Journal of Integral Equations and Applications Articles (Project Euclid)
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The latest articles from Journal of Integral Equations and Applications on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2010 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Thu, 05 Aug 2010 15:41 EDTTue, 22 Mar 2011 10:06 EDThttp://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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The determination of boundary coefficients from far field measurements
http://projecteuclid.org/euclid.jiea/1277125620
<strong>Fioralba Cakoni</strong>, <strong>David Colton</strong>, <strong>Peter Monk</strong><p><strong>Source: </strong>J. Integral Equations Appl., Volume 22, Number 2, 167--191.</p>projecteuclid.org/euclid.jiea/1277125620_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTA well-conditioned boundary integral equation for transmission problems of electromagnetismhttp://projecteuclid.org/euclid.jiea/1450388943<strong>David Levadoux</strong>, <strong>Florence Millot</strong>, <strong>Sébastien Pernet</strong>. <p><strong>Source: </strong>Journal of Integral Equations and Applications, Volume 27, Number 3, 431--453.</p><p><strong>Abstract:</strong><br/>
We propose a new well-conditioned boundary integral equation to solve
transmission problems of electromagnetism. This equation is well posed and
appears as a compact perturbation of the identity leading to fast iterative solutions
without the help of any preconditioner. Some numerical experiments
confirm this result.
</p>projecteuclid.org/euclid.jiea/1450388943_20151217164901Thu, 17 Dec 2015 16:49 ESTThe oscillation of solutions of Volterra integral and integro-differential equations with highly oscillatory kernelshttp://projecteuclid.org/euclid.jiea/1454939250<strong>Hermann Brunner</strong>, <strong>Yunyun Ma</strong>, <strong>Yuesheng Xu</strong>. <p><strong>Source: </strong>Journal of Integral Equations and Applications, Volume 27, Number 4, 455--487.</p><p><strong>Abstract:</strong><br/>
We study the oscillatory structures of solutions of Volterra integral and integro-differential equations (VIEs, VIDEs) with highly oscillatory kernels. Based on the structured oscillatory spaces introduced in Wang and Xu \cite {OPG}, we first analyze the degree of oscillation of the solution of VIEs associated with the oscillatory kernels belonging to a certain structured oscillatory space by using the resolvent representation of the solution. According to a decomposition of the oscillatory integrals in the complex plane, we prove that the Volterra integral operator reduces the oscillatory order of the functions in the structured oscillatory spaces corresponding to the oscillatory structure of the kernel. The analogous oscillatory structure of solutions of VIDEs is then analyzed by representing the solution of the VIDEs by the differential resolvent kernel and by exploiting the relationship between the VIDEs and the equivalent VIE. We conclude that the solutions of the VIEs and VIDEs preserve the oscillatory components of the kernel.
</p>projecteuclid.org/euclid.jiea/1454939250_20160209085918Tue, 09 Feb 2016 08:59 ESTExistence of solution of impulsive second order neutral integro- differential equations with state delayhttp://projecteuclid.org/euclid.jiea/1454939251<strong>Sanjukta Das</strong>, <strong>D.N. Pandey</strong>, <strong>N. Sukavanam</strong>. <p><strong>Source: </strong>Journal of Integral Equations and Applications, Volume 27, Number 4, 489--520.</p><p><strong>Abstract:</strong><br/>
This paper consists of two parts. The first part deals with the existence of a mild solution of a class of instantaneous impulsive second order partial neutral differential equations with state dependent delay. The second part studies the non-instantaneous impulsive conditions on the same problem. The Kuratowski measure of noncompactness and M\'onch fixed point theorem are used to prove the existence of the mild solution. We remove the restrictive conditions on the priori estimation available in literature. The compactness assumption on the associated cosine or sine family of operators, nonlinear terms and associated impulsive term are also not required in this paper. The noncompactness measure estimation, the Lipschitz conditions and compactness on the nonlinear functions are replaced by simple and natural assumptions. We introduce new non-instantaneous impulses with fixed delays. In the last section, we study examples to illustrate the result presented.
</p>projecteuclid.org/euclid.jiea/1454939251_20160209085918Tue, 09 Feb 2016 08:59 ESTApplication of a global implicit function theorem to a general fractional integro-differential system of Volterra typehttp://projecteuclid.org/euclid.jiea/1454939252<strong>Dariusz Idczak</strong>, <strong>Stanislaw Walczak</strong>. <p><strong>Source: </strong>Journal of Integral Equations and Applications, Volume 27, Number 4, 521--554.</p><p><strong>Abstract:</strong><br/>
In this paper, we use a global implicit function theorem for the investigation of the existence and uniqueness of a solution as well as the sensitivity of a Cauchy problem for a general integro-differential system of order $\alpha \in (0,1)$ of Volterra type, involving two functional parameters nonlinearly.
</p>projecteuclid.org/euclid.jiea/1454939252_20160209085918Tue, 09 Feb 2016 08:59 ESTA multiple nonlinear Abel type integral equationhttp://projecteuclid.org/euclid.jiea/1454939253<strong>W. Mydlarczyk</strong>. <p><strong>Source: </strong>Journal of Integral Equations and Applications, Volume 27, Number 4, 555--572.</p><p><strong>Abstract:</strong><br/>
We discuss a multiple nonlinear Abel type integral equation. The basic results provide criteria for the existence of nontrivial everywhere positive solutions. They are expressed in terms of the generalized Osgood condition. The global behavior of the solution, especially the conditions when it experiences blow-up, is also considered.
</p>projecteuclid.org/euclid.jiea/1454939253_20160209085918Tue, 09 Feb 2016 08:59 ESTGlobal existence and blow-Ups for certain ordinary integro-differential equationshttp://projecteuclid.org/euclid.jiea/1454939254<strong>Martin Saal</strong>. <p><strong>Source: </strong>Journal of Integral Equations and Applications, Volume 27, Number 4, 573--602.</p><p><strong>Abstract:</strong><br/>
We will study ordinary integro-differential equations of second order with nonlinearity given as a convolution, but differently from the widely investigated cases. In addition, the kernel depends on the solution. Such equations play a key role in the theory of glass-forming liquids, and we will establish results on global existence and investigate the long-term behavior. In contrast, we give examples where blow-ups occur.
