Equations In Spaces Research Articles

Exact and Paraxial Broadband Airy Wave Packets in Free Space and a Temporally Dispersive Medium

A question of physical importance is whether finite-energy spatiotemporally localized (i.e., pulsed) generalizations of monochromatic accelerating Airy beams are feasible. For luminal solutions, this question has been answered within the framework of paraxial geometry. The time-diffraction technique that has been motivated by the Lorentz invariance of the equation governing the narrow angular spectrum and narrowband temporal spectrum paraxial approximation has been used to derive finite-energy spatiotemporally confined subluminal, luminal, and superluminal Airy wave packets. The goal in this article is to provide novel exact finite-energy broadband spatio-temporally localized Airy solutions (a) to the scalar wave equation in free space; (b) in a dielectric medium moving at its phase velocity; and (c) in a lossless second-order temporally dispersive medium. Such solutions can be useful in practical applications involving broadband (few-cycle) wave packets.

The Navier–Stokes equations in mixed-norm time-space parabolic Morrey spaces

The Navier–Stokes equations in mixed-norm time-space parabolic Morrey spaces

Generalized sequential state equation method for moving subsystem-induced structural parametric resonance

Parametric excitation is a unique type of excitation that arises from the time-varying parameters of a dynamical system, typically induced by the periodic movements of the system components. However, the structure-moving subsystems problem creates a new type of parametric excitation where the subsystems do not have exact periodic movements. Under the circumstances, dimension of the mathematical model of such a system exhibits time dependency, which prohibits the employment of conventional analytical methods to predict its stability. In this paper, we propose a novel Generalized Sequential State Equation Method that provides a distinct solution framework for analysis of the subsystem-induced parametric resonance. With the time-domain system partition and state mapping, the proposed method reconfigures the coupled system with a sequence of state space equations and solves them chronologically. The proposed method relaxes the single-subsystem-assumption in its original form and applies to a general scenario where the subsystems travel with arbitrary characteristic period. With the proposed method, occurrences of the subsystem-induced parametric resonance are precisely predicted via a set of analytical stability criteria and the steady state response is determined in an exact analytical form.

On covariant perturbations with scalar field in modified Gauss–Bonnet gravity

We investigate cosmological perturbations of f(G) gravity in the presence of a scalar field. Using the 1+3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1+3$$\\end{document} covariant formalism, we present the energy overdensity perturbation equations responsible for large scale structure formation. After applying harmonic decomposition method together with the redshift transformation technique, we obtain the fully perturbed equations in redshift space. The equations are solved to study the growth of matter overdensities contrast with redshift. For both short- and long-wavelength modes, we obtain numerical results for particular functional form f(G) models and scalar field. We find that, for this choice the energy overdensity perturbations decay with increase in redshift. However, for both short- and long-wavelength modes, the perturbations which include amplitude effects due to the f(G) models with a scalar field do differ remarkably from those in Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document}CDM. The results reduce to GR results in the limit of f(G)→G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(G)\\rightarrow G$$\\end{document} and in the absence of scalar field.

A new approach to abstract linear viscoelastic equation in Hilbert space

A new approach to abstract linear viscoelastic equation in Hilbert space

Hausdorff and fractal dimensions of attractors for functional differential equations in Banach spaces

The main objective of this paper is to obtain estimations of Hausdorff dimension as well as fractal dimension of global attractors and pullback attractors for both autonomous and nonautonomous functional differential equations (FDEs) in Banach spaces. New criterions for the finite Hausdorff dimension and fractal dimension of attractors in Banach spaces are firslty proposed by combining the squeezing property and the covering of finite subspace of Banach spaces, which generalize the method established in Hilbert spaces. In order to surmount the barrier caused by the lack of orthogonal projectors with finite rank, which is the key tool for proving the squeezing property of partial differential equations in Hilbert spaces, we adopt the state decomposition of phase space based on the exponential dichotomy of the studied FDEs to obtain similar squeezing property. The theoretical results are applied to a retarded nonlinear reaction-diffusion equation and a non-autonomous retarded functional differential equation in the natural phase space, for which explicit bounds of dimensions that do not depend on the entropy number but only depend on the spectrum of the linear parts and Lipschitz constants of the nonlinear parts are obtained.

