Simplified Integral Equation Research Articles

Mathematical Modeling of Cell Growth via Inverse Problem and Computational Approach

A simple cell population growth model is proposed, where cells are assumed to have a physiological structure (e.g., a model describing cancer cell maturation, where cells are structured by maturation stage, size, or mass). The main question is whether we can guarantee, using the death rate as a control mechanism, that the total number of cells or the total cell biomass has prescribed dynamics, which may be applied to modeling the effect of chemotherapeutic agents on malignant cells. Such types of models are usually described by partial differential equations (PDE). The population dynamics are modeled by an inverse problem for PDE in our paper. The main idea is to reduce this model to a simplified integral equation that can be more easily studied by various analytical and numerical methods. Our results were obtained using the characteristics method.

Quasi-Static Surface-PEEC Modeling of Electromagnetic Problem With Finite Dielectrics

This paper presents a quasi-static surface-based partial element equivalent circuit (PEEC) model for electromagnetic problems consisting of coupled conductors and heterogeneous dielectrics of finite size. The PEEC model is based on the surface equivalent principle. Unlike the traditional surface-based PEEC models, in which the integral equations are set up by enforcing the field continuity in the true field region, the homogeneous integral equations in this model are obtained by enforcing the tangential null field condition in the null field regions. Simplified integral equations are obtained under the quasi-static assumption, in which the circuit elements carrying electric currents and magnetic charges on the dielectric surfaces are vanished. Consequently, the size of the subcircuit describing dielectrics is reduced significantly compared to the full-wave surface-PEEC (S-PEEC) model. Since the quasi-static S-PEEC model only contains frequency-independent circuit elements, it can be conveniently used for time-domain simulation. Three numerical examples are presented to validate the new PEEC model for typical embedded RF passive components, packaging and interconnection problems in both the frequency domain and the time domain. Excellent agreement is observed between the results of the proposed PEEC model, S-PEEC model, and those of commercial software within the quasi-static frequency range.

The quark anti-quark potential and the cusp anomalous dimension from a TBA equation

We derive a set of integral equations of the TBA type for the generalized cusp anomalous dimension, or the quark antiquark potential on the three sphere, as a function of the angles. We do this by considering a family of local operators on a Wilson loop with charge L. In the large L limit the problem can be solved in terms of a certain boundary reflection matrix. We determine this reflection matrix by using the symmetries and the boundary crossing equation. The cusp is introduced through a relative rotation between the two boundaries. Then the TBA trick of exchanging space and time leads to an exact equation for all values of L. The L = 0 case corresponds to the cusped Wilson loop with no operators inserted. We then derive a slightly simplified integral equation which describes the small angle limit. We solve this equation up to three loops in perturbation theory and match the results that were obtained with more direct approaches.

A Theoretical Approach to Room Acoustic Simulations Based on a Radiative Transfer Model

A theoretical approach to room acoustic simulations based on a radiative transfer model is developed by adapting the classical radiative transfer theory from optics to acoustics. The proposed acoustic radiative transfer model expands classical geometrical room acoustic modeling algorithms by incorporating a propagation medium that absorbs and scatters radiation, handling both diffuse and non-diffuse reflections on boundaries and objects in the room. The main scope of this model is to provide a proper foundation for a wide number of room acoustic simulation models, in order to establish and unify their principles. It is shown that this room acoustic modeling technique establishes the basis of two recently proposed algorithms, the acoustic diffusion equation and the room acoustic rendering equation. Both methods are derived in detail using an analytical approximation and a simplified integral equation of the proposed method, respectively, allowing a clear definition of the underlying assumptions, limitations, advantages and disadvantages.

Feasibility of simplified integral equation modeling of low-frequency marine CSEM with a resistive target

We have assessed the accuracy of a simplified integral equation (SIE) modeling approach for marine controlled-source electromagnetics (CSEM) with low applied frequencies and a resistive target. The most computationally intensive part of rigorous integral equation (IE) modeling is the computation of the anomalous electric field within the target itself. This leads to a matrix problem with a dense coefficient matrix. It is well known that, in general, the presence of many grid cells creates a computational disadvantage for dense-matrix methods compared to sparse-matrix methods. The SIE approach replaces the dense-matrix part of rigorous IE modeling by sparse-matrix calculations based on an approximation of Maxwell’s equations. The approximation is justified theoretically if a certain dimensionless parameter [Formula: see text] is small. As opposed to approximations of the Born type, the validity of the SIE approach does not rely on the anomalous field to be small in comparison with the background field in the target region. We have calculated [Formula: see text] for a range of parameter values typical for marine CSEM, and compared the SIE approach numerically to the rigorous IE method and to the quasi-linear (QL) and quasi-analytic (QA) approximate solutions. It is found that the SIE approach is very accurate for small [Formula: see text], corresponding to frequencies in the lower range of those typical for marine CSEM for petroleum exploration. In addition, the SIE approach is found to be significantly more accurate than the QL and QA approximations for small [Formula: see text].

