Calculation of the stress state of a noncircular helicoidal tube based on the tensor theory of shells

The purpose of the study was to develop an algorithm for calculating helical-symmetric shells with a closed contour in oblique Gaussian coordinates. The twist and length of the shell were taken unchanged. The method is based on the representation of the generating contour of the helicoidal surface by a discrete set of points with the replacement of differentiation along the angular coordinate by finite differences. The unknown were the displacement vectors at the indicated points of the contour. Due to the helicoidal symmetry, the differentiation of vector quantities with respect to the helical coordinate was replaced by vector multiplication. The tensor of deformations and the tensor of the parameters of the change in curvature were calculated using the nabla operator, represented in oblique Gaussian coordinates. Integration over the contour coordinate was replaced by summation over discrete points. The tensors found, which characterize the deformed state, were used to calculate the strain energy of one period of the helicoidal shell, and then the total potential of the mechanical system was compiled. The unknown displacements were determined by minimizing the total potential, taking into account the constraints that prohibit the displacement of the shell as a rigid whole. The study gives a numerical example of the application of the developed approach.

Spherical One-Way Wave Equation

The coordinate-free one-way wave equation is transferred in spherical coordinates. Therefore it is necessary to achieve consistency between gradient, divergence and Laplace operators and to establish, beside the conventional radial Nabla operator ∂Φ/∂r, a new variant ∂rΦ/r∂r. The two Nabla operator variants differ in the near field term Φ/r whereas in the far field r≫0 there is asymptotic approximation. Surprisingly, the more complicated gradient ∂rΦ/r∂r results in unexpected simplifications for – and only for – spherical waves with the 1/r amplitude decrease. Thus the calculation always remains elementary without the wattless imaginary near fields, and the spherical Bessel functions are obsolete.

Pharmacokinetics and Pharmacodynamics Models of Tumor Growth and Anticancer Effects in Discrete Time

We study the h-discrete and h-discrete fractional representation of a pharmacokinetics-pharmacodynamics (PK-PD) model describing tumor growth and anticancer effects in continuous time considering a time scale h𝕅0, where h > 0. Since the measurements of the drug concentration in plasma were taken hourly, we consider h = 1/24 and obtain the model in discrete time (i.e. hourly). We then continue with fractionalizing the h-discrete nabla operator in the h-discrete model to obtain the model as a system of nabla h-fractional difference equations. In order to solve the fractional h-discrete system analytically we state and prove some theorems in the theory of discrete fractional calculus. After estimating and getting confidence intervals of the model parameters, we compare residual squared sum values of the models in one table. Our study shows that the new introduced models provide fitting as good as the existing models in continuous time.

A study on discrete and discrete fractional pharmacokinetics-pharmacodynamics models for tumor growth and anti-cancer effects

We study the discrete and discrete fractional representation of a pharmacokinetics - pharmacodynamics (PK-PD) model describing tumor growth and anti-cancer effects in continuous time considering a time scale $h\mathbb{N}_0^h$, where h > 0. Since the measurements of the tumor volume in mice were taken daily, we consider h = 1 and obtain the model in discrete time (i.e. daily). We then continue with fractionalizing the discrete nabla operator to obtain the model as a system of nabla fractional difference equations. The nabla fractional difference operator is considered in the sense of Riemann-Liouville definition of the fractional derivative. In order to solve the fractional discrete system analytically we state and prove some theorems in the theory of discrete fractional calculus. For the data fitting purpose, we use a new developed method which is known as an improved version of the partial sum method to estimate the parameters for discrete and discrete fractional models. Sensitivity analysis is conducted to incorporate uncertainty/noise into the model. We employ both frequentist approach and Bayesian method to construct 90 percent confidence intervals for the parameters. Lastly, for the purpose of practicality, we test the discrete models for their efficiency and illustrate their current limitations for application.

On discrete fractional solutions of the hydrogen atom type equations

Discrete fractional calculus deals with sums and differences of arbitrary orders. In this study, we acquire new discrete fractional solutions of hydrogen atom type equations by using discrete fractional nabla operator ??(0 < ? < 1). This operator is applied homogeneous and non-homogeneous hydrogen atom type equations. So, we obtain many particular solutions of these equations.

On Discrete Fractional Solutions of Non-Fuchsian Differential Equations

In this article, we obtain new fractional solutions of the general class of non-Fuchsian differential equations by using discrete fractional nabla operator ∇ η ( 0 < η < 1 ) . This operator is applied to homogeneous and nonhomogeneous linear ordinary differential equations. Thus, we obtain new solutions in fractional forms by a newly developed method.