Approximate Integral Equation Research Articles

New momentum integral equation applicable to boundary layer flows under arbitrary pressure gradients

By incorporating the traditionally overlooked advective term in the wall-normal momentum equation, a new momentum integral equation is developed for two-dimensional incompressible turbulent boundary layers under arbitrary pressure gradients. The classical Kármán's integral arises as a special instance of the new momentum integral equation when the pressure gradient is weak. The new momentum integral equation's validity is substantiated by direct numerical simulation data. Unlike the classical Kármán's integral, which is limited to predicting wall shear stress within mild pressure gradients, the new momentum integral equation accurately computes wall shear stress across a broad range of pressure gradients, even in the presence of strong adverse pressure gradients that lead to flow separation. Moreover, a new pressure parameter $\beta _\kappa$ is introduced through examining terms in the new momentum integral equation. This parameter naturally quantifies the pressure gradient's influence on turbulent boundary layers and offers guidance for applying the classical Kármán's integral. Additionally, to facilitate experimental determination of wall shear stress under strong pressure gradients, an approximate integral equation is proposed that relies solely on easily measurable variables. Validation against direct numerical simulation data demonstrates that this simplified equation provides reasonably accurate estimates of wall shear stress in turbulent boundary layers experiencing strong pressure gradients.

Error asymptotic expansion for the numerical approximation of logarithmic-kernel integral equations on closed curves

Error asymptotic expansion for the numerical approximation of logarithmic-kernel integral equations on closed curves

Reconstruction of Two Functions in the Model of Vibrations of a String One End of Which Is Placed in a Moving Medium

The paper considers an inverse problem of determining the coefficients in the model of small transverse vibrations of a homogeneous finite string one end of which is placed in a moving medium and the other is free. The vibrations are simulated by a hyperbolic equation on an interval. One boundary condition has a nonclassical form. Additional data for solving the inverse problem are the values of the solution of the forward problem with a known fixed value of the spatial argument. In the inverse problem, it is required to determine the function in the nonclassical boundary condition and a functional factor on the right-hand side of the equation. Uniqueness and existence theorems for the inverse problem are proved. For the forward problem, conditions for unique solvability are established in a form that simplifies the analysis of the inverse problem. For the numerical solution of the inverse problem, an algorithm is proposed for the stage-by-stage separate reconstruction of the sought-for functions using the method of successive approximations for integral equations.

High-Accuracy Numerical Schemes for Solving Plane Boundary Problem for a Polyharmonic Equation and Their Application to Problems of Hydrodynamics

Boundary value problems are considered for harmonic, biharmonic equations, as well as the general polyharmonic equation for multiply connected domains on the plane. The problems are reduced to solving linear integral equations on boundary contours, which are assumed to be smooth. An algorithm for deriving an approximation of integral equations by a linear system is presented, taking into account the logarithmic singularities of the kernels of integral operators, through which integral equations are expressed. The algorithm uses the periodicity of functions defined on closed boundary contours. As the number of grid points increases, the approximation error decreases faster than the grid spacing to any fixed degree. Applications to solving problems of hydrodynamics, filtration and other problems of theoretical physics are considered.

The characteristics of laminar film condensation of pure vapor flow in a vertical tube

In this study, we investigate the characteristics of laminar film condensation of pure vapor flow in a vertical tube, taking into account (i) the vapor flow rate reduction owing to condensation, (ii) the finiteness of the vapor boundary-layer thickness, (iii) the curvature of the vapor layer, and (iv) the pressure variation. An approximate integral two-phase boundary-layer equation model is derived, and a condition that pressure recovery occurs is determined from the asymptotic analysis around the tube inlet. Furthermore, the present model is further simplified to obtain an analytical solution if the degree of subcooling is sufficiently large. The asymptotic solution for sufficiently large degrees of subcooling gives a good approximation to the numerical solution. The tube length of complete condensation is estimated. Comparison with vertical plane condensation demonstrates that the film thickness and the Nusselt number increased and decreased, respectively, by the vapor flow reduction for large degrees of subcooling.

Variational atomic model of plasma accounting for ion radial correlations and electronic structure of ions.

We propose a model of ion-electron plasma (or nucleus-electron plasma) that accounts for the electronic structure around nuclei (i.e., ion structure) as well as for ion-ion correlations. The model equationsare obtained through the minimization of an approximate free-energy functional, and it is shown that the model fulfills the virial theorem. The main hypotheses of this model are (1) nuclei are treated as classical indistinguishable particles, (2) electronic density is seen as a superposition of a uniform background and spherically symmetric distributions around each nucleus (system of ions in a plasma), (3) free energy is approached using a cluster expansion (nonoverlapping ions), and (4) resulting ion fluid is modeled through an approximate integral equation. In the present paper, the model is described only in its average-atom version.

A numerical simulation of non-linear stochastic Itô-Volterra integral equations driven by multi- fractional Gaussian noise

In this article, a discussion about stochastic integral equations driven by multi-fractional Gaussian (MFGN) noise has been done which determines the stochastic solutions to fundamental equations to the study of diffusion dynamics in vibrated granular systems. A Bernstein polynomials (BP) approach has been implemented to find an approximation of the stochastic integral equations under study which is efficient and has less computational cost. An example based on the integral equation has been solved to further more establish the efficiency of the polynomial scheme.

Bernstein polynomial approximation of non-linear stochastic Itô-Volterra integral equations driven by multi-fractional Gaussian noise

In this article, a discussion about stochastic integral equations driven by multi-fractional Gaussian (MFGN) noise has been done which determines the stochastic solutions to fundamental equations based on effective elastic properties of nanomaterials. A Bernstein polynomials (BP) approach has been implemented to find an approximation of the stochastic integral equations under study which is efficient and has less computational cost. An example based on the integral equation has been solved to further more establish the efficiency of the polynomial scheme.

Existence, uniqueness and approximation of nonlocal fractional differential equation of sobolev type with impulses

<abstract><p>This paper is concerned with the study of nonlocal fractional differential equation of sobolev type with impulsive conditions. An associated integral equation is obtained and then considered a sequence of approximate integral equations. By utilizing the techniques of Banach fixed point approach and analytic semigroup, we obtain the existence and uniqueness of mild solutions to every approximate solution. Then, Faedo-Galerkin approximation is used to establish certain convergence outcome for approximate solutions. In order to illustrate the abstract results, we present an application as a conclusion.</p></abstract>

Charge asymmetric electrolytes around a rigid cylindrical polyelectrolyte: A generalization of the capacitive compactness

The surface mean electrostatic potential and capacitive compactness of the electrical double layer surrounding an infinite rigid cylindrical polyelectrolyte are analysed for equally-sized -1:z+ aqueous electrolytes. Monte Carlo simulations, the non-linear Poisson–Boltzman equation, the modified Poisson–Boltzmann theory, and the hypernetted chain/mean spherical approximation integral equation are used to examine the role of multivalent cations (coions) when the properties of monovalent anions (counterions) are fixed. A non-zero mean electrostatic potential in the neighbourhood of an uncharged polyelectrolyte is predicted by the simulations and the modified Poisson–Boltzmann theory in the presence of multivalent cations. The concept of capacitive compactness is generalised to overcome the divergence found in the classical definition, when a non-zero potential at the point of zero charge is present. With a highly electrified colloidal surface an inversion is observed, as a function of colloidal charge, in the precedence of the mean electrostatic potential near the polyelectrolyte’s surface and of the capacitive compactness between -1:+1 and -1:z+ electrolytes.

Positivity of a weakly singular operator and approximation of wave scattering from the sphere

We investigate properties of a family of integral operators ℬ with a weakly singular compactly supported zonal kernel function on the surface S of the unit 3D sphere. The support is over a spherical cap of height h∈(0,2]. Operators like this arise in some common types of approximations of time domain boundary integral equations (TDBIE) describing the scattering of acoustic waves from the surface of the sphere embedded in an infinite homogeneous medium where h is directly related to the time step size. We show that the Legendre polynomials of degree ℓ≥0 satisfy ∫0h Pℓ(1−z2∕2)dz>0 for all h∈(0,2] and, using spherical harmonics and the Funk–Hecke formula for the eigenvalues of ℬ, that this is a key to unlocking positivity results for a subfamily of these operators. As well as positivity results we give detailed upper and lower bounds on the eigenvalues of ℬ and on ∫S u(x)(ℬu)(x)dx. We give various examples of where these results are useful in numerical approximations of the TDBIE on the sphere and show that positivity of ℬ is a necessary condition for these approximation schemes to be well defined. We also show the connection between the results for eigenvalues and the separation of variables solution of the TDBIE on the sphere. Finally we show how this relates to scattering from an infinite flat surface and Cooke’s 1937 result ∫0r J0(z)dz>0 for all r>0.

Analysis of direct current resistivity data using continuous wavelet transform

We have developed a new method for the direct current resistivity interpretation, based on the continuous wavelet transform (CWT) of electric potential-difference data. It exploits the main properties of the CWT, such as stability versus noise, and does not require a starting model or other a priori information such as a model weighting function or constraints. Because the approximate integral equation of the resistivity problem has the same form as the forward problem for potential fields, the authors analyze geoelectric data (with dipole-dipole configuration) using the wavelets belonging to the Poisson kernel semigroup. They find that the CWT analysis of the measured electric potential difference is able to identify buried bodies, defining their depth, position, and extent. Such parameters are estimated with no prior knowledge of the resistivity contrast between the bodies and the background. We consider several synthetic models, such as dikes, compact bodies, and contacts. In general, the depth and the lateral thickness of the buried bodies are estimated with good accuracy, using a diagram relating the singular point estimations to the different values of the dipole separation factor n. Thanks to the good results obtained from synthetic data, we test the method with data generated during laboratory experiments. In two laboratory-scale models, our method displays a better precision compared with smoothness-constrained least-squares inversion in identifying the exact position of the edges of a buried body. Finally, we find that combining CWT and inversion is advantageous: after constraining the inverse problem with a priori information from the CWT analysis, we obtain an improved inverse model.

Quadratic non-polynomial spline approximation for non-linear Volterra-Fredholm integral equations of the second kind

A quadratic non-polynomial spline is studied in depth in this paper in order to solve non-linear Volterra-Fredholm integral equations of type 2. This section introduces an algorithm, followed by example. As a result of a comparison of exact and approximate solutions, numerical results show convergence between the two. To demonstrate the technique’s feasibility and capability, numerical example is presented.

New existence results on nonlocal neutral fractional differential equation in concepts of Caputo derivative with impulsive conditions

In this article, we examine a nonlinear impulsive neutral fractional differential equation with nonlocal condition in an arbitrary Hilbert space. We obtain an associated integral equation and then consider a sequence of approximate integral equations by the projection of considered associated nonlocal neutral fractional integral equation onto finite dimensional space. The existence and uniqueness of an approximate solution by using analytic semigroup theory and the Fixed-point method are demonstrated. Convergence of the solutions of the approximate integral equation is proved. The Faedo-Galerkin approximation of the solution is studied and demonstrated some convergence results. Lastly, the application is presented to illustrate the theory of the main results.

Results on neutral differential equation of sobolev type with nonlocal conditions

In this work, we analyse the study of neutral fractional differential equation in an arbitrary Hilbert space. An associated integral equation is studied and approximate integral equation is obtained. We demonstrate the existence and uniqueness of an approximate solution by using analytic semigroup theory and the Fixed point method. In the application part, we discuss the approximation and the convergence results for such an approximation.

Initial value problem for elastic system in transversely isotropic inhomogeneous media

In this paper, we consider an initial value problem (IVP) for three dimensional elasticity system in a transversely isotropic inhomogeneous media. We will rewrite the problem in the form of Fourier images by means of Fourier transform method. After some arrangements, the problem is reduced to integral equations in the vector form. Using the properties of the vector integral equation and successive approximations method, an explicit formula for the solution of the IVP in transversely isotropic inhomogeneous media is constructed, and existence and uniqueness of the solution is stated. By a computational example, we illustrate the robustness of the method.

Extremely correlated superconductors

Superconductivity in the t-J model is studied by extending the recently introduced extremely correlated fermi liquid theory. Exact equations for the Greens functions are obtained by generalizing Gor’kov’s equations to include extremely strong local repulsion between electrons of opposite spin. These equations are expanded in a parameter λ representing the fraction of double occupancy, and the lowest order equations are further simplified near Tc, resulting in an approximate integral equation for the superconducting gap. The condition for Tc is studied using a model spectral function embodying a reduced quasiparticle weight Z near half-filling, yielding an approximate analytical formula for Tc. This formula is evaluated using parameters representative of single layer High-Tc systems. In a narrow range of electron densities that is necessarily separated from the Mott–Hubbard insulator at half filling, we find a typical Tc∼102 K.

Displacements of a surface of FGM-coated half-space heated in a circular area taking into account imperfect coating-substrate interface

The paper presents a solution to the axisymmetric problem of heating of a half-space with a functionally graded coating. The coating is assumed to be imperfectly bonding to the substrate that makes it possible to model various defects arising on the interface. The problem is reduced to solution of a dual integral equation. To solve the dual integral equation approximations for the kernel transform and all the compliance functions of a thermoelastic coated half-space was used. The solution was obtained in an analytical form. For the first time, analytical expressions are obtained for surface displacements and temperature distribution over the surface of a functionally graded coating both in the heating zone and outside it. Due to the use of the bilateral asymptotic method, the expressions are asymptotically exact for coatings of small and large relative thickness and of high accuracy for coatings of intermediate relative thickness including those constructed for complicated variation of thermoelastic properties in depth of the coating.

The rate of convergence of an iterative‐computational algorithm for second‐kind nonlinear Volterra integral equations with weakly singular kernels

This paper attempts to present an iterative‐computational method for solving weakly singular nonlinear Volterra integral equations (WSNVIEs) of the second type based on rationalized Haar wavelets (RHWs). The proposed algorithm does not need to solve any linear or nonlinear system for evaluating the wavelet coefficients. After a brief introduction of rationalized Haar functions (RHFs), the computational matrices of integration and fractional integral are applied to reduce the approximation of integral equations to some matrix algebraic equations. Next, the error analysis of the problem by using the two‐dimensional iterated projection operator is offered. We will prove that the rate of convergence of the proposed algorithm is , where h is the iteration number and β is the contraction constance. The method for any WSNVIE of the second kind with 0 ≤ β < 1 is convergent. The proposed method is computationally attractive, and comparing the results obtained of the known technique is more efficient.

An analytical approach to reconstruction of axisymmetric defects in pipelines using T(0, 1) guided waves

Torsional guided waves have been widely utilized to inspect the surface corrosion in pipelines due to their simple displacement behaviors and the ability of longrange transmission. Especially, the torsional mode T(0, 1), which is the first order of torsional guided waves, plays the irreplaceable position and role, mainly because of its non-dispersion characteristic property. However, one of the most pressing challenges faced in modern quality inspection is to detect the surface defects in pipelines with a high level of accuracy. Taking into account this situation, a quantitative reconstruction method using the torsional guided wave T(0, 1) is proposed in this paper. The methodology for defect reconstruction consists of three steps. First, the reflection coefficients of the guided wave T(0, 1) scattered by different sizes of axisymmetric defects are calculated using the developed hybrid finite element method (HFEM). Then, applying the boundary integral equation (BIE) and Born approximation, the Fourier transform of the surface defect profile can be analytically derived as the correlative product of reflection coefficients of the torsional guided wave T(0, 1) and the fundamental solution of the intact pipeline in the frequency domain. Finally, reconstruction of defects is precisely performed by the inverse Fourier transform of the product in the frequency domain. Numerical experiments show that the proposed approach is suitable for the detection of surface defects with arbitrary shapes. Meanwhile, the effects of the depth and width of surface defects on the accuracy of defect reconstruction are investigated. It is noted that the reconstructive error is less than 10%, providing that the defect depth is no more than one half of the pipe thickness.