New entropic bounds on time scales via Hermite interpolating polynomial

Hermite’s interpolation is utilized to establish a new generalization of an inequality for higher order convex functions containing Csiszár divergence on time scales. New entropic bounds in q-calculus and h-discrete calculus are also deduced. Some estimates for Zipf–Mandelbrot entropy are also given.

Direct integrator of block type methods with additional derivative for general third order initial value problems

This paper presents the construction of the two-point and three-point block methods with additional derivatives for directly solving [Formula: see text]. The proposed block methods are formulated using Hermite Interpolating Polynomial and approximate the solution of the problem at two or three-point concurrently. The block methods obtain the numerical solutions directly without reducing the equation into the first order system of initial value problems (IVPs). The order and zero-stability of the proposed methods are also investigated. Numerical results are presented and comparisons with other existing block methods are made. The performance shows that the proposed methods are very efficient in solving the general third order IVPs.

Implicit Two-Point Block Method for Solving Fourth-Order Initial Value Problem Directly with Application

This paper proposes an implicit block method with two-point to directly solve the fourth-order Initial Value Problems (IVPs). The implicit block method is derived by adopting Hermite interpolating polynomial as the basis function, incorporating the first derivative of ft,y,y′,y′′,y′′′ to enhance the solution’s accuracy. A block formulation is presented to acquire the numerical approximation at two points simultaneously. The introduced method’s basic properties, including order, zero stability, and convergence, are presented. Numerical experiments are carried out to verify the accuracy and efficiency of the proposed method compared with those of the several existing methods. Application in ship dynamics is also presented which yield impressive results for the proposed two-point block method.

Two Interpolation Methods Using Multiply-Rotated Piecewise Cubic Hermite Interpolating Polynomials

Two interpolation methods are presented, both of which use multiple Piecewise Cubic Hermite Interpolating Polynomials (PCHIPs). The first method is based on performing 16 PCHIPs on 8 rotated versions of the plot of the data versus an independent variable (such as pressure or time). These 16 PCHIPs are then used to form 8 interpolations of the original data, and finally, these 8 are averaged. When the original data are unevenly spaced with respect to the independent variable, we show that it is best to perform the Multiply-Rotated PCHIP (MR-PCHIP) method using the “data index” as the independent variable, and then to subsequently perform one last PCHIP of the data index with respect to the original independent variable. This MR-PCHIP method avoids the flat spots that are a feature of the PCHIP method when the data have multiple values approximately equal to a local extreme value. The MR-PCHIP interpolated data have continuous first derivatives at the data points. This method also avoids the unrealistic overshoots that can occur when using the standard cubic spline interpolation procedure. The second interpolation method is designed specifically for hydrographic data with the aim of minimizing the formation of unrealistic water masses by the interpolation procedure. This is achieved by applying a Piecewise Cubic Hermite Interpolating Polynomial to each of 8 rotations of the salinity versus temperature plot (Multiply-Rotated Salinity–Temperature PCHIP, MRST-PCHIP) with bottle number (that is, data index) as the vertical interpolating coordinate, thereby making the MRST-PCHIP method independent of the heave of a water column. This method is equivalent to interpolating in the salinity–temperature diagram, and MRST-PCHIP proves very effective at avoiding the production of anomalous water masses that otherwise occur when interpolating temperature and salinity separately.

RRT-A*-BT approach for optimal collision-free path planning for mobile robots

This paper deals with the problem of optimal collision-free path planning for mobile robots evolving inside indoor cluttered environments. Addressing this challenge, a hybrid approach is proposed combining Rapidly-exploring Random Trees (RRT), A-Star (A*) and Back-Tracking (BT) algorithms (RRT-A*-BT). Thus, a vision system is used for a nearly-exact modeling of the environment through image processing. Moreover, each iteration of the basic RRT approach is guided by A* algorithm while trying to take the shortest path linking the robot current position to target . In case of a blockage, BT algorithm is used to get out the robot from this situation. Finally, Piecewise Cubic Hermite Interpolating Polynomial (PCHIP) is used to smooth the generated optimal path. RRT-A*-BT approach is validated through different scenarios; obtained results are compared with previous works on same environments with same conditions. The results prove that RRT-A*-BT is better and faster than other algorithms of the literature, such as Genetic Algorithms and Conventional RRT, in terms of (i) computation time,(ii) path length and (iii) transfer time..

Bigeometric Cesàro difference sequence spaces and Hermite interpolation

In this paper, we introduce some difference sequence spaces in bigeometric calculus. We determine the [Formula: see text]-duals of these sequence spaces and study their matrix transformations. We also develop an interpolating polynomial in bigeometric calculus which is analogous to the classical Hermite interpolating polynomial.

Bearing Capacity Model of Corroded RC Eccentric Compression Columns Based on Hermite Interpolation and Fourier Fitting

With an extension in service years, bridges inevitably suffer from performance deterioration. Columns are the main components of bridge structures, which support the superstructure. The damage of pier columns is often more harmful to bridges than that of other components. To accurately evaluate the time-varying characteristics of corroded columns, this paper proposes a new model for the bearing capacity evaluation of deteriorated reinforced concrete (RC) eccentric compression columns based on the Hermite interpolation and Fourier function. Firstly, the axial compression point, the pure bending point and the balanced failure point were selected as the basic points, and the deteriorated strength of these basic points was calculated by considering factors such as concrete cracking, reduction of reinforcement area, buckling of the steel bar, bond slip and strength reduction of confined concrete. After that, the interpolation points were generated by a piecewise cubic Hermite interpolating polynomial, and the explicit expression of the interpolation points fitting function was realized by the trigonometric Fourier series model. Finally, comparison studies based on measured data from forty-five corroded RC eccentric compression columns were conducted to investigate the accuracy and efficiency of the proposed method. The results show that: (1) the prediction results for bearing capacity of corroded RC columns are in good agreement with the measured data, with the average ratio of predicted results to test results at 1.06 and the standard deviation at 0.14; (2) the proposed model unifies the three stress states of axial compression, eccentric compression and pure bending, and is consistent with the continuum mechanics characteristics; (3) the decrements of axial load carrying capacity for 10% and 50% of the corrosion rate are 31.4% and 45.2%, while in flexure they are 25.4% and 77.4%, respectively; and (4) the test data of small-scale specimens may overestimate the negative effect of corrosion on the bearing capacity of actual structures. The findings in this paper could lay a solid starting point for structural life prediction technologies based on nondestructive testing.