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In this paper, we propose a sharp and quantitative criterion, which focuses solely on Q curvature, to demonstrate the Chern-Gauss-Bonnet integral. In contrast to the previous results [6,7,11] , we use a new approach that involves estimating the singular integral. Furthermore, we derive the asymptotic formula for the solution to the general Q curvature equation. [ABSTRACT FROM AUTHOR]

This paper is devoted to a class of integral type Brezis-Nirenbreg problems on the Heisenberg group. They are a class of nonlinear integral equations on the bounded domains of Heisenberg group and related to the CR Yamabe problems on CR manifold. Based on the sharp Hardy-Littlewood-Sobolev inequalities on the Heisenberg group, the nonexistence and existence results are obtained. [ABSTRACT FROM AUTHOR]

In a recent article the authors showed that the radiative Transfer equations with multiple frequencies and scattering can be formulated as a nonlinear integral system. In the present article, the formulation is extended to handle reflective boundary conditions. The fixed point method to solve the system is shown to be monotone. The discretization is done with a P 1 Finite Element Method. The convolution integrals are precomputed at every vertex of the mesh and stored in compressed hierarchical matrices, using Partially Pivoted Adaptive Cross-Approximation. Then the fixed point iterations involve only matrix vector products. The method is O (N N 3 ln ⁡ N) , with respect to the number of vertices, when everything is smooth. A numerical implementation is proposed and tested on two examples. As there are some analogies with ray tracing the programming is complex. • We present a method and a an algorithm using an H-matrix compression scheme to compute the temperature of a gas under electromagnetic radiations. • The method is now capable of handling reflective boundaries. • The paper improves on previously existing results and the numerical methods use state of the art computational tools. [ABSTRACT FROM AUTHOR]

In this paper we introduce and study some nonlinear integral equations on bounded domains that are related to the sharp Hardy–Littlewood–Sobolev inequality. Existence results as well as nonexistence results are obtained. [ABSTRACT FROM AUTHOR]

Abstract We consider the Banach function space consisting of real functions defined on the half-axis and having increments tempered by a given modulus of continuity. Next, we formulate and prove a sufficient condition for relative compactness of a bounded subset of the mentioned Banach space. On the basis of that condition we construct a measure of noncompactness in the function space in question. Using that measure we prove a theorem on the existence of solutions of a quadratic integral equation in the space of functions satisfying the Hölder condition on the real half-axis. [ABSTRACT FROM AUTHOR]

Abstract Consider the integral equation f q − 1 (x) = ∫ Ω f (y) | x − y | n − α d y , f (x) > 0 , x ∈ Ω ‾ , where Ω ⊂ R n is a smooth bounded domain. For 1 < α < n , the existence of energy maximizing positive solution in the subcritical case 2 < q < 2 n n + α , and nonexistence of energy maximizing positive solution in the critical case q = 2 n n + α are proved in [6]. For α > n , the existence of energy minimizing positive solution in the subcritical case 0 < q < 2 n n + α , and nonexistence of energy minimizing positive solution in the critical case q = 2 n n + α are also proved in [4]. Based on these, in this paper, the blowup behaviour of energy maximizing positive solution as q → (2 n n + α) + (in the case of 1 < α < n), and the blowup behaviour of energy minimizing positive solution as q → (2 n n + α) − (in the case of α > n) are analyzed. We see that for 1 < α < n the blowup behaviour obtained is quite similar to that of the elliptic equation involving the subcritical Sobolev exponent. But for α > n , different phenomena appear. [ABSTRACT FROM AUTHOR]

Highlights • Solving the anisotropic acoustic equation without S-wave artifacts. • Correct the dispersion error and phase error of the finite-difference method. • Deal with free surface and absorbing boundary condition in a direct way. Abstract Numerical solutions of the acoustic wave equation, especially in anisotropic media, is crucial to seismic modeling, imaging and inversion as it provides efficient, practical, and stable approximate representation of the medium. However, a clean implementation (free of shear wave artifacts and dispersion) of wave propagation, especially in anisotropic media, requires an integral operator, the direct evaluation of which is extremely expensive. Recently, the low-rank method was proposed to provide a good approximation to the integral operator utilizing Fourier transforms. Thus, we propose to split the integral operator into two terms. The first term provides a differential operator that approximates that can be approximated with a standard finite-difference method. We, then, apply the low-rank approximation on the residual term of the finite-difference operator. We implement the two terms in two complementing steps, in which the spectral step corrects for any errors admitted by the finite difference step. Even though we utilize finite-difference approximations, the resulting algorithm admits spectral accuracy. Also, through the finite difference step, the method can deal approximately with the free surface and absorbing boundary conditions in a straight forward manner. Numerical examples show that the method is of high accuracy and efficiency. [ABSTRACT FROM AUTHOR]