</p>projecteuclid.org/euclid.jiea/1454939254_20160209085918Tue, 09 Feb 2016 08:59 ESTVolume Index for Volume 27http://projecteuclid.org/euclid.jiea/1454939255<strong> </strong>. <p><strong>Source: </strong>Journal of Integral Equations and Applications, Volume 27, Number 4, 603--604.</p>projecteuclid.org/euclid.jiea/1454939255_20160209085918Tue, 09 Feb 2016 08:59 ESTControllability of fractional integrodifferential equations with start-dependent delayhttp://projecteuclid.org/euclid.jiea/1450389205<strong> Aissani, Benchohra</strong>. <p><strong>Source: </strong>Journal of Integral Equations and Applications, Forthcoming articles, --.</p>projecteuclid.org/euclid.jiea/1450389205_20160209100311Tue, 09 Feb 2016 10:03 ESTSolvability of a Volume Integral Equation Formulation for Anisotropic Elastodynamic Scatteringhttp://projecteuclid.org/euclid.jiea/1450389206<strong> Bonnet</strong>. <p><strong>Source: </strong>Journal of Integral Equations and Applications, Forthcoming articles, --.</p>projecteuclid.org/euclid.jiea/1450389206_20160209100311Tue, 09 Feb 2016 10:03 ESTVolterra Integral Equations on Variable Exponent Lebesque Spaceshttp://projecteuclid.org/euclid.jiea/1450389208<strong> Castillo, Ramos-Fernandez, Rojas</strong>. <p><strong>Source: </strong>Journal of Integral Equations and Applications, Forthcoming articles, --.</p>projecteuclid.org/euclid.jiea/1450389208_20160209100311Tue, 09 Feb 2016 10:03 ESTProbablistic regularization of Fredholm integral equations of the first kindhttp://projecteuclid.org/euclid.jiea/1450389210<strong> DeMicheli, Viano</strong>. <p><strong>Source: </strong>Journal of Integral Equations and Applications, Forthcoming articles, --.</p>projecteuclid.org/euclid.jiea/1450389210_20160209100311Tue, 09 Feb 2016 10:03 ESTOn a nonlinear abstract Volterra equationhttp://projecteuclid.org/euclid.jiea/1450389211<strong> Emmrich, Vallet</strong>. <p><strong>Source: </strong>Journal of Integral Equations and Applications, Forthcoming articles, --.</p>projecteuclid.org/euclid.jiea/1450389211_20160209100311Tue, 09 Feb 2016 10:03 ESTApproximate solution of Urysohn integral equations with non-smooth kernelshttp://projecteuclid.org/euclid.jiea/1455030187<strong> Kulkarni, Nidhin</strong>. <p><strong>Source: </strong>Journal of Integral Equations and Applications, Forthcoming articles, --.</p>projecteuclid.org/euclid.jiea/1455030187_20160209100311Tue, 09 Feb 2016 10:03 ESTA Collocation Method Solving Integral Equation Models for Image Restorationhttp://projecteuclid.org/euclid.jiea/1455030188<strong> Liu, Shen, Xu, Yang</strong>. <p><strong>Source: </strong>Journal of Integral Equations and Applications, Forthcoming articles, --.</p>projecteuclid.org/euclid.jiea/1455030188_20160209100311Tue, 09 Feb 2016 10:03 ESTA Mode III Interface Crack with Surface Strain Gradient Elasticityhttp://projecteuclid.org/euclid.jiea/1450389214<strong> Wang, Schiavone</strong>. <p><strong>Source: </strong>Journal of Integral Equations and Applications, Forthcoming articles, --.</p>projecteuclid.org/euclid.jiea/1450389214_20160209100311Tue, 09 Feb 2016 10:03 ESTVolterra integral equations on variable exponent Lebesgue spaceshttp://projecteuclid.org/euclid.jiea/1460727503<strong>R.E. Castillo</strong>, <strong>J.C. Ramos-Fernández</strong>, <strong>E.M. Rojas</strong>. <p><strong>Source: </strong>Journal of Integral Equations and Applications, Volume 28, Number 1, 1--29.</p><p><strong>Abstract:</strong><br/>
In this paper, in the framework of Lebesgue spaces with variable exponent, we are going to provide conditions for the existence and uniqueness of the solutions of a class of Volterra integral equations induced by Carath\'eodory functions having diverse growth behaviors. To attain our goals, we will use topological degree theory for condensing maps and fixed point results for the sum of mappings of contractive type.
</p>projecteuclid.org/euclid.jiea/1460727503_20160415093830Fri, 15 Apr 2016 09:38 EDTProbabilistic regularization of Fredholm integral equations of the first kindhttp://projecteuclid.org/euclid.jiea/1460727504<strong>Enrico De Micheli</strong>, <strong>Giovanni Alberto Viano</strong>. <p><strong>Source: </strong>Journal of Integral Equations and Applications, Volume 28, Number 1, 31--74.</p><p><strong>Abstract:</strong><br/>
The main purpose of this paper is to focus on various issues inherent to the regularization theory of Fredholm integral equations of the first kind. Particular attention is devoted to the probabilistic approach to regularization, and a regularizing algorithm based on statistical methods is then proposed and tested on examples. The information theory approach is studied from two different viewpoints: the first approach is the standard one based on probability theory; the second one is formulated, in analogy with communication theory, in terms of the $\varepsilon $-capacity in the sense elaborated by Kolmogorov and his school. The classical problem of the resolving power in optics is then used to exemplify the relation between these two approaches.
</p>projecteuclid.org/euclid.jiea/1460727504_20160415093830Fri, 15 Apr 2016 09:38 EDTOn a nonlinear abstract Volterra equationhttp://projecteuclid.org/euclid.jiea/1460727505<strong>Etienne Emmrich</strong>, <strong>Guy Vallet</strong>. <p><strong>Source: </strong>Journal of Integral Equations and Applications, Volume 28, Number 1, 75--89.</p><p><strong>Abstract:</strong><br/>
Existence of solutions is shown for equations of the type $Av + B( KGv,v) = f$, where $A$, $B$ and $G$ are possibly nonlinear operators acting on a Banach space $V$, and $K$ is a Volterra operator of convolution type. The proof relies on the convergence of a suitable time discretization scheme.