On wave-like differential equations in general Hilbert space with application to Euler–Bernoulli bending vibrations of a beam

Wave-like differential equations occur in many engineering applications. Here the engineering setup is embedded into the framework of functional analysis of modern mathematical physics. After an overview, the L2–Hilbert space approach to free Euler–Bernoulli bending vibrations of a beam in one spatial dimension is investigated. We analyze in detail the corresponding positive, selfadjoint differential operators of 4-th order associated to the boundary conditions in statics. A comparison with free string wave swinging is outlined.

A 1D model for the dynamic instability of magnetized beams in leakage flow energy harvesters

This paper presents a one-dimensional fluid–structure interaction model for investigating the dynamic stability of elastic mounted beams under the influence of magnetic forces in a leakage flow energy harvesting environment. The interaction between the beam and external magnetic fields is represented by a nonlinear long range force distribution that is non linearly dependent on the beam displacement. The formulation results in a two way coupled fluid–structure interaction system that is one way coupled with the electromagnetic system. A perturbation technique is utilized to linearize the second order equilibrium equations of the coupled system, which are then transformed into a set of first-order ordinary differential equations in the state space. The effect of the linearized magnetic force coefficient on the eigenvalues and dynamic stability of the system is analyzed through a parametric study. The results indicate the onset of oscillatory dynamics in the coupled magneto-fluid-structural system is a non monotonic function of the magnetic force.

Generalized Convergence for Multi-Step Schemes under Weak Conditions

We have developed a local convergence analysis for a general scheme of high-order convergence, aiming to solve equations in Banach spaces. A priori estimates are developed based on the error distances. This way, we know in advance the number of iterations required to reach a predetermined error tolerance. Moreover, a radius of convergence is determined, allowing for a selection of initial points assuring the convergence of the scheme. Furthermore, a neighborhood that contains only one solution to the equation is specified. Notably, we present the generalized convergence of these schemes under weak conditions. Our findings are based on generalized continuity requirements and contain a new semi-local convergence analysis (with a majorizing sequence) not seen in earlier studies based on Taylor series and derivatives which are not present in the scheme. We conclude with a good collection of numerical results derived from applied science problems.

Voltage Equalizing Strategy for Cascaded Constant Current Power Supply

In order to detect the underground medium with large depth, the transmitter of electromagnetic sounding instrument has higher and higher power and voltage. A novel voltage equalizing strategy for cascaded constant current power supply using in electromagnetic transmitter was proposed in this paper. The cascaded topology of the transmitter has two isolated full‐bridge AC/DC converters, which had same input source and were connected in series on the output side. The mathematical model based on state space average method was analyzed. After simplifying the circuit, the state space equation was used to describe the circuit model. The stability and frequency characteristics of the system were analyzed. Aiming at the constant current output and the output voltage balance of the two converters, the system coupling model was established. The constant current controller and voltage equalizing controller were designed with decoupling method. A 300 W prototype was built and the experimental results verified the performance of the proposed method. © 2024 Institute of Electrical Engineer of Japan and Wiley Periodicals LLC.