Abstract In this paper we present an efficient and accurate integral equation method to compute the scattering resonances for a subwavelength metallic slit structure. A new boundary integral equation is derived for the scattering problem, and the computation of scattering resonances is reduced to solving the eigenvalues of the corresponding homogeneous formulation over the complex plane. The integral operators are evaluated with high-order precisions by accurate calculations of the Green's functions for the layered medium and accelerated computation of the slit Green's function. The Newton's method is employed for solving the eigenvalues of the boundary integral formulation. We propose an effective strategy for obtaining the initial guesses of scattering resonances by introducing an approximate model for the scattering problem, for which the leading orders of the resonances are derived by asymptotic analysis. Numerical experiments are provided to demonstrate the accuracy, efficiency, and robustness of the method. [ABSTRACT FROM AUTHOR]

Abstract The article suggests a new approach using the so-called consistency method for the inversion of the spherical mean Radon transform in 2D with detectors on a line. We present a new iterative formula which gives an algorithm to recover an unknown function supported completely on one side of a line L from its spherical means over circles centered on the line L. Our reconstruction formula has the benefit of being local. Such an inversion is required in problems of thermo- and photo-acoustic tomography, ultrasound reflection tomography, radar imaging, and a few other. [ABSTRACT FROM AUTHOR]

Abstract In the first part of this work we show the convergence with respect to an asymptotic parameter ε of a delayed heat equation. It represents a mathematical extension of works considered previously by the authors [14–16]. Namely, this is the first result involving delay operators approximating protein linkages coupled with a spatial elliptic second order operator. For the sake of simplicity we choose the Laplace operator, although more general results could be derived. The main arguments are (i) new energy estimates and (ii) a stability result extended from the previous work to this more involved context. They allow to prove convergence of the delay operator to a friction term together with the Laplace operator in the same asymptotic regime considered without the space dependence in [14]. In a second part we extend fixed-point results for the fully non-linear model introduced in [16] and prove global existence in time. This shows that the blow-up scenario observed previously does not occur. Since the latter result was interpreted as a rupture of adhesion forces, we discuss the possibility of bond breaking both from the analytic and numerical point of view. [ABSTRACT FROM AUTHOR]

For refracted spectrally negative Lévy processes, we identify expressions for several quantities related to the Laplace transforms of their weighted occupation times until first exit times. Such quantities are expressed in terms of the unique solutions to integral equations involving weight functions and scale functions for the associated spectrally negative Lévy processes. Previous results on refracted Lévy processes are recovered. [ABSTRACT FROM AUTHOR]

In this work, a modified Particle Method in fluid mechanics involving irregular distribution and discontinuity issues is proposed. Due to the approximation accuracy, momentum conservation and governing equation feasibility, traditional Particle Method, Moving Particle Semi-implicit (MPS) method in this paper, cannot guarantee the accuracy and stability of computation, and the divergence calculation in Navier–Stokes equation may cause numerical error in computation domain with discontinuity. To enhance the computation accuracy of Particle Method, we modify the gradient operator with a corrected tensor emanating from minimizing the local error of the first-order Taylor Expansion approximation. Besides, inspired by Peridynamic which is mostly used in solid mechanics, a new governing equation in an integral form which maintains the momentum conservation is obtained. Combining the modified gradient and new governing equation, we obtain a new particle-based method. The numerical results verify its feasibility and illustrate a good computational performance of proposed Method in the fluid dynamics involving non-uniform particle distribution. [ABSTRACT FROM AUTHOR]

