Solution Of Equation Research Articles

Regularizing effect for unbounded flux-limited viscosity solutions of a discontinuous Hamilton-Jacobi equation on junction

Regularizing effect for unbounded flux-limited viscosity solutions of a discontinuous Hamilton-Jacobi equation on junction

Development of an innovative diffraction scattering theory of X-rays and electrons in imperfect crystals.

Fundamental equations describing the X-ray and electron diffraction scattering in imperfect crystals have been derived in the form of the matrix Fredholm-Volterra integral equation of the second kind. A theoretical approach has been developed using the perfect-crystal Green function formalism. In contrast, another approach utilizes the wavefield eigenfunctions related to the diagonalized matrix propagators of the conventional Takagi-Taupin and Howie-Whelan equations. Using the Liouville-Neumann-type series formalism for building up the matrix Fredholm-Volterra integral equation solutions, the general resolvent function solutions of the X-ray and electron diffraction boundary-valued Cauchy problems have been obtained. Based on the resolvent-type solutions, the aim is to reveal the features of the diffraction scattering onto the crystal lattice defects, including the mechanisms of intra- and interbranch wave scattering in the strongly deformed regions in the vicinity of crystal lattice defect cores. Using the two-stage resolvent solution of the second order, this approach has been supported by straightforward calculation of the electron bright- and dark-field contrasts of an edge dislocation in a thick foil. The results obtained for the bright- and dark-field profiles of the edge dislocation are discussed and compared with analogous ones numerically calculated by Howie & Whelan [Proc. R. Soc. A (1962), 267, 206].

Qualitative Analysis for the Solutions of Fractional Stochastic Differential Equations

Fractional pantograph stochastic differential equations (FPSDEs) combine elements of fractional calculus, pantograph equations, and stochastic processes to model complex systems with memory effects, time delays, and random fluctuations. Ensuring the well-posedness of these equations is crucial as it guarantees meaningful, reliable, and applicable solutions across various disciplines. In differential equations, regularity refers to the smoothness of solution behavior. The averaging principle offers an approximation that balances complexity and simplicity. Our research contributes to establishing the well-posedness, regularity, and averaging principle of FPSDE solutions in Lp spaces with p≥2 under Caputo derivatives. The main ingredients in the proof include the use of Hölder, Burkholder–Davis–Gundy, Jensen, and Grönwall–Bellman inequalities, along with the interval translation approach. To understand the theoretical results, we provide numerical examples at the end.

Existence for fractional evolutionary inclusions involving nonlinear weakly continuous operators with applications

In this paper, our primary purpose is to settle the existence for a type of fractional-order evolutionary inclusions. By utilizing the Rothe method and the surjective result of weakly continuous operators, we show that the solution of the evolutionary equation exists under several kinds hypotheses on the data. Finally, the obtained results are applied to verify the existence of solutions for a time-fractional evolutionary hemivariational inequality, a time-fractional Navier–Stokes–Voigt equation and a quasistatic friction contact problem.

Subharmonic functions with separated variables and their connection with generalized convex functions

In this paper, we consider the necessary and sufficient conditions for the subharmonicity of functions of two variables, representable as a product of two functions of one variable in the Cartesian coordinate system or in the polar coordinate system in domains on the plane. We establish a connection of such functions with functions that are convex with respect to solutions of second-order linear differential equations, i.e., convex with respect to two functions.

Self-consistent calculation of the optical emission spectrum of an argon capacitively coupled plasma based on the coupling of particle simulation with a collisional-radiative model

We report the development of a computational framework for the calculation of the optical emission spectrum of a low-pressure argon capacitively coupled plasma (CCP), which is based on the coupling of a particle-in-cell/Monte Carlo collision simulation code with a diffusion-reaction-radiation code for Ar I excited levels. In this framework, the particle simulation provides the rates of the direct and stepwise electron-impact excitation and electron-impact de-excitation for 30 excited levels, as well as the rates of electron-impact direct and stepwise ionization. These rates are used in the solutions of the diffusion equations of the excited species in the second code, along with the radiative rates for a high number of Ar-I transitions. The calculations also consider pooling ionization, quenching reactions, and radial diffusion losses. The electron energy distribution function and the population densities of the 30 excited atomic levels are computed self-consistently. The calculations then provide the emission intensities that reproduce reasonably well the experimentally measured optical emission spectrum of a symmetric CCP source operated at 13.56 MHz with 300 V peak-to-peak voltage, in the 2–100 Pa pressure range. The accuracy of the approach appears to be limited by the one-dimensional nature of the model, the treatment of the radiation trapping through the use of escape factors, and the effects of radiative cascades from higher excited levels not taken into account in the model.