</p>projecteuclid.org/euclid.jiea/1460727505_20160415093830Fri, 15 Apr 2016 09:38 EDTThe direct scattering problem of obliquely incident electromagnetic waves by a penetrable homogeneous cylinderhttp://projecteuclid.org/euclid.jiea/1460727506<strong>Drossos Gintides</strong>, <strong>Leonidas Mindrinos</strong>. <p><strong>Source: </strong>Journal of Integral Equations and Applications, Volume 28, Number 1, 91--122.</p><p><strong>Abstract:</strong><br/>
In this paper we consider the direct scattering problem of obliquely incident time-harmonic electromagnetic plane waves by an infinitely long dielectric cylinder. We assume that the cylinder and the outer medium are homogeneous and isotropic. From the symmetry of the problem, Maxwell's equations are reduced to a system of two 2D Helmholtz equations in the cylinder and two 2D Helmholtz equations in the exterior domain where the fields are coupled on the boundary. We prove uniqueness and existence of this differential system by formulating an equivalent system of integral equations using the direct method. We transform this system into a Fredholm type system of boundary integral equations in a Sobolev space setting. To handle the hypersingular operators we take advantage of Maue's formula. Applying a collocation method we derive an efficient numerical scheme and provide accurate numerical results using as test cases transmission problems corresponding to analytic fields derived from fundamental solutions.
</p>projecteuclid.org/euclid.jiea/1460727506_20160415093830Fri, 15 Apr 2016 09:38 EDTA mode III interface crack with surface strain gradient elasticityhttp://projecteuclid.org/euclid.jiea/1460727507<strong>Xu Wang</strong>, <strong>Peter Schiavone</strong>. <p><strong>Source: </strong>Journal of Integral Equations and Applications, Volume 28, Number 1, 123--148.</p><p><strong>Abstract:</strong><br/>
We study the contribution of surface strain gradient elasticity to the anti-plane deformations of an elastically isotropic bimaterial containing a mode~III interface crack. The surface strain gradient elasticity is incorporated using an enriched version of the continuum-based surface/interface model of Gurtin and Murdoch. We obtain a complete semi-analytic solution valid everywhere in the solid (including at the crack tips) by reducing the boundary value problem to two coupled hyper-singular integro-differential equations which are solved numerically using Chebyshev polynomials and a collocation method. Our solution demonstrates that the presence of surface strain gradient elasticity on the crack faces leads to bounded stresses at the crack tips.
</p>projecteuclid.org/euclid.jiea/1460727507_20160415093830Fri, 15 Apr 2016 09:38 EDTControllability of fractional integrodifferential equations with state-dependent delayhttp://projecteuclid.org/euclid.jiea/1467399273<strong>Khalida Aissani</strong>, <strong>Mouffak Benchohra</strong>. <p><strong>Source: </strong>Journal of Integral Equations and Applications, Volume 28, Number 2, 149--167.</p><p><strong>Abstract:</strong><br/>
According to fractional calculus theory and Sadovskii's fixed point theorem, we establish sufficient conditions for controllability of the fractional integro-differential equation with state-dependent delay. An example is provided to illustrate the theory.
</p>projecteuclid.org/euclid.jiea/1467399273_20160701145435Fri, 01 Jul 2016 14:54 EDTSolvability of a volume integral equation formulation for anisotropic elastodynamic scatteringhttp://projecteuclid.org/euclid.jiea/1467399274<strong>Marc Bonnet</strong>. <p><strong>Source: </strong>Journal of Integral Equations and Applications, Volume 28, Number 2, 169--203.</p><p><strong>Abstract:</strong><br/>
This article investigates the solvability of volume integral equations arising in elastodynamic scattering by penetrable obstacles. The elasticity tensor and mass density are allowed to be smoothly heterogeneous inside the obstacle and may be discontinuous across the background-obstacle interface, the background elastic material being homogeneous. Both materials may be anisotropic, within certain limitations for the background medium. The volume integral equation associated with this problem is first derived, relying on known properties of the background fundamental tensor. To avoid difficulties associated with existing radiation conditions for anisotropic elastic media, we also propose a definition of the radiating character of transmission solutions. The unique solvability of the volume integral equation (and of the scattering problem) is established. For the important special case of isotropic background properties, our definition of a radiating solution is found to be equivalent to the Sommerfeld-Kupradze radiation conditions. Moreover, solvability for anisotropic elastostatics, directly related to known results on the equivalent inclusion method, is recovered as a by-product.
</p>projecteuclid.org/euclid.jiea/1467399274_20160701145435Fri, 01 Jul 2016 14:54 EDTCompositions of pseudo almost automorphic functions via measure theory and applicationshttp://projecteuclid.org/euclid.jiea/1467399275<strong>Zhenbin Fan</strong>, <strong>Qixiang Dong</strong>, <strong>Gang Li</strong>. <p><strong>Source: </strong>Journal of Integral Equations and Applications, Volume 28, Number 2, 205--219.</p><p><strong>Abstract:</strong><br/>
In this paper, we establish some composition theorems of $\mu $-pseudo almost automorphic functions via measure theory, then derive sufficient conditions for the existence and uniqueness of pseudo almost automorphic mild solutions to fractional differential equations with Caputo derivatives.
</p>projecteuclid.org/euclid.jiea/1467399275_20160701145435Fri, 01 Jul 2016 14:54 EDTApproximate solution of Urysohn integral equations with non-smooth kernelshttp://projecteuclid.org/euclid.jiea/1467399276<strong>Rekha P. Kulkarni</strong>, <strong>T.J. Nidhin</strong>. <p><strong>Source: </strong>Journal of Integral Equations and Applications, Volume 28, Number 2, 221--261.</p><p><strong>Abstract:</strong><br/>
Consider a nonlinear operator equation $ x - K ( x )=f$, where $K$ is a Urysohn integral operator with a kernel of the type of Green's function and defined on $L^\infty [0, 1]$. For $ r \geq 0$, we choose the approximating space to be a space of discontinuous piecewise polynomials of degree $\leq r$ with respect to a quasi-uniform partition of $[0, 1]$ and consider an interpolatory projection at $r+1$ Gauss points. Previous authors have proved that the orders of convergence in the collocation and the iterated collocation methods are $ r+1$ and $r + 2 + \min \{ r, 1 \}$, respectively. We show that the order of convergence in the iterated modified projection method is $4$ if $ r = 0$ and is $ 2 r + 3$ if $ r \geq 1$. This improvement in the order of convergence is achieved while retaining the size of the system of equations that needs to be solved, the same as in the case of the collocation method. Numerical results are given for specific examples.
