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In this research, two systems of nonlinear Volterra integral equation and Volterra integro‐differential equation were considered. New results in sense of Ulam stabilities in relation to these two systems were proved on a finite interval. The proof of the results on the Ulam stabilities of that classes of the equations are based on the nonlinear alternative related to Banach's contraction principle. The outcomes of this research give new contribution to the theory of Ulam stabilities. [ABSTRACT FROM AUTHOR]

In many physical problems, it is important to capture exponentially small effects that lie beyond‐all‐orders of an algebraic asymptotic expansion; when collected, the full asymptotic expansion is known as a trans‐series. Applied exponential asymptotics has been enormously successful in developing practical tools for studying the leading exponentials of a trans‐series expansion, typically for singularly perturbed nonlinear differential or integral equations. Separately to applied exponential asymptotics, there exists a related line of research known as Écalle's theory of resurgence, which, via Borel resummation, describes the connection between trans‐series and a certain class of holomorphic functions known as resurgent functions. Most applications and examples of Écalle's resurgence theory focus mainly on nonparametric asymptotic expansions (i.e., differential equations without a parameter). The relationships between these latter areas with applied exponential asymptotics have not been thoroughly examined—largely due to differences in language and emphasis. In this work, we establish these connections as an alternative framework to the factorial‐over‐power ansatz procedure in applied exponential asymptotics and clarify a number of aspects of applied exponential asymptotic methodology, including Van Dyke's rule and the universality of factorial‐over‐power ansatzes. We provide a number of useful tools for probing more pathological problems in exponential asymptotics and establish a framework for future applications to nonlinear and multidimensional problems in the physical sciences. [ABSTRACT FROM AUTHOR]

Two weakly nonlinear integral equations of motion commonly used in the literature to study the nonlinear dynamics of straight and initially not rectilinear Euler‐Bernoulli beams, respectively, are further investigated. Attention is focused on the process known as "static condensation", which consists of neglecting the axial inertia in the exact, fully nonlinear system of equations of motion to determine the axial displacement as a function of the transversal one. The novelty of the paper relies on showing that, contrarily to expectation and somehow surprisingly, the integral equation for beams with a not rectilinear initial configuration cannot be obtained by the static condensation process starting from the exact, fully nonlinear, equations of motion, apart from a very particular and specific case. On the contrary, it is confirmed the well‐known result that for rectilinear beams the integral equation can be obtained by the static condensation. This highlights a major difference between the two integral equations in terms of reliability and allows us a better understanding of the integral equation of rectilinear beams, underlying its stronger mathematical background than the classical counterpart for not straight beams (i.e., its being obtainable from the exact, fully nonlinear, equations of motion via static condensation), which provides it with a "special" behavior and makes it more trustworthy. [ABSTRACT FROM AUTHOR]

In this paper, a spectral method is implemented to solve a nonlinear multidimensional Volterra integral equation. Under some condition, the spectral convergence analysis of the solution u$$ u $$ is investigated in both L∞$$ {L}^{\infty } $$ norm and Lωα,β2$$ {L}_{\omega^{\alpha, \beta}}^2 $$ norm. The results of a numerical example confirm the theoretical prediction. [ABSTRACT FROM AUTHOR]

Ocean geochemical tracers such as radiocarbon, protactinium and thorium isotopes, and noble gases are widely used to constrain a range of physical and biogeochemical processes in the ocean. However, their routine simulation in global ocean circulation and climate models is hindered by the computational expense of integrating them to a steady state. Here, a new approach to this long‐standing "spin‐up" problem is introduced to efficiently compute equilibrium distributions of such tracers in seasonally‐forced models. Based on "Anderson Acceleration," a sequence acceleration technique developed in the 1960s to solve nonlinear integral equations, the new method is entirely "black box" and offers significant speed‐up over conventional direct time integration. Moreover, it requires no preconditioning, ensures tracer conservation and is fully consistent with the numerical time‐stepping scheme of the underlying model. It thus circumvents some of the drawbacks of other schemes such as matrix‐free Newton Krylov that have been proposed to address this problem. An implementation specifically tailored for the batch HPC systems on which ocean and climate models are typically run is described, and the method illustrated by applying it to a variety of geochemical tracer problems. The new method, which provides speed‐ups by over an order of magnitude, should make simulations of such tracers more feasible and enable their inclusion in climate change assessments such as IPCC. Plain Language Summary: Radiocarbon and other geochemical tracers have provided great insight into the workings of the ocean but are prohibitively expensive to simulate in climate models. This study introduces a new computational method that can be applied to any model to greatly speed‐up simulations of such tracers, enabling their routine inclusion in climate models and thus more effective use of those tracers. Key Points: Geochemical tracers have provided great insight into oceanic processes but are prohibitively expensive to simulate in climate modelsA new "sequence acceleration" method is introduced offering speed‐ups of 10–25 times for a range of typical geochemical tracer problemsThe new method is completely "black box" and can be applied to any model [ABSTRACT FROM AUTHOR]