Uniformly-like convex solutions for the polynomial-like iterative equation in Banach spaces

Uniformly-like convex solutions for the polynomial-like iterative equation in Banach spaces

Polynomial Solutions of Electric Field Equations in Anisotropic Media

The time-dependent system of partial differential equations of the second order describing the electric wave propagation in electrically and magnetically anisotropic homogeneous media is considered in the paper. A method for the computation of the polynomial solutions of the initial value problem for the considered system is proposed. Symbolic computations are used and these symbolic computations are implemented in Maple. It is proved also that these polynomial solutions are approximate solutions of the considered initial value problem with smooth initial data and the inhomogeneous term. The computational experiments confirm the robustness of the suggested method for the computation of electric fields in general electrically and magnetically anisotropic media.

Wave interaction with multiple thin flexible porous barriers in water of uniform finite depth

The interaction of water waves with multiple flexible porous barriers partially immersed in a uniform finite-depth fluid is investigated in the present study. Using linear theory, the physical problem is mathematically modeled as a boundary value problem. The eigenfunction expansion approach is used to generate the velocity potential function in terms of orthogonal eigenfunctions. The Havelock’s inversion formula is then used to convert the boundary value problem into a set of Fredholm-type integral equations. By choosing a suitable weight function, the multi-term Galerkin’s approximation technique is used to obtain the mathematical solutions of these integral equations in terms of the Chebyshev polynomial. Numerical solutions regarding the reflection and transmission coefficients, hydrodynamic forces, and dissipated wave energy at the barriers are obtained and graphically represented in several figures for various non-dimensional parameters. It is observed from the graphical representations that the permeability and flexibility of the barriers play a crucial role in the modeling of efficient breakwaters.

Asymptotic Behavior of <i>n</i>th Order Impulsive Differential Equations

This paper devotes to the asymptotic behavior of all solutions of nth order impulsive differential equations. Based on impulsive differential inequality, boundedness and zero tendency of every solution for nth order impulsive differential equations are obtained. In addition, we derive globally uniformly exponential stability of every solution under Lyapunov function and impulsivetechnique, and these results are extend to <i>n</i>th order differential equations with periodic coefficient and periodic impulse. Meanwhile, an example with simulations are provided to verify the conclusion.

Bridging trails in reflectionless potential deformation: Two paths and one horizon

In contrast to the conventional one-parameter class of isospectral deformation using translation, we calculate the two- and three-parameter classes of isospectral deformation of the well-known reflectionless potential by utilizing a far different approach of scaling methodology. Subsequently, using these results, we find that this more general class of deformations is not unique but instead subsume in the same class of conventional one-parameter translational deformation. We also provide a theoretical foundation for how these two incredibly different approaches converge at the same destination. Finally, we show that the most generic class of potentials, obtained by scaling deformation, are solutions of the nonlinear KdV equation.

Bounded solutions for semilinear differential equations

Bounded solutions for semilinear differential equations

Koopman neural operator as a mesh-free solver of non-linear partial differential equations