Over the recent decades, a number of fast approximate solutions of Lippmann–Schwinger equation, which are more accurate than classic Born and Rytov approximations, were proposed in the field of electromagnetic modeling. Those developments could be naturally extended to acoustic and elastic fields; however, until recently, they were almost unknown in seismology. This paper presents several solutions of this kind applied to acoustic modeling for both lossy and lossless media. We evaluated the numerical merits of those methods and provide an estimation of their numerical complexity. In our numerical realization we use the matrix-free implementation of the corresponding integral operator. We study the accuracy of those approximate solutions and demonstrate, that the quasi-analytical approximation is more accurate, than the Born approximation. Further, we apply the quasi-analytical approximation to the solution of the inverse problem. It is demonstrated that, this approach improves the estimation of the data gradient, comparing to the Born approximation. The developed inversion algorithm is based on the conjugate-gradient type optimization. Numerical model study demonstrates that the quasi-analytical solution significantly reduces computation time of the seismic full-waveform inversion. We also show how the quasi-analytical approximation can be extended to the case of elastic wavefield. [ABSTRACT FROM AUTHOR]

Scattering from large, open cavity structures is of importance in a variety of electromagnetic applications. In this paper, we propose a new well conditioned integral equation for scattering from general open cavities embedded in an infinite, perfectly conducting half-space. The integral representation permits the stable evaluation of both the electric and magnetic field, even in the low-frequency regime, using the continuity equation in a post-processing step. We establish existence and uniqueness results, and demonstrate the performance of the scheme in the cavity-of-revolution case. High-order accuracy is obtained using a Nyström discretization with generalized Gaussian quadratures. [ABSTRACT FROM AUTHOR]

We solve the 2D transverse-electrically polarized domain-integral equation in a layered background medium by applying a Gabor frame as a projection method. This algorithm employs both a spatial and a spectral discretization of the electric field and the contrast current in the direction of the layer extent. In the spectral domain we use a representation on the complex plane that avoids the poles and branchcuts found in the Green function. Because of the special choice of the complex-plane path in the spectral domain and because of the choice to use a Gabor frame to represent functions on this path, fast algorithms based on FFTs are available to transform to and from the spectral domain, yielding an O ( N log ⁡ N ) scaling in computation time. [ABSTRACT FROM AUTHOR]

To study the temperature in a gas subjected to electromagnetic radiations, one may use the Radiative Transfer equations coupled with the Navier-Stokes equations. The problem has 7 dimensions; however with minimal simplifications it is equivalent to a small number of integro-differential equations in 3 dimensions. We present the method and a numerical implementation using an H -matrix compression scheme. The result is very fast: 240K physical points, all directions of radiation and 683 frequencies require less than 35 minutes on an Apple M1 Laptop. The method is capable of handling variable absorption and scattering functions of spatial positions and frequencies. The implementation is done using htool,1 a matrix compression library interfaced with the PDE solver freefem++. Applications to the temperature in the French Chamonix valley are presented at different hours of the day with and without snow or cloud and with a variable absorption coefficient taken from the Gemini measurements. The software is precise enough to assert temperature differences due to increased absorption in the vibrational frequency subrange of greenhouse gases. [ABSTRACT FROM AUTHOR]

Let N be an integral operator of the form ( N u ) ( x ) = ∫ R c n ( x , x − y ) u ( y ) d y acting in L p ( R c ) with a measurable kernel n satisfying the estimate | n ( x , y ) | ≤ β ( y ) , where β ∈ L 1 . It is proved that if the function t ↦ n ( t , ⋅ ) is continuous in the norm of L 1 and the operator 1 + N has an inverse, then ( 1 + N ) − 1 = 1 + M , where M is an integral operator possessing the same properties. [ABSTRACT FROM AUTHOR]

We apply boundary integral equations for the first time to the two-dimensional scattering of time-harmonic waves from a smooth obstacle embedded in a continuously-graded unbounded medium. In the case we solve, the square of the wavenumber (refractive index) varies linearly in one coordinate, i.e. ( Δ + E + x 2 ) u ( x 1 , x 2 ) = 0 where E is a constant; this models quantum particles of fixed energy in a uniform gravitational field, and has broader applications to stratified media in acoustics, optics and seismology. We evaluate the fundamental solution efficiently with exponential accuracy via numerical saddle-point integration, using the truncated trapezoid rule with typically 10 2 nodes, with an effort that is independent of the frequency parameter E . By combining with a high-order Nyström quadrature, we are able to solve the scattering from obstacles 50 wavelengths across to 11 digits of accuracy in under a minute on a desktop or laptop. [ABSTRACT FROM AUTHOR]