We consider the solvability of fractional differential equations involving the Riesz fractional derivative. Our approach basically relies on the reduction of the problem considered to the equivalent nonlinear mixed Volterra and Cauchy‐type singular integral equation and on the theory of fractional calculus. By establishing a compactness property of the Riemann–Liouville fractional integral operator on Lebesgue spaces and using the well‐known Krasnoselskii's fixed point theorem, an existence of at least one solution is gleaned. An example is finally included to show the applicability of the theory. [ABSTRACT FROM AUTHOR]

In this paper, an iterative algorithm is established to solve discrete Lyapunov matrix equations. In this algorithm, a tuning parameter is introduced such that the iterative solution can be updated by using a combination of the information in the last step and the previous step. Some conditions for the convergence of the proposed algorithm are given. In addition, an approach is also developed to choose the optimal tuning parameter such that the algorithm achieves its fastest convergence rate. A numerical example is employed to illustrate the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

Based on the least‐squares method, we proposed a new algorithm to obtain the solution of the second kind of regular and weakly singular Volterra‐Fredholm integral equations in reproducing kernel spaces. The stability and uniform convergence of the algorithm are investigated in detail. Numerical experiments verify the theoretical findings. Meanwhile, this method is also applicable to the nonlinear Volterra integral equations. Test problems which have non‐smooth solutions are also considered, and our proposed method is efficient as some recent Krylov subspace methods such as LSQR and LSMR. [ABSTRACT FROM AUTHOR]

A parameter‐dependent perturbation of the spectrum of the scalar Laplacian is studied for a class of nonlocal and non‐self‐adjoint rank one perturbations. A detailed description of the perturbed spectrum is obtained both for Dirichlet boundary conditions on a bounded interval as well as for the problem on the full real line. The perturbation results are applied to the study of a related parameter‐dependent nonlinear and nonlocal parabolic equation. The equation models a feedback system that admits an interpretation as a thermostat device or in the context of an agent‐based price formation model for a market. The existence and the stability of periodic self‐oscillations of the related nonlinear and nonlocal heat equation that arise from a Hopf bifurcation are proved. The bifurcation and stability results are obtained both for the nonlinear parabolic equation with Dirichlet boundary conditions and for a related problem with nonlinear Neumann boundary conditions that model feedback boundary control. They follow from a Popov criterion for integral equations after reducing the stability analysis for the nonlinear parabolic equation to the study of a related nonlinear Volterra integral equation. While the problem is studied in the scalar case only, it can be extended naturally to arbitrary Euclidean dimension and to manifolds. [ABSTRACT FROM AUTHOR]

The aim of this research is to present a new iterative procedure in approximating nonlinear system of algebraic equations with applications in integral equations as well as partial differential equations (PDEs). The presented scheme consists of several steps to reach a high rate of convergence and also an improved index of efficiency. The theoretical parts are furnished, and several computational tests mainly arising from practical problems are given to manifest its applicability. [ABSTRACT FROM AUTHOR]

From decomposition method for operators, we consider a Newton‐Steffensen iterative scheme for approximating a solution of nonlinear Fredholm integral equations with non‐differentiable Nemystkii operator. By means of a convergence study of the iterative scheme applied to this type of nonlinear Fredholm integral equations, we obtain domains of existence and uniqueness of solution for these equations. In addition, we illustrate this study with a numerical experiment. [ABSTRACT FROM AUTHOR]

We show that the isometry group of the bounded Urysohn space is a simple group. [ABSTRACT FROM AUTHOR]

