Integration Constants Research Articles

One- and Two-Dimensional Analytical Solutions of Thermal Stress for Bimodular Functionally Graded Beams under Arbitrary Temperature Rise Modes

In this study, we analytically solved the thermal stress problem of a bimodular functionally graded bending beam under arbitrary temperature rise modes. First, based on the strain suppression method in a one-dimensional case, we obtained the thermal stress of a bimodular functionally graded beam subjected to bending moment under arbitrary temperature rise modes. Using the stress function method based on compatibility conditions, we also derived two-dimensional thermoelasticity solutions for the same problem under pure bending and lateral-force bending, respectively. During the solving, the number of unknown integration constants is doubled due to the introduction of bimodular effect; thus, the determination for these constants depends not only on the boundary conditions, but also on the continuity conditions at the neutral layer. The comparisons indicate that the one- and two-dimensional thermal stress solutions are consistent in essence, with a slight difference in the axial stress, which exactly reflects the distinctions of one- and two-dimensional problems. In addition, the temperature rise modes in this study are not explicitly indicated, which further expands the applicability of the solutions obtained. The originality of this work is that the one- and two-dimensional thermal stress solutions for bimodular functionally graded beams are derived for the first time. The results obtained in this study may serve as a theoretical reference for the analysis and design of beam-like structures with obvious bimodular functionally graded properties in a thermal environment.

A Novel Approach to the Integration for Generating Consistent Ground Acceleration, Velocity and Displacement Time Histories

The velocity and the displacement time history obtained by directly integrating the acceleration time history will drift unrealistically because of the over-determinacy of the integration constants. Applying baseline corrections to resolve drift disturbs the frequency content and renders the corrected processes mutually inconsistent. In this study, eigenfunctions of sixth-order eigenvalue problem are introduced as basis functions for decomposing recorded ground motion time history. Taking advantage of the eigenfunctions and their first two order of differentiations are drift-free and consistent, the decomposed and reconstructed acceleration, velocity, and displacement time histories can be drift-free and mutually consistent without direct integration and baseline correction. Two earthquake ground motions are presented as examples, which show that the decomposed acceleration, velocity, and displacement time histories are drift-free, mutually consistent, and almost the same as the seed motion. The new approach can replace integral in the generation of velocity and displacement time histories.

Multi-Parameter Reaction–Diffusion Systems with Quadratic Nonlinearity and Delays: New Exact Solutions in Elementary Functions

The study considers a nonlinear multi-parameter reaction–diffusion system of two Lotka–Volterra-type equations with several delays. It treats both cases of different diffusion coefficients and identical diffusion coefficients. The study describes a few different techniques to solve the system of interest, including (i) reduction to a single second-order linear ODE without delay, (ii) reduction to a system of three second-order ODEs without delay, (iii) reduction to a system of three first-order ODEs with delay, (iv) reduction to a system of two second-order ODEs without delay and a linear Schrödinger-type PDE, and (v) reduction to a system of two first-order ODEs with delay and a linear heat-type PDE. The study presents many new exact solutions to a Lotka–Volterra-type reaction–diffusion system with several arbitrary delay times, including over 50 solutions in terms of elementary functions. All of these are generalized or incomplete separable solutions that involve several free parameters (constants of integration). A special case is studied where a solution contains infinitely many free parameters. Along with that, some new exact solutions are obtained for a simpler nonlinear reaction–diffusion system of PDEs without delays that represents a special case of the original multi-parameter delay system. Several generalizations to systems with variable coefficients, systems with more complex nonlinearities, and hyperbolic type systems with delay are discussed. The solutions obtained can be used to model delay processes in biology, ecology, biochemistry and medicine and test approximate analytical and numerical methods for reaction–diffusion and other nonlinear PDEs with delays.