Let R + n be the upper half Euclidean space and let α be any real number between 0 and 2. Consider the following Dirichlet problem involving the fractional Laplacian: (1) { ( − Δ ) α / 2 u = u p , x ∈ R + n , u ≡ 0 , x ∉ R + n . Instead of using the conventional extension method of Caffarelli and Silvestre [8] , we employ a new and direct approach by studying an equivalent integral equation (2) u ( x ) = ∫ R + n G ( x , y ) u p ( y ) d y . Applying the method of moving planes in integral forms , we prove the non-existence of positive solutions in the critical and subcritical cases under no restrictions on the growth of the solutions. [ABSTRACT FROM AUTHOR]

In this paper we deal with the inverse scattering problem for an impedance obstacle. The aim is to recover both the impedance function and the scatterer simultaneously. Based on boundary integral equations, our method splits the inverse problem into a well-posed direct problem followed by a smaller ill-posed problem which has advantages both in understanding the inverse problem and in the numerical reconstructions. [ABSTRACT FROM AUTHOR]

Abstract: This article deals with boundary integral equation preconditioning for the multiple scattering problem. The focus is put on the single scattering preconditioner, corresponding to the diagonal part of the integral operator, for which two results are proved. Indeed, after applying this geometric preconditioner, it appears that, firstly, every direct integral equations become identical to each other, and secondly, that the indirect integral equation of Brakhage–Werner becomes equal to the direct integral equations, up to a change of basis. These properties imply in particular that the convergence rate of a Krylov subspaces solver will be exactly the same for every preconditioned integral equations. To illustrate this, some numerical simulations are provided at the end of the paper. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the following integral system which is related to the weak type convolution-Youngʼs inequality. Under the assumption of that and , we show that system (0.1) doesnʼt have a positive solution in . Furthermore, we prove that as and , system (0.1) does not admit positive solution in ( ), which implies that the maximizing pair of the weak type convolution-Youngʼs inequality with kernel function does not exist. Meanwhile, for and , we also show that the system (0.1) doesnʼt admit non-negative Lebesgue measurable solution. This is distinct from the original conformal invariant integral system. [Copyright &y& Elsevier]

Abstract: We present a frequency-domain renormalized integral equation formulation for solving a three-dimensional visco-acoustic medium using an iterative solver. Upon applying this special renormalization, the resulting integral equation operator can be proven to have a contraction property. Hence, solving the linear-system of equations using a Krylov optimization method, will result in a good convergence rate. Furthermore since the matrix–vector multiplication can be done using a Fast-Fourier transform (FFT) technique, its operation is of the order of , where N is the size of the discretization grid. This technique also allows us to use matrix-free implementation. Hence, the memory usage is about . Numerical tests show that the computational time and memory usage of this renormalized integral equation approach can be quite competitive with the frequency-domain finite difference iterative solver. Further, the numerical examples demonstrate that it is possible to solve a problem with over 100 million unknowns using an integral equation approach. [Copyright &y& Elsevier]

Abstract: In this paper we study the quadratic integral equation of the form Several existence theorems for a.e. monotonic solutions in Orlicz spaces are proved for strongly nonlinear functions f. The presented method of the proof can be easily extended to different classes of solutions. [Copyright &y& Elsevier]

Abstract: An integral equation based scheme is presented for the fast and accurate computation of effective conductivities of two-component checkerboard-like composites with complicated unit cells at very high contrast ratios. The scheme extends recent work on multi-component checkerboards at medium contrast ratios. General improvement include the simplification of a long-range preconditioner, the use of a banded solver, and a more efficient placement of quadrature points. This, together with a reduction in the number of unknowns, allows for a substantial increase in achievable accuracy as well as in tractable system size. Results, accurate to at least nine digits, are obtained for random checkerboards with over a million squares in the unit cell at contrast ratio 106. Furthermore, the scheme is flexible enough to handle complex valued conductivities and, using a homotopy method, purely negative contrast ratios. Examples of the accurate computation of resonant spectra are given. [Copyright &y& Elsevier]

Abstract: In this paper we apply fixed point theorems for increasing mappings in ordered normed spaces to prove existence and comparison results for solutions of discontinuous functional differential and integral equations containing Henstock–Kurzweil integrable functions. [Copyright &y& Elsevier]

Abstract: In this paper, we generalize some integral inequalities to more general situations, and the inequalities of Pachpatte type are corollaries of our''s. We establish bounds on the solutions, and we show the usefulness of our results in investigating the asymptotic behavior and the stability on the solutions of integral equations, differential equations and integro-differential equations with time delay. [Copyright &y& Elsevier]