Ion temperature gradient driven instability in high beta plasmas of a sheared slab

A set of simplified integral eigenmode equations with neglecting of the magnetic gradient drift effects are derived and employed to investigate the ion temperature gradient modes in plasmas of sheared slab geometry. Perturbations of both perpendicular and parallel components of magnetic vector potential are considered, as well as the perturbation of the electrostatic potential. The characteristics of the ion temperature gradient instability are studied with the equations and the corresponding computer code, and compared with those with the drift effects included. The results show that the magnetic gradient drift has a block effect on ion temperature gradient mode stabilization, and the perpendicular component of the perturbed magnetic vector potential has strong stabilizing effects on ion temperature gradient modes. Nevertheless, the results from the simplified equations are qualitatively in agreement with those from the complete equations and the differences between them are not significant quantitatively. The simplified equations have the advantages of saving computer time and being well suited to analytic methods.

Simplified integral equation for analysing the printed strip dipole antenna

A new simple space-domain integral equation for calculating the current distribution of a printed strip dipole antenna is presented. By taking into account the effect of the dielectric substrate on the integral equation of the isolated strip dipole antenna in free space, and applying the equivalent radius technique, a very simple space-domain integral equation for a printed strip dipole antenna can be obtained. The validity of the equation is demonstrated through the calculation of the input impedance of a printed-strip monopole antenna, both with and without the parasitic element, on an infinite ground-plane and on a finite circular ground-plane. The theoretical results are compared with those from experiment, and also with data published in the literature. Good agreement is obtained.

The boundary element modelling of the human body exposed to the ELF electromagnetic fields

In this paper a cylindrical model of human body exposed to the extremely low frequency (ELF) electromagnetic field is presented. The analysis is based on the solution of the simplified integral equation for thick wires. The numerical solution of the integral equations is performed by the Galerkin–Bubnov variant of the boundary element method. Several numerical results for the ELF exposures are presented.

Contact angles

Young [Philos. Trans. (1805)] derived an equation for the contact angle between a liquid–gas interface and a solid boundary, but doubts have been raised about its validity. This issue is reexamined on the basis of a new integral equation for the interface [J. B. Keller and G. J. Merchant, J. Stat. Phys. 63, 1039 (1991)]. The equation is solved asymptotically by the method of matched asymptotic expansions for small values of the range of intermolecular forces divided by a typical macroscopic length. The leading term in the outer expansion satisfies the Young–Laplace partial differential equation for the interface. The leading term in the boundary-layer expansion satisfies a simplified integral equation. Matching the solutions of these two equations shows that the slope angle at the solid boundary, of the leading term in the outer expansion, is indeed given by the Young equation. Numerical solutions of the boundary-layer integral equation are presented to show how the interface varies near the solid boundary.

Doppler-shift decoupling of radiation reabsorption in expanding laser-produced plasmas

To investigate the influence of radiation reabsorption on the level populations of ions in an expanding laser-produced plasma of intermediate density, we start with the system of rate equations for the population densities coupled with the line-radiation transport equation, the dynamical Doppler effect due to the differential macroscopic velocity field included. In a physically motivated approximation, for spatially varying absorption and emission, and general three-dimensional plasma geometry, an integral equation describing the effect of Lyman-a radiation reabsorption on the spatial behavior of the population density of the upper resonance level is derived. Assuming a sufficiently large velocity gradient so that the Doppler-induced frequency shift dominates the linewidth, after asymptotically evaluating the frequency integral involved in the kernel we are led to a simplified integral equation exhibiting the reduction of radiation reabsorption by Doppler decoupling. In particular, in the case of a cylindrical, radially expanding laser plasma we discuss this Fredholm equation for the reabsorption-influenced population density.

A simplified integral equation for adsorption of gas mixtures on heterogeneous surfaces

The multiple integral representing the overall isotherm for adsorption of gas mixtures on heterogeneous surfaces is transformed to a single integral, which is promising for predicting the mixed-gas adsorption by means of single-gas adsorption parameters. This transformation is possible when the adsorption energies of components for various adsorption sites show a certain type of correlation.

