Solution Of Equation Research Articles

Creep stress analysis of Transversely Isotropic Rotating Disc Composed of Piezoelectric Material

If a crystal undergoes mechanical stress, then an electrical potential is developed across its sides; this phenomenon is called piezoelectricity, or the piezoelectric effect. Piezoelectric materials include quartz, zinc oxide (ZnO), lead zirconate titanate (PZT4), and Rochelle salt. There are numerous uses for piezoelectricity in signal and electrical transducers. In this work, transitional and creep stresses are evaluated in a thin disc made up of piezoelectric transversely isotropic solids subjected to rotation. The mathematical expressions of stress components are computed for the rotating disc by using Seth’s transition theory considering the rotation effect in circular disc. The electric displacement relations and various stress components are computed using Hooke’s Law. A non-homogeneous differential equation is obtained by using mathematical relations and the equilibrium equation. The theoretical solution of the differential equation is obtained under specified boundary conditions. Creep stresses are studied for the considered material. The obtained results are shown graphically, analyzed numerically, and concluded with a discussion of the results for transversely isotropic piezoelectric materials such as BaTiO3 and PZT-4, among others. On the basis of all the numerical discussions and graphs, it is concluded that discs composed of transversely isotropic piezoelectric (PZT4 and BaTiO3) materials are more stable than discs composed of transversely isotropic (magnesium and beryl).

Solutions for Singular Quasilinear Equations of N-Laplacian Type with Critical Exponential Growth

Solutions for Singular Quasilinear Equations of N-Laplacian Type with Critical Exponential Growth

Existence solutions for nondensely integro-differential equations with state-dependent delay

Existence solutions for nondensely integro-differential equations with state-dependent delay

Legendre Transformation in the Born–Infeld Model, Monge–Ampere Equation and Exact Solutions

Legendre Transformation in the Born–Infeld Model, Monge–Ampere Equation and Exact Solutions

Qualitative Outcomes on Monotone Iterative Technique and Quasilinearization Method on Time Scale

In this paper, a nonlinear dynamic equation with an initial value problem (IVP) on a time scale is considered. First, applying comparison results with a coupled lower solution (LS) and an upper solution (US), we improved the quasilinearization method (QLM) for the IVP. Unlike other studies, we consider the LS and US pair of the seventh type instead of the natural type. It was determined that the solutions of the dynamic equation converge uniformly and monotonically to the unique solution of the IVP, and the convergence is quadratic. Moreover, we will use the delta derivative (Δγ) instead of the classical derivative (dγ) in the proof because it studies a time scale. In the second part of the paper, we applied the monotone iterative technique (MIT) coupled with the LS and US, which is an effective method, proving a clear analytical representation for the solution of the equation when the relevant functions are monotonically non-decreasing and non-increasing. Then an example is given to illustrate the results obtained.

A novel approach for the numerical solution of nonlinear Fredholm integral equations using Hosoya polynomial method

Abstract In this paper, we study the graph theoretical polynomial known as the Hosoya polynomial obtained from one of the standard classes of graphs called path. Using this polynomial applied for the numerical solution of the nonlinear Fredholm integral equation, which reduces in the algebraic system of equation with collocation points, then solving this system using Newton’s iterative with the help of MATLAB, we obtain the required approximate solution. The desired results in terms of a set of continuous polynomials over a closed interval [0, 1]. Illustrative applications show the efficiency, accuracy and validity of the proposed technique.

A robust septic hermite collocation technique for dirichlet boundary condition Heat conduction equation

Abstract In the current manuscript, approximate solution for 1D heat conduction equation will be sought with the Septic Hermite Collocation Method (SHCM). To achieve this goal, by means of the roots of both Chebyschev and Legendre polynomials used at the inner collocation points, the pseudo code of the method is found out and applied using Matlab which is one of the widely utilized symbolic programming platforms. The unconditional stability of the scheme is shown by the traditional von-Neumann stability technique. To illustrate the accuracy and effectiveness of this newly current numerical scheme, a comparison among analytical and the computed numerical results is presented in tabular forms. It has been illustrated that the scheme is both accurate and effective one and at the same time can be used in a successful way for finding out numerical solutions of several nonlinear problems as well as linear ones.

Hosoya polynomial method for the numerical solution of Volterra integral equations

AbstractThe numerical solution of Volterra integral equation using one of the graph theoretic polynomial is Hosoya polynomial. To reduce the VIEs to a system of algebraic equations by substituting collocation points. To simplify these system by the help of Matlab using Newton’s iteration technique, we get the Hosoya coefficients and substitute these coefficients in function approximation to get the required solutions as shown in tables and graphically represent the figures. The error analysis demonstrates the accuracy, stability, and consistency. The proposed method shows efficiency and validity as compared to the existing methods.