Abstract: In this paper we prove monotone convergence theorems for Henstock–Kurzweil integrable functions from a compact real interval to an ordered Banach space. These theorems are then applied to prove existence results for solutions of a discontinuous functional integral equation containing Henstock–Kurzweil integrable functions. [ABSTRACT FROM AUTHOR]

Abstract: An algorithm is presented for the fast and accurate solution of the electrostatic equation on multi-component random checkerboards. It relies on a particular choice of integral equation, extended as to separate ill-conditioning due to singular fields in corners from ill-conditioning due to interaction of clusters of well-conducting squares at large distances. Two separate preconditioners take care of the two separate phenomena. In a series of numerical examples, effective conductivities are computed for random checkerboards containing up to 104 squares with conductivity ratios of up to 106. The achievable relative precision in these examples is on the order of 10−11. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we generalize some integral inequalities to more general situations. We establish bounds on the solutions and, by means of examples, we show the usefulness of our results in investigating the global existence of the solutions of integral equations and differential equations. [Copyright &y& Elsevier]

Abstract: In this paper, we consider band structure calculations governed by the Helmholtz or Maxwell equations in piecewise homogeneous periodic materials. Methods based on boundary integral equations are natural in this context, since they discretize the interface alone and can achieve high order accuracy in complicated geometries. In order to handle the quasi-periodic conditions which are imposed on the unit cell, the free-space Green’s function is typically replaced by its quasi-periodic cousin. Unfortunately, the quasi-periodic Green’s function diverges for families of parameter values that correspond to resonances of the empty unit cell. Here, we bypass this problem by means of a new integral representation that relies on the free-space Green’s function alone, adding auxiliary layer potentials on the boundary of the unit cell itself. An important aspect of our method is that by carefully including a few neighboring images, the densities may be kept smooth and convergence rapid. This framework results in an integral equation of the second kind, avoids spurious resonances, and achieves spectral accuracy. Because of our image structure, inclusions which intersect the unit cell walls may be handled easily and automatically. Our approach is compatible with fast-multipole acceleration, generalizes easily to three dimensions, and avoids the complication of divergent lattice sums. [ABSTRACT FROM AUTHOR]

Abstract: Automatic Chebyshev spectral collocation methods for Fredholm and Volterra integral and integro-differential equations have been implemented as part of the chebfun software system. This system enables a symbolic syntax to be applied to numerical objects in order to pose and solve problems without explicit references to discretization. The same objects can be used in matrix-free iterative methods in linear algebra, in order to avoid very large dense matrices or allow application to problems with nonsmooth coefficients. As a further application of the ability to implement operator equations, a method of Greengard for the recasting of differential equations as integral equations is generalized to mth order boundary value and generalized eigenvalue problems. In the integral form, large condition numbers associated with differentiation matrices in high-order problems are avoided. The ability to implement the recasting process generally follows from implementation of the operator expressions in chebfun. The integral method also can be extended to first-order systems, although chebfun syntax does not currently allow easy implementation in this case. [Copyright &y& Elsevier]

Abstract: We use Newton’s method to approximate a locally unique solution of an equation in a Banach space setting. We introduce recurrent functions to provide a weaker semilocal convergence analysis for Newton’s method than before [J. Appell, E. De Pascale, J.V. Lysenko, P.P. Zabrejko, New results on Newton–Kantorovich approximations with applications to nonlinear integral equations, Numer. Funct. Anal. Optim. 18 (1997) 1–17; I.K. Argyros, The theory and application of abstract polynomial equations, in: Mathematics Series, St. Lucie/CRC/Lewis Publ., Boca Raton, Florida, USA, 1998; I.K. Argyros, Concerning the “terra incognita” between convergence regions of two Newton methods, Nonlinear Anal. 62 (2005) 179–194; I.K. Argyros, Convergence and Applications of Newton-Type Iterations, Springer-Verlag Publ., New York, 2008; S. Chandrasekhar, Radiative Transfer, Dover Publ., New York, 1960; F. Cianciaruso, E. De Pascale, Newton–Kantorovich approximations when the derivative is Hölderian: Old and new results, Numer. Funct. Anal. Optim. 24 (2003) 713–723; N.T. Demidovich, P.P. Zabrejko, Ju.V. Lysenko, Some remarks on the Newton–Kantorovich method for nonlinear equations with Hölder continuous linearizations, Izv. Akad. Nauk Belorus 3 (1993) 22–26. (in Russian); E. De Pascale, P.P. Zabrejko, Convergence of the Newton–Kantorovich method under Vertgeim conditions: A new improvement, Z. Anal. Anwendvugen 17 (1998) 271–280; L.V. Kantorovich, G.P. Akilov, Functional Analysis, Pergamon Press, Oxford, 1982; J.V. Lysenko, Conditions for the convergence of the Newton–Kantorovich method for nonlinear equations with Hölder linearizations, Dokl. Akad. Nauk BSSR 38 (1994) 20–24. (in Russian); B.A. Vertgeim, On conditions for the applicability of Newton’s method, (Russian), Dokl. Akad. Nauk., SSSR 110 (1956) 719–722; B.A. Vertgeim, On some methods for the approximate solution of nonlinear functional equations in Banach spaces, Uspekhi Mat. Nauk 12 (1957) 166–169. (in Russian); English transl.:; Amer. Math. Soc. Transl. 16 (1960) 378–382] provided that the Fréchet-derivative of the operator involved is -Hölder continuous (). Numerical examples involving integral and differential equations are also provided in this study. [Copyright &y& Elsevier]

