Analytical solution of a fractional differential equation in the theory of viscoelastic fluids

The aim of this paper is to present analytical solutions of fractional delay differential equations (FDDEs) of an incompressible generalized Oldroyd-B fluid with fractional derivatives of Caputo type. Using a modification of the method of separation of variables the main equation with non-homogeneous boundary conditions is transformed into an equation with homogeneous boundary conditions, and the resulting solutions are then expressed in terms of Green functions via Laplace transforms. This results presented in two condition, in first step when 0 ≤ α, β ≤ 1/2 and in the second step we considered 1/2 ≤ α, β ≤ 1, for each step 1,2 for the unsteady flows of a generalized Oldroyd-B fluid, including a flow with a moving plate, are considered via examples.

Cold emission optimization of a diesel- and alternative fuel-driven CI engine

This paper deals with the emission optimization of a compression ignition (CI) engine during cold ambient operation. Hence, in the present study, the effect of different injector nozzle geometries and pilot injection strategies, but also the influence of intake swirl, rail pressure, exhaust gas recirculation (EGR) as well as EGR cooling on the emission behavior during cold run are investigated. Therefore, test bed experiments under steady-state cold conditions are conducted on a state-of-the-art CI single cylinder research engine (SCRE) with approximately 0.5 l swept volume representing the typical passenger car (PC) cylinder size. The cold charge air temperature of down to −8 $$^{\circ }\hbox { C}$$ ∘ C and a low engine coolant and lube oil temperature represent a cold run close to reality. For emulating the higher friction of a typical 4-cylinder PC engine during cold run, the indicated mean effective pressure (IMEP) is increased according to a specifically developed equation and the turbocharger main equation is solved permanently to adjust the gas exchange loss. To take account of a potential future tightening of emission legislation, in addition to limited exhaust gas emissions, non-limited emissions such as carbonyls are measured as well. Since alternative fuels are able to make a significant contribution to the defossilisation of transportation, an oxygen-containing fuel, consisting of 100 % renewable blend components (HVO, ethers and alcohols) and fulfilling the EN 590 legislation is investigated under the same cold conditions in addition to the research on conventional diesel fuel.

Phase Change and Space Travel

Here we discuss the implications of phase change equations and what bearing they might have on interstellar space travel. The phase change equations are derived from either thermodynamics or statistical mechanics and have a similarity. Then, the main equation for limit of superheat is posited to be a solution to the problem of propulsion in space travel. There are two matter-antimatter systems considered: electron – positron and hydrogen – antihydrogen. What is involved in the space travel problem is harvesting of antimatter in magnetic bottles and keeping it separate.

Symmetry and Its Importance in the Oscillation of Solutions of Differential Equations

Oscillation and symmetry play an important role in many applications such as engineering, physics, medicine, and vibration in flight. The purpose of this article is to explore the oscillation of fourth-order differential equations with delay arguments. New Kamenev-type oscillatory properties are established, which are based on a suitable Riccati method to reduce the main equation into a first-order inequality. Our new results extend and simplify existing results in the previous studies. Examples are presented in order to clarify the main results.

On the existence of the resolvent and separability of a class of the Korteweg-de Vriese type linear singular operators

Partial differential equations of the third order are the basis of mathematical models of many phenomena and processes, such as the phenomenon of energy transfer of hydrolysis of adenosine triphosphate molecules along protein molecules in the form of solitary waves, i.e. solitons, the process of transferring soil moisture in the aeration zone, taking into account its movement against the moisture potential. In particular, this class includes the nonlinear Korteweg-de Vries equation, which is the main equation of modern mathematical physics. It is known that various problems have been studied for the Korteweg-de Vries equation and many fundamental results obtained. In this paper, issues about the existence of a resolvent and separability (maximum smoothness of solutions) of a class of linear singular operators of the Korteweg-de Vries type in the case of an unbounded domain with strongly increasing coefficients are investigated.

