Uniformly Convergent Numerical Scheme for Singularly Perturbed Parabolic PDEs with Shift Parameters

In this article, singularly perturbed parabolic differential difference equations are considered. The solution of the equations exhibits a boundary layer on the right side of the spatial domain. The terms containing the advance and delay parameters are approximated using Taylor series approximation. The resulting singularly perturbed parabolic PDEs are solved using the Crank–Nicolson method in the temporal discretization and nonstandard finite difference method in the spatial discretization. The existence of a unique discrete solution is guaranteed using the discrete maximum principle. The uniform stability of the scheme is investigated using solution bound. The uniform convergence of the scheme is discussed and proved. The scheme converges uniformly with the order of convergence O N − 1 + Δ t 2 , where N is number of subintervals in spatial discretization and Δ t is mesh length in temporal discretization. Two test numerical examples are considered to validate the theoretical findings of the scheme.

Numerical study of time delay singularly perturbed parabolic differential equations involving both small positive and negative space shifts

A time dependent singularly perturbed differential-difference equation is considered. The problem involves time delay and general small space shift terms. Taylor series approximation is used to expand the space shift term. A robust numerical scheme based on the backward Euler scheme for the time and classical upwind scheme for space is proposed. The convergence analysis is carried out. It is observed that the proposed scheme converges almost first order up to a logarithm term and optimal first order in space on the Shishkin and Bakhvalov–Shishkin mesh, respectively. Numerical results confirm the efficiency of the proposed scheme, which are in agreement with the theoretical bounds.

Exact and inexact decompositions of trade price indices

A reasonable concept for the true trade price index in situations where low-price countries capture market shares from high-price countries is the average price paid by importers for the same quality of good or service from all exporting countries. However, decompositions of trade price indices are usually inexact in the sense that the average price used as the underlying aggregator formula is not exactly reproduced. In this paper, we compare analytically exact and inexact decompositions of trade price indices, paying particular attention to the bias in aggregate inflation occurring from applying the first-order Taylor series approximation and not the quadratic approximation lemma to a geometric average price. Our calculations, using the Norwegian clothing industry as an illustration, reveal that the bias in aggregate inflation over the sample period of 1997–2016 is quite substantial and as much as 0.6 percentage point in some years. We therefore conclude that the quadratic approximation lemma should be used in practice to exactly reproduce the underlying aggregator formula.

Logarithmic Generalization of the Lambert W Function and Its Applications to Adiabatic Thermostatistics of the Three-Parameter Entropy

A generalization of the Lambert W function called the logarithmic Lambert function is introduced and is found to be a solution to the thermostatistics of the three-parameter entropy of classical ideal gas in adiabatic ensembles. The derivative, integral, Taylor series, approximation formula, and branches of the function are obtained. The heat functions and specific heats are computed using the “unphysical” temperature and expressed in terms of the logarithmic Lambert function.

Squaring Technique using Vedic Mathematics

Vedic Mathematics provides an interesting approach to modern computing applications by offering an edge of time and space complexities over conventional techniques. Vedic Mathematics consists of sixteen sutras and thirteen sub-sutras, to calculate problems revolving around arithmetic, algebra, geometry, calculus and conics. These sutras are specific to the decimal number system, but this can be easily applied to binary computations. This paper presented an optimised squaring technique using Karatsuba-Ofman Algorithm, and without the use of Duplex property for reduced algorithmic complexity. This work also attempts Taylor Series approximation of basic trigonometric and inverse trigonometric series. The advantage of this proposed power series approximation technique is that it provides a lower absolute mean error difference in comparison to previously existing approximation techniques.

Practical Considerations with Radar Data Height and Great Circle Distance Determination

Owing to their ease of use, “simplified” propagation models, like the Equivalent Earth model, are commonly employed to determine radar data locations. With the assumption that electromagnetic rays follow paths of constant curvature, which is a fundamental assumption in the Equivalent Earth model, propagation equations that do not depend upon the spatial transformation that is utilized in the Equivalent Earth model are derived. This set of equations provides the true constant curvature solution and is less complicated, conceptually, as it does not depend upon a spatial transformation. Moreover, with the assumption of constant curvature, the relations derived herein arise naturally from ray tracing relations.Tests show that this new set of equations is more accurate than the Equivalent Earth equations for a “typical” propagation environment in which the index of refraction n decreases linearly at the rate dn/dh = -1/4a, where h is height above ground and a is the Earth’s radius. Moreover, this new set of equations performs better than the Equivalent Earth equations for an exponential reference atmosphere, which provides a very accurate representation of the average atmospheric n structure in the United States. However, with this n profile the equations derived herein, the Equivalent Earth equations, and the relation associated with a flat Earth constant curvature model produce relatively large height errors at low elevations and large ranges.Taylor series approximations of the new equations are examined. While a second-order Taylor series approximation for height performs well under “typical” propagation conditions, a convenient Taylor series approximation for great circle distance was not obtained.

A note on the estimation of variance for big BAF sampling

Background The double sampling method known as “big BAF sampling” has been advocated as a way to reduce sampling effort while still maintaining a reasonably precise estimate of volume. A well-known method for variance determination, Bruce’s method, is customarily used because the volume estimator takes the form of a product of random variables. However, the genesis of Bruce’s method is not known to most foresters who use the method in practice. Methods We establish that the Taylor series approximation known as the Delta method provides a plausible explanation for the origins of Bruce’s method. Simulations were conducted on two different tree populations to ascertain the similarities of the Delta method to the exact variance of a product. Additionally, two alternative estimators for the variance of individual tree volume-basal area ratios, which are part of the estimation process, were compared within the overall variance estimation procedure. Results The simulation results demonstrate that Bruce’s method provides a robust method for estimating the variance of inventories conducted with the big BAF method. The simulations also demonstrate that the variance of the mean volume-basal area ratios can be computed using either the usual sample variance of the mean or the ratio variance estimators with equal accuracy, which had not been shown previously for Big BAF sampling. Conclusions A plausible explanation for the origins of Bruce’s method has been set forth both historically and mathematically in the Delta Method. In most settings, there is evidently no practical difference between applying the exact variance of a product or the Delta method—either can be used. A caution is articulated concerning the aggregation of tree-wise attributes into point-wise summaries in order to test the correlation between the two as a possible indicator of the need for further covariance augmentation.

A Novel Iterative Method of Determining the Pose Error of Planar Clearance-Affected Flexible Parallel Mechanisms Under Loaded Mode

Clearance and flexibility play an essential role in determining the accuracy of a planar parallel mechanism. However, previous accuracy prediction methods either considered only one of them or combined them in linear superposition. Therefore, this study proposes a novel iterative method for determining the pose error by considering clearance and flexibility simultaneously. First, the rigid-flexible model of the mechanism with clearances is developed based on the virtual joint method, in which the equilibrium conditions under the external load are established via the virtual work principle and differential forward kinematics. Then, using a Taylor series approximation, the “instant” stiffness matrix corresponding to a specific load is deduced. On this basis, an iterative scheme is explored to search for the final equilibrium pose, in which a child iterative scheme is constructed to determine the joint variables and suffered wrench of the single chain given a pose. Finally, the developed method is demonstrated by calculating the comparative pose errors of the planar five-bar mechanism and 3-RPR robot.