</p>projecteuclid.org/euclid.jiea/1467399276_20160701145435Fri, 01 Jul 2016 14:54 EDTA collocation method solving integral equation models for image restorationhttp://projecteuclid.org/euclid.jiea/1467399277<strong>Yuzhen Liu</strong>, <strong>Lixin Shen</strong>, <strong>Yuesheng Xu</strong>, <strong>Hongqi Yang</strong>. <p><strong>Source: </strong>Journal of Integral Equations and Applications, Volume 28, Number 2, 263--307.</p><p><strong>Abstract:</strong><br/>
We propose a collocation method for solving integral equations which model image restoration from out-of-focus images. Restoration of images from out-of-focus images can be formulated as an integral equation of the first kind, which is an ill-posed problem. We employ the Tikhonov regularization to treat the ill-posedness and obtain results of a well-posed second kind integral equation whose integral operator is the square of the original operator. The present of the square of the integral operator requires high computational cost to solve the equation. To overcome this difficulty, we convert the resulting second kind integral equation into an equivalent system of integral equations which do not involve the square of the integral operator. A multiscale collocation method is then applied to solve the system. A truncation strategy for the matrices appearing in the resulting discrete linear system is proposed to design a fast numerical solver for the system of integral equations. A quadrature method is used to compute the entries of the resulting matrices. We estimate the computational cost of the numerical method and its approximate accuracy. Numerical experiments are presented to demonstrate the performance of the proposed method for image restoration.
</p>projecteuclid.org/euclid.jiea/1467399277_20160701145435Fri, 01 Jul 2016 14:54 EDTNumerical methods for systems of nonlinear integro-parabolic equations of Volterra typehttp://projecteuclid.org/euclid.jiea/1476706344<strong>Igor Boglaev</strong>. <p><strong>Source: </strong>Journal of Integral Equations and Applications, Volume 28, Number 3, 309--342.</p><p><strong>Abstract:</strong><br/>
This paper deals with the numerical solution of systems of nonlinear integro-parabolic problems of Volterra type. The numerical approach is based on the method of upper and lower solutions. A monotone iterative method is constructed. Existence and uniqueness of a solution to the nonlinear difference scheme are established. An analysis of convergence rates of the monotone iterative method is given. Construction of initial upper and lower solutions is discussed. Numerical experiments are presented.
</p>projecteuclid.org/euclid.jiea/1476706344_20161017081229Mon, 17 Oct 2016 08:12 EDTBoundary integral operator for the fractional Laplacian on the boundary of a bounded smooth domainhttp://projecteuclid.org/euclid.jiea/1476706345<strong>Tongkeun Chang</strong>. <p><strong>Source: </strong>Journal of Integral Equations and Applications, Volume 28, Number 3, 343--372.</p><p><strong>Abstract:</strong><br/>
We introduce the boundary integral operator induced from the fractional Laplace equation on the boundary of a bounded smooth domain. For~$\frac 12\lt \alpha \lt 1$, we show the bijectivity of the boundary integral operator~$S_{2\alpha }:L^p(\partial \Omega )\to H^{2\alpha -1}_p(\partial \Omega )$ for $1 \lt p \lt \infty $. As an application, we demonstrate the existence of the solution of the Dirichlet boundary value problem of the fractional Laplace equation.
</p>projecteuclid.org/euclid.jiea/1476706345_20161017081229Mon, 17 Oct 2016 08:12 EDT$C^\sigma ,\alpha $ estimates for concave, non-local parabolic equations with critical drifthttp://projecteuclid.org/euclid.jiea/1476706346<strong>Héctor Chang Lara</strong>, <strong>Gonzalo Dávila</strong>. <p><strong>Source: </strong>Journal of Integral Equations and Applications, Volume 28, Number 3, 373--394.</p><p><strong>Abstract:</strong><br/>
Given a concave integro-differential operator $I$, we study regularity for solutions of fully nonlinear, nonlocal, parabolic equations of the form $u_t-Iu=0$. The kernels are assumed to be smooth but non necessarily symmetric, which accounts for a critical non-local drift. We prove a $C^{\sigma +\alpha }$ estimate in the spatial variable and $C^{1,\alpha }$ estimates in time assuming time regularity for the boundary data. The estimates are uniform in the order of the operator $I$, hence allowing us to extend the classical Evans-Krylov result for concave parabolic equations.
</p>projecteuclid.org/euclid.jiea/1476706346_20161017081229Mon, 17 Oct 2016 08:12 EDTWell-posed boundary integral equation formulations and Nyström discretizations for the solution of Helmholtz transmission problems in two-dimensional Lipschitz domainshttp://projecteuclid.org/euclid.jiea/1476706347<strong>Víctor Domínguez</strong>, <strong>Mark Lyon</strong>, <strong>Catalin Turc</strong>. <p><strong>Source: </strong>Journal of Integral Equations and Applications, Volume 28, Number 3, 395--440.</p><p><strong>Abstract:</strong><br/>
We present a comparison among the performance of solvers based on Nystr\"om discretizations of several well-posed boundary integral equation formulations of Helmholtz transmission problems in two-dimensional Lipschitz domains. Specifically, we focus on the following four classes of boundary integral formulations of Helmholtz transmission problems: (1)~the classical first kind integral equations for transmission problems~\cite {costabel-stephan}, (2)~the classical second kind integral equations for transmission problems~\cite {KressRoach}, (3)~the single integral equation formulations~\cite {KleinmanMartin}, and (4)~certain direct counterparts of recently introduced generalized combined source integral equations \cite {turc2, turc3}. The former two formulations were the only formulations whose well-posedness in Lipschitz domains was rigorously established \cite {costabel-stephan, ToWe:1993}. We establish the well-posedness of the latter two formulations in appropriate functional spaces of boundary traces of solutions of transmission Helmholtz problems in Lipschitz domains. We give ample numerical evidence that Nystr\"om solvers based on formulations (3) and (4) are computationally more advantageous than solvers based on the classical formulations (1) and (2), especially in the case of high-contrast transmission problems at high frequencies.
