scholarly journals AN APPROXIMATE METHOD FOR CALCULATING GRAIN FLOW IN A VERTICAL CYLINDRICAL VIBROSIEVE

2021 ◽  
pp. 57-65
Author(s):  
Stanislav Olshanskiy ◽  
Maksym Slipchenko ◽  
Sergiy Kharchenko ◽  
Yurii Polievoda
Keyword(s):  
Differential Equation ◽  
Special Functions ◽  
Type Equation ◽  
Asymptotic Formulas ◽  
Practical Implementation ◽  
Radial Coordinate ◽  
Elementary Functions ◽  
Grain Flow ◽  
Theoretical Results ◽  
Approximate Formulas

A modified hydrodynamic model of a stable grain flow of an inhomogeneous mixture over the surface of a vertical cylindrical vibrating sieve is proposed under the assumption that the porosity of the mixture in the moving annular layer depends on the velocity of movement. A linear dependence of the porosity of the mixture on the velocity of movement are accepted, where higher speed corresponds to higher porosity.. The calculation of the velocity is reduced to solving an inhomogeneous differential equation of the Bessel type. Further, by "freezing" the variable coefficient of this equation, the problem has been simplified. This simplification is permissible due to the fact that the thickness of the moving layer of the mixture is much less than the radius of the vibrating sieve. As a result, the dependence of the velocity on the radial coordinate is expressed through the elementary functions. A compact formula for determining the maximum grain flow velocity is obtained. By integrating, in elementary functions, the formula for the average velocity in the layer is obtained. An approximate formula for the performance of the vibrating sieve by the mass of the exit fraction is derived. For this, it is proposed to calculate the corresponding integral approximately by the Simpson formula, so as not to calculate the values of special functions of large argument using the asymptotic formulas. It is shown that the named productivity significantly depends on the porosity of the grain mixture. In order to obtain information about the actual errors of the approximate formulas, we additionally carried out the numerical integration of the original non-simplified Bessel-type equation on a computer. A comparative analysis of the calculation results confirmed the small errors of the simplifications introduced into the equation of motion, as well as the adequacy of the theoretical results obtained. By passing to a simplified differential equation, approximate formulas were derived and tested for calculating the main characteristics of the grain flow along a vertical cylindrical vibrating sieve, taking into account the change in porosity in the grain mixture layer from the velocity of movement. The work summarizes the known theoretical results obtained using hydrodynamic models of the motion of grain mixtures fluidized by vibrations. The generalization carried out slightly complicated the theory, because the final calculation formulas are quite compact and convenient in practical implementation.

scholarly journals Computation of Stray Losses in Transformer Bushing Regions Considering Harmonics in the Load Current

2020 ◽  
Vol 10 (10) ◽  
pp. 3527
Author(s):  
Sohail Khan ◽  
Serguei Maximov ◽  
Rafael Escarela-Perez ◽  
Juan Carlos Olivares-Galvan ◽  
Enrique Melgoza-Vazquez ◽  
...  
Keyword(s):  
Differential Equation ◽  
Finite Element ◽  
Electromagnetic Field ◽  
Boundary Conditions ◽  
Eddy Current ◽  
Special Functions ◽  
Separation Of Variables ◽  
Practical Implementation ◽  
Load Current ◽  
Good Agreement

The presence of harmonics in the load current considerably increases stray losses in electric transformers. In this research paper, a new model for computing the electromagnetic field (EMF) and eddy current (EC) losses in transformer tank covers is derived considering harmonics. Maxwell’s equations are solved with their corresponding boundary conditions. The differential equation thus obtained is solved using the method of separation of variables. The obtained expressions do not require the use of special functions, accommodating them for practical implementation in the industry. The obtained formulas are evaluated for different spectrum contents of the load current and losses. The results are in good agreement with simulations carried out using the Altair Flux finite element (FE) software.

scholarly journals Simple Equations Method and Non-Linear Differential Equations with Non-Polynomial Non-Linearity

Entropy ◽  
2021 ◽  
Vol 23 (12) ◽  
pp. 1624
Author(s):  
Nikolay K. Vitanov ◽  
Zlatinka I. Dimitrova
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Special Functions ◽  
Linear Differential Equations ◽  
Elementary Functions ◽  
Composite Function ◽  
Linear Algebraic Equations ◽  
Non Linear ◽  
Linear Differential ◽  
Simple Equations

We discuss the application of the Simple Equations Method (SEsM) for obtaining exact solutions of non-linear differential equations to several cases of equations containing non-polynomial non-linearity. The main idea of the study is to use an appropriate transformation at Step (1.) of SEsM. This transformation has to convert the non-polynomial non- linearity to polynomial non-linearity. Then, an appropriate solution is constructed. This solution is a composite function of solutions of more simple equations. The application of the solution reduces the differential equation to a system of non-linear algebraic equations. We list 10 possible appropriate transformations. Two examples for the application of the methodology are presented. In the first example, we obtain kink and anti- kink solutions of the solved equation. The second example illustrates another point of the study. The point is as follows. In some cases, the simple equations used in SEsM do not have solutions expressed by elementary functions or by the frequently used special functions. In such cases, we can use a special function, which is the solution of an appropriate ordinary differential equation, containing polynomial non-linearity. Specific cases of the use of this function are presented in the second example.

