermediate and present voltages. When integrating differential equations with Runge Kutta method of order 4, a constant step size (ℎ) is used until a testing procedure confirms that the discontinuity occurs in the present integration interval. This step size function calculations would take place at the end of the functional calculations, but before the dependent variables were updated. Runge Kutta methods along with convolution are given by array interpretation (Butcher matrix) representation, this leads to identify the equilibrium state. The input parameters indicate the voltage coefficient controlled by current sources and measures it a random periodic time. The output parameters provide stable independent values and calculated from past voltage and current values. Finally solutions are compared with exact values and RK method of order 4 along with Heun, Midpoint and Taylors’s method with various ℎ values.

Rail impedance directly affects the transmission performance of track circuit . Considering the condition of earth stratification, for the difficult to calculate the rail impedance due to the semi-infinite integration interval and the oscillation of the integrand by using the Carson formula, The truncation method is proposed to divide the impedance formula is divided into definite integral and tail integral. The integral is approximated by the spline function, and the tail integral is calculated by using the exponential integral and Euler formula. Based on it, the rail impedance calculation formula of track circuit is obtained. The electromagnetic field model of track circuit with earth stratification is simulated by finite element method, and the correctness of the method is verified. Based on the formula, the influence of current frequency, soil depth and conductivity on rail impedance is studied. The relative error between the calculated results of rail impedance and the simulation results of finite element is within 5%. It can be seen that the formula has high accuracy and correctly reflects the law of rail impedance variation with current frequency, soil depth and resistivity. It provides a reliable reference for the theoretical calculation of rail impedance of track circuit.

Abstract. Reliable, sensitive, and widely available hydrogen chloride (HCl) measurements are important for understanding oxidation in many regions of the troposphere. We configured a commercial HCl cavity ring-down spectrometer (CRDS) for sampling HCl in the ambient atmosphere and developed validation techniques to characterize the measurement uncertainties. The CRDS makes fast, sensitive, and robust measurements of HCl in a high-finesse optical cavity coupled to a laser centred at 5739 cm−1. The accuracy was determined to reside between 5 %–10 %, calculated from laboratory and ambient air intercomparisons with annular denuders. The precision and limit of detection (3σ) in the 0.5 Hz measurement were below 6 and 18 pptv, respectively, for a 30 s integration interval in zero air. The response time of this method is primarily characterized by fitting decay curves to a double exponential equation and is impacted by inlet adsorption/desorption, with these surface effects increasing with relative humidity and decreasing with decreasing HCl mixing ratios. The minimum 90 % response time was 10 s and the equilibrated response time for the tested inlet was 2–6 min under the most and least optimal conditions, respectively. An intercomparison with the EPA compendium method for quantification of acidic atmospheric gases showed good agreement, yielding a linear relationship statistically equivalent to unity (slope of 0.97 ± 0.15). The CRDS from this study can detect HCl at atmospherically relevant mixing ratios, often performing comparably or better in sensitivity, selectivity, and response time than previously reported HCl detection methods.

This manuscript presents an efficient pair of hybrid Nyström techniques to solve second-order Lane–Emden singular boundary value problems directly. One of the proposed strategies uses three off-step points. The obtained formulas are paired with an appropriate set of formulas implemented for the first step to avoid singularity at the left end of the integration interval. The fundamental properties of the proposed scheme are analyzed. Some test problems, including chemical kinetics and physical model problems, are solved numerically to determine the efficiency and validity of the proposed approach.

In this paper, we continue to solve as accurately as possible singular eigenvalues problems attached to the Schrödinger equation. We use the conventional ChC and SiC as well as Chebfun. In order to quantify the accuracy of our outcomes, we use the drift with respect to some parameters, i.e., the order of approximation N, the length of integration interval X, or a small parameter ε, of a set of eigenvalues of interest. The deficiency of orthogonality of eigenvectors, which approximate eigenfunctions, is also an indication of the accuracy of the computations. The drift of eigenvalues provides an error estimation and, from that, one can achieve an error control. In both situations, conventional spectral collocation or Chebfun, the computing codes are simple and very efficient. An example for each such code is displayed so that it can be used. An extension to a 2D problem is also considered.

Abstract. Reliable, sensitive, and widely available hydrogen chloride (HCl) measurements are important for understanding oxidation in many regions of the troposphere. We configured a commercial HCl cavity ring-down spectrometer (CRDS) for sampling HCl in the ambient atmosphere and developed calibration and validation techniques to characterize the measurement uncertainties. The CRDS makes fast, sensitive, and robust measurements of HCl in a high finesse optical cavity coupled to a laser centered at 5739 cm−1. The accuracy was determined to reside between 5–10 %, calculated from laboratory calibrations and an ambient air intercomparison with annular denuders. The precision and limit of detection (3σ) in the 0.5 Hz measurement were below 6 pptv and 18 pptv, respectively for a 30 second integration interval in zero air. The response time of this method is primarily characterized by fitting decay curves to a double exponential equation and is impacted by inlet adsorption/desorption, with these surface effects increasing with RH and decreasing with decreasing HCl mixing ratios. The response time for the tested inlet was 2–6 minutes under the most and least optimal conditions, respectively. An intercomparison with the EPA compendium method for quantification of acidic atmospheric gases showed good agreement, yielding a linear relationship statistically equivalent to unity (slope of 0.97 ± 0.15). The CRDS from this study can detect HCl at atmospherically relevant mixing ratios, often performing comparable or better in sensitivity, selectivity, and response-time from previously reported HCl detection methods.

This paper presents the development of a new numerical scheme for the solution of exponential growth and decay models emanated from biological sciences. The scheme has been derived via the combination of two interpolants namely, polynomial and exponential functions. The analysis of the local truncation error of the derived scheme is investigated by means of the Taylor’s series expansion. In order to test the performance of the scheme in terms of accuracy in the context of the exact solution, four biological models were solved numerically. The absolute error has been computed successfully at each mesh point of the integration interval under consideration. The numerical results generated via the scheme agree with the exact solution and with the fifth order convergence based upon the analysis carried out. Hence, the scheme is found to be of order five, accurate and is a good approach to be included in the class of linear explicit numerical methods for the solution of initial value problems in ordinary differential equations.