</p>projecteuclid.org/euclid.jiea/1476706347_20161017081229Mon, 17 Oct 2016 08:12 EDTApplication of measure of noncompactness to Volterra equations of convolution typehttp://projecteuclid.org/euclid.jiea/1481792835<strong>Edgardo Alvarez</strong>, <strong>Carlos Lizama</strong>. <p><strong>Source: </strong>Journal of Integral Equations and Applications, Volume 28, Number 4, 441--458.</p><p><strong>Abstract:</strong><br/>
Sufficient conditions for the existence of at least one solution of a nonlinear integral equation with a general kernel are established. The existence result is proved in $C([0,T],E)$, where $E$ denotes an arbitrary Banach space. We use the Darbo-Sadovskii fixed point theorem and techniques of measure of noncompactness. We extend and generalize results obtained by other authors in the context of fractional differential equations. One example illustrates the theoretical results.
</p>projecteuclid.org/euclid.jiea/1481792835_20161215040727Thu, 15 Dec 2016 04:07 ESTOn some regular fractional Sturm-Liouville problems with generalized Dirichlet conditionshttp://projecteuclid.org/euclid.jiea/1481792836<strong>Fatima-Zahra Bensidhoum</strong>, <strong>Hacen Dib</strong>. <p><strong>Source: </strong>Journal of Integral Equations and Applications, Volume 28, Number 4, 459--480.</p><p><strong>Abstract:</strong><br/>
The present work deals with some spectral properties of the problem
\medskip $(\mathcal{P} )$ $\Bigg \{$\vbox {$D^{\alpha }_{b,-}(p(x)D^{\alpha }_{a,+}y)(x)+\lambda q(x)\,y(x)=0$,\quad $a\lt x\lt b$,
\vspace {-2pt} \qquad \quad $\displaystyle \lim _{\stackrel {x\rightarrow a}{>}}(x-a)^{1-\alpha }y(x)=0=y(b)$,} \smallskip
\noindent where $p,q \in C([a,b])$, $p(x)>0$, $q(x)>0$, for all $x \in [a,b]$ and ${1}/{2} \lt \alpha \lt 1$. $D^{\alpha }_{b,-}$ and $D^{\alpha }_{a,+}$ are the right- and left-sided Riemann-Liouville fractional derivatives of order $\alpha \in (0,1)$, respectively. $\lambda $ is a scalar parameter.
First, we prove, using the spectral theory of linear compact operators, that this problem has an infinite sequence of real eigenvalues and the corresponding eigenfunctions form a complete orthonormal system in the Hilbert space $L^{2}_q[a,b]$. Then, we investigate some asymptotic properties of the spectrum as $\alpha \underset {\lt }{\rightarrow } 1$. We give, in particular, the asymptotic expansion of the first eigenvalue.
</p>projecteuclid.org/euclid.jiea/1481792836_20161215040727Thu, 15 Dec 2016 04:07 ESTApproximation of solutions to a delay equation with a random forcing term and non local conditionshttp://projecteuclid.org/euclid.jiea/1481792837<strong>Renu Chaudhary</strong>, <strong>Dwijendra N. Pandey</strong>. <p><strong>Source: </strong>Journal of Integral Equations and Applications, Volume 28, Number 4, 481--507.</p><p><strong>Abstract:</strong><br/>
The existence and approximation of a solution to a delay equation with a random forcing term and non local conditions is studied by using a stochastic version of the well-known Banach fixed point theorem and semigroup theory. Moreover, the convergence of Faedo-Galerkin approximations of the solution is shown. An example is given which illustrates the results.
</p>projecteuclid.org/euclid.jiea/1481792837_20161215040727Thu, 15 Dec 2016 04:07 ESTPerturbed Hammerstein integral equations with sign-changing kernels and applications to nonlocal boundary value problems and elliptic PDEshttp://projecteuclid.org/euclid.jiea/1481792838<strong>Christopher S. Goodrich</strong>. <p><strong>Source: </strong>Journal of Integral Equations and Applications, Volume 28, Number 4, 509--549.</p><p><strong>Abstract:</strong><br/>
We demonstrate the existence of at least one positive solution to the perturbed Hammerstein integral equation \[ y(t)=\gamma _1(t)H_1\big (\varphi _1(y)\big )+\gamma _2(t)H_2\big (\varphi _2(y)\big )\] \[\qquad \qquad \qquad \quad +\lambda \int _0^1G(t,s)f\big (s,y(s)\big )\, ds,\] where certain asymptotic growth properties are imposed on the functions $f$, $H_1$ and $H_2$. Moreover, the functionals $\varphi _1$ and $\varphi _2$ are realizable as Stieltjes integrals with signed measures, which means that the nonlocal elements in the Hammerstein equation are possibly of a very general, sign-changing form. We focus here on the case where the kernel $(t,s)\mapsto G(t,s)$ is allowed to change sign and demonstrate the existence of at least one positive solution to the integral equation. As applications, we demonstrate that, by choosing $\gamma _1$ and $\gamma _2$ in particular ways, we obtain positive solutions to boundary value problems, both in the ODEs and elliptic PDEs setting, even when the Green's function is sign-changing, and, moreover, we are able to localize the range of admissible values of the parameter~$\lambda $. Finally, we also provide a result that for each $\lambda >0$ yields the existence of at least one positive solution.
</p>projecteuclid.org/euclid.jiea/1481792838_20161215040727Thu, 15 Dec 2016 04:07 ESTExistence of solutions and controllability for impulsive fractional order damped systemshttp://projecteuclid.org/euclid.jiea/1481792839<strong>Zhenhai Liu</strong>, <strong>Xuemei Li</strong>. <p><strong>Source: </strong>Journal of Integral Equations and Applications, Volume 28, Number 4, 551--579.</p><p><strong>Abstract:</strong><br/>
In this paper, we are concerned with the controllability of linear and nonlinear Caputo impulsive fractional order damped systems in Banach spaces. Our main purpose is to establish some necessary and sufficient conditions for controllability for this kind of impulsive control system by using Mittag-Leffler matrix functions and the Schauder fixed point theorem.