Estimation of the time for steam generator trip due to cyber intrusions

The time required to trip a pressurized water reactor (PWR) by inserting malicious signals into its steam generator (SG) control system has been studied using the Generic PWR (GPWR) Simulator. A semi-analytical model is developed to approximately reproduce the simulator response and understand the dynamics of the control unit. A series of two proportional-integral controllers determines control action according to preset constants, the readings from the feedwater level sensor, and those from feedwater and steam flowrate transmitters. It is observed that the most important factor that determines whether a trip will occur is how much additional water is added to or withheld from the SG over time compared to normal operating conditions. In order to determine the effects of control action on the SG, changes in mass inventory are considered. This approach models the SG water level as a function of mass inventory and has a backward temporal memory. A Python interface is developed for the GPWR framework to automatically simulate different spoofing scenarios and post-process the related data. We observe that the trip times predominantly depend on flow mismatch and/or level errors. Controller parameters, including the integral time and gain constants, either speed up or slow down the rate of progression to a trip setpoint but do not cause a trip by themselves. The reactor can trip on a high-level signal when the reading crosses above 78%, increased from its reference level of 57%, or a low-level reading when it is below 25%. The present results show roughly how long the operators would have to respond to an attack, given a specific set of spoofing signals within the issue space analyzed. We have generated a simple surface by fitting a combination of exponential functions to the data obtained from the GPWR Simulator. In general, trips on a low level have been observed to occur faster than those on a high level.

Extracting the exact solitons of time-fractional three coupled nonlinear Maccari’s system with complex form via four different methods

This current research is considered some new exact soliton solutions to the time-fractional three coupled non-linear Maccari’s system in complex form with novel truncated M-fractional derivative. The obtained results may be used in the description of the model in fruitful way. The novel derivative operator is applied to study the aforementioned model. In addition, the achieved results are included in the form of dark, bright and combo optical solitons. The achieved solutions are also verified by using the MATHEMATICA software. Modified integration methods, namely, the Kudryashov method, the Expa function method, the extended (G′/G)−expansion method and the extended ShGEEM are applied to achieve the results. Obtained solutions are remodeled in several hyperbolic and trigonometric forms based on different restrictions between parameters involved in equations and integration constants that appear in the solutions. A few significant ones among the reported solutions are pictured to perceive the physical utility and peculiarity of the considered model using mathematical software. These optical solitons suggest that these three methods are useful, easy to use and effective than other methods.

The Walsh series discretization method for free vibration analysis of composite spherical shells based on the shear deformation theory

A Walsh series Discretization (WSD) method-based solution approach is presented to conduct free vibration analysis of composite spherical shells with general boundary conditions in this paper. The first-order shear deformation theory considering the shear effect and rotary inertia is used to establish the theoretical model. Walsh series and Fourier series are chosen as the basis functions of the meridional and circumferential directions of the shell, and the integral constants generated in this process are determined by the boundary conditions. As an innovation of this model, weight parameters are introduced to obtain general boundary conditions. By using the separation of variables, the complex multivariable coupled equations can be transformed into simple linear algebraic equations of single variable. On this basis, the frequency parameters and model diagram of composite laminated spherical shell can be easily obtained by solving algebraic equations. Compared with the existing literature, the numerical results show that this method has good convergence and accuracy. On this basis, the influence of geometric parameters and material parameters on the free vibration characteristics of composite laminated spherical shell is also discussed. The relevant numerical results can be used as the basic data for designers.

Nonconservative Unimodular Gravity: Gravitational Waves

Unimodular gravity is characterized by an extra condition with respect to general relativity, i.e., the determinant of the metric is constant. This extra condition leads to a more restricted class of invariance by coordinate transformation: The symmetry properties of unimodular gravity are governed by the transverse diffeomorphisms. Nevertheless, if the conservation of the energy–momentum tensor is imposed in unimodular gravity, the general relativity theory is recovered with an additional integration constant which is associated to the cosmological term Λ. However, if the energy–momentum tensor is not conserved separately, a new geometric structure appears with potentially observational signatures. In this text, we consider the evolution of gravitational waves in a nonconservative unimodular gravity, showing how it differs from the usual signatures in the standard model. As our main result, we verify that gravitational waves in the nonconservative version of unimodular gravity are strongly amplified during the evolution of the universe.

ELLIPTIC ASYMPTOTIC REPRESENTATION OF THE FIFTH PAINLEVÉ TRANSCENDENTS

For the fifth Painlevé transcendents, an asymptotic representation by the Jacobi sn-function is presented in cheese-like strips along generic directions near the point at infinity. Its elliptic main part depends on a single integration constant, which is the phase shift and is parametrized by monodromy data for the associated isomonodromy deformation. In addition, under a certain supposition, the error term is also expressed by an explicit asymptotic formula, whose leading term is written in terms of integrals of the sn-function and the ϑ-function, and contains the other integration constant. Instead of the justification scheme for asymptotic solutions of Riemann-Hilbert problems by the Brouwer fixed point theorem, we begin with a boundedness property of a Lagrangian function, which enables us to determine the modulus of the sn-function satisfying the Boutroux equations and to construct deductively the elliptic representation.

