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Motivated by applications in population models, we consider S -asymptotically periodic solution of fractional differential equations with periodic environment forces or asymptotically periodic ones. The system is quasi-monotone, and the existence of positive S -asymptotically periodic solution is established by using upper and lower solutions. The sufficient conditions that ensure the uniqueness of positive S -asymptotically periodic solution are also established on the basis of theory of sublinear operator. The applications of the general conclusions to classical population models yield the global convergence of positive S -asymptotically periodic solution in logistic equation with or without weak Allee effect, and the model of two cooperative populations. • Existence of asymptotically periodic solution. • Uniqueness of asymptotically periodic solution. • Upper and lower solutions. • Monotonic operator. • Ecological applications. [ABSTRACT FROM AUTHOR]

Abstract In this paper, some new theorems for various cyclic contractions are established in G b -metric spaces to discuss the existence and uniqueness of the solutions for a class of the operator equations, which generalize many known results in corresponding literatures. On the other hand, we introduce the notion of the cyclic α - ψ φ-contractive mappings in G b -metric spaces and establish a new fixed point theorem, which is applied to consider the existence of the solutions of the integral equations and ordinary differential equations. Moreover, some examples are given to support our main results. [ABSTRACT FROM AUTHOR]

• Use of the FFT for fast numerical derivative computation. • Use of the SVE of the integral kernel associated to derivative operator of a generic order. • Proof of the order of convergence of the proposed method. • Development of a robust algorithm for first order derivative. • Generalization of the algorithm to higher order derivatives. • Generalization of the algorithm to derivatives of multivariate functions. We consider the numerical differentiation of a function tabulated at equidistant points. The proposed method is based on the Fast Fourier Transform (FFT) and the singular value expansion of a proper Volterra integral operator that reformulates the derivative operator. We provide the convergence analysis of the proposed method and the results of a numerical experiment conducted for comparing the proposed method performance with that of the Neville Algorithm implemented in the NAG library. [ABSTRACT FROM AUTHOR]

We build a numerical approximation method, for Fredholm integral equation solution of the second type. This method is based on the regularization by convolution and Fourier series expansion. It provides a better convergence order. [ABSTRACT FROM AUTHOR]

In this paper we obtain some equivalences between cyclic contractions and non-cyclic contractions in the framework of b -metric spaces. Our results improve and complement several recent fixed point results for cyclic contractions in b -metric spaces established by George et al. (2015) and Nashine et al. (2014). Moreover, all the results are with much shorter proofs. In addition, an application to integral equations is given to illustrate the usability of the obtained results. [ABSTRACT FROM AUTHOR]

This paper deals with a new approach to study the nonlinear inviscid flow over arbitrary bottom topography. The problem is formulated as a nonlinear boundary value problem which is reduced to a Dirichlet problem using certain transformations. The Dirichlet problem is solved by applying Plemelj–Sokhotski formulae and it is noticed that the solution of the Dirichlet problem depends on the solution of a coupled Fredholm integral equation of the second kind. These integral equations are solved numerically by using a modified method. The free-surface profile which is unknown at the outset is determined. Different kinds of bottom topographies are considered here to study the influence of bottom topography on the free-surface profile. The effects of the Froude number and the arbitrary bottom topography on the free-surface profile are demonstrated in graphical forms for the subcritical flow. Further, the nonlinear results are validated with the results available in the literature and compared with the results obtained by using linear theory. [ABSTRACT FROM AUTHOR]

In this paper, we use a modification of Krasnoselskii's fixed point theorem introduced by Burton (2002) (see [6] Theorem 3) to obtain stability results of the zero solution of the totally nonlinear neutral differential equations with functional delay x′(t)=-a(t)h(x(t-τ(t)))+d/dtQ(t,x(t-τ(t)))+G(t,x(t-τ(t))). The stability of the zero solution of this equation provided that h(0)=Q(t,0)=G(t,0,0)=0. The Caratheodory condition is used for the functions Q and G. [ABSTRACT FROM AUTHOR]

The presented paper deals with an approach to analytical periodic solution and to stability assessment of one-degree-of-freedom linear vibrating systems. It is supposed that these systems are excited by the time periodic force and contain time periodic stiffness. The periodic Green's function determined as a response to a Dirac chain of unit impulses repeating with period of excitation is used to transform the equation of motion into the Fredholm integral equation with degenerated kernel. If the Dirac chain is expressed as a Fourier series and a limited number of terms is taken into account, the solution of the integral equation can also be obtained in a series form. It has been found that the real eigenvalues of the system matrix determine the critical values of the fluctuation stiffness parameter. The values of this real parameter correspond to the borders of (in)stability in the plane given by the variation of the angle frequency and of the fluctuation stiffness parameter. Moreover, very interesting property of the system matrix was observed. The positive sign of the real valued determinant of the system matrix means the existence of periodic solution (system is stable). In the opposite case, the periodic solution does not exist (system is unstable). The verification of obtained results was performed on two case studies. The Floquet method was used to validate the stability assessment. Presented analytical periodic solution was compared with steady state obtained by the Runge-Kutta continuation. A very good agreement was achieved in both cases. [ABSTRACT FROM AUTHOR]