</p>projecteuclid.org/euclid.jiea/1481792839_20161215040727Thu, 15 Dec 2016 04:07 ESTEssential norm of a Volterra-type integral operator from Hardy spaces to some analytic function spaceshttp://projecteuclid.org/euclid.jiea/1481792840<strong>Jizhen Zhou</strong>, <strong>Xiangling Zhu</strong>. <p><strong>Source: </strong>Journal of Integral Equations and Applications, Volume 28, Number 4, 581--593.</p><p><strong>Abstract:</strong><br/>
In this paper, we obtain some estimates of essential norm of the Volterra-type integral operator $T_g$, where \[ T_gf(z)=\int ^z_0f(\zeta )g'(\zeta )\,d\zeta , \] from Hardy spaces to the BMOA space, Besov spaces, Berg\-man spaces and Bloch-type spaces.
</p>projecteuclid.org/euclid.jiea/1481792840_20161215040727Thu, 15 Dec 2016 04:07 ESTVolume Index for Volume 28http://projecteuclid.org/euclid.jiea/1484276414<strong> </strong>. <p><strong>Source: </strong>Journal of Integral Equations and Applications, Volume 28, Number 4, 595--596.</p>projecteuclid.org/euclid.jiea/1484276414_20170112220027Thu, 12 Jan 2017 22:00 ESTRecent progress in time domain boundary integral equationshttp://projecteuclid.org/euclid.jiea/1490583469<strong>Víctor Domínguez</strong>, <strong>Nicolas Salles</strong>, <strong>Francisco-Javier Sayas</strong>. <p><strong>Source: </strong>Journal of Integral Equations and Applications, Volume 29, Number 1, 1--4.</p><p><strong>Abstract:</strong><br/>
Introduction to this special issue on recent progress in time domain boundary integral equations.
</p>projecteuclid.org/euclid.jiea/1490583469_20170326225756Sun, 26 Mar 2017 22:57 EDTComparison between numerical methods applied to the damped wave equationhttp://projecteuclid.org/euclid.jiea/1490583470<strong>A. Aimi</strong>, <strong>M. Diligenti</strong>, <strong>C. Guardasoni</strong>. <p><strong>Source: </strong>Journal of Integral Equations and Applications, Volume 29, Number 1, 5--40.</p><p><strong>Abstract:</strong><br/>
For the numerical solution of Dirichlet-Neumann problems related to 1D damped wave propagation, from a numerical point of view, we compare the so-called energetic approach, considered here separately for boundary and finite element methods with classical finite difference schemes, both explicit and implicit. The analysis reveals the superiority of energetic approximations with respect to unconditional stability and accuracy with respect to any choice of discretization parameters.
</p>projecteuclid.org/euclid.jiea/1490583470_20170326225756Sun, 26 Mar 2017 22:57 EDTNumerical approximation of first kind Volterra convolution integral equations with discontinuous kernelshttp://projecteuclid.org/euclid.jiea/1490583471<strong>Penny J. Davies</strong>, <strong>Dugald B. Duncan</strong>. <p><strong>Source: </strong>Journal of Integral Equations and Applications, Volume 29, Number 1, 41--73.</p><p><strong>Abstract:</strong><br/>
The cubic ``convolution spline'' method for first kind Volterra convolution integral equations was introduced in P.J. Davies and D.B. Duncan, $\mathit {Convolution\ spline\ approximations\ of\ Volterra\ integral\ equations}$, Journal of Integral Equations and Applications \textbf {26} (2014), 369--410. Here, we analyze its stability and convergence for a broad class of piecewise smooth kernel functions and show it is stable and fourth order accurate even when the kernel function is discontinuous. Key tools include a new discrete Gronwall inequality which provides a stability bound when there are jumps in the kernel function and a new error bound obtained from a particular B-spline quasi-interpolant.
</p>projecteuclid.org/euclid.jiea/1490583471_20170326225756Sun, 26 Mar 2017 22:57 EDTAdaptive time domain boundary element methods with engineering applicationshttp://projecteuclid.org/euclid.jiea/1490583472<strong>Heiko Gimperlein</strong>, <strong>Matthias Maischak</strong>, <strong>Ernst P. Stephan</strong>. <p><strong>Source: </strong>Journal of Integral Equations and Applications, Volume 29, Number 1, 75--105.</p><p><strong>Abstract:</strong><br/>
Time domain Galerkin boundary elements provide an efficient tool for numerical solution of boundary value problems for the homogeneous wave equation. We review recent advances in their a~posteriori error analysis and the resulting adaptive mesh refinement procedures, as well as basic algorithmic aspects of these methods. Numerical results for adaptive mesh refinements are discussed in two and three dimensions, as are benchmark problems in a half-space related to the transient emission of traffic noise.
</p>projecteuclid.org/euclid.jiea/1490583472_20170326225756Sun, 26 Mar 2017 22:57 EDTA new and improved analysis of the time domain boundary integral operators for the acoustic wave equationhttp://projecteuclid.org/euclid.jiea/1490583473<strong>Matthew E. Hassell</strong>, <strong>Tianyu Qiu</strong>, <strong>Tonatiuh Sánchez-Vizuet</strong>, <strong>Francisco-Javier Sayas</strong>. <p><strong>Source: </strong>Journal of Integral Equations and Applications, Volume 29, Number 1, 107--136.</p><p><strong>Abstract:</strong><br/>
We present a novel analysis of the boundary integral operators associated to the wave equation. The analysis is done entirely in the time-domain by employing tools from abstract evolution equations in Hilbert spaces and semi-group theory. We prove a single general theorem from which well-posedness and regularity of the solutions for several boundary integral formulations can be deduced as specific cases. By careful choices of continuous and discrete spaces, we are able to provide a concise analysis for various direct and indirect formulations, both for their Galerkin in space semi-discretizations and at the continuous level. Some of the results here are improvements on previously known results, while other results are equivalent to those in the literature. The methodology presented greatly simplifies analysis of the operators of the Calder\'on projector for the wave equation and can be generalized to other relevant boundary integral equations.