Quantum focusing conjecture in two-dimensional evaporating black holes

We consider the quantum focusing conjecture (QFC) for two-dimensional evaporating black holes in the Russo-Susskind-Thorlacius (RST) model. The QFC is closely related to the behavior of the generalized entropy. In the context of the black hole evaporation, the entanglement entropy of the Hawking radiation is decreasing after the Page time, and therefore it is not obvious whether the QFC holds. One of the present authors previously addressed this problem in a four-dimensional spherically symmetric dynamical black hole model and showed that the QFC is satisfied. However, the background spacetime considered was approximated by the Vaidya metric, and quantum effects of matters in the semiclassical regime were not fully taken into consideration. It remains to be seen if the QFC in fact holds for exact solutions of the semiclassical Einstein equations. In this paper, we address this problem in the RST model, which allows us to solve the semiclassical equations of motion exactly. We prove that the QFC is satisfied for evaporating black holes in the RST model with the island formation taken into account.

A Fast Solution of the Dynamic Rate Equation for a High-Power Fiber Laser

A Fast Solution of the Dynamic Rate Equation for a High-Power Fiber Laser

Gravitational production of heavy particles during and after inflation

We investigate the gravitational production of a scalar field χ with a mass exceeding the Hubble scale during inflation mχ ≳ HI, employing both analytical and numerical approaches. We demonstrate that the steepest descent method effectively captures the epochs and yields of gravitational production in a compact and simple analytical framework. These analytical results align with the numerical solutions of the field equation. Our study covers three spacetime backgrounds: de Sitter, power-law inflation, and the Starobinsky inflation model. Within these models, we identify two distinct phases of particle production: during and after inflation. During inflation, we derive an accurate analytic expression for the particle production rate, accounting for a varying Hubble rate. After inflation, the additional burst of particle production depends on the inflaton mass around its minimum. When this mass is smaller than the Hubble scale during inflation, HI, there is no significant extra production. However, if the inflaton mass is larger, post-inflation production becomes the dominant contribution. Furthermore, we explore the implications of gravitationally produced heavy fields for dark matter abundance, assuming their cosmological stability.

Extension of Meir-Keeler-Khan (ψ − α) Type Contraction in Partial Metric Space

In numerous scientific and engineering domains, fractional-order derivatives and integral operators are frequently used to represent many complex phenomena. They also have numerous practical applications in the area of fixed point iteration. In this article, we introduce the notion of generalized Meir-Keeler-Khan-Rational type (ψ−α)-contraction mapping and propose fixed point results in partial metric spaces. Our proposed results extend, unify, and generalize existing findings in the literature. In regards to applicability, we provide evidence for the existence of a solution for the fractional-order differential operator. In addition, the solution of the integral equation and its uniqueness are also discussed. Finally, we conclude that our results are superior and generalized as compared to the existing ones.

Determination of dynamic loads in the crane suspension when lifting a load from a rigid base

Loading and unloading works are an integral part of the construction process. Cranes of various types were mostly use to perform these works. To ensure trouble-free operation and increase the reliability of cranes, when calculating structures and components of their working equipment, it is important to take into account dynamic loads, which are several times higher than static loads. Elements of dynamic loads in the crane suspension are its elastic components (flexible traction bodies) - ropes. The process of lifting a load from a rigid base and picking it up is considered, which divided into three stages: the first is the selection of clearances and the tension of the ropes; the second is the pre-opening stage of lifting the load; the third is the post-detachment stage of lifting the load. For each stage, the initial conditions accepted, the differential equations of the movement of loads compiled, their solution given taking into account many factors, and expressions derived for determining the forces in the load suspension. At the first stage, the duration of the gap selection (tension of the ropes) is determined, at the second stage, the speed of separation of the load from the base is determined, at the third stage, the maximum force in the elastic element determined. The method of determining the forces in the suspension of the load, the duration of the selection of clearances (tension of the ropes), the speed of separation of the load from the base, and the maximum force in the elastic element presented in the work allows you to significantly simplify the solution of complex equations, to determine simple expressions and to determine them with sufficient accuracy for practical calculations values.

Quantum Bernoulli noises approach to quantum master equations and applications to Ehrenfest-type theorems

This paper investigates the regularity properties of quantum master equations with unbounded coefficients within the space of square-integrable complex-valued Bernoulli functionals. Initially, we demonstrate the existence of a unique regular solution to the linear stochastic Schrödinger equation driven by cylindrical Brownian motions. Subsequently, we establish the existence of regular solutions for the autonomous linear quantum master equation and provide a probabilistic representation of this solution in terms of the stochastic Schrödinger equation. Finally, by taking the expectation of some observables with respect to the solution of the quantum master equation, we derive a system of differential equations that describe the evolution of the mean values of certain quantum observables.

