System Of Linear Integral Equations Research Articles

Sign-Definiteness of the Integral Cost on a System of Linear Integral Equations of Volterra Type

We study a question of sign-definiteness of a quadratic cost subject to the system of linear integral equations which has no Legendre term, i.e. is totally degenerate. Therefore, it cannot be studied by the clasical theory of quadratic forms. However, applying some generalizations of the known Goh transform, it is possible to reduce the given functional to a nondegenerate one, and hence, to obtain new necessary conditions of its nonnegativity. Here we reduce the original problem to a form which differs from the “classical” one only by parameter which defines endpoint value of a new control, by using Goh transformation (Goh (1966)). Thus, a new class of problems with reduced cost is investigated and corresponding necessary optimality conditions are obtained.

Метод описания немарковских процессов, задаваемых системой линейных интегральных уравнений

Метод описания немарковских процессов, задаваемых системой линейных интегральных уравнений

On systems of linear functional equations of the second kind in L 2

We consider the systems of linear functional equations of the third kind with measurecompact operators in L 2 and the general systems of linear integral equations of the third kind in L 2. We propose a method for reducing these systems either to equivalent integral equations of the first kind with nuclear operators or to equivalent integral equations of the second kind with quasidegenerate Carleman kernels. Various exact and approximate methods for solving the arising integral equations are applicable.

Computational Analysis of a Nuclear Reactor Cell in the Linear-Anisotropic Approximation by the Generalized Method of First Collisions Probabilities

This work is devoted to the development of a method for solving the Boltzmann kinetic equation for neutron transport. The method is focused on solving an integral equation for neutron transport in a reactor cell with a complex geometry and different boundary conditions. An algorithm is proposed for solving the problem taking account of anisotropic scattering in the linear-anisotropic approximation. The approach is based on expanding the neutron flux in a system of orthogonal two-dimensional polynomials in each uniform zone of a heterogeneous cell. This expansion reduces the system of linear integral equations to a system of linear algebraic equations. The relations required to calculate the coefficients in the equations (six-fold integrals) as well as the computational algorithm are presented. Calculations of cylindrical and cluster cells are presented. The calculations are compared with the surface pseudosource method. It is shown that the results have advantages over the method of first collisions probabilities.

NUMERICAL METHODS FOR SOLVING FUZZY FREDHOLM INTEGRAL EQUATION OF THE SECOND KIND

In this article some numerical methods, namely: the Taylor expan- sion method and the Trapezoidal method, have been investigated and imple- mented to solve a fuzzy Fredholm integral equation of the second kind. Conse- quently, we convert a linear fuzzy Fredholm integral equation of the second kind into a linear system of integral equations of the second kind in crisp case. To demonstrate the credibility of these numerical schemes we consider a numerical test example. The numerical results show to be in a close agreement with the exact solution.

On some transmission problems in a plane corner

We consider the transmission problem in Sobolev Slobodetskii spaces for two model general elliptic pseudo differential equations in a plane, where a one equation is given inside a corner, and another one is outside. Starting from general solutions of these equations we consider certain transmission conditions on angle sides. For some cases this transmission problem was reduced to an equivalent system of one-dimensional linear integral equations.

Analytical model of enemy's artillery battery engagement with account for timeliness of its detection

The article presents mathematical model of self-propelled artillery battery on fire position functioning with account for timeliness of its detection, description of its functioning is given by semi-Markov processes and system of linear integral equations. General formula for defining timeliness of fire engagement is obtained, design ratios on condition that target time on fire position is distributed by normal law, and time for engagement preparation by exponential rule are found. Calculation of effectiveness indicator of enemy’s self-propelled artillery battery damage level is done.

Analysis and Computation of the Pressure Field of Preatherosclerotic Transport in the Artery Wall

Prerequisite to atherogenesis is low-density lipoprotein (LDL) cholesterol transport from the blood into the artery wall across its endothelial cell (EC) lining. Whereas most EC junctions are tight enough to preclude LDL passage, some $({\approx}1/2000-6000)$ rare ECs have temporarily-widened junctions (typically ${\approx}10^{-5}$ of the problem's radial scale) that allow passage of large molecular aggregates like LDL. In a purely diffusive transport model, Laplace's equation describes the steady tracer profile throughout the two major subendothelial wall layers, connected by matching conditions. In a diffusive and transmural-pressure-driven advective transport model, Laplace’s equation describes the steady pressure distribution in the wall, whose gradient is proportional to the advecting velocity. Both problems yield the same boundary value problem that has mixed Dirichlet/ Robin boundary conditions. We construct a linear system of integral equations that approximates the series expansion coefficients o...

Vector estimators of the Monte Carlo method with a finite variance

In this paper we study problems related to the finiteness of the variance of weighted Monte Carlo estimators for solving a system of second-kind integral equations. Modifications of the finiteness criterion for the variance of a weighted scalar estimator are based on the construction of an appropriate system of linear integral equations with ‘majorant’ kernels. A majorant criterion of the finiteness of a vector weighted estimator is given. This criterion uses a comparison of the estimator with a scalar estimator obtained by the method of randomization. It is shown that for the corresponding randomized algorithm the variance of the scalar estimator and the mean simulation time for a single trajectory are bounded if the original solution is bounded. The ability to use branching to construct a vector estimator of the solution to a system of integral equations is also studied.

Integral Operators Acting as Variables of the Matrix Polynomial‎: ‎Application to System of Integral Equations

‎The aim of this work is to clarify a new viewpoint of connection between system of integral‎ ‎equations and matrix polynomials‎. ‎A procedure is described for transforming a‎ ‎linear system of integral equations to an independent‎ ‎system‎. ‎The latter is converted to equalities by considering its‎ ‎equivalent matrix polynomial equation which employs the integral operator as its variable‎, ‎and admits a normal form‎ ‎for simplifying the system‎. ‎We will show that under certain suitable‎ ‎conditions‎, ‎an independent reduced system is obtained‎, ‎which can be shown to have the same unknowns as the main system‎, ‎and‎ ‎has only one unknown in each equation‎. ‎In fact‎, ‎the basic idea enables us to develop a methodology to solve general systems of linear integral equations‎.

A Comparison Study Between a Chebyshev Collocation Method and the Adomian Decomposition Method for Solving Linear System of Fredholm Integral Equations of the Second Kind

In this paper, we obtain the approximate solutions of a system of linear integral equations by using both Chebyshev polynomials method and Adomian decomposition method. Also, a comparison between the two methods is discussed.

Hydrofoil flow over the interface of a two-layer heavy fluid with a free surface and rigid bottom

The theory of small-amplitude waves is used to analyze the hydrofoil flow of a two-layer heavy fluid. The upper layer is bounded by a free surface, while the lower layer is bounded by a horizontal bottom. The fluid layers have different densities and flow velocities. The problem is solved via the simulation of boundaries by singularities. Due to this method, the boundary condition specified on the contour is satisfied analytically exactly. By using the interface conditions, the problem is reduced to two systems of three singular integrodifferential equations. A special regularization technique gives systems of linear integral equations, which are solved numerically by applying the method of successive approximations with the use of a specially developed algorithm and a FORTRAN program. The numerical-analytical method developed applies to a wing section of arbitrary, including actual, shape placed in a fluid flow with interfaces of various types. The computations were performed for a NACA 66mod hydrofoil. The influence exerted by the angle of attack and the interfaces on the hydrodynamic hydrofoil characteristics is investigated in different ranges of Froude numbers. Shapes of internal and surface waves are obtained. Hydrodynamic effects associated with the dead water phenomenon are detected.

Harmonic Solutions of a Mixed Boundary Problem Arising in the Modeling of Macromolecular Transport into Vessel Walls.

The earliest events leading to atherosclerosis involve the transport of low density lipooprotein (LDL) cholesterol from the blood across endothelial cells that line the artery wall. Laplace's equation describes the steady state diffusion profile of a tracer through the vessel wall. This gives rise to a boundary value problem with mixed Dirichlet and Robin conditions. We construct a linear system of integral equations that approximate the coefficients of the series expansion of the solution. We prove the existence of the solution to this problem analytically by using Gershgorin's theorem on the location of the eigenvalues of the corresponding matrix. We give a uniqueness proof using Miranda's theorem [1]. The analytical construction method forms the basis for a numerical calculation algorithm. We apply our results to the transport problem above and use them to interpret experimental observations of the growth of localized tracer leakage spots with tracer circulation time.

Approximate solution of a system of linear integral equations by the Taylor expansion method

A simple and efficient approximate technique is developed to obtain the solution to a system of linear integral equations. This technique is based on the Taylor expansion. The method has been successfully applied to determine approximate solutions of a system of Fredholm integral equations and Volterra integral equations of not only the second kind but also the first kind. The mth order approximation of the solution is exact up to a polynomial of degree equal to or less than m. Several illustrative examples are presented to show the effectiveness and accuracy of this method.

Numerical expansion methods for solving systems of linear integral equations using interpolation and quadrature rules

In this paper we introduce a numerical method for solving a system of linear integral equations. The main idea is based on the interpolations of unknown functions at some interpolation points chosen in advance. We then use Clenshaw–Curtis quadrature formulae to approximate the integrals appearing in the system of equations. The technique is very effective and simple, and the performance of this suggested method is illustrated by means of a few significant examples.

The Onset of Oscillations in Microvascular Blood Flow

We explore the stability of equilibrium solution(s) of a simple model of microvascular blood flow in a two-node network. The model takes the form of convection equations for red blood cell concentration, and contains two important rheological effects—the Fahraeus–Lindqvist effect, which governs viscosity of blood flow in a single vessel, and the plasma skimming effect, which describes the separation of red blood cells at diverging nodes. We show that stability is governed by a linear system of integral equations, and we study the roots of the associated characteristic equation in detail. We demonstrate using a combination of analytical and numerical techniques that it is the relative strength of the Fahraeus–Lindqvist effect and the plasma skimming effect which determines the existence of a set of network parameter values which lead to a Hopf bifurcation of the equilibrium solution. We confirm these predictions with direct numerical simulation and suggest several areas for future research and application.

Analytical solution of the cable equation with synaptic reversal potentials for passive neurons with tip-to-tip dendrodendritic coupling

A passive cable model is presented for a pair of electrotonically coupled neurons in order to investigate the effects of tip-to-tip dendrodendritic gap junctions on the interaction between excitation and either pre or postsynaptic inhibition. The model represents each dendritic tree by a tapered equivalent cylinder attached to an isopotential soma. Analytical solution of the cable equation with synaptic reversal potentials is considered for each neuron to yield a system of Volterra integral equations for the voltage. The solution to the system of linear integral equations (expressed as a Neumann series) is used to determine the current spread within the two coupled neurons, and to re-examine the sensitivity of the soma potentials (in particular) to the coupling resistance for various loci of synaptic inputs. The model is actually posed generally, so that active as well as passive properties could be considered. In the active case, a system of non-linear integral equations is derived for the voltage.

Numerical solution of linear Fredholm fuzzy integral equations of the second kind by Adomian method

Using parametric form of fuzzy numbers we convert a linear fuzzy Fredholm integral equation of the second kind to a linear system of integral equations of the second kind in crisp case. We use Adomian method and find the approximate solution of this system and hence obtain an approximation for fuzzy solution of the linear fuzzy Fredholm integral equation of the second kind. We apply the method to some examples.

Chebyshev polynomial solutions of systems of linear integral equations

A Chebyshev collocation method has been presented to solve systems of linear integral equations in terms of Chebyshev polynomials. This method transforms the integral system into the matrix equation with the help of Chebyshev collocation points and the unknown of this equation is a Chebyshev coefficient matrix. They are easily acquired by using the last matrix equation, which corresponds to the system of linear algebraic equations. As a result, the finite Chebyshev series approach is obtained.

Long-term stability of cylindrical shells of viscoelastic composites

On the grounds of Hamilton's principle the system of linear integral equations has been ob­tained for the solution of stability problems of cylindrical shells made of viscoelastic composite materials by the finite element method.