</p>projecteuclid.org/euclid.jiea/1490583473_20170326225756Sun, 26 Mar 2017 22:57 EDTMathematical aspects of variational boundary integral equations for time dependent wave propagationhttp://projecteuclid.org/euclid.jiea/1490583474<strong>Patrick Joly</strong>, <strong>Jerónimo Rodríguez</strong>. <p><strong>Source: </strong>Journal of Integral Equations and Applications, Volume 29, Number 1, 137--187.</p><p><strong>Abstract:</strong><br/>
In this work, we provide a review of recent results on the mathematical analysis of space-time variational bilinear forms associated to transient boundary integral operators for the wave equation. Most of the results will be proven directly in the time domain and compared to similar results (most of them obtained in the Laplace domain) that can be found in the literature.
</p>projecteuclid.org/euclid.jiea/1490583474_20170326225756Sun, 26 Mar 2017 22:57 EDTRunge-Kutta convolution quadrature and FEM-BEM coupling for the time-dependent linear Schrödinger equationhttp://projecteuclid.org/euclid.jiea/1490583475<strong>Jens Markus Melenk</strong>, <strong>Alexander Rieder</strong>. <p><strong>Source: </strong>Journal of Integral Equations and Applications, Volume 29, Number 1, 189--250.</p><p><strong>Abstract:</strong><br/>
We propose a numerical scheme to solve the time-dependent linear Schr\"odinger equation. The discretization is carried out by combining a Runge-Kutta time stepping scheme with a finite element discretization in space. Since the Schr\"odinger equation is posed on the whole space $\mathbb{R}^d$, we combine the interior finite element discretization with a convolution quadrature based boundary element discretization. In this paper, we analyze the resulting fully discrete scheme in terms of stability and convergence rate. Numerical experiments confirm the theoretical findings.
</p>projecteuclid.org/euclid.jiea/1490583475_20170326225756Sun, 26 Mar 2017 22:57 EDT$L^p$-solutions for a class of fractional integral equationshttp://projecteuclid.org/euclid.jiea/1497664828<strong>Ravi P. Agarwal</strong>, <strong>Asma Asma</strong>, <strong>Vasile Lupulescu</strong>, <strong>Donal O'Regan</strong>. <p><strong>Source: </strong>Journal of Integral Equations and Applications, Volume 29, Number 2, 251--270.</p><p><strong>Abstract:</strong><br/>
This paper considers the existence of $L^{p}$-solutions for a class of fractional integral equations involving abstract Volterra operators in a separable Banach space. Some applications for the existence of $L^{p}$-solutions for different classes of fractional differential equations are given.
</p>projecteuclid.org/euclid.jiea/1497664828_20170616220033Fri, 16 Jun 2017 22:00 EDTA modified volume integral equation for anisotropic elastic or conducting inhomogeneities: Unconditional solvability by Neumann serieshttp://projecteuclid.org/euclid.jiea/1497664829<strong>Marc Bonnet</strong>. <p><strong>Source: </strong>Journal of Integral Equations and Applications, Volume 29, Number 2, 271--295.</p><p><strong>Abstract:</strong><br/>
This work addresses the solvability and solution of volume integrodifferential equations (VIEs) associated with 3D free-space transmission problems (FSTPs) involving elastic or conductive inhomogeneities. A modified version of the singular volume integral equation (SVIE) associated with the VIE is introduced and shown to be of second kind involving a contraction operator, i.e., solvable by Neumann series, implying the well-posedness of the initial VIE. Then, the solvability of VIEs for frequency-domain FSTPs (modelling the scattering of waves by compactly-supported inhomogeneities) follows by a compact perturbation argument. This approach extends work by Potthast \cite {potthast:99} on 2D electromagnetic problems (transverse-electric polarization conditions) involving orthotropic inhomogeneities in a isotropic background and contains recent results on the solvability of Eshelby's equivalent inclusion problem as special cases. The proposed modified SVIE is also useful for iterative solution methods, as Neumannn series converge (i)~unconditionally for static problems and (ii)~on some inhomogeneity configurations for which divergence occurs with the usual SVIE for wave scattering problems.
</p>projecteuclid.org/euclid.jiea/1497664829_20170616220033Fri, 16 Jun 2017 22:00 EDTWell-posedness of fractional degenerate differential equations with infinite delay in vector-valued functional spaceshttp://projecteuclid.org/euclid.jiea/1497664830<strong>Shangquan Bu</strong>, <strong>Gang Cai</strong>. <p><strong>Source: </strong>Journal of Integral Equations and Applications, Volume 29, Number 2, 297--323.</p><p><strong>Abstract:</strong><br/>
We study the well-posedness of degenerate fractional differential equations with infinite delay $(P_\alpha ): D^\alpha (Mu)(t) =Au(t)+\int _{-\infty }^t a(t-s)Au(s)\,ds+f(t)$, $0\leq t\leq 2\pi $, in Lebesgue-Bochner spaces $L^p(\mathbb {T}; X)$ and Besov spaces $B_{p,q}^s(\mathbb {T}; X)$, where $A$ and $M$ are closed linear operators on a Banach space~$X$ satisfying $D(A)\subset D(M)$, $\alpha >0$ and $a\in L^1(\mathbb {R}_+)$ are fixed. Using well known operator-valued Fourier multiplier theorems, we completely characterize the well-posedness of $(P_\alpha )$ in the above vector-valued function spaces on $\mathbb {T}$.