Asymptotic behavior of solutions of half-linear differential equations and generalized Karamata functions

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Kusano Takaŝi ◽  
Jelena V. Manojlović
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Continuous Function ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Regular Variation ◽  
Asymptotic Formulas ◽  
Asymptotic Behavior Of Solutions ◽  
Positive Continuous Function ◽  
Linear Differential

AbstractWe study the asymptotic behavior of eventually positive solutions of the second-order half-linear differential equation(p(t)\lvert x^{\prime}\rvert^{\alpha}\operatorname{sgn}x^{\prime})^{\prime}+q(% t)\lvert x\rvert^{\alpha}\operatorname{sgn}x=0,where q is a continuous function which may take both positive and negative values in any neighborhood of infinity and p is a positive continuous function satisfying one of the conditions\int_{a}^{\infty}\frac{ds}{p(s)^{1/\alpha}}=\infty\quad\text{or}\quad\int_{a}^% {\infty}\frac{ds}{p(s)^{1/\alpha}}<\infty.The asymptotic formulas for generalized regularly varying solutions are established using the Karamata theory of regular variation.

scholarly journals A Convergent Collocation Approach for Generalized Fractional Integro-Differential Equations Using Jacobi Poly-Fractonomials

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 979
Author(s):  
Sandeep Kumar ◽  
Rajesh K. Pandey ◽  
H. M. Srivastava ◽  
G. N. Singh
Keyword(s):  
Differential Equation ◽  
Fractional Derivative ◽  
Collocation Method ◽  
Caputo Fractional Derivative ◽  
Fractional Derivatives ◽  
Special Cases ◽  
Collocation Approach ◽  
Linear And Nonlinear ◽  
Simulation Results ◽  
Theoretical Results

In this paper, we present a convergent collocation method with which to find the numerical solution of a generalized fractional integro-differential equation (GFIDE). The presented approach is based on the collocation method using Jacobi poly-fractonomials. The GFIDE is defined in terms of the B-operator introduced recently, and it reduces to Caputo fractional derivative and other fractional derivatives in special cases. The convergence and error analysis of the proposed method are also established. Linear and nonlinear cases of the considered GFIDEs are numerically solved and simulation results are presented to validate the theoretical results.

Applications of Erdélyi-Kober fractional integral for solving time-fractional Tricomi-Keldysh type equation

2020 ◽  
Vol 23 (5) ◽  
pp. 1381-1400 ◽  
Author(s):  
Kangqun Zhang
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Integral Operator ◽  
Nonlinear Equations ◽  
Existence And Uniqueness ◽  
Fractional Integral ◽  
Formal Solution ◽  
Type Equation ◽  
Multiplier Theorem ◽  
Problem Of Time

Abstract In this paper we consider Cauchy problem of time-fractional Tricomi-Keldysh type equation. Based on the theory of a Erdélyi-Kober fractional integral operator, the formal solution of the inhomogeneous differential equation involving hyper-Bessel operator is presented with Mittag-Leffler function, then nonlinear equations are considered by applying Gronwall-type inequalities. At last, we establish the existence and uniqueness of L p -solution of time-fractional Tricomi-Keldysh type equation by use of Mikhlin multiplier theorem.

scholarly journals Exponential Stability of Stochastic Differential Equation with Mixed Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Wenli Zhu ◽  
Jiexiang Huang ◽  
Xinfeng Ruan ◽  
Zhao Zhao
Keyword(s):  
Differential Equation ◽  
Time Delay ◽  
Stochastic Differential Equation ◽  
Exponential Stability ◽  
Sufficient Conditions ◽  
Lyapunov Stability Theory ◽  
Martingale Inequality ◽  
Mixed Delay ◽  
Exponentially Stable ◽  
Theoretical Results

This paper focuses on a class of stochastic differential equations with mixed delay based on Lyapunov stability theory, Itô formula, stochastic analysis, and inequality technique. A sufficient condition for existence and uniqueness of the adapted solution to such systems is established by employing fixed point theorem. Some sufficient conditions of exponential stability and corollaries for such systems are obtained by using Lyapunov function. By utilizing Doob’s martingale inequality and Borel-Cantelli lemma, it is shown that the exponentially stable in the mean square of such systems implies the almost surely exponentially stable. In particular, our theoretical results show that if stochastic differential equation is exponentially stable and the time delay is sufficiently small, then the corresponding stochastic differential equation with mixed delay will remain exponentially stable. Moreover, time delay upper limit is solved by using our theoretical results when the system is exponentially stable, and they are more easily verified and applied in practice.