On Bethe equations of 2d conformal field theory

We study the higher spin algebras of two-dimensional conformal field theory from the perspective of quantum integrability. Starting from Maulik-Okounkov instanton R-matrix and applying the procedure of algebraic Bethe ansatz, we obtain infinite commuting families of Hamiltonians of quantum ILW hierarchy parametrized by the shape of the auxiliary torus. We calculate explicitly the first five of these Hamiltonians. Then, we numerically verify that their joint spectrum can be parametriezed by solutions of Litvinov’s Bethe ansatz equations and we conjecture a general formula for the joint spectrum of all ILW Hamiltonians, based on results of Feigin, Jimbo, Miwa and Mukhin.There are two interesting degeneration limits, the infinitely thick and the infinitely thin auxiliary torus. In one of these limits, the ILW hierarchy degenerates to Yangian or Benjamin-Ono hiearchy and the Bethe equations can be easily solved. The other limit is singular but we can nevertheless extract local Hamiltonians corresponding to quantum KdV or KP hierarchy. Litvinov’s Bethe equations in this local limit provide an alternative to Bethe ansatz equations of Bazhanov, Lukyanov and Zamolodchikov, but are more transparent, work at any rank and are manifestly symmetric under triality symmetry of W\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{W} $$\\end{document}1+∞. Finally, we illustrate analytic properties of the solutions of Bethe equations in minimal models, in particular for Lee-Yang CFT. The analytic structure of Bethe roots is very rich as it captures the representation theory of W\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{W} $$\\end{document}N minimal models.

Conformal Killing gravity: pp-waves; multiple matter sources

Abstract Recently Harada proposed a gravitational theory which is of third order in the derivatives of the metric tensor. This attracted some attention particularly as it predicts a late-time transition from cosmological deceleration to accelerated expansion without assuming the presence of dark energy or a non-zero cosmological constant. This theory has been dubbed conformal Killing gravity (CKG). The most general exact solutions of the Harada field equations are known for a number of important physical situations: homogeneous and isotropic cosmological models, static spherically symmetric vacuum and electrovac spacetimes. These are analogues of the well-known FRWL, Schwarzschild and Reissner–Nordström metrics of general relativity (GR). In this paper a subclass of pp-waves are studied and the most general exact solution obtained together with its specialization for plane waves. The generalization from GR to Harada’s theory is straightforward. The solutions have Petrov type N or 0 and the Ricci tensor is either zero or the Segré type is [(211)] with zero eigenvalue. For any metric in CKG it is shown that more than one possible matter source can generate the solution. If the metric admits one or more Killing vectors or Killing tensors, the ambiguity in the possible matter sources increases.

To the Theory of the Lamb Shift in the Relativistic Hydrogen Atom

Radiative corrections which remove the accidental degeneracy in the spectrum of the relativistic hydrogen atom and lead to the modification of the Coulomb law, are calculated within the novel approach, based on the exact solution of the Dirac equation with the Coulomb potential. The energy spectrum of the hydrogen atom is obtained with account for these corrections, and the Lamb shift is calculated for the lowest energy states.

Enhanced framework for solving general energy equations based on metropolis-hasting Markov chain Monte Carlo

Due to the widespread presence of heat and mass transfer phenomena in industrial applications, numerous studies have been devoted to the accurate solution of energy equations, providing a foundation for the analysis of heat and mass transfer processes in practical applications. In this study, a Green's Function Markov Superposition Monte Carlo (GMSMC) for solving general energy equations has been developed based on probability and statistical principles owing to its advantageous features of insensitivity towards dimension and geometric complexity as well as the capability to handle multiple integrals in complex domains. The energy equation is first decomposed, and corresponding probability models are established for each component, considering their interrelationships. Subsequently, a solution framework for solving the general energy equation is constructed by integrating these probability models based on a Markov chain structure. The mathematical principles and formulae of the proposed method are derived in detail. The performance of the proposed method is validated by several heat transfer systems with different combinations of boundary conditions and features, which mainly include the distribution of the internal heat source and whether the convection or transient term is included. Results of the validation show that the temperatures obtained by the proposed method are in good agreement with the FEM based on a fine grid, no matter whether the calculation is for a single point or a distribution.

Fluctuations and Power Low Distribution Function in Nonequilibrium Systems

The Fokker–Planck equation is formulated for the distribution functions of macroscopic open systems in the space of slowly changing physical variables (energy, adiabatic invariants, etc.). The stationary solution of such equations determines a quasi-equilibrium distribution function in the relevant space. The proposed approach involves the evolution of systems under the action of dissipation and diffusion in the space of the appropriate variables. It is shown that the well-known power law distribution can be obtained by considering internal and external fluctuations in statistical systems.

Symmetric structures and dynamic analysis of a (2+1)-dimensional generalized Benny-Luke equation

We study the symmetric structures and dynamic analysis of a (2 + 1)-dimensional generalized Benny-Luke (GBL) equation based on the Lie symmetry method, the GBL equation is an important non-integrable model of water waves. Specifically, we construct multiple exact solutions of the GBL equation and obtain its nonlocally related systems. Firstly, the Lie point symmetries and conservation laws of the GBL equation are computed, and then we get the reduced ordinary differential equation from one of the conservation laws. Multiple methods, for example, the dynamical systems method, the power series method, the homogeneous balancing method and generalized variable separation method, are used to solve the ordinary differential equation and abundant exact solutions of the GBL equation are got. Finally, we extend these exact solutions by discrete symmetries, and give three-dimensional graphs of partial exact solutions. In addition, we construct the nonlocally related PDE systems, which contains the potential systems from the conservation laws and an inverse system from a Lie point symmetry of the GBL equation. These findings reveal the dynamical behavior behind the GBL equation and broaden the range of nonlinear water wave model solutions.