</p>projecteuclid.org/euclid.jiea/1497664830_20170616220033Fri, 16 Jun 2017 22:00 EDTGlobal existence and asymptotic stability of mild solutions for stochastic evolution equations with nonlocal initial conditionshttp://projecteuclid.org/euclid.jiea/1497664831<strong>Pengyu Chen</strong>, <strong>Ahmed Abdelmonem</strong>, <strong>Yongxiang Li</strong>. <p><strong>Source: </strong>Journal of Integral Equations and Applications, Volume 29, Number 2, 325--348.</p><p><strong>Abstract:</strong><br/>
The aim of this paper is to discuss the global existence, uniqueness and asymptotic stability of mild solutions for a class of semilinear evolution equations with nonlocal initial conditions on infinite interval. A sufficient condition is given for judging the relative compactness of a class of abstract continuous family of functions on infinite intervals. With the aid of this criteria the compactness of the solution operator for the problem studied on the half line is obtained. The theorems proved in this paper improve and extend some related results in this direction. Discussions are based on stochastic analysis theory, analytic semigroup theory, relevant fixed point theory and the well known Gronwall-Bellman type inequality. An example to illustrate the feasibility of our main results is also given.
</p>projecteuclid.org/euclid.jiea/1497664831_20170616220033Fri, 16 Jun 2017 22:00 EDT$L^p$-approximation by truncated max-product sampling operators of Kantorovich-type based on Fejér kernelhttp://projecteuclid.org/euclid.jiea/1497664832<strong>Lucian Coroianu</strong>, <strong>Sorin G. Gal</strong>. <p><strong>Source: </strong>Journal of Integral Equations and Applications, Volume 29, Number 2, 349--364.</p><p><strong>Abstract:</strong><br/>
By use of the so-called max-product method, in this paper we associate to the truncated linear sampling operators based on the Fej\'er-type kernel, nonlinear sampling operators of Kantorovich type, for which we prove convergence results in the $L^{p}$-norm, $1\le p\le +\infty $, with quantitative estimates.
</p>projecteuclid.org/euclid.jiea/1497664832_20170616220033Fri, 16 Jun 2017 22:00 EDTNecessary Fredholm conditions for weighted singular integral operators with shifts and slowly oscillating datahttp://projecteuclid.org/euclid.jiea/1502676095<strong>Alexei Yu. Karlovich</strong>, <strong>Yuri I. Karlovich</strong>, <strong>Amarino B. Lebre</strong>. <p><strong>Source: </strong>Journal of Integral Equations and Applications, Volume 29, Number 3, 365--399.</p><p><strong>Abstract:</strong><br/>
We extend the main result of {KKL11b} to the case of more general weighted singular integral operators with two shifts of the form \[ (aI-b U_\alpha )P_\gamma ^++(cI-dU_\beta )P_\gamma ^-, \] acting on the space $L^p(\mathbb{R} _+)$, $1\lt p\lt \infty $, where \[ P_\gamma ^\pm =(I\pm S_\gamma )/2 \] are operators associated with the weighted Cauchy singular integral operator $S_\gamma $, given by \[ (S_\gamma f)(t)=\frac {1}{\pi i}{\int _{\mathbb{R} _+}} \bigg (\frac {t}{\tau }\bigg )^\gamma \frac {f(\tau )}{\tau -t}\,d\tau \] with $\gamma \in \mathbb{C} $ satisfying $0\lt 1/p+\Re \gamma \lt 1$, and $U_\alpha ,U_\beta $ are the isometric shift operators given by \[ U_\alpha f=(\alpha ')^{1/p}(f\circ \alpha ), \qquad U_\beta f=(\beta ')^{1/p}(f\circ \beta ), \] generated by diffeomorphisms $\alpha ,\beta $ of $\mathbb{R} _+$ onto itself having only two fixed points at the endpoints $0$ and $\infty $, under the assumptions that the coefficients $a,b,c,d$ and the derivatives $\alpha ',\beta '$ of the shifts are bounded and continuous on $\mathbb{R} _+$ and admit discontinuities of slowly oscillating type at $0$ and $\infty $.
</p>projecteuclid.org/euclid.jiea/1502676095_20170813220140Sun, 13 Aug 2017 22:01 EDTCentral part interpolation schemes for integral equations with singularitieshttp://projecteuclid.org/euclid.jiea/1502676096<strong>Kerli Orav-Puurand</strong>, <strong>Arvet Pedas</strong>, <strong>Gennadi Vainikko</strong>. <p><strong>Source: </strong>Journal of Integral Equations and Applications, Volume 29, Number 3, 401--440.</p><p><strong>Abstract:</strong><br/>
Two high order methods are constructed and analyzed for a class of Fredholm integral equations of the second kind with kernels that may have weak boundary and diagonal singularities. The proposed methods are based on improving the boundary behavior of the exact solution with the help of a change of variables, and on central part interpolation by polynomial splines on the uniform grid. A detailed error analysis for the proposed numerical schemes is given. This includes, in particular, error bounds under various types of assumptions on the equation, and shows that the proposed central part collocation approach has accuracy and numerical stability advantages compared with standard piecewise polynomial collocation methods, including the collocation at Chebyshev knots.
</p>projecteuclid.org/euclid.jiea/1502676096_20170813220140Sun, 13 Aug 2017 22:01 EDTWell-conditioned boundary integral equation formulations and Nyström discretizations for the solution of Helmholtz problems with impedance boundary conditions in two-dimensional Lipschitz domainshttp://projecteuclid.org/euclid.jiea/1502676097<strong>Catalin Turc</strong>, <strong>Yassine Boubendir</strong>, <strong>Mohamed Kamel Riahi</strong>. <p><strong>Source: </strong>Journal of Integral Equations and Applications, Volume 29, Number 3, 441--472.</p><p><strong>Abstract:</strong><br/>
We present a regularization strategy that leads to well-conditioned boundary integral equation formulations of Helm\-holtz equations with impedance boundary conditions in two-dimensional Lipschitz domains. We consider both the case of classical impedance boundary conditions, as well as that of transmission impedance conditions wherein the impedances are certain coercive operators. The latter type of problem is instrumental in the speed up of the convergence of Domain Decomposition Methods for Helmholtz problems. Our regularized formulations use as unknowns the Dirichlet traces of the solution on the boundary of the domain. Taking advantage of the increased regularity of the unknowns in our formulations, we show through a variety of numerical results that a graded-mesh based Nystr\"om discretization of these regularized formulations leads to efficient and accurate solutions of interior and exterior Helmholtz problems with impedance boundary conditions.
</p>projecteuclid.org/euclid.jiea/1502676097_20170813220140Sun, 13 Aug 2017 22:01 EDT