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We present a computational method for reconstructing the refractive index of an unknown complex-shaped two-dimensional medium with embedded metal inclusions from its transverse magnetic electromagnetic scattering properties. We present a novel hybrid surface-volume integral equation that generalises the Lippmann–Schwinger equation to media containing embedded scatterers. Using this hybrid equation, we show that the Frechet derivative of the forward-mapping satisfies a particular inhomogeneous wave scattering problem. We solve the inhomogeneous wave scattering problem using a novel coupled FEM-BEM formulation. Our numerical scheme is based on a thin plate spline ansatz for the refractive index, which can be constructed using a relatively small number of control points, and can be efficiently constructed and evaluated even for the complex-shaped multiply-connected domains of interest. Numerical experiments demonstrate the effectiveness of our method by reconstructing several challenging media. [ABSTRACT FROM AUTHOR]

The paper is concerned with the wavelet-Galerkin method for the numerical solution of Fredholm linear integral equations and second-order integro-differential equations. We propose a construction of a quadratic spline-wavelet basis on the unit interval, such that the wavelets have three vanishing moments and the shortest support among such wavelets. We prove that this basis is a Riesz basis in the space L 2 0 , 1. We adapt the basis to homogeneous Dirichlet boundary conditions, and using a tensor product we construct a wavelet basis on the hyperrectangle. We use the wavelet-Galerkin method with the constructed bases for solving integral and integro-differential equations, and we show that the matrices arising from discretization have uniformly bounded condition numbers and that they can be approximated by sparse matrices. We present numerical examples and compare the results with the Galerkin method using other quadratic spline wavelet bases and other methods. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of solutions for the following fractional boundary value problem c D α u (t) = f (t , u (t)) , α ∈ (n , n + 1) , 0 ≤ t < + ∞ , u (0) = 0 , u ′ ′ (0) = 0 , ... , u (n) (0) = 0 , lim t → + ∞ c D α − 1 u (t) = β u (ξ). The considerations of this paper are based on the concept of a new family of measures of noncompactness in the space of functions C n , γ (R +) satisfying the Hölder condition and a fixed point theorem of Darbo type. We also provide an illustrative example in support of our existence theorems. Finally, to credibility, we apply successive approximation and homotopy perturbation method to find solution of the above problem with high accuracy. [ABSTRACT FROM AUTHOR]

Abstract In this study, we derive an asymptotic solution of the integral equation satisfied by the second moment function M 2 t , a. We first find the Laplace transform M 2 L s , a and then obtain M 2 t , a asymptotically by inversion. Further, we have derived the asymptotic expressions of M 2 t , a for some special lifetime distributions such as exponential, gamma, Weibull, lognormal and truncated normal. Finally, the asymptotic solution is compared with the numerical solution to evaluate its performance. [ABSTRACT FROM AUTHOR]

In this paper, we propose a center Lipschitz condition for the second Fréchet derivative together with the use of restricted domains in order to improve the domain of starting points for Newton’s method. In addition, we compare the new result with an older one and see that the former improves the latter. [ABSTRACT FROM AUTHOR]

An error in the demonstration process in literature Alvarez and Lizama (2016) is pointed out and a theorem is proposed to correct the corresponding conclusion. [ABSTRACT FROM AUTHOR]

We analyze the semilocal convergence of Newton’s method under a center Lipschitz condition for the second derivative of the operator involved different from that used by other authors until now. In particular, we propose to center the Lipschitz condition for the second derivative in a different point from that where Newton’s method starts. This allows us to obtain different starting points for Newton’s method and modify the domain of starting points. [ABSTRACT FROM AUTHOR]

This paper presents our study of American floating strike lookback options written on dividend-paying assets. The valuation of these options can be mathematically formulated as a free boundary inhomogeneous Black–Scholes PDE with a Neumann boundary condition, which we, by using a Mellin transform, convert into a relatively simple ordinary differential equation with Dirichlet boundary conditions. We then use these results to derive an integral equation that can be used to calculate the price of American floating strike lookback options. In addition, we also used Mellin transforms to derive the closed-form of the perpetual case. [ABSTRACT FROM AUTHOR]