Abstract: Using Helmholtz’s wake model, we reduce the study of the free boundary flow past an obstacle consisting of an arc of circle to the investigation of a Hammerstein nonlinear integral equation depending on a real parameter . The papers dedicated to this problem investigated the case which corresponds to a convex obstacle with respect to the incoming fluid. Herein, we apply for the first time in the literature the arclength continuation method for the case corresponding to a concave arc of a circle. For the existence and the uniqueness of the solution was demonstrated, but for , depending on its value compared to the one of a turning point, the integral equation has either no solution or two distinct solutions corresponding to two different obstacles. We numerically calculate the free lines, the velocity field and the stream lines. A diagram of the drag coefficient versus the arc measure for both convex and concave obstacles suggests us to draw some conclusions concerning the optimization of the blades of a vertical axis (Savonius) wind turbine. [Copyright &y& Elsevier]

Abstract: In order to compute an approximate solution of a weakly singular integral equation, we first regularize the kernel and then truncate the associated Fourier series. Applications to Green and Abel operators are given. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the existence of integral solutions to a general nonlinear integral equationWith the help of Krasnoselskii’s fixed point theorem and the theory of measure of weak noncompactness, we establish a new and general existence theorem for the nonlinear functional integral equation. Moreover, an example, which can not be treated by the related theorems in [5,20,22], is given to illustrate the new existence theorem. [Copyright &y& Elsevier]

Abstract: By using difference equations, we give some easily applicable sufficient conditions for the non-existence of nonnegative solutions satisfying some additional conditions, of the integral equationwhere and X is a measurable space with probability measure μ. [Copyright &y& Elsevier]

Abstract: In this paper, a novel hybrid method is presented for solving the second kind linear Volterra integral equations. Due to the powerful regression ability of least squares support vector regression (LS-SVR), we approximate the unknown function of integral equations by using LS-SVR in intervals with known numerical solutions. The trapezoid quadrature is used to approximate subsequent integrations in intervals with unknown numerical solutions. The feasibility of the proposed method is examined on some integral equations. Experimental results of comparison with analytic and repeated modified trapezoid quadrature method’s solutions show that the proposed algorithm could reach a very high accuracy. The proposed algorithm could be a good tool for solving the second kind linear Volterra integral equations. [Copyright &y& Elsevier]

Abstract: Using the setting of cone metric space, a fixed point theorem is proved for two maps, and several corollaries are obtained. In these cases, the cone does not need to be normal. These results generalize several well known compatible recent and classical results in the literature. As an application, the existence of solution of an integral equation is presented. [Copyright &y& Elsevier]

Abstract: In this paper we consider fast numerical solution methods for two dimensional Fredholm integral equation of the second kindwhere a(x, y, u, v) is smooth and g(x, y) is in L 2[α, β]2. Discretizing the integral equation by certain quadrature rule, we get a linear system. To deduce fast approximate solution methods for the resulted linear system, we study the approximation of the four-variable kernel function a(x, y, u, v) by piecewise polynomial: partition the domain [α, β]4 into subdomains of the same size and interpolate the kernel function a(x, y, u, v) in each subdomain. Fast matrix–vector multiplication algorithms and efficient iterative methods are derived. Numerical results are given to illustrate the efficiency of our methods. [Copyright &y& Elsevier]

Abstract: In this paper, we provide a semilocal convergence analysis for a family of Newton-like methods, which contains the best-known third-order iterative methods for solving a nonlinear equation in Banach spaces. It is assumed that the operator F is twice Fréchet differentiable and satisfies a Lipschitz type condition but it is unbounded. By using majorant sequences, we provide sufficient convergence conditions to obtain cubic semilocal convergence. Results on existence and uniqueness of solutions, and error estimates are also given. Finally, a numerical example is provided. [Copyright &y& Elsevier]

Abstract: In this paper, an algorithm is proposed for the solution of second-order boundary value problems with two-point boundary conditions. The Green’s function method is applied first to transform the ordinary differential equation into an equivalent integral one, which has already satisfied the boundary conditions. And then, the homotopy perturbation method is used to the resulting equation to construct the numerical solution for such problems. Numerical examples demonstrate the efficiency and reliability of the algorithm developed, it is quite accurate and readily implemented for both linear and nonlinear differential equations with homogeneous and nonhomogeneous boundary conditions. Furthermore, the lower order approximation is of higher accuracy for most cases. Some other extended applications of this algorithm are also exhibited. [Copyright &y& Elsevier]

Abstract: This note is concerned with the problem of determining approximate solutions of Fredholm integral equations of the second kind. Approximating the solution of a given integral equation by means of a polynomial, an over-determined system of linear algebraic equations is obtained involving the unknown coefficients, which is finally solved by using the least-squares method. Several examples are examined in detail. [Copyright &y& Elsevier]

Abstract: The main objective of this 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: In this paper first, Fredholm integral equation of the first kind is introduced. Then we decide to solve this kind of equations by projection methods such as Galerkin and collocation methods. Then we use Haar Wavelet as a base function to estimate the solution of the integral equation of the first kind. Finally by some numerical examples the efficiency of these methods are discussed. [Copyright &y& Elsevier]

Abstract: We present an implicit Simpson’s rule for the numerical computation of quantum stochastic differential equations. We derive the method, present convergence results and a numerical example. Our method avoid the high cost of high-order methods as well the cost of obtaining starting values for multi-step methods. [Copyright &y& Elsevier]

Abstract: When wavelets are used as basis functions in Galerkin approach to solve the integral equations, Integrals of the form occur. By a change of variable, these integrals can be translated into integrals involving only θ. In this paper, we find quadