General Integral Equation Research Articles

LP solutions to the parameterized Fredholm integral equations associated with Chandrasekhar kernels

ABSTRACT In this article, we study the solution of the Fredholm integral equation with parameters. On the one hand, we use Hilbert-type inequality to study Chandrasekhar-type integral operators, generalize the reachability results for the norm of the Chandrasekhar integral operator, and establish the existence and uniqueness results of the solutions of Chandrasekhar-type integral equations with parameters. On the other hand, we develop an iterative kernel technique for integral equations in infinite intervals, and construct series solutions of general Fredholm integral equations with parameters in different spaces.

Results on solvability of nonlinear quadratic integral equations of fractional orders in Banach algebra

Here, we investigate the existence result for a nonlinear quadratic functional integral equation of fractional order using a fixed point theorem of Dhage. The continuous dependence of solution on the delay functions will be studied. As an application, an existence theorem for the fractional hybrid differential equations is proved. Also, we study a general quadratic integral equation of fractional order.

The Parabolic Transform and Some Singular Integral Evolution Equations

Some singular integral evolution equations with wide class of closed operators are studied in Banach space. The considered integral equations are investigated without the existence of the resolvent of the closed operators. Also, some non-linear singular evolution equations are studied. An abstract parabolic transform is constructed to study the solutions of the considered ill-posed problems. Applications to fractional evolution equations and Hilfer fractional evolution equations are given. All the results can be applied to general singular integro-differential equations. The Fourier Transform plays an important role in constructing solutions of the Cauchy problems for parabolic and hyperbolic partial differential equations. This means that the Fourier transform is suitable but under conditions on the characteristic forms of the partial differential operators. Also, the Laplace transform plays an important role in studying the Cauchy problem for abstract differential equations in Banach space. But in this case, we need the existence of the resolvent of the considered abstract operators. This note is devoted to exploring the Cauchy problem for general singular integro-partial differential equations without conditions on the characteristic forms and also to study general singular integral evolution equations. Our approach is based on applying the new parabolic transform. This transform generalizes the methods developed within the regularization theory of ill-posed problems.

Solution of fuzzy singular integral equation with Abel\u2019s type kernel using a novel hybrid method

In this paper, we are applying a novel analytical hybrid method to find the solution of a fuzzy Volterra Abel’s integral equation of the second kind. The fuzzy number is used in its parametric form under which the fuzzy Volterra Abel’s integral equation will be converted into a system of integral equations as in a crisp case. Moreover, to solve the general fuzzy Volterra integral equation with Abel’s type kernel, and to show that the proposed method is efficient, a few accurate and simple examples are given for the demonstration of our results.

A hybrid Legendre block-pulse method for mixed Volterra–Fredholm integral equations

Mixed Volterra–Fredholm integral equations can arise from mathematical modelling of the spatio-temporal development of an epidemic. In this paper a hybrid Legendre block-pulse method is used to provide solutions to a general integral equation of this type. The main idea of this approach is to reduce a Volterra–Fredholm integral equation to a system of algebraic equations. An order convergence analysis is provided. The accuracy of the proposed method is then confirmed by solving some problems, the first being a nonlinear epidemic problem.

Necessary conditions of Pontraygin’s type for general controlled stochastic Volterra integral equations

This article is addressed to giving a solution to a unsolved problem, i.e., to establish the necessary optimality conditions of Pontraygin’s type for controlled stochastic Volterra integral equations (SVIEs) when the control region is non-convex and the control variable enters into the diffusion. This problem has been open since [J. Yong, Stochastic Process Appl. 116 (2006) 779–795] obtained the analogue result for the case of convex control region. The key is to introduce a pair of suitable second-order adjoint processes (SOAPs). It is found that the usual way of using only one SOAP in the maximum condition for the classical setting of controlled stochastic differential equations does not work here.

An Analytic Approximation for Valuation of the American Option Under the Heston Model in Two Regimes

This paper studies the valuation of the American call-option under the Heston model in two regimes, i.e., fast-mean reverting and slow-mean reverting regimes. In the case of the European-style option under the Heston model, a closed-form solution for one-dimensional integration can be derived. However, in the case of the American-style option, it is impossible to obtain a general analytic integral equation for the price. By using singular and regular perturbation techniques introduced by Fouque et al. (Multiscale stochastic volatility for equity, interest-rate and credit derivative, Cambridge University Press, Cambridge, 2011) and the maturity randomization method introduced by Carr (Rev Financ Stud 11:597–626, 1998), we provide an approximate analytic solution of the American call-option and describe a numerical scheme to evaluate the value of this solution. Numerical results show that our method is accurate and efficient compared to the finite-difference method and the Longstaff and Schwartz (Rev Financ Stud 14(1):113–147, 2001) method.

Krasnosel'skii-Schaefer type method in the existence problems

We consider a general integral equation satisfying algebraic conditions in a Banach space. Using Krasnosel'skii-Schaefer type method and technical assumptions, we prove an existence theorem producing a periodic solution of some nonlinear integral equation.

A Potential-Based Formalism for Modeling Local and Hydrodynamic Nonlocal Responses From Plasmonic Waveguides

In this paper, we propose a general boundary integral equation (BIE) approach for solving both exterior (e.g., scattering) and interior (e.g., guided wave propagation) problems involving general plasmonic waveguiding structures with arbitrary cross-sectional geometries and with a continuous translational symmetry in the direction of wave propagation. Contrary to the field-based approach which deals with the electric and magnetic fields, we employ a potential-based formalism instead, involving vector and scalar potentials, and match them at the media interfaces. The proposed approach can handle not only conventional plasmonic waveguides on the order of a few hundred nanometers but also those whose critical sizes are a few nanometers and in which the nonlocal hydrodynamic effects need to be accounted for. The BIEs describing the interaction of light with the plasmonic waveguide are solved by the method of moments (MoM) algorithm. Two illustrative examples, the first one of which deals with the scattering problem, while the second computes the dispersion diagram of a plasmonic cylinder, are considered for both local and nonlocal cases. An excellent agreement is observed between the numerical and theoretical results, with the latter being derived from the generalized Mie theory.

Analytically derived matrix end-form elastic-forces equations for a low-order cable element using the absolute nodal coordinate formulation

In everyday, practical, multibody, dynamic simulations, any savings in computational time are welcome. The aim of this paper is to develop an analytical mathematical approach to be used on a low-order element to reduce the computational costs. In this paper the Bernoulli-Euler beam energy formulation is expressed with an absolute nodal coordinate formulation (ANCF) and the general integral equation due to internal strain and bend generalized elastic forces is derived and expressed with the pre-computed matrices. The derived matrices, together with the Gauss quadrature integration, minimize the number of numerical operations when performing a dynamic time simulation. The equations are general and can be used for a cable element, without any restrictions on the strain-bend-displacement relationship. The proposed equations are implemented in the ANCF dynamic model, and the presented numerical results show that savings in the computational time can be achieved.

On existence and uniqueness of $L_1$-solutions for quadratic integral equations via a Krasnoselskii-type fixed point theorem

Using a Krasnoselskii-type fixed point theorem due to Burton, we discuss the existence of integrable solutions of general quadratic-Urysohn integral equations on a bounded interval $(a,b)$. Uniqueness of the solution is also studied. An example to illustrate our theory is also included.

Interface integral technique for the thermoelasticity of random structure matrix composites

We consider linear thermoelastic composite media, which consist of a homogeneous matrix containing a statistically homogeneous random set of aligned homogeneous heterogeneities of non-canonical (i.e. non-ellipsoidal) shape. The representations of the effective properties (effective moduli, thermal expansion, and stored energy) are expressed through the statistical averages of the interface polarization tensors introduced apparently for the first time. The properties of the interface polarization tensors are described. The new general integral equations connecting the stress and strain fields in the point being considered with the stress and strain fields in the surrounding points are obtained for the random fields of heterogeneities. The method is based on a recently developed centering procedure where the notion of a perturbator is introduced in terms of boundary interface integrals estimated by the method of fundamental solution for a single inclusion inside the infinite matrix. This enables us to reconsider basic concepts of micromechanics such as effective field hypothesis, quasi-crystalline approximation, and the hypothesis of “ellipsoidal symmetry.” Effective properties (such as effective moduli, thermal expansion, and stored energy) as well as the first statistical moments of stresses in the phases are estimated for statistically homogeneous composites with the general case of the inclusion shape. The results of this reconsideration are quantitatively estimated for some modeled statistically homogeneous composites reinforced by aligned homogeneous heterogeneities of non-canonical shape. The explicit new representations of the effective thermoelastic properties and stress concentration factor are expressed through some building blocks ( perturbators) described by numerical solutions for one heterogeneity inside the infinite medium subjected to the homogeneous remote loading. Some new effects are detected that are impossible in the framework of a classical background of micromechanics.

On the classical dynamics of charged particle in special class of spatially non-uniform magnetic field

Motion of a charged particle in uniform magnetic field has been studied in detail, classically as well as quantum mechanically. However, classical dynamics of a charged particle in non-uniform magnetic field is solvable only for some specific cases. We present in this paper, a general integral equation for some specific class of non-uniform magnetic field and its solutions for some of them. We also examine the supersymmetry of Hamiltonians in exponentially decaying magnetic field with radial dependence and conclude that this kind of non-uniformity breaks supersymmetry.

Heat Conduction in Multilayered Thermoelectric Materials with Multiple Cracks

Multilayer thin-film thermoelectric materials are of technological importance. This paper describes a method to analyze the heat conduction in a multilayered thermoelectric plate containing some non-collinear cracks. The material properties in one layer may be different from those in another even though each layer may still be homogeneous. Using the Fourier integral transforms, the boundary value problem is reduced to a system of general singular integral equations. The model is sufficiently general to account for any number of layers and any number of cracks. As a numerical illustration, the electric flux intensity factor, energy flux intensity factor and thermal flux intensity factor for a three-layer plate specimen with two cracks are presented. The effects of strip width on the electric flux intensity factor and thermal flux intensity factor are studied.

Joint moments of the total discounted gains and losses in the renewal risk model with two-sided jumps

This paper considers a renewal insurance risk model with two-sided jumps (e.g. Labbé et al., 2011), where downward and upward jumps typically represent claim amounts and random gains respectively. A generalization of the Gerber–Shiu expected discounted penalty function (Gerber and Shiu, 1998) is proposed and analyzed for sample paths leading to ruin. In particular, we shall incorporate the joint moments of the total discounted costs associated with claims and gains until ruin into the Gerber–Shiu function. Because ruin may not occur, the joint moments of the total discounted claim costs and gain costs are also studied upon ultimate survival of the process. General recursive integral equations satisfied by these functions are derived, and our analysis relies on the concept of ‘moment-based discounted densities’ introduced by Cheung (2013). Some explicit solutions are obtained in two examples under different cost functions when the distribution of each claim is exponential or a combination of exponentials (while keeping the distributions of the gains and the inter-arrival times between successive jumps arbitrary). The first example looks at the joint moments of the total discounted amounts of claims and gains whereas the second focuses on the joint moments of the numbers of downward and upward jumps until ruin. Numerical examples including the calculations of covariances between the afore-mentioned quantities are given at the end along with some interpretations.

Stationary statistical theory of two-surface multipactor regarding all impacts for efficient threshold analysis

Statistical multipactor theories are critical prediction approaches for multipactor breakdown determination. However, these approaches still require a negotiation between the calculation efficiency and accuracy. This paper presents an improved stationary statistical theory for efficient threshold analysis of two-surface multipactor. A general integral equation over the distribution function of the electron emission phase with both the single-sided and double-sided impacts considered is formulated. The modeling results indicate that the improved stationary statistical theory can not only obtain equally good accuracy of multipactor threshold calculation as the nonstationary statistical theory, but also achieve high calculation efficiency concurrently. By using this improved stationary statistical theory, the total time consumption in calculating full multipactor susceptibility zones of parallel plates can be decreased by as much as a factor of four relative to the nonstationary statistical theory. It also shows that the effect of single-sided impacts is indispensable for accurate multipactor prediction of coaxial lines and also more significant for the high order multipactor. Finally, the influence of secondary emission yield (SEY) properties on the multipactor threshold is further investigated. It is observed that the first cross energy and the energy range between the first cross and the SEY maximum both play a significant role in determining the multipactor threshold, which agrees with the numerical simulation results in the literature.

Generalized Matrix Exponential Solutions to the AKNS Hierarchy

Generalized matrix exponential solutions to the AKNS equation are obtained by the inverse scattering transformation (IST). The resulting solutions involve six matrices, which satisfy the coupled Sylvester equations. Several kinds of explicit solutions including soliton, complexiton, and Matveev solutions are deduced from the generalized matrix exponential solutions by choosing different kinds of the six involved matrices. Generalized matrix exponential solutions to a general integrable equation of the AKNS hierarchy are also derived. It is shown that the general equation and its matrix exponential solutions share the same linear structure.

Effective field hypothesis in Hashin–Shtrikman bounds estimations for effective moduli of composites with noncanonical shape of inclusions

One consider linear elastic composite materials (CMs), which consist of a homogeneous matrix containing a statistically homogeneous random set of identical aligned inhomogeneous inclusions of non-canonical (i.e. nonellipsoidal) shape. Estimates of the Hashin-Shtrikman type are developed by extremizing of the classical variational functional involving a general integral equation (GIE) connecting the strain fields in the point being considered with the strain fields in the surrounding points. An improvement of the approach proposed is based on the use of a piece wise constant approximation of the trial effective field rather than the trial polarisation tensor that, in its turn, allows one to take into account any inhomogeneity of the strain distributions inside the inclusions. Due to a sensitivity of the proposed approach to the inhomogeneity of these strain distributions, a final Hashin-Shtrikman bounds representation demonstrates more significant dependence on the shape of noncanonical inclusions than the corresponding classical bounds (that is shown by the concrete numerical examples).

CORDIAL VOLTERRA INTEGRAL EQUATIONS AND SINGULAR FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS IN SPACES OF ANALYTIC FUNCTIONS∗

We study general cordial Volterra integral equations of the second kind and certain singular fractional integro-differential equation in spaces of analytic functions. We characterize properties of the cordial Volterra integral operator in these spaces, including compactness and describe its spectrum. This enables us to obtain conditions under which these equations have a unique analytic solution. We also consider approximate solution of these equations and prove exponential convergence of approximate solutions to the exact solution.

Method of fundamental solutions in micromechanics of elastic random structure composites

One considers linearly elastic composite media, which consist of a homogeneous matrix containing a statistically homogeneous random set of aligned homogeneous heterogeneities of non-canonical (i.e. nonellipsoidal) shape. The new general integral equations connecting the stress and strain fields in the point being considered with the stress and strain fields in the surrounding points are obtained for the random fields of heterogeneities. The method is based on a recently developed centering procedure where the notion of a perturbator is introduced in terms of boundary interface integrals that makes it possible to reconsider basic concepts of micromechanics such as effective field hypothesis, quasi-crystalline approximation, and the hypothesis of ellipsoidal symmetry. The results of this reconsideration are quantitatively estimated for some modeled composite reinforced by aligned homogeneous heterogeneities of non canonical shape. Some new effects are detected that are impossible in the framework of a classical background of micromechanics.