In this article, we establish some fixed-point results of Krasnoselskii type for the sum of two weakly sequentially continuous mappings that extend previous ones. In the last section, we apply such results to study the existence of solutions to a nonlinear integral equation modelled in a Banach space. [ABSTRACT FROM AUTHOR]

We consider the inverse scattering problem for the shape determination of a rigid scatterer or a cavity in a homogeneous and isotropic elastic medium. Both problems are formulated in R for time-harmonic fields and longitudinal or transversal incident plane waves. Representation of the scattered field as a single- or a double-layer potential, equivalently, leads to a system of two nonlinear integral equations for the density and the parametrization of the boundary. A detailed numerical implementation is presented for computing the corresponding solutions of both systems and numerical reconstructions are given to show the effectiveness of the method. The authors consider the inverse scattering problem for the shape determination of a rigid scatterer or a cavity in a homogeneous and isotropic elastic medium. Both problems are formulated in R for time-harmonic fields and longitudinal or transversal incident plane waves. Representation of the scattered field as a single- or a double-layer potential, equivalently, leads to a system of two nonlinear integral equations for the density and the parametrization of the boundary. A detailed numerical implementation is presented for computing the corresponding solutions of both systems and numerical reconstructions are given to show the effectiveness of the method. [ABSTRACT FROM AUTHOR]

This article presents a computational approach to the imaging of a partially immersed imperfectly conducting cylinder. An imperfectly conducting cylinder of unknown shape and conductivity scatters the incident transverse magnetic (TM) wave in free space while the scattered field is recorded outside. Based on the boundary condition and the measured scattered field, a set of nonlinear integral equations, and the inverse scattering problem are reformulated into an optimization problem. We use genetic algorithm (GA) to reconstruct the shape and the conductivity of a partially immersed imperfectly conducting cylinder. The genetic algorithm is then used to find out the global extreme solution of the cost function. Numerical results demonstrated that, even when the initial guess is far away from the exact one, good reconstruction can be obtained. In such a case, the gradient-based methods often get trapped in a local extreme. In addition, the effect of random noise on the reconstruction is investigated. © 2009 Wiley Periodicals, Inc. Int J Imaging Syst Technol, 19, 299–305, 2009 [ABSTRACT FROM AUTHOR]

Recent trends in compact device modelling and circuit simulation suggest a growing movement towards standardization of Verilog-A as a vehicle for semiconductor device specification and model interchange among commercial and open source simulators. This paper introduces a nonlinear equation-defined device (EDD) characterized by current, voltage and charge equations with a similar syntax to Verilog-A. The EDD has been implemented in Qucs and used extensively as a central feature in an interactive modelling system that allows straightforward prototyping of compact device models prior to translation into Verilog-A. To illustrate the properties and the use of the Qucs EDD a number of examples centred on well-known SPICE models are described. Copyright © 2008 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Pointwise bounds are obtained for the solution of an initial boundary value problem for the resonant nonlinear Schrödinger equations. The context is that of a straight-line region with prescribed moving boundaries, expanding or noncontracting, upon which zero (Dirichlet) conditions are imposed. [ABSTRACT FROM AUTHOR]

The present investigation considers the problem of finite deformation of slender, elastic beams with constant cross-section. The pertinent non-linear equations characterizing the deformation of the beam are derived using a variational principle which also yields the associated natural and imposed boundary conditions. The mathematical model of the beam accounts for the (coupled) flexure, torsion, and extension (contraction) of the beam but it neglects deformations due to shear. Two example problems are presented, each considering the deformation of a cantilever beam subjected to oblique (unsymmetrical) loading. Example 1 treats the case of an end loaded beam, whereas Example 2 treats the same beam subjected to a uniformly distributed load. In both examples results are compared to the equivalent cases treated within the framework of (conventional) linear theory. [ABSTRACT FROM AUTHOR]

Positive solutions are established for the singular integral equation $y(t) = f(t) + {\int ^1 _0} k(t, s) \left [{ {{1} \over {[y(s)] ^{\alpha}}} + h(y(s))} \right ] ds, t \in [0, 1]$ and α > 0 fixed. The case f ≡ h ≡ 0, α =1 arises in communication theory. [ABSTRACT FROM AUTHOR]