Abstract: In this paper we introduce a Conjugate Gradient Fast Fourier Transform (CG-FFT) scheme for the numerical solution of the integral equation formulating three-dimensional elastic scattering problems. The formulation is in terms of the stress tensor and particle velocities as the unknown field variables. In contrast with the formulation based on particle displacements, this approach leads to integral representations that do not involve derivatives of the unknown fields, thus resulting in simplified and more stable numerics. The numerical procedure is based on suitable quadrature formulas that provide (second order) accurate approximations while retaining the convolution nature of the relevant integrals that make them amenable to efficient evaluation via FFTs. The scheme is further improved through the introduction of (approximation-based) pre-conditioners that are shown to accelerate the convergence of the CG iterations. Numerical results are presented that demonstrate the accuracy and efficiency of the proposed methodology. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the numerical solution of the integral equation of the second kind reduced by acoustic scattering in shallow oceans with Dirichlet condition. Based on analyzing the singularity of the truncating kernel with a sum of infinite series, using our trigonometric interpolatory wavelets and collocation method, we obtain the numerical solution which possesses a fast convergence rate like . Moreover, the entries of the stiffness matrix can be obtained by FFT, which lead the computational complexity to decrease obviously. [Copyright &y& Elsevier]

Abstract: We take a fairly comprehensive approach to the problem of solving elliptic partial differential equations numerically using integral equation methods on domains where the boundary has a large number of corners and branching points. Use of non-standard integral equations, graded meshes, interpolatory quadrature, and compressed inverse preconditioning are techniques that are explored, developed, mixed, and tested on some familiar problems in materials science. The recursive compressed inverse preconditioning, the major novelty of the paper, turns out to be particularly powerful and, when it applies, eliminates the need for mesh grading completely. In an electrostatic example for a multiphase granular material with about two thousand corners and triple junctions and a conductivity ratio between phases up to a million we compute a common functional of the solution with an estimated relative error of . In another example, five times as large but with a conductivity ratio of only a hundred, we achieve an estimated relative error of . [Copyright &y& Elsevier]

Abstract: The conventional magnetic induction equation that governs hydromagnetic dynamo action is transformed into an equivalent integral equation system. An advantage of this approach is that the computational domain is restricted to the region occupied by the electrically conducting fluid and to its boundary. This integral equation approach is first employed to simulate kinematic dynamos excited by Beltrami-like flows in a finite cylinder. The impact of externally added layers around the cylinder on the onset of dynamo actions is investigated. Then it is applied to simulate dynamo experiments within cylindrical geometry including the “von Kármán sodium” (VKS) experiment and the Riga dynamo experiment. A modified version of this approach is utilized to investigate magnetic induction effects under the influence of externally applied magnetic fields which is also important to measure the proximity of a given dynamo facility to the self-excitation threshold. [Copyright &y& Elsevier]