This work is to propose high asymptotic order methods for highly oscillatory Volterra integral equations. According to the error of the Filon quadrature method, the piecewise Hermite collocation method is considered. For the computation, the highly oscillatory integrals in the collocation equations are treated efficiently. Then, both the classical order and asymptotic order are analyzed in detail with the help of some estimates on highly oscillatory integrals. At last, some numerical examples are provided to verify the sharpness of the estimate in the final part. [ABSTRACT FROM AUTHOR]

The aim of this paper is to establish the existence and uniqueness of the common solution for the system of nonlinear Fredholm integral equations, nonlinear Volterra integral equations and nonlinear fractional differential equations using the common fixed point results equipped with illustrative examples. Some common fixed point results satisfying the generalized contraction condition involving w -distance and weak altering distance functions are proved. Then, an example is provided to support the usability of our result along with numerical experiments. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider a Sparre Andersen risk model perturbed by a Brownian motion, where the individual claim sizes are dependent on the interclaim times. We assume that dividends are paid off under a threshold strategy. Integral and integro-differential equations satisfied by the Gerber–Shiu functions are obtained, and a solution procedure is also proposed. [Copyright &y& Elsevier]

Abstract: The aim of this paper is to derive spectral and condition number estimates of the single-layer operator for low-frequency multiple scattering problems. This work extends to dense media the analysis initiated in Antoine and Thierry (in press) [19]. Estimates are obtained first in the case of circular cylinders by Fourier analysis and are next formally adapted to disks, ellipses and rectangles in the framework of boundary element methods. Numerical simulations validating the approach are also given. [Copyright &y& Elsevier]

Abstract: Two integral equations, representing the mechanical response of a 2D infinite plate supported along a line and subject to a transverse concentrated force, are examined. The kernels of the integral operators are of the type and . In spite of the fact that these are only weakly singular, the two equations are studied in a more general framework, which allows us to consider also solutions having non-integrable endpoint singularities. The existence and uniqueness of solutions of the equations are discussed and their endpoint singularities detected. Since the two equations are of interest in their own right, some properties of the associated integral operators are examined in a scale of weighted Sobolev type spaces. Then, new results on the existence and uniqueness of integrable solutions of the equations that in some sense are complementary to those previously obtained are derived. [Copyright &y& Elsevier]

Abstract: In the study of iterative methods with high order of convergence, Gander provides a general expression for iterative methods with order of convergence at least three in the scalar case. Taking into account an extension of this result, we define a family of iterations in Banach spaces with -order of convergence at least four for quadratic equations. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the renewal risk process under a threshold dividend payment strategy. For this model, the expected discounted dividend payments and the Gerber–Shiu expected discounted penalty function are investigated. Integral equations, integro-differential equations and some closed form expressions for them are derived. When the claims are exponentially distributed, it is verified that the expected penalty of the deficit at ruin is proportional to the ruin probability. [Copyright &y& Elsevier]

The one dimensional Stefan problem is reformulated as a shape optimization problem for the position of the phase transition as a function of time. The functional to be minimized is the mismatch of the Dirichlet to Neumann map at the moving interface. We show that the minimizer is the only stationary point of the shape functional. A gradient based optimization method is derived using shape calculus. The state and adjoint equations of the heat equation are solved with integral equation techniques which avoid a discretization in the domain. A Nyström quadrature method is analyzed and numerical results are presented. [ABSTRACT FROM AUTHOR]

Abstract: The purpose of the present note is to establish some new delay integral inequalities, which provide explicit bounds on unknown functions and generalize some results of Li et al. [Some new delay integral inequalities and their applications, J. Comput. Appl. Math. 180 (2005) 191–200]. The inequalities given here can be used to investigate the qualitative properties of certain delay differential equations and delay integral equations. [Copyright &y& Elsevier]

