Equations In Spaces Research Articles

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.

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

Solvability of product of $ n $-quadratic Hadamard-type fractional integral equations in Orlicz spaces

<abstract><p>The current study demonstrated and studied the existence of monotonic solutions, as well as the uniqueness of the solutions for a general and abstract form of a product of $ n $-quadratic fractional integral equations of Hadamard-type in Orlicz spaces $ L_\varphi $. We utilized the analysis of the measure of non-compactness associated with Darbo's fixed-point theorem and fractional calculus to obtain the results.</p></abstract>

The Double‐Logarithmic Regularity Criterion of Pressure for the 3D Navier–Stokes Equations in the Besov Space

In the paper, we consider the regularity of the weak solutions for the incompressible 3D Navier–Stokes (N–S) equations. Our main result is the double‐logarithmic regularity criterion of pressure for the 3D Navier–Stokes equations in the Besov space. Our results extend and generalize previous works.

On the Existence and Uniqueness of Fuzzy Mild Solution of Fractional Evolution Equations

In this paper, the nonlocal Cauchy problem is discussed for the fuzzy fractional evolution equations in an arbitrary Banach space for order q ∈ (1, 2) and the criteria on the existence and uniqueness of mild fuzzy solutions are obtained by using Schauder’s fixed point theorem. An example to illustrate the applications of main results is also given.

ON THE CORRECT STATEMENTS OF BOUNDARY VALUE PROBLEMS FOR PDE'S IN BANACH-SOBOLEV SPACES

Abstract. In this report we consider the Banach Functional Space    X , defined on measure space   m ;  , where n R   is bounded domain, m is the Lebesgue measure in n R . It is defined the Banach Sobolev Space    m X W , generated by norm of    X . Regarding these spaces the elliptic equation is considered and the concepts of classical, local, weak and strong solutions are given. One method for defining the trace, trace operator and trace space generated by space    1 X W are introduced. These concepts allow us to consider the boundary value problems for differential equations regarding the spaces    m X W . Finally, it is considered the Dirichlet problem for polyharmonic equations in Sobolev spaces, generated by norm of symmetric spaces and the correct solvability of the corresponding problem is proved.

Existence and upper semicontinuity of pullback attractors for Kirchhoff wave equations in time-dependent spaces

Existence and upper semicontinuity of pullback attractors for Kirchhoff wave equations in time-dependent spaces

Solutions and anti-periodic solutions for impulsive differential equations and inclusions containing Atangana-Baleanu fractional derivative of order $ \zeta \in (1, 2) $ in infinite dimensional Banach spaces

<abstract><p>In this paper, we improved recent results on the existence of solutions for nonlinear fractional boundary value problems containing the Atangana-Baleanu fractional derivative of order $ \zeta \in (1, 2) $. We also derived the exact relations between these fractional boundary value problems and the corresponding fractional integral equations in infinite dimensional Banach spaces. We showed that the continuity assumption on the nonlinear term of these equations is insufficient, give the derived expression for the solution, and present two results about the existence and uniqueness of the solution. We examined the case of impulsive impact and provide some sufficiency conditions for the existence and uniqueness of the solution in these cases. We also demonstrated the existence and uniqueness of anti-periodic solution for the studied problems and considered the problem when the right-hand side was a multivalued function. Examples were given to illustrate the obtained results.</p></abstract>

Numerical-mathematical methods for Hyperbolic-Parabolic Systems: Investigation of Volter-Gursat Equation and Green's Function in Three-Dimensional Spaces

Numerical-mathematical methods for Hyperbolic-Parabolic Systems: Investigation of Volter-Gursat Equation and Green's Function in Three-Dimensional Spaces

The vanishing learning rate asymptotic for linear L2-boosting

We investigate the asymptotic behaviour of gradient boosting algorithms when the learning rate converges to zero and the number of iterations is rescaled accordingly. We mostly consider L2-boosting for regression with linear base learner as studied in P. Bühlmann and B. Yu, J. Am. Statist. Assoc. 98 (2003) 324–339 and analyze also a stochastic version of the model where subsampling is used at each step (J.H. Friedman, Computat. Statist. Data Anal. 38 (2002) 367–378). We prove a deterministic limit in the vanishing learning rate asymptotic and characterize the limit as the unique solution of a linear differential equation in an infinite dimensional function space. Besides, the training and test error of the limiting procedure are thoroughly analyzed. We finally illustrate and discuss our result on a simple numerical experiment where the linear L2-boosting operator is interpreted as a smoothed projection and time is related to its number of degrees of freedom.

Well-posedness and blow-up results for a time-space fractional diffusion-wave equation

<abstract><p>In this paper, we demonstrate the local well-posedness and blow up of solutions for a class of time- and space-fractional diffusion wave equation in a fractional power space associated with the Laplace operator. First, we give the definition of the solution operator which is a noteworthy extension of the solution operator of the corresponding time-fractional diffusion wave equation. We have analyzed the properties of the solution operator in the fractional power space and Lebesgue space. Next, based on some estimates of the solution operator and source term, we prove the well-posedness of mild solutions by using the contraction mapping principle. We have also investigated the blow up of solutions by using the test function method. The last result describes the properties of mild solutions when $ \alpha\rightarrow1^- $. The main feature of the proof is the reasonable use of continuous embedding between fractional space and Lebesgue space.</p></abstract>

ON THE SOLUTION OF p,q TYPE EQUATIONS IN GRAND SOBOLEV SPACES WITH DOMINANT MIXED DERIVATIVES

Abstract. In this abstract with we study the help of variational method existence and uniqueness of solution of p,q type equations in grand Sobolev spaces with dominant mixed derivatives.