Abstract: We study the travelling wave problem with an asymmetric kernel J and a monostable nonlinearity. We prove the existence of a minimal speed, and under certain hypothesis the uniqueness of the profile for . For we show examples of nonuniqueness. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to present some fixed point results for self-generalized contractions in ordered metric spaces. Our results generalize and extend some recent results of A.C.M. Ran, M.C. Reurings [A.C.M. Ran, M.C. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc. 132 (2004) 1435–1443], J.J. Nieto, R. Rodríguez-López [J.J. Nieto, R. Rodríguez-López, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order 22 (2005) 223–239; J.J. Nieto, R. Rodríguez-López, Existence and uniqueness of fixed points in partially ordered sets and applications to ordinary differential equations, Acta Math. Sin. (Engl. Ser.) 23 (2007) 2205–2212], J.J. Nieto, R.L. Pouso, R. Rodríguez-López [J.J. Nieto, R.L. Pouso, R. Rodríguez-López, Fixed point theorem theorems in ordered abstract sets, Proc. Amer. Math. Soc. 135 (2007) 2505–2517], A. Petruşel, I.A. Rus [A. Petruşel, I.A. Rus, Fixed point theorems in ordered L-spaces, Proc. Amer. Math. Soc. 134 (2006) 411–418] and R.P. Agarwal, M.A. El-Gebeily, D. O''Regan [R.P. Agarwal, M.A. El-Gebeily, D. O''Regan, Generalized contractions in partially ordered metric spaces, Appl. Anal., in press]. As applications, existence and uniqueness results for Fredholm and Volterra type integral equations are given. [Copyright &y& Elsevier]

Abstract: The plan of this paper is to obtain one-point iterative methods with any R-order of convergence, when they are applied to approximate solutions of quadratic equations in Banach spaces. To do this, we consider real Cauchy''s method and, under certain natural modifications, it is extended to Banach spaces. Some applications are also provided. [Copyright &y& Elsevier]

Abstract: This article in devoted to the study of the nonlocal dispersal equation and its stationary counterpart. We prove global existence for the initial value problem, and under suitable hypothesis on g and J, we prove that positive bounded stationary solutions exist. We also analyze the asymptotic behavior of the finite mass solutions as , showing that they converge locally to zero. [Copyright &y& Elsevier]

Abstract: In this paper we investigate the exponential dichotomy of linear skew-product semiflows over semiflows by considering the operators generated by the integral equation related to strongly continuous cocycles over metric spaces acting on Banach bundles. We characterize the existence of exponential dichotomy by properties of these operators and use this characterization to prove the robustness of exponential dichotomy. [Copyright &y& Elsevier]

Abstract: We study the existence of periodic solutions of the nonlinear neutral system of differential equations of the form In the process we use the fundamental matrix solution of and convert the given neutral differential equation into an equivalent integral equation. Then we construct appropriate mappings and employ Krasnoselskii''s fixed point theorem to show the existence of a periodic solution of this neutral differential equation. We also use the contraction mapping principle to show the existence of a unique periodic solution of the equation. [Copyright &y& Elsevier]

Abstract: This paper is devoted to answering a question asked recently by Y. Li regarding geometrically interesting integral equations. The main result is to give a necessary and sufficient condition on the parameters so that the integral equation with parameters to be discussed in this paper have regular solutions. In the case such condition is satisfied, we will write down the exact solution. As its application of our method, we should show that the non-existence theory of the solutions of prescribed scalar curvature equation on can be generalized to that of prescribed Branson–Paneitz Q-curvature equations on . [Copyright &y& Elsevier]

Abstract: The aim of this paper is to give characterizations for uniform and exponential dichotomies of evolution families on the half-line. We associate with a discrete evolution family the subspace . Supposing that is closed and complemented, we prove that the admissibility of the pair implies the uniform dichotomy of Φ. Under the same hypothesis on , we obtain that the admissibility of the pair with is a sufficient condition for the exponential dichotomy of Φ, which becomes necessary when Φ is with exponential growth. We apply our results in order to deduce new characterizations for exponential dichotomy of evolution families in terms of the solvability of associated difference and integral equations. [Copyright &y& Elsevier]

Abstract: In this paper, we describe an integral equation approach for simulating diffusion problems with moving interfaces. The solutions are represented as moving layer potentials where the unknowns are only defined on the interfaces. The resulting integro-differential equation (IDE) system is solved using spectral deferred correction (SDC) techniques developed for general differential algebraic equations (DAEs), and the time dependent potentials are evaluated efficiently using fast convolution algorithms. The numerical solver is applied to the BCF model for the epitaxial step-flow growth of crystals, for which the solutions are calculated accurately instead of using quasi-static approximations. Numerical results in 1+1 dimensions are compared with available results in the literature. [Copyright &y& Elsevier]