Solution of the Percus–Yevick equation for hard spherocylinders. I. The entire pair correlation function

The Percus–Yevick equation for hard spherocylinders has been numerically solved using a recent algorithm to calculate the shortest distances between rods proposed by ourselves. The equation is solved for four different reduced densities to η=0.3879 and length-to-breadth ratio L*=1.0. Comparison with available Monte Carlo results for some selected orientations shows moderate agreement. Dependence of the pair correlation function and direct correlation function on mutual orientations and density are analyzed for these orientations. We found that none of the current approximations used in perturbation theories or in simplified integral equations can give even a qualitative description of the behavior of correlation functions for the above elongation L*.

Three-dimensional unsteady transonic flow: An integral equation formulation

The unsteady transonic small perturbation differential equation has been converted into an integro-differential equation by application of the classical Green's function method. After assuming that the motion consists of infinitesimal perturbations around a thin, nearly-planar body a simplified integral equation is obtained for the streamwise velocity component, which is suitable for fast numerical computations.

Simplified transonic integral equations for lifting profiles and wings

The basic small perturbation integral equations for steady inviscid transonic flow past a thin cambered profile at small incidence have been simplified here using Oswatitsch substitution for the velocity field in terms of that on the profile axis. The resulting equations contain only one-dimensional singular integrals. These equations represent simplification of the equations of Nixon and Hancock [3], from which the two-dimensional camber integrals have been eliminated. Also the corresponding simplified equations for a thin lifting wing have been derived, which contain only two-dimensional singular integrals.

Statistical Mechanical Theory of Rod-Like Molecular Fluids with Kihara-Type Interactions. I. Molecular Distribution Functions

Abstract A simplified integral equation for the pair distribution function of linear molecules interacting via a potential depending only on the shortest distance between molecular cores, is written down. The theory is developed starting from the Ornstein-Zernike equation and the Percus-Yevick closure relation, both for non-spherical particles. Two basic assumptions are made. Firstly the molecular pair distribution function is supposed to depend explicitly on the shortest distance but not on molecular orientations; secondly the direct correlation function inside the integral in the Ornstein-Zernike equation is approximated by some average value conveniently defined. The new integral equation is numerically solved for CO2 and N2 assuming a Kihara potential. The main features of the distribution function, particularly its behaviour with temperature and density, are also investigated. The results obtained seem to agree well with a very probable structural model for liquid CO2 and a reasonable model for N2.

On the analysis of the umbrella antennas using the Hallen-type integral equation

Umbrella antennas of heights 0.125\lambda and 0.5141\lambda are numerically and experimentally analyzed, and the effect of the junction on the current distribution is discussed. A simplified Hallen-type integral equation and an integral equation for the scalar potential of the antenna composed of straight wires are formulated and applied to umbrella antennas. The integral terms in these equations are analytically reduced to single integrals, and the extended boundary condition that the electric field vanishes on the axis of the antenna is used.

A simplified integral equation method for the calculation of the eigenvalues of Helmholtz equation

AbstractAn integral equation method is presented for the numerical calculation of the eigenvalues of the scalar Helmholtz equation. By using a particular solution instead of Green's function, the calculations could be simplified due to the elimination of complex numbers.

Contribution to the Theory of Weak Coupling Superconductors

We have managed to simplify the Eliashberg equations for the case of weak coupling superconductors. Further, an explicit approximate analytic solution of the simplified equations has been obtained for the zero temperature gap edge. An expression for the critical temperature is also given. It is found to be of the BCS form and allows the BCS parameter N(0)V to be identified with a simple function of the normal state properties. The validity of our simplified integral equations and of our analytic solution is established by comparison with numerical solutions of the complete Eliashberg equations. The equations are used to discuss the effect of pressure on the gap, the isotope effect, as well as other properties.

Comparisons Between New Methods for Reduction of Ionograms to Electron Density Profiles

The uniform lamination analysis method is used for the solution of the simplified integral equation which relates the electron density profiles in the ionosphere to the average measured virtual heights, taking the wave refractive index as the kernel function. The arbitrary auxiliary function is chosen to be ϕ(fN) = µ(fN), which simplifies the calculation as compared to taking ϕ(fN) = fN. Four examples are calculated for both cases and compared with the original assumed electron density profiles. Accuracy and required computer times are discussed.

Simplified analysis of ionospheric h'( f) records, using wave refractive index

A new simplified method for the analysis of the ionospheric h'( f) records is developed, using the wave refractive index μ instead of the group refractive index μ'. The average virtual height curves are the required experimental data for this method. The new simplified integral equation applies for the ordinary and the extraordinary waves. The final solution involves expansions of the functions in power series.