The lacking of analytic solutions of diverse partial differential equations (PDEs) gives birth to a series of computational techniques for numerical solutions. Although numerous latest advances are accomplished in developing neural operators, a kind of neural-network-based PDE solver, these solvers become less accurate and explainable while learning long-term behaviors of non-linear PDE families. In this paper, we propose the Koopman neural operator (KNO), a new neural operator, to overcome these challenges. With the same objective of learning an infinite-dimensional mapping between Banach spaces that serves as the solution operator of the target PDE family, our approach differs from existing models by formulating a non-linear dynamic system of equation solution. By approximating the Koopman operator, an infinite-dimensional operator governing all possible observations of the dynamic system, to act on the flow mapping of the dynamic system, we can equivalently learn the solution of a non-linear PDE family by solving simple linear prediction problems. We validate the KNO in mesh-independent, long-term, and zero-shot predictions on five representative PDEs (e.g., the Navier-Stokes equation and the Rayleigh-Bénard convection) and three real dynamic systems (e.g., global water vapor patterns and western boundary currents). In these experiments, the KNO exhibits notable advantages compared with previous state-of-the-art models, suggesting the potential of the KNO in supporting diverse science and engineering applications (e.g., PDE solving, turbulence modeling, and precipitation forecasting).

DISSIPATION AND DISPERSION IN FRACTIONAL NONLINEAR WAVE DYNAMICS: ANALYTICAL AND NUMERICAL EXPLORATIONS

The fractional nonlinear generalized [Formula: see text]-dimensional Chaffee–Infante (FGCI) equation is a nonlinear evolution equation that governs the propagation of waves in dispersive media, exhibiting both dissipative and dispersive effects. It finds applications in various fields such as fluid mechanics, plasma physics, and nonlinear optics, where the interplay between nonlinearity, dispersion, and dissipation plays a crucial role in determining the dynamics of the system. This study aims to obtain analytical and numerical solutions for the FGCI equation, which is a generalization of the Chaffee–Infante equation. The analytical approach employed is the Khater II (Khat II) method, an efficient technique for constructing approximate analytical solutions to nonlinear fractional differential equations. Additionally, He’s variational iteration (HVI) method is implemented as a numerical scheme to validate the accuracy of the analytical solutions obtained from the Khat II method. The results obtained through these methods demonstrate the effectiveness of the proposed techniques in solving the FGCI equation and provide valuable insights into the dynamics of wave propagation phenomena governed by the equation. The significance of this study lies in its contribution to the understanding of nonlinear wave propagation phenomena and the development of analytical and numerical techniques for solving fractional differential equations. The novelty of the research lies in the application of the Khat II method to the FGCI equation, which has not been previously explored, and the validation of the analytical solutions through the HVI method. The study is conducted in the field of applied mathematics, with a focus on nonlinear fractional differential equations and their applications in various physical systems.

Mathieu–Hill Equation Stability Analysis for Trapped Ions: Anharmonic Corrections for Nonlinear Electrodynamic Traps

The stability properties of the Hill equation are discussed, especially those of the Mathieu equation that characterize ion motion in electrodynamic traps. The solutions of the Mathieu-Hill equation for a trapped ion are characterized by employing the Floquet theory and Hill’s method solution, which yields an infinite system of linear and homogeneous equations whose coefficients are recursively determined. Stability is discussed for parameters a and q that are real. Characteristic curves are introduced naturally by the Sturm–Liouville problem for the well-known even and odd Mathieu equations cem(z,q) and sem(z,q). In the case of a Paul trap, the stable solution corresponds to a superposition of harmonic motions. The maximum amplitude of stable oscillations for ideal conditions (taken into consideration) is derived. We illustrate the stability diagram for a combined (Paul and Penning) trap and represent the frontiers of the stability domains for both axial and radial motion, where the former is described by the canonical Mathieu equation. Anharmonic corrections for nonlinear Paul traps are discussed within the frame of perturbation theory, while the frontiers of the modified stability domains are determined as a function of the chosen perturbation parameter and we demonstrate they are shifted towards negative values of the a parameter. The applications of the results include but are not restricted to 2D and 3D ion traps used for different applications such as mass spectrometry (including nanoparticles), high resolution atomic spectroscopy and quantum engineering applications, among which we mention optical atomic clocks and quantum frequency metrology.

