## The boundary-value problem is reduced to a Fredholm integral equation of the first kind in the velocity magnitude on the free surface.

Impulsive impact of a submerged body

## We study how a boundary value problem with a nonlocal integral condition of the first kind for a mixed type equation with a singular coefficient in a rectangular domain depends on a numerical parameter occurring in the equation.

Nonlocal Boundary Value Problem with an Integral Condition for a Mixed Type Equation with a Singular Coefficient

## As it is well known the problem of solving the Fredholm integral equation of the first kind belongs to the class of ill-posed problems.

On the Solution of Fredholm Integral Equations of the First Kind

## Since the solution of the equation obtained by the ultrasonic attenuation model produces a Fredholm integral equation of the first kind, an inversion algorithm combining simulated annealing with genetic algorithm based on ultrasonic attenuation mechanism is proposed to solve the ill-posed problem in the inversion calculation of particle concentration.

Measurement of Particle Concentration by Multifrequency Ultrasound Attenuation in Liquid–Solid Dispersion

## The heat conduction problem can be formulated as Fredholm integral first kind equations.

Development the Numerical Method to Solve the Inverse Initial Value Problem for the Thermal Conductivity Equation of Composite Materials

## This paper presents an approximate analytical solution of the heat conduction problem for a two-layer plate under symmetric boundary conditions of the first kind.

Further boundary conditions in heat conduction problems in multilayer structures

## In the space H, a nonlinear operator equation of the first kind is studied, when the linear, nonlinear operator and the right-hand side of the equation are given approximately.

SOLUTION OF A NONLINEAR OPERATOR EQUATION OF THE FIRST KIND WITH A CONTINUOUS POSITIVE LINEAR OPERATOR

## We consider solving a system of semi-discrete first kind integral equations with a right-hand-side being a finite dimensional vector of sampling values and propose a regularization method for the system in a functional reproducing kernel Hilbert space (FRKHS), where the linear functionals that define the semi-discrete integral operator are continuous.

Regularization in a functional reproducing kernel Hilbert space

## The results show that: for the first kind of subsynchronous oscillation, the dominant influencing factors are the power grid strength and the current inner loop control parameters; with the decrease of the power grid strength, the current inner loop control parameters decrease, weakening the system damping, and the oscillation frequency gradually decreases; for the second kind of subsynchronous oscillation, the dominant influencing factors are the current inner loop control parameters, with the decrease of the power grid strength The damping of the system is weakened and the oscillation frequency increases slightly.

Comparative analysis of two kinds of subsynchronous oscillation of direct drive PMSG based wind farm dominated by inner current loop

## In this paper, we present Padé approximations of some functions involving complete elliptic integrals of the first kind K ( x ) $K(x)$ , and motivated by these approximations we also present the following double inequality: 1 − x 2 1 − x 2 + x 4 62 < 2 e 2 π K ( x ) − 1 ( 1 + 1 1 − x 2 ) < 1 − 96 100 x 2 1 − 96 100 x 2 + x 4 64 , x ∈ ( 0 , 1 ).

Some Padé approximations and inequalities for the complete elliptic integrals of the first kind

## We consider solving a system of semi-discrete first kind integral equations with a right-hand-side being a finite dimensional vector of sampling values and propose a regularization method for the system in a functional reproducing kernel Hilbert space (FRKHS), where the linear functionals that define the semi-discrete integral operator are continuous.

Regularization in a functional reproducing kernel Hilbert space

## Finally, some numerical experiments with a Fredholm first kind integral equation are carried out to demonstrate the performance of our designed hypothesis testing statistics in an ill-posed model.

The hypothesis testing statistics in linear ill-posed models

## The heat conduction problem can be formulated as Fredholm integral first kind equations.

Development the Numerical Method to Solve the Inverse Initial Value Problem for the Thermal Conductivity Equation of Composite Materials

## However, there are no relations binding the values of all components of the stress-strain state at the half-plane boundary This paper shows that arbitrary formulation of the boundary conditions considerably simplifies integral equations and transfers them to the class of Fredholm’s first kind equations.

Solution of three basic problems of elasticity for half-plane

## For each configuration of the barrier, the problem is reduced to solving an integral equation or a coupled integral equation of first kind involving horizontal component of velocity below or above the barrier and above the step.

Water wave propagation over an infinite step in the presence of a thin vertical barrier

## For each position of the barrier, the problem is reduced to solving three weakly singular integral equations of first kind involving horizontal component of velocity above the two edges of the trench and below the barrier.

Wave propagation over a rectangular trench in the presence of a partially immersed barrier

## The RRA-algorithm is a variation of the Tikhonov method and it is suggested as a regularization method for first kind Fredholm integral equation.

Verification of the RRA-Algorithm Regularization for the Analysis of Stochastic Structures in Bioinformatic Intelligent Systems

## The downward continuation (DWC) of airborne gravity data usually adopts the Poisson integral equation, which is the first kind Fredholm integral equation.

Downward continuation of airborne gravity data based on iterative methods

## In specific cases the Legendre wavelet Galerkin method is used for the numerical solution when surface is subjected to the first kind boundary condition with constant latent heat and thus the results obtained are compared with Turkyilmazoglu (2018) results and found in good compliance.

Wavelet based numerical approach of non-classical moving boundary problem with convection effect and variable latent heat under the most generalized boundary conditions

## In both cases our numerical simulation strategy involves approximating the fractal screen $\Gamma$ by a sequence of smoother "prefractal" screens, for which we compute the scattered field using boundary element methods that discretise the associated first kind boundary integral equations.

Boundary element methods for acoustic scattering by fractal screens

## In this paper, under certain assumptions and transformations, the spacecraft relative equations of motion, in terms of orbital element differences, is approximated into the nonlinear first kind Abel-type and Riccati-type differential equations.

Closed form parametric solutions of nonlinear Abel-type and Riccati-type spacecraft relative motion

## Of the integral equations considered here are first kind Abel integral equation and integral equation with log kernel and hypersingular integral equations of first and second kind.

Line element method of solving singular integral equations

10.1155/2021/9260828

## Finally, an application from our results using the modified Bessel function of the first kind is established as well.

Fractional Hermite–Jensen–Mercer Integral Inequalities with respect to Another Function and Application

10.1016/J.ENSM.2021.06.006

## Herein, spinel Zn3V3O8 is presented as the first kind of vanadium-based material as zinc supplied cathode for AZIBs.

Spinel Zn3V3O8: A High-Capacity Zinc Supplied Cathode for Aqueous Zn-Ion Batteries

10.1080/09720502.2020.1860290

## This research investigates two methods which use accelerating of Al Tememe’s inverse triangular acceleration of the first kind [1], by using “inverse sine triangular acceleration rule for Al-Tememe of the first kind” in the first method, and “inverse tangent triangular acceleration rule for Al-Tememe of the first kind” in the second method ; both of them have a major error of the first kind and these are beneficial to find double integrations with the continuous integrands specified numerically by the midpoint rule on both internal and external dimensions x and t respectively by above methods.

Accelerations Asi(MM) and Ati(MM) to find double integrals numerically

10.1155/2021/3115424

## We give some relationships between degenerate poly-Frobenius-Euler polynomials and degenerate Whitney numbers and Stirling numbers of the first kind.

Construction on the Degenerate Poly-Frobenius-Euler Polynomials of Complex Variable

## The latter provides evidence about further generalizations of coherent states, built from the Susskind-Glogower ones by extending the indexes of the Bessel functions of the first kind and, alternatively, by employing the modified Bessel functions of the second kind.

Generalized Susskind–Glogower coherent states

10.26886/2414-634x.8(44)2020.2

## The given conditions and criteria, the fulfillment of which reduces the problem of heat transfer under boundary conditions of the third kind to a problem under boundary conditions of the first kind.

NUMERICAL AND ANALYTICAL SOLUTION OF THE HEAT EXCHANGE PROBLEM

10.1088/1742-6596/1909/1/012059

## The resulting excitation force is expressed as a series of Bessel functions of the first kind.

Reduction of the unbalance excited vibrations during the resonance passage of a rotor

10.3390/MATH9030281

## Then, some useful relations including the Stirling numbers of the second and the first kinds, the usual Fubini polynomials, and the higher-order Bernoulli polynomials are derived.

Two-Variable Type 2 Poly-Fubini Polynomials

10.3934/math.2021660

## The purpose of the present paper is to give sufficient conditions for the families of integral operators, which involve the normalized forms of the generalized Lommel functions of the first kind to be univalent in the open unit disk.

Univalence and convexity conditions for certain integral operators associated with the Lommel function of the first kind

10.1016/j.apm.2021.06.022

## The relations among individuals are divided into three categories: strong links (SLs), the first kind of weak links (FWLs) and the second kind of weak links (SWLs).

Dynamical analysis of the SIS epidemic model in cluster events

10.1093/qjmam/hbab007

## A convolution integral equation of the first kind and integro-differential equation of the second kind with the kernel $e^{-\gamma |y-\eta|}$ on a finite and semi-infinite interval are analyzed.

Integral and integro-differential equations with an exponential kernel and applications

10.1155/2021/7513406

## Also, we study the necessary conditions for finding the optimal strategies for each government to fight terrorism; we discuss the existence of the solution of the formulated problem and the stability set of the first kind of the optimal strategies.

A Study on Nash-Collative Differential Game of N -Players for Counterterrorism

## The first kind of method modifies the baseline method by adding the road centerline extraction task branch based on ordinal regression.

10.5194/essd-2021-143-rc2

## The second comparison is of the first kind where NO2 products from Brewer MK IV and Pandora were compared over a year-long time period.

Comment on essd-2021-143

10.1088/1757-899X/1150/1/012008

## High-temperature gas-cooled reactor (HTGR) is the first kind of power plant with fourth-generation features in the world, and a large amount of graphite and carbon material is adopted as structural material and fuel matrix material, including boron-containing carbon (BC) and isostatic pressure graphite (IG-110).

Study on the water vapor adsorption Characteristics of Carbon materials in HTGR

10.54113/j.sust.2021.000001

## Base Isolation (BI) systems represent the first kind of control devices applied to civil structures.

Design issues for smart isolation of structures: past and recent research

10.3390/MATH9080819

## Moreover, the closer the point of application of the pulsed force to the middle of the elastic body under boundary conditions of the first kind is greater (for boundary conditions of the second kind closer to the end).

Influence of Impulse Disturbances on Oscillations of Nonlinearly Elastic Bodies

10.15407/fmmit2021.33.012

## A method is proposed for constructing a spline function, which has on the indicated lines the same discontinuities of the first kind as the approximated discontinuous function, and a method for finding the Fourier coefficients of the indicated continuous or differentiable function.

Analysis of the results of a computational experiment to restore the discontinuous functions of two variables using projections

10.3390/math9010083

## Consequently, this new procedure is especially useful for solving non-homogeneous Fredholm integral equations of the first kind.

A Picard-Type Iterative Scheme for Fredholm Integral Equations of the Second Kind

10.1016/j.apnum.2020.07.005

## Numerical schemes for nonlinear weakly singular Volterra and Fredholm integral equations of the first kind are rarely investigated in the literature.

Approximate solution of some nonlinear classes of Abel integral equations using hybrid expansion

10.17721/2706-9699.2021.1.17

## This system of boundary integral equations consists of singular integral equations of the first kind and integral equations of the second kind with a logarithmic singularity.

MATHEMATICAL MODEL OF WAVE DIFFRACTION BY THE SYSTEM OF STRIPES WITH DIFFERENT VALUES OF SURFACE IMPEDANCE

10.1016/J.IJMECSCI.2021.106414

## To deal with the spatial discretization of the displacement field, the displacements of plate are expressed as the expansions of Chebyshev polynomials of the first kind (CPOFK).

Dynamic analysis of arbitrarily restrained stiffened plate under moving loads

10.29235/1561-2430-2021-57-1-23-32

## On the bottom of the boundary, the Cauchy conditions are specified, and the second of them has a discontinuity of the first kind at a point.

The classical solution of the mixed problem for the one-dimensional wave equation with the nonsmooth second initial condition

10.1016/J.SSI.2021.115699

## According to Wagner and Schmalzried, one has to distinguish between rate constants of the first kind (“practical” Tammann constant), in the case of simultanous and coupled growth of multiple product phases, and rate constants of the second kind (“true” Tammann constant), in the case of the uncoupled and only growth of one product phase in equilibrium with the adjacent phases.

Reaction kinetics in the system Y2O3/Al2O3 – A solid state reaction forming multiple product phases investigated by using thin film techniques

10.1038/s41598-021-91871-3

## Here we use a fast and simple L-curve method to estimate the TR parameter (λ) for regularization of the Fredholm integral equations of first kind in impedance.

Computation of distribution of relaxation times by Tikhonov regularization for Li ion batteries: usage of L-curve method

10.21883/FTT.2021.07.51050.272

## Based on the three-phase model of simple matter reviewed to the appearance and disappearance of the S-loop of a phase transition (PT) of the first kind on isotherme of state equation in the range of PT crystal-liquid (C-L).

О критической точке плавления простого вещества

10.3103/S0025654421010131

## Since in this model of friction there are no discontinuities of the first kind in velocity, the problem is mathematically simpler than problems with one-dimensional (according to Coulomb) dry friction.

Flat Dynamics of a Homogeneous Parallelepiped with Dry Friction

10.3390/PROCEEDINGS2021077016

## We then discuss some modern methods for screening through the analysis of online data streams, either through active or passive data collection, the difference being that in the first kind, the subject is active in the data collection process; in the second kind, secondary data streams are used, such as Twitter tweets, Facebook posts, or Google searches.

Testing and Screening

10.3997/2214-4609.202150054

## As a rule, it is the product of a smooth function of bounded variation by a Bessel function of the first kind.

Intrusive Magmatism, Influence on The Formation of Generation-Accumulation Hydrocarbon Systems of The Baikit Anteclise (Eastern Siberia)

10.1108/MEQ-01-2021-0002

## Originality/valueThis is first kind of study to identify CSFs for implementing environmentally sustainable practices in the utilities sector.

Modeling and analyzing critical success factors for implementing environmentally sustainable practices in a public utilities organization: a case study

10.1016/J.COMPELECENG.2021.107008

## For the first kind of scenario, OS-ELM is inserted an alternative output node which can be extended whenever the new class instances are received.

Online sequential extreme learning machine with the increased classes

10.1016/J.JQSRT.2021.107889

## Here, we address the theory of photophoresis for a more general class of on-axis axisymmetric beams of the first kind.

Generalized Lorenz-Mie theory in the analysis of longitudinal photophoresis of arbitrary-index particles: On-axis axisymmetric beams of the first kind

10.1109/ICPR48806.2021.9413124

## The first kind is obtained by imposing the $\ell_{2,1}$ constraint on the projection matrix in order to perform feature ranking.

Feature Extraction by Joint Robust Discriminant Analysis and Inter-class Sparsity

10.1088/1742-6596/1999/1/012136

## The Sturm-Liouville method was used to solve this problem by reduce it to the integral equation for first kind.

Development the Regularization Computing Method for Solving Boundary Value Problem to Heat Equation in the Composite Materials

10.33042/2522-1809-2021-1-161-8-18

## For the elastic modulus of the first kind, the excess varies from 1.

ANALYSIS OF ANALYTICAL MODELS AND THE DEPENDENCES REALIZED BY THEM FOR DEFINITION OF MECHANICAL CHARACTERISTICS OF COMPOSITE FILLERS

10.1134/S2517751621050061

## 5) in the limiting current density due to electroconvection, which occurs according to the mechanism of electroosmosis of the first kind.

Influence of Titanium Dioxide Particles Percentage in Modifying Layer on Surface Properties and Current-Voltage Characteristics of Composite Cation-Exchange Membranes

## We review some structural details of Promise Theory, applied to Promises of the First Kind, to assist in the comparison of Promise Theory with other forms of physical and mathematical modelling, including Category Theory and Dynamical Systems.