Unification proposals for fatigue crack propagation laws

2017 ◽  
Vol 13 (2) ◽  
pp. 262-283 ◽  
Author(s):  
Vladimir Kobelev
Keyword(s):  
Differential Equation ◽  
Crack Propagation ◽  
Crack Length ◽  
Closed Form ◽  
Crack Growth ◽  
Stress Ratio ◽  
Elementary Functions ◽  
Content Type ◽  
Number Of Cycles ◽  
Cycles To Failure

Purpose The purpose of this paper is to propose the new dependences of cycles to failure for a given initial crack length upon the stress amplitude in the linear fracture approach. The anticipated unified propagation function describes the infinitesimal crack-length growths per increasing number of load cycles, supposing that the load ratio remains constant over the load history. Two unification functions with different number of fitting parameters are proposed. On one hand, the closed-form analytical solutions facilitate the universal fitting of the constants of the fatigue law over all stages of fatigue. On the other hand, the closed-form solution eases the application of the fatigue law, because the solution of nonlinear differential equation turns out to be dispensable. The main advantage of the proposed functions is the possibility of having closed-form analytical solutions for the unified crack growth law. Moreover, the mean stress dependence is the immediate consequence of the proposed law. The corresponding formulas for crack length over the number of cycles are derived. Design/methodology/approach In this paper, the method of representation of crack propagation functions through appropriate elementary functions is employed. The choice of the elementary functions is motivated by the phenomenological data and covers a broad region of possible parameters. With the introduced crack propagation functions, differential equations describing the crack propagation are solved rigorously. Findings The resulting closed-form solutions allow the evaluation of crack propagation histories on one hand, and the effects of stress ratio on crack propagation on the other hand. The explicit formulas for crack length over the number of cycles are derived. Research limitations/implications In this paper, linear fracture mechanics approach is assumed. Practical implications Shortening of evaluation time for fatigue crack growth. Simplification of the computer codes due to the elimination of solution of differential equation. Standardization of experiments for crack growth. Originality/value This paper introduces the closed-form analytical expression for crack length over number of cycles. The new function that expresses the damage growth per cycle is also introduced. This function allows closed-form analytical solution for crack length. The solution expresses the number of cycles to failure as the function of the initial size of the crack and eliminates the solution of the nonlinear ordinary differential equation of the first order. The different common expressions, which account for the influence of the stress ratio, are immediately applicable.

scholarly journals Self-Inductance of the Circular Coils of the Rectangular Cross-Section with the Radial and Azimuthal Current Densities

Physics ◽  
2020 ◽  
Vol 2 (3) ◽  
pp. 352-367
Author(s):  
Slobodan Babic ◽  
Cevdet Akyel
Keyword(s):  
Cross Sections ◽  
Special Functions ◽  
Rectangular Cross Section ◽  
The Self ◽  
Thin Wall ◽  
Elementary Functions ◽  
Special Cases ◽  
Current Densities ◽  
New Formula ◽  
Azimuthal Current

In this paper, we give new formulas for calculating the self-inductance for circular coils of the rectangular cross-sections with the radial and the azimuthal current densities. These formulas are given by the single integration of the elementary functions which are integrable on the interval of the integration. From these new expressions, we can obtain the special cases for the self-inductance of the thin-disk pancake and the thin-wall solenoids that confirm the validity of this approach. For the asymptotic cases, the new formula for the self-inductance of the thin-wall solenoid is obtained for the first time in the literature. In this paper, we do not use special functions such as the elliptical integrals of the first, second and third kind, nor Struve and Bessel functions because that is very tedious work. The results of this work are compared with already different known methods and all results are in excellent agreement. We consider this approach novel because of its simplicity in the self-inductance calculation of the previously-mentioned configurations.

scholarly journals Limiting Values and Functional and Difference Equations

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 407 ◽  
Author(s):  
N.-L. Wang ◽  
Praveen Agarwal ◽  
S. Kanemitsu
Keyword(s):  
Difference Equations ◽  
Functional Equations ◽  
Boundary Behavior ◽  
Special Functions ◽  
Asymptotic Formulas ◽  
Limit Values ◽  
Summatory Function ◽  
The Difference ◽  
Limiting Values ◽  
Mathematics And Science

Boundary behavior of a given important function or its limit values are essential in the whole spectrum of mathematics and science. We consider some tractable cases of limit values in which either a difference of two ingredients or a difference equation is used coupled with the relevant functional equations to give rise to unexpected results. As main results, this involves the expression for the Laurent coefficients including the residue, the Kronecker limit formulas and higher order coefficients as well as the difference formed to cancel the inaccessible part, typically the Clausen functions. We establish these by the relation between bases of the Kubert space of functions. Then these expressions are equated with other expressions in terms of special functions introduced by some difference equations, giving rise to analogues of the Lerch-Chowla-Selberg formula. We also state Abelian results which not only yield asymptotic formulas for weighted summatory function from that for the original summatory function but assures the existence of the limit expression for Laurent coefficients.

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