Search Results (7)

In this paper, the distributed order calculus was used to establish a creep damage theoretical model to accurately describe the creep properties of viscoelastic materials. Firstly, the definition and basic properties in math of the distributed order calculus were given. On this basis, the mechanical elements of the distributed order damper were built to describe the viscoelastic properties. Then, the distributed order damper was introduced into the three-parameter solid model to establish the distributed order three-parameter solid model. The inverse Laplace transform derived the operator’s contour integrals and the path integrals along Hankel’s path. The integral properties were analysed. Next, the creep properties and relaxation characteristics of the distributed order three-parameter solid model were studied in detail. Finally, taking the rock materials as an example, the distributed order damage damper model was established. Its operator integrals were calculated, and the properties were discussed. Meanwhile, taking the integer-order Nishihara model as the standard, the distributed order damage creep combined model of the rock mass was constructed. The calculation examples were given to study the damage creep properties of the rock mass. Full article

This paper presents a rigorous solution of the Helmholtz equation for regular waveguide structures with the finite sizes of all cross-section elements that may have an arbitrary shape. The solution is based on the theory of Singular Integral Equations (SIE). The SIE method proposed here is used to find a solution to differential equations with a point source. This fundamental solution of the equations is then applied in an integral representation of the general solution for our boundary problem. The integral representation always satisfies the differential equations derived from the Maxwell’s ones and has unknown functions ${\mu }_e$ and ${\mu }_h$ that are determined by the implementation of appropriate boundary conditions. The waveguide structures under consideration may contain homogeneous isotropic materials such as dielectrics, semiconductors, metals, and so forth. The proposed algorithm based on the SIE method also allows us to compute waveguide structures containing materials with high losses. The proposed solution allows us to satisfy all boundary conditions on the contour separating materials with different constitutive parameters and the condition at infinity for open structures as well as the wave equation. In our solution, the longitudinal components of the electric and magnetic fields are expressed in the integral form with the kernel consisting of an unknown function ${\mu }_e$ or ${\mu }_h$ and the Hankel function of the second kind. It is important to note that the above-mentioned integral representation is transformed into the Cauchy type integrals with the density function ${\mu }_e$ or ${\mu }_h$ at certain singular points of the contour of integration. The properties and values of these integrals are known under certain conditions. Contours that limit different materials of waveguide elements are divided into small segments. The number of segments can determine the accuracy of the solution of a problem. We assume for simplicity that the unknown functions ${\mu }_e$ and ${\mu }_h$, which we are looking for, are located in the middle of each segment. After writing down the boundary conditions for the central point of every segment of all contours, we receive a well-conditioned algebraic system of linear equations, by solving which we will define functions ${\mu }_e$ and ${\mu }_h$ that correspond to these central points. Knowing the densities ${\mu }_e$, ${\mu }_h$, it is easy to calculate the dispersion characteristics of the structure as well as the electromagnetic (EM) field distributions inside and outside the structure. The comparison of our calculations by the SIE method with experimental data is also presented in this paper. Full article

While browsing through the famous book of Bierens de Haan, we came across a table with some very interesting integrals. These integrals also appeared in the book of Gradshteyn and Ryzhik. Derivation of these integrals are not listed in the current literature to best of our knowledge. The derivation of such integrals in the book of Gradshteyn and Ryzhik in terms of closed form solutions is pertinent. We evaluate several of these definite integrals of the form ${\int }_0^{\infty }\frac{{(a+y)}^k-{(a-y)}^k}{e^{by}-1}dy$, ${\int }_0^{\infty }\frac{{(a+y)}^k-{(a-y)}^k}{e^{by}+1}dy$, ${\int }_0^{\infty }\frac{{(a+y)}^k-{(a-y)}^k}{sinh\left(by\right)}dy$ and ${\int }_0^{\infty }\frac{{(a+y)}^k+{(a-y)}^k}{cosh\left(by\right)}dy$ in terms of a special function where k, a and b are arbitrary complex numbers. Full article

In this paper we study the time-fractional wave equation of order $1<\nu <2$ and give a probabilistic interpretation of its solution. In the case $0<\nu <1$ , $d=1$ , the solution can be interpreted as a time-changed Brownian motion, while for $1<\nu <2$ it coincides with the density of a symmetric stable process of order $2/\nu$ . We give here an interpretation of the fractional wave equation for $d>1$ in terms of laws of stable d−dimensional processes. We give a hint at the case of a fractional wave equation for $\nu >2$ and also at space-time fractional wave equations. Full article

The derivation of integrals in the table of Gradshteyn and Ryzhik in terms of closed form solutions is always of interest. We evaluate several of these definite integrals of the form ${\int }_0^{\infty }log(1\pm e^{-\alpha y})R(k,a,y)dy$ in terms of a special function, where $R(k,a,y)$ is a general function and k, a and $\alpha$ are arbitrary complex numbers, where $Re\left(\alpha \right)>0$ . Full article

We present a method using contour integration to evaluate the definite integral of the form ${\int }_0^{\infty }{log}^k\left(ay\right)R\left(y\right)dy$ in terms of special functions, where $R\left(y\right)=\frac{y^m}{1+\alpha y^n}$ and $k,m,a,\alpha$ and n are arbitrary complex numbers. We use this method for evaluation as well as to derive some interesting related material and check entries in tables of integrals. Full article

The purpose of this note is to provide an expository introduction to some more curious integral formulas and transformations involving generating functions. We seek to generalize these results and integral representations which effectively provide a mechanism for converting between a sequence’s ordinary and exponential generating function (OGF and EGF, respectively) and vice versa. The Laplace transform provides an integral formula for the EGF-to-OGF transformation, where the reverse OGF-to-EGF operation requires more careful integration techniques. We prove two variants of the OGF-to-EGF transformation integrals from the Hankel loop contour for the reciprocal gamma function and from Fourier series expansions of integral representations for the Hadamard product of two generating functions, respectively. We also suggest several generalizations of these integral formulas and provide new examples along the way. Full article