Analytical and numerical techniques for solving a fractional integro-differential equation in complex space

<p>In this article, we describe the existence and uniqueness of a solution to the nonlinear fractional Volterra integro differential equation in complex space using the fixed-point theory. We also examine the remarkably effective Euler wavelet method, which converts the model to a matrix structure that lines up with a system of algebraic linear equations; this method then provides approximate solutions for the given problem. The proposed technique demonstrates superior accuracy in numerical solutions when compared to the Euler wavelet method. Although we provide two cases of computational methods using MATLAB R2022b, which could be the final step in confirming the theoretical investigation.</p>

Describing geophysical turbulence with a Schrödinger–Coriolis equation in velocity space

In this paper, we examine the predictions of the scale-relativity approach for a turbulent fluid in rotation. We first show that the time derivative of the governing Navier–Stokes equation in the usual x-space can be transformed into a Schrödinger-like equation in velocity space with an external vectorial field to account for the rotation, together with a local velocity harmonic oscillator (VHO) potential in the v-space. The coefficients of this VHO are given by the second order x-derivatives of the pressure. We can then give formulas for the velocity and acceleration probability distribution functions (PDF). Using a simple model of anisotropic harmonic oscillator, we compare our predictions with relevant data from both direct numerical simulations (DNS) and oceanic drifter velocity measurements. We find a good agreement of the predicted acceleration PDF with that observed from drifters and some possible support in DNS for the existence of gaps in the local velocity PDF, expected in the presence of a Coriolis force.

Generalized variational principles for the modified Benjamin-Bona-Mahony equation in the fractal space

Because variational principles are very important for some methods to get the numerical or exact solutions, it is very important to seek explicit variational formulations for the non-linear PDE. At first, this paper describes the modified Benjamin-Bona-Mahony equation in fractal porous media or with irregular boundaries. Then, by designing skillfully the trial-Lagrange functional, variational principles are successfully established for the modified Benjamin-Bona-Mahony equation in the fractal space, respectively. Furthermore, the obtained variational principles are proved correct by minimizing the functionals with the calculus of variations.

New results for fractional ordinary differential equations in fuzzy metric space

<abstract><p>In this paper, we primarily focused on the existence and uniqueness of the initial value problem for fractional order fuzzy ordinary differential equations in a fuzzy metric space. First, definitions and relevant properties of the Gamma function and Beta function within a fuzzy metric space were provided. Second, by employing the principle of fuzzy compression mapping and Choquet integral of fuzzy numerical functions, we established the existence and uniqueness of solutions to initial value problems for fuzzy ordinary differential equations. Finally, several examples were presented to demonstrate the validity of our obtained results.</p></abstract>

Fuzzy stability of mixed type functional equations in Modular spaces

In this article, our aim is to find some stability results for mixed type cubic and quartic functional equations in fuzzy modular spaces using the fundamental results of fixed-point theory. The fixed point method provides one of the effective techniques that can be used to investigate the fuzzy stability of a mixed type cubic and quartic functional equations.

Local well-posedness and blow-up criterion to a nonlinear shallow water wave equation

<abstract><p>The initial data problem to a nonlinear shallow water wave equation in nonhomogeneous Besov space is discussed. Using the decomposition of Littlewood-Paley and the properties of nonhomogeneous Besov space, we establish the well-posedness of short time solutions for the equation in the Besov space. A blow-up criterion of solutions is also obtained.</p></abstract>

Breakdown of $ C^{\infty} $-smoothing effects of solutions to the semilinear heat equation in the whole space

Breakdown of $ C^{\infty} $-smoothing effects of solutions to the semilinear heat equation in the whole space

Two regularity criteria of the 3D magneto-micropolar equations in Vishik spaces

Two regularity criteria of the 3D magneto-micropolar equations in Vishik spaces

Optimal quadrature formulas in Sobolev space for solving the generalized Abel integral equation

In this article, a composite optimal quadrature formula is constructed for an approximate analytical solution of the generalized integral Abel equation in the Sobolev functional space. The optimal coefficients of this quadrature formula have been found. In addition, using the constructed composite optimal quadrature formula, numerical results of examples of the generalized Abel integral equation were obtained and compared with the exact solution.

On the global existence of mild solutions to binary-ternary Boltzmann equation in an integrable space

On the global existence of mild solutions to binary-ternary Boltzmann equation in an integrable space