Initial value problem for fractional Volterra integrodifferential pseudo-parabolic equations

In this paper, we investigate the initial boundary value problem for the Caputo time-fractional pseudo-parabolic equations with fractional Laplace of order $ 0<\nu\le1 $ and the nonlinear memory source term. For $ 0<\nu<1 $, the Problem will be considered on a bounded domain of $ \R^d $. By some Sobolev embeddings and the properties of Mittag-Lefler function, we will give some results on the existence and the uniqueness of mild solution for the Problem \eqref{Main-Equation} below. When $ \nu=1 $, we will introduce some $ L^p-L^q $ estimates, and base on them we derive the global existence of a mild solution in the whole space $ \R^d. $

Non-Linear Neutral Differential Equations with Damping: Oscillation of Solutions

The oscillation of non-linear neutral equations contributes to many applications, such as torsional oscillations, which have been observed during earthquakes. These oscillations are generally caused by the asymmetry of the structures. The objective of this work is to establish new oscillation criteria for a class of nonlinear even-order differential equations with damping. We employ different approach based on using Riccati technique to reduce the main equation into a second order equation and then comparing with a second order equation whose oscillatory behavior is known. The new conditions complement several results in the literature. Furthermore, examining the validity of the proposed criteria has been demonstrated via particular examples.

Forecasting the parameters of building structures using regression equations. Part 1. Forecasting with independent action of variables and constancy of operating factors

The article presents a method for forecasting the parameters of building structures using regression equations. The most commonly used regression equations were used, namely, linear, polynomial, power, exponential, exponential, logarithmic, semi-logarithmic, hyperbolic and logistic ones. The authors propose to use one-factor regression equations in which the variable value is time, and the dependent value is the parameter of the building structure, the changes of which the researcher needs to determine. Thereafter, the authors present the basic equations as multi-factorial ones. This is achieved by means of replacing the coefficients of the main equation with regression equations which are obtained after carrying out a series of tests with variable values of the selected input parameters (such as environmental conditions, material of construction, etc.). Regression equations are derived for each state, and, as a result, a number of parameter values for the basic regression equations are available. Then, a repeated regression analysis is carried out and a regression equation is set up for the coefficients of the main regression equation, which depends on the value of the specified parameters. Such equations are called secondary equations. Examples are given for conditional linear regression, where it is demonstrated how the coefficients of the main regression equation are replaced and what final form the equation achieves after replacing the coefficients with secondary equations. The presented method allows, in the case of expression from the basic equation of the time parameter, to forecast the residual resource, with a large number of parameters of building structures determined.

ADA Solve the Cubic Equation in a New Method with Engineering Application

Up to date the cubic equation or matrix tensor is consisting of nine values ​​such as stress tensor that turns into the cubic equation which has been used for solving classic method. This is to impose an initial root several times to get it when achieves the equation and any other party is zero. Then dividing the cubic equation on the equation of the root. After that dividing the cubic equation on the equation of the root and using the classical method to find the rest of the roots. This is a very difficult issue, especially if the roots are secret or large for those who are looking in a difficult field or even for those who are in the examination room. In this research, two equations were reached, one that calculates the angle and the other that calculates the three roots at high accuracy without any significant error rate. By taking advantage of the traditional method, not by imposing a value to get the root of that equation, but by imposing an equation to get the solution equation that gives the value of that root. After imposing that equation, the general equation was derived from which that calculated the three roots directly and without any attempts. The angle that was implicitly derived during the derive of the main equation is calculated by taking advantage of the constants that do not change (invariants) for the matrix tensor (T).

Inverse spectral problem of an anharmonic oscillator on a half-axis with the Neumann boundary condition

An anharmonic oscillator {T(q)=-\frac{d^{2}}{dx^{2}}+x^{2}+q(x)} on the half-axis {0\leq x<\infty} with the Neumann boundary condition is considered. By means of transformation operators, the direct and inverse spectral problems are studied. We obtain the main integral equations of the inverse problem and prove that the main equation is uniquely solvable. An effective algorithm for reconstruction of perturbed potential is indicated.