Correction to: Bifurcations, chaotic behavior, sensitivity analysis and new optical solitons solutions of Sasa-Satsuma equation

Correction to: Bifurcations, chaotic behavior, sensitivity analysis and new optical solitons solutions of Sasa-Satsuma equation

Planar Lorentz invariant velocities with a wave equation application

In this paper we determine the functional form of those planar velocity fields for which the associated system of two ordinary differential equations are automatically invariant under a Lorentz transformation. For planar motion we determine first order partial differential equations for the velocity components u(x,y,t) and w(x,y,t) in the x− and y−directions respectively and their general solutions in terms of two arbitrary functions. These partial differential equations and the associated partial differential relations connecting energy and momentum are fully compatible with the Lorentz invariant energy–momentum relations and appear not to have been given previously in the literature. For a particular special relativistic model, one example is given involving similarity solutions of the wave equation. An interesting special case gives rise to a family of particle paths which are characterized by a single arbitrary function, and for which the magnitude of their velocities is the speed of light. This is indicative of the abundant possibilities existing in the “fast-lane”.

Energetic processes and thermodynamic analysis of the spinning Kiselev black hole with cloud of strings

Here we study the impact of the string clouds, the quintessence dark energy and other spacetime parameters on the center-of-mass energy, Penrose-process and thermodynamic properties of a stationary black hole solution of the Einstein field equations. We observe that the three distinct horizons i.e the Cauchy horizon, the event horizon and the cosmological horizon exist when the spin parameter a of the black hole is less than its extremal value aE. Interestingly we see that the infinite redshift surface of the black hole is not much sensitive to the quintessence parameter γ. It is sensitive to the parameter a and it increases as the parameter a increases. Further, we notice that an increase in the string cloud parameters b, quintessence parameter γ and a make the radius of the ergosphere of the black hole larger. We also notice that a rise in the parameter a makes the innermost stable circular orbits (ISCO) smaller and for an increase in the parameter b and γ the ISCO becomes larger than that for the Kerr spacetime. For the center-of-mass energy of the spinning black hole with quintessence and cloud of strings we observe that it can be arbitrarily high only for the extremal case of the black hole for a specific range of values of the angular momentum of the colliding particles. We notice that the Penrose-process is more efficient for smaller values of the string cloud parameter b and the parameter γ has no significant impact on this process. Moreover we see that an increase in the parameters b, γ and a cause the Helmholtz free energy to increase and opposite is true for the temperature of this type of rotating black hole. We learn that the area and entropy of the black hole are also depend on the parameters b, γ and a, and their effect make the entropy smaller. Interestingly we find a range of the horizon radius for the thermodynamic stability of the black hole. To get more information about the thermodynamic quantities of the rotating black hole with quintessence and cloud of strings, we plot them for different ranges of the spacetime parameters.

Withdrawn: Transient modeling of natural gas in pipeline networks by two non-iterative explicit and implicit finite volume methods

This paper describes two non-iterative explicit and implicit finite volume methods (FVM) to calculate the unsteady characteristics of natural gas in distribution systems. The proposed finite volume model could be used to solve the gas network as a whole for each time step. The main advantage of this approach is that it can analyze gas distribution networks with less computational cost. In addition, contrary to the conventional iterative algorithms, it does not consist of any divergence problems (Elman et al., 1996). To achieve this goal, the explicit FVM is derived from the governing equations. Then, the linearization process is performed, and the numerical solution of the linear equations is developed by using the implicit FVM. Three classic test cases are exemplified to check the performance of the proposed methods. The results confirm that both explicit and implicit methods, even in complicated distribution networks, can predict the elements of unsteady natural gas flows. Furthermore, when the governing equations are linearized and the implicit FVM is applied, satisfactory results with less computational cost, compared to the explicit one, are produced.

Inverses of fuzzy relation matrices with addition-min composition

This paper mainly considers the post-inverse matrix of a fuzzy relation matrix in terms of addition-min composition. A necessary and sufficient condition for the consistency of the inverse matrix problem is given by transforming the problem into a series of particular fuzzy relation equations. The uniqueness of the post-inverse is also investigated. Furthermore, it is proved that the search for the minimal solutions of the particular fuzzy relation equations can be converted into solving a linear system. Based on these discussions, an algorithm is constructed for solving a post-inverse of a given fuzzy relation matrix.