Inverse scattering of an imperfectly conducting cylinder buried in a half-space is presented. A conducting cylinder of unknown shape and conductivity is buried in one half-space and scatters the incident field from another half-space. Based on the measured scattered field and the boundary condition, a set of nonlinear integral equations is derived and the inverse problem is reformulated into an optimization problem. The genetic algorithm is then employed to find the global extreme solution of the object function. As a result, the shape and the conductivity of the scatterer can be reconstructed. Even when the initial guess is far away from the exact one, the genetic algorithm can avoid the local extreme and converge to a global extreme solution. In such a case, the gradient-based method often gets stuck in a local extreme. Numerical results are given to show the effectiveness of the genetic algorithm. Multiple incident directions permit good reconstruction of shape and, to a lesser extent, conductivity in the presence of noise in measured data. © 2001 John Wiley & Sons, Inc. Int J Imaging Syst Technol 11, 355–360, 2000 [ABSTRACT FROM AUTHOR]

In the paper we address the problem of 2D adaptive quadrilateral mesh generation by using the variational principles. We first find the variational integral which generates the known grid system of curve-by-curve error equidistributions. We then use the same integral to generate a new adaptive grid system which is superior to the known system in that the new system produces smoother meshes. Moreover, the new integral may be combined linearly with existing smoothness control integrals to yield more robust adaptive grid systems. Numerical results and comparisons are reported. [ABSTRACT FROM AUTHOR]

This paper describes an improvement of the solution by Krwczyk, Moore and Jones (KMJ algorithm) for nonlinear equations based on the interval analysis. When the KMJ algorithm is applied to practical problems such as the determination of the operatingpoint of a multistable electronic circuit, a large computation time is required. The reasom for large computation is as follows: (i) The existence of the solutions is determined using the direct-replacement type interval extension of functions, which enlargest the search regions for the solutions; (ii) the region partitioning is repeated until a region X is obtained such that k((x) sx, where k(x) is Krawczyk's interval function and X is the interval region. The situation is demonstrated by an example. As a solution for these problems, the following method is proposed. [ABSTRACT FROM AUTHOR]

As a method f or analysis of second- order nonlinear interactions of optical waves in an optical waveguide, a simple analysis is proposed in which the dependency of the wave on the direction perpendicular to the propagation direction is processed by the Galerkirt method. In this method, the spacing of the nodes in the direction perpendicular to the propagation direction can be made arbitrary so that an optical wave with a narrow width can be treated. It is also unnecessary to analyze successively in the propagation direction and the solution can be obtained directly. Therefore, the computational labor is small. As an example, the optical second harmonic generation (SHG) device in which the phase matching is accomplished by the Čerenkov radiation is analyzed numerically. [ABSTRACT FROM AUTHOR]

In previous works we reported the development of symbolic computation tools to automate the design of nonlinear state feedback controllers [1], nonlinear PID controllers [2], nonlinear lag-lead compensators [3], and nonlinear obsevers [4] using the extended linearization method [5]. In this paper we show that a careful analysis of the state variables representation of these classical controllers indicates that all of them are particular cases of the nonlinear extension of a mth order linear filter, consisting of a kth order input derivative operator followed by an output mth order linear dynamical system. Using this two blocks decomposition approach, the design of nonlinear extensions of nth order controllers can be decomposed into two independent subalgorithms: a kth order PD controller algorithm, and a mth order state vector feedback algorithm. Hence, an appropriate, assembly of our symbolic computation tools NL Feedback and NLPID could, in principle, allow to use the extended linearization method to synthesize nonlinear extension of arbitrary nth order linear filter (controllers). [ABSTRACT FROM AUTHOR]

A nonlinear integrodifferential equation governing the amplitude evolution of a wavepacket near the critical value of the beta parameter is derived. The basic velocity profile is a hyperbolic tangent shear layer and although the neutral eigensolution is regular, all higher-order terms in the expansion of the stream function are singular at the critical point. The analysis is inviscid and in the critical layer both wave packet effects and nonlinearity are present, but the former are taken to be slightly larger. Unlike the Stuart-Watson theory, the critical layer analysis dictates the form of the amplitude equation, the outer expansion being relatively passive. A secondary instability analysis shows that the packet is unstable to sideband perturbations, but the instability is weak so its main consequence would be to produce some modulation of the packet without destroying its coherence. [ABSTRACT FROM AUTHOR]