Ajuste e parâmetros genéticos da curva de crescimento em bovinos da raça Nelore Mocho

ABSTRACT This study was undertaken to compare different non-linear models for fitting growth curves of Polled Nellore animals as well as to estimate genetic parameters for the components of the growth curve. The study involved body weight-age data of 6,717 Polled Nellore cattle from birth to 650 days of age, which belonged to the Brazilian Association of Zebu Breeders (ABCZ), corresponding to the period from 1980 to 2011. Four non-linear models (Brody, Bertalanffy, Logistic, and Gompertz) were fitted and compared by the adjusted coefficient of determination (R2adj), mean absolute deviation of residuals (MAD), root mean square error (RMSE), Akaike information criterion (AIC), and Bayesian information criterion (BIC). To estimate the genetic parameters and genetic values of asymptotic weight (A), integration constant (B), and maturation rate (K), the Bayesian inference method was adopted. The Brody model showed the lowest values of MAD, RMSE, AIC, and BIC and the highest R2adj. Heritability estimates for parameters A, B, and K were 0.11, 0.16, and 0.30, respectively, whereas genetic correlations were 0.01 (A-B), -0.91 (A-K), and 0.24 (B-K). The Brody model provided the best fit. The K parameter shows enough genetic variability for selection in the herd. Heavier animals in adulthood tend to exhibit lower growth rates. Despite the low heritability estimate of parameter A, there were genetic gains, indicating that selection is being efficient on asymptotic weight.

Rational solutions to the Painlevé II equation from particular polynomials

The Painlevé equations were derived by Painlevé and Gambier in 1895-1910. Given a rational function R in w, w′ and analytic in z, they searched what were the second order ordinary differential equations of the form w″ = R(z, w, w′) with the properties that the singularities other than poles of any solution or this equation depend on the equation only and not of the constants of integration. They proved that there are 50 equations of this type, and the Painlevé II is one of these. Here, we construct solutions to the Painlevé II equation (PII) from particular polynomials. We obtain rational solutions written as a derivative with respect to the variable x of a logarithm of a quotient of a determinant of order n + 1 by a determinant of order n. We obtain an infinite hierarchy of rational solutions to the PII equation. We give explicitly the expressions of these solutions solution for the first orders.

White dwarfs with rapid rotation

A new analytical approach for calculation of white dwarfs characteristics that accounts for two important competing factors — axial rotation and Coulomb interparticle interactions, is proposed. The feature of our approach is simultaneous usage of differential and integral forms of equilibrium equation. In dimensionless form the differential equilibrium equation is strongly nonlinear inhomogeneous equation of the second order in partial derivatives with two dimensionless parameters — the relativistic parameter in stellar center $x_0$ and dimensionless angular velocity $\Omega$. In inner stellar region, rotation is taken into account as perturbation in the linear approximation for $\Omega^2$. In stellar periphery rotation is considered as the main factor. Usage of the integral equation provides correct calculations of integration constants. Dwarf's mass, moment of inertia relative to the axis of rotation, equatorial and polar radii, equatorial gravity in the following parameter space $1\leq x_0\leq24$, $0\leq\Omega<\Omega_{\rm max}(x_0)$ have been calculated based on the solutions of equilibrium equation. For the first time it was calculated the total energy of dwarf as function of these parameters. By the extrapolation, it was calculated the maximal values $\Omega_{\rm max}(x_0)$, as well as the observed angular velocity $\omega_{\rm max}(x_0)$. The considered model is generalized by taking into account Coulomb interparticle interactions. Also, we provide the examples of application of obtained results. It was shown that the characteristics of observed massive dwarfs do not contradict the calculated values for the model with consideration of solid body rotation and Coulomb interparticle interactions.

