Fitted Numerical Scheme for Second-Order Singularly Perturbed Differential-Difference Equations with Mixed Shifts

This paper presents the study of singularly perturbed differential-difference equations of delay and advance parameters. The proposed numerical scheme is a fitted fourth-order finite difference approximation for the singularly perturbed differential equations at the nodal points and obtained a tridiagonal scheme. This is significant because the proposed method is applicable for the perturbation parameter which is less than the mesh size , where most numerical methods fail to give good results. Moreover, the work can also help to introduce the technique of establishing and making analysis for the stability and convergence of the proposed numerical method, which is the crucial part of the numerical analysis. Maximum absolute errors range from 10 − 03 up to 10 − 10 , and computational rate of convergence for different values of perturbation parameter, delay and advance parameters, and mesh sizes are tabulated for the considered numerical examples. Concisely, the present method is stable and convergent and gives more accurate results than some existing numerical methods reported in the literature.

MATHEMATICAL MODELING OF ERRORS IN DETERMINING THE TIME OF DEATH ON THE BASIS OF THE NEWTON’S - RICHMAN’S COOLING LAW

A mandatory element of the development and implementation of diagnostic technologies for determining the time of death by the thermal method is the assessment of their possible errors. For equations of cadaver cooling that have a deterministic character, the estimation of errors is possible on the basis of a mathematical model of indirect measurement. In this article, a mathematical model is proposed for estimating the maximum absolute errors in determining the prescription of death on the basis of the Newtons - Richmans cooling law under conditions of constant and changing ambient temperature. Aim. To develop, on the basis of a mathematical model of indirect measurement, a method for estimating errors in determining the prescription of death on the basis of the Newtons - Richmans cooling law. Material and methods. Mathematical modeling of errors in determining the time of death in conditions of constant and changing ambient temperature is carried out on the basis of the Newtons - Richmans law. The computer program code is written in the C# programming language using the Microsoft Visual Studio 2019 application. Results. On the basis of the indirect measurement model, a method for estimating the maximum absolute errors in determining the time of death during cooling according to the Newtons - Richmans law under conditions of constant and changing ambient temperature is developed. The results obtained allow us to carry out an analytical determination of the errors in time the prescription of death in the early postmortem period. Conclusions. A mathematical model is developed for estimating the maximum absolute errors in determining the time of death on the basis of the Newtons - Richmans cooling law under conditions of constant and changing ambient temperature. The developed mathematical model is implemented in the format of the application program Warm Bodies NRN. The proposed method is advisable to use in forensic medical expert practice in determining the time of death.

Exponentially Fitted Numerical Method for Singularly Perturbed Differential-Difference Equations

This paper presents a numerical method to solve singularly perturbed differential-difference equations. The solution of this problem exhibits layer or oscillatory behavior depending on the sign of the sum of the coefficients in reaction terms. A fourth-order exponentially fitted numerical scheme on uniform mesh is developed. The stability and convergence of the proposed method have been established. The effect of delay parameter (small shift) on the boundary layer(s) has also been analyzed and depicted in graphs. The applicability of the proposed scheme is validated by implementing it on four model examples. Maximum absolute errors in comparison with the other numerical experiments are tabulated to illustrate the proposed method.

Attitude stabilization control for quadrotor using self-tuning fuzzy-PD

This research aims to develop a quadrotor control system for maintaining its position and balance from disturbance while hovering. A fast and reliable control technique is required to respond to high maneuverability and high non-linearity of six degrees of freedom system. Hence, this research focuses on designing a Self-Tuning Fuzzy-PD control system for quadrotor’s attitude. The designed control system utilizes input data from the Inertial Navigation System (INS). Then the quadrotor’s attitude is controlled by passing the PWM signal to the flight controller APM 2.6. The result shows that the average absolute error for the roll, pitch, and yaw angles are relatively small, as mentioned consecutively 2.0790, 2.2660, and 1.5280, while the maximum absolute errors are 6.3140, 6.7220, and 3.820.

Solving Singularly Perturbed Differential-Difference Equations with Dual Layer using Fourth Order Numerical Method

In this paper, we presented a fourth-order numerical method to solve SPDDE with the dual-layer. The answer to the problem shows dual-layer behavior. A fourth-order finite difference plan on a uniform mesh is developed. The result of the delay and also advance parameters on the boundary layer(s) has likewise been evaluated as well as represented in charts. The applicability of the planned plan is actually confirmed through executing it on model examples. To show the accuracy of the method, the results are presented in terms of maximum absolute errors.

The Convergence of Geometric Mesh Cubic Spline Finite Difference Scheme for Nonlinear Higher Order Two-Point Boundary Value Problems

An efficient algorithm for the numerical solution of higher (even) orders two-point nonlinear boundary value problems has been developed. The method is third order accurate and applicable to both singular and nonsingular cases. We have used cubic spline polynomial basis and geometric mesh finite difference technique for the generation of this new scheme. The irreducibility and monotone property of the iteration matrix have been established and the convergence analysis of the proposed method has been discussed. Some numerical experiments have been carried out to demonstrate the computational efficiency in terms of convergence order, maximum absolute errors, and root mean square errors. The numerical results justify the reliability and efficiency of the method in terms of both order and accuracy.

A Miniature Four-Hole Probe for Measurement of Three-Dimensional Flow with Large Gradients

A miniature four-hole probe with a sensing area of 1.284 mm2to minimise the measurement errors due to the large pressure and velocity gradients that occur in highly three-dimensional turbomachinery flows is designed, fabricated, calibrated, and validated. The probe has good spatial resolution in two directions, thus minimising spatial and flow gradient errors. The probe is calibrated in an open jet calibration tunnel at a velocity of 50 m/s in yaw and pitch angles range of ±40 degrees with an interval of 5 degrees. The calibration coefficients are defined, determined, and presented. Sensitivity coefficients are also calculated and presented. A lookup table method is used to determine the four unknown quantities, namely, total and static pressures and flow angles. The maximum absolute errors in yaw and pitch angles are 2.4 and 1.3 deg., respectively. The maximum absolute errors in total, static, and dynamic pressures are 3.4, 3.9, and 4.9% of the dynamic pressures, respectively. Measurements made with this probe, a conventional five-hole probe and a miniature Pitot probe across a calibration section, demonstrated that the errors due to gradient and surface proximity for this probe are considerably reduced compared to the five-hole probe.

Quantitative Determination of Lactic Acid and Succinic Acid in Foods by Gas Chromatography

Lactic and succinic acids are liberated from the sample with sulfuric and phosphotungstic acids, extracted with absolute ether, csterified with boron trifluoride-npropyl alcohol, and gas chromatographed on a diethylene glycol succinate column. Acctophenone is used as internal standard, and a dilution procedure is described for accommodating samples having high concentrations of lactic and succinic acids. Recovery data were obtained on whole egg samples with acids added in concentrations representative of decomposed eggs: up to 100 mg lactic acid and up to 60 mg succinic acid/100 g egg. Maximum absolute errors were 7.1 mg for lactic acid and 4.4 ing for succinic acid. Maximum percentage errors were 7.1 and 13.3, respectively. The method was also applied to beef, shrimp, and cottage cheese whey.