Abstract: Finding a formulation for electromagnetic scattering of surfaces which is both well-posed and produces a well-conditioned linear system is still a challenging problem. We here propose one such formulation valid in the high-frequency regime. The mathematical analysis is provided and numerical results on rather complex geometries show the performance of the method. [Copyright &y& Elsevier]

Abstract: We examine the use of wavelet packets for the fast solution of integral equations with a highly oscillatory kernel. The redundancy of the wavelet packet transform allows the selection of a basis tailored to the problem at hand. It is shown that a well chosen wavelet packet basis is better suited to compress the discretized system than wavelets. The complexity of the matrix–vector product in an iterative solution method is then substantially reduced. A two-dimensional wavelet packet transform is derived and compared with a number of one-dimensional transforms that were presented earlier in literature. By means of some numerical experiments we illustrate the improved efficiency of the two-dimensional approach. [Copyright &y& Elsevier]

Abstract: The aim of the present paper is to establish some new delay integral inequalities, which provide explicit bounds on unknown functions. The inequalities given here can be used as tools in the qualitative theory of certain delay differential equations and delay integral equations. [Copyright &y& Elsevier]

Abstract: Numerical solution of linear integral equations by means of the Sinc collocation method based on the double exponential transformation, abbreviated the DE transformation, is considered. We first apply the method to the Volterra integral equation of the second kind and then to the Volterra equation of the first kind. This method is also applied to the Fredholm integral equation of the second kind. For the Volterra equations we employed a formula for numerical indefinite integration developed by Muhammad and Mori obtained by applying the DE transformation incorporated into the Sinc expansion of the integrand, while for the Fredholm equation we employed the conventional DE transformation for definite integrals. An error analysis of the method is given and in every case a convergence rate of for the error is established where N is a parameter representing the number of terms of the Sinc expansion. Also, the condition of the matrix of the main system of linear equations is watched through an estimate of the condition number returned by the program. Numerical examples show the convergence rate mentioned above and confirm the high efficiency of the present method. [Copyright &y& Elsevier]

The authors propose a simple numerical method to approximate the solution of CSIE. The convergence and the stability of the procedure are proved and some numerical examples are shown. [Copyright &y& Elsevier]

For the scattering problem for time-harmonic waves from an impedance crack in two dimensions, we give a uniqueness and existence analysis via a combined single- and double-layer potential approach in a Ho¨lder space setting leading to a system of integral equations that contains a hypersingular operator. For its numerical solution we describe a fully discrete collocation method based on trigonometric interpolation and interpolatory quadrature rules including a convergence analysis and numerical examples. [Copyright &y& Elsevier]

We propose a new system of integral equations for the exterior time harmonic Maxwell''s equation. This system is derived first from elementary manipulations of classical equations then by the minimization of a quadratic functional associated to incoming and outgoing electromagnetic waves. We analyze the inf–sup condition and various penalized problems related to this system. Then we prove that an iterative algorithm for the solution of the system of integral equations is convergent. Other numerical issues are also discussed. [Copyright &y& Elsevier]

In this paper, the pricing of an American-style down-and-out call option with debates is explored under the Black–Scholes model. This is a complex problem due to the combination of the knock-out feature and early-exercise feature, which requires the unknown optimal exercise boundary to be determined simultaneously with the option price. Despite all the challenges, we are still able to present a semi-analytical solution by deriving an integral equation representation for the target option price using the incomplete Fourier transform, after innovatively introducing another unknown function for the first-order derivative of the option price at the barrier. Numerical schemes are then established to implement the integral equations. Remarkably, it turns out that only one single nonlinear equation for the optimal exercise boundary needs to be solved numerically, while the introduced unknown function has an analytical form in the recursive process, which guarantees the efficiency of our approach. [ABSTRACT FROM AUTHOR]

In this manuscript, we initiated to study a notion of graphical rectangular metric space and established some fixed point theorems in the aforesaid space. We also discuss the structure of an open ball and Hausdorffness of the new metric. Suitable examples and graphs are presented to demonstrate the novelty of the results established. In the end, we apply the obtained result to show the existence of the solution of a particular type of integral equation. [ABSTRACT FROM AUTHOR]