Generic Three-Parameter Wormhole Solution in Einstein-Scalar Field Theory

An exact analytical, spherically symmetric, three-parametric wormhole solution has been found in the Einstein-scalar field theory, which covers the several well-known wormhole solutions. It is assumed that the scalar field is massless and depends on the radial coordinate only. The relation between the full contraction of the Ricci tensor and Ricci scalar has been found as RαβRαβ=R2. The derivation of the Einstein field equations have been explicitly shown, and the exact analytical solution has been found in terms of the three constants of integration. The several wormhole solutions have been extracted for the specific values of the parameters. In order to explore the physical meaning of the integration constants, the solution has been compared with the previously obtained results. The curvature scalar has been determined for all particular solutions. Finally, it is shown that the general solution describes naked singularity characterized by the mass, the scalar quantity and the throat.

Microscopic origin of Einstein's field equations and the raison d'être for a positive cosmological constant

In the paradigm of effective field theory, one hierarchically obtains the effective action Aeff[q,⋯] for some low(er) energy degrees of freedom q, by integrating out the high(er) energy degrees of freedom ξ, in a path integral, based on an action A[q,ξ,⋯]. We show how one can integrate out a vector field va in an action A[Γ,v,⋯] and obtain an effective action Aeff[Γ,⋯] which, on variation with respect to the connection Γ, leads to the Einstein's field equations and a metric compatible with the connection. The derivation predicts a non-zero, positive, cosmological constant, which arises as an integration constant. The Euclidean action A[Γ,v,⋯], has an interpretation as the heat density of null surfaces, when translated into the Lorentzian spacetime. The vector field va can be interpreted as the Euclidean analogue of the microscopic degrees of freedom hosted by any null surface. Several implications of this approach are discussed.

The Λ and the CDM as Integration Constants

Notoriously, the two main problems of the standard ΛCDM model of cosmology are the cosmological constant Λ and the cold dark matter, CDM. This essay shows that both the Λ and the CDM arise as integration constants in a careful derivation of Einstein’s equations from first principles in a Lorentz gauge theory. The dark sector of the universe might only reflect the geometry of a spontaneous symmetry breaking that is necessary for the existence of spacetime and an observer therein.

Coupled approximate long and short solutions versus exact Navier and Galerkin ones for cylindrical shell under radial load

Cylindrical shell under concentrated force is a fundamental problem, which provides the understanding of general patterns of deformation and allow to consider many other types of loading and geometries. The common idea of existing approaches is expansion of the looking for solutions into Fourier series with respect to nφ, where φ is a circumferential coordinate, and n is the circumferential wave number. This reduces the problem to the 8th order even differential equation as to axial coordinate. Yet the finding of relevant 8 eigenfunctions and exact relations of 8 constants of integration with boundary conditions are still beyond the possibilities of analytical treatment. Most analytical solutions are based on Vlasov-like solution of simplified 4th order equations with unknown ranges of their accuracies. The goal of paper is threefold. First, to give the exact solutions for infinite shell, which are based on: (a) the well-known Navier approach, and (b) the Galerkin methods with application of decaying exponential functions. Second, to modernize the hypotheses of Vlasov’s theory (long-wave axial solution) and supplement it with the short solution, and commonly use both in formulation of boundary conditions, i.e., to take into account that their interaction occurs not due to shear force only but other force and displacement parameters may interact too. Third, to perform the tedious comparison of exact Galerkin and Navier solutions with approximate ones for the whole range of circumferential waves n, assess their accuracy and establish the accuracy of plate-like solution for the larger number of n.

Exact closed-form solution for free vibration of Euler-Bernoulli and Timoshenko beams with intermediate elastic supports

This paper presents exact closed-form solutions for free vibration of discretely supported Euler-Bernoulli (DEB) and Timoshenko beams (DTB) in the presence of an arbitrary number of intermediate elastic constraints. The exact eigenvalue equations and mode shapes of the DEB and DTB are derived using the generalized function method for the first time, which expands the general solution of mode shapes as a combination of the standard trigonometric/hyperbolic functions with integration constants extended to generalized functions. The second-order eigenvalue equation is formulated from the perspective of the entire domain of the beam without enforcement of any continuity conditions. Several typical constraints of interest in practical engineering are investigated and other cases can be achieved by following similar methodologies elucidated in this paper. The accuracy of present solutions is validated by numerical results obtained from the finite element method (FEM) and previous literature adopting the transfer matrix method (TMM) and Green's function method (GFM). The exiting frequency equation and mode shape of a beam with elastic end constraints are proved to be the degenerate solutions of current study. Courant's maximum-minimum principle is reproduced with detailed discussions on the support stiffnesses and positions, bimodal behavior, and sensitivity to geometrical variations. As an extended application, a new method based on rigorous mathematical deduction is proposed to efficiently determine the critical stiffness leading to the maximum fundamental frequency of a beam with multiple intermediate supports. This study is of significance to the support optimization and dynamic analysis of discretely supported beam-like structures.

Application of superposition method to study the mechanical behaviour of overlying strata in longwall mining

The sudden failure of roof strata is one of the fiercest dynamic instability events in underground longwall mining. To understand the mechanical behaviour of strata, researchers simplified the strata as a beam over an elastic foundation. However, some reported models involved a big equation system with unknown constants of integration. Solving such equations is a complicated mathematical task. As an improvement, this study adopted the superposition method to avoid arduous mathematical processes. In addition, the partially yielded coal seam and non-uniformly distributed overburden pressure were both considered in the proposed model. A set of case studies were carried out to investigate the influence of support scheme and yielded coal seam on strata behaviour. Results suggested the partially yielded coal seam is an essential condition to reproduce a reasonable strata behaviour. The strata can fail in front of the working face or over goaf controlled by the yield distance and yield degree of the coal seam. The bending moment and strain energy density are significantly reduced by increasing the yield distance and support capacity rather than yield degree. In addition to the above, results such as the deflection, slope, bending moment, shear force and strain energy density of strata and bearing pressure of coal seam were presented. This study demonstrated the superposition method is an easy tool to solve the mechanical model of overlying strata. By adopting this model, the failure patterns and mechanical state of overlying strata in longwall mining can be analytically investigated.

New singular wave solutions to the KdV‐Sawada‐Kotera‐Ramani equation by the modified simple equation method

The singular solutions and some special solutions of the KdV‐Sawada‐Kotera‐Ramani equation which have been analyzed in detail with solution functions supplemented by the relevant 3‐dimensional presentations of them. In the paper, very large number of possible concrete solutions are presented and discussed, based on the very various choices of possible values of the integration constants existing in the general solution function forms of the relevant class of problems being investigated. The contents of the main chapters have been classified and represented in a more concise and informative manner by pointing out the most important characteristics of the relevant solution sub‐results related by the actually chosen sub‐system of the general system of coupled ordinary differential equations and this process motivated this very detailed study. Finally, this paper obtains the unusually widespread collection of all these relatively laborsome solutions' set which can promote the application of the equation.

New insights into singularity analysis

Abstract In this work, we emphasize the use of singularity analysis in obtaining analytic solutions for equations for which standard Lie point symmetry analysis fails to make any lucid decision. We study the higher-dimensional Kadomtsev–Petviashvili, Boussinesq, and Kaup–Kupershmidt equations in a more general sense. With higher-order equations, there can be a commensurate number of resonances and when consistency for the full equation is examined at each resonance the constant of integration is supposed to vanish from the expression so that it remains arbitrary, but if there is an instance of this not happening, the consistency can be partially established by giving the offending constant the value from the defining equation. If consistency is otherwise not compromised, the equation can be said to be partially integrable, i.e., integrable on a surface of the complex space. Furthermore, we propose an approach that is meant to magnify the scope of singularity analysis for equations admitting higher values for resonances or positive leading-order exponent.

A Novel Version of HPM Coupled with the PSEM Method for Solving the Blasius Problem

This work studies the nonlinear differential equation that models the Blasius problem (BP) which is of great importance in fluid dynamics. The aim is to obtain an approximate analytical expression that adequately describes the phenomenon considered. To find such approximation, we propose a new method denominated powered homotopy perturbation (PHPM). Unlike HPM algorithm, the successive integration process generated by PHPM will consider zero the constants of integration in each approximation, except the last one. In the same way, PHPM will propose an adequate initial trial function provided of some unknown parameters in such a way that it will not evaluate the initial conditions in the iterations of the process; therefore, this set of parameters will be employed with the purpose of adjusting in the best accurate way the proposed approximation at the final part of the process. As a matter of fact, we will note from this analysis that the proposed solution is compact and easy to evaluate and involves a sum of five exponential functions plus a linear part of two terms, which is ideal for practical applications. With the purpose to get a better approximation, we find useful to combine PHPM with the power series extender method (PSEM) which implies to add to the PHPM solution one rational function with parameters to adjust. From this proposal, we find an approximate solution competitive with others from the literature.