Promise Theory and the Alignment of Context, Processes, Types, and Transforms

10.1109/ACCESS.2020.3047627

## Subsequently, a combination of the nonnegative least squares (NNLS) algorithm and post facto smoothing method has been employed to derive the discretized solution of the Fredholm integral equation of the first kind.

A Build-In Data Inversion Method to Retrieve Aerosol Size Distributions for a Portable Ultrafine Particle Sizer (PUPS)

10.3390/ecerph-3-09109

## It is the first kind of study to analyze the facial expressions and behavioural measures (coughing, sneezing, flu and hand movements).

The Development of Deep Learning AI based Facial Expression Recognition Technique for Identifying the Patients With Suspected Coronavirus

## The method used in the code is based on the Bessel function of the first kind of order zero.

WEST tokamak hard x-ray tomography inversion

10.1155/2021/6615976

## Finally, to validate the superiority of the obtained results, an application is provided to solve a first kind of Fredholm type integral equations.

On Fuzzy b-Metric-Like Spaces

10.32523/2306-6172-2021-9-1-69-86

## We consider the Dirichlet condition at the left end of the rod (at x = 0) corresponding to the heating of this end and the homogeneous condition of the first kind at the right end (at x = 1) corresponding to cooling during interaction with the environment as boundary conditions.

An inverse heat conduction boundary problem for a two-part rod with different thermal conductivity

10.20310/2686-9667-2021-26-133-35-43

## On the lateral surface of a cylidrical region, the homogeneous boundary conditions of the first kind are given.

On an ill-posed boundary value problem for the Laplace equation in a circular cylinder

## A method is presented for computing exponential spectro-temporal modulation, showing that it can be expressed analytically as a sum over linearly offset sidebands with component amplitudes equal to the values of the modified Bessel function of the first kind.

Exponential spectro-temporal modulation generation

10.1016/J.PHYSA.2021.126208

## We give sufficient conditions so that the ground states of a given Hamiltonian are stable under perturbations of the first kind in terms of order preservation.

Stability of energy landscape for Ising models

10.1007/s13398-021-01027-1

## In this paper, we obtained a simple rational approximation for $${\mathcal {K}}_{a}(r)$$ K a ( r ) : \begin{aligned} \frac{2}{\pi }{\mathcal {K}}_{a}(r) \, \mathbf {>}\frac{\left( 1-3a+3a^{2}\right) r\mathbf {^{\prime }}+\left( 1+3a-3a^{2}\right) }{\left( 1+a-a^{2}\right) r\mathbf {^{\prime }}+\left( 1-a+a^{2}\right) }\; \end{aligned} 2 π K a ( r ) > 1 - 3 a + 3 a 2 r ′ + 1 + 3 a - 3 a 2 1 + a - a 2 r ′ + 1 - a + a 2 holds for all $$r\in (0,1),$$ r ∈ ( 0 , 1 ) , where $${\mathcal {K}}_{a}(r)=$$ K a ( r ) = $$\frac{\pi }{2}F\left( a,1-a;1;r^{2}\right) =\frac{\pi }{2}\sum _{n=0}^{\infty }\frac{ \left( a\right) _{n}\left( 1-a\right) _{n}}{(n!)^{2}}r^{2n}$$ π 2 F a , 1 - a ; 1 ; r 2 = π 2 ∑ n = 0 ∞ a n 1 - a n ( n ! ) 2 r 2 n is the generalized elliptic integral of the first kind, and $$r\mathbf {^{\prime }=} \sqrt{1-r^{2}}$$ r ′ = 1 - r 2.

A simple rational approximation to the generalized elliptic integral of the first kind

10.15507/2079-6900.23.202101.28-42

## In the article, the conditions for the existence of limit cycles of the first kind are obtained for self-tuning systems with delay, which, in turn, determine the conditions for the occurrence of hidden synchronization modes in such systems.

Dynamics of the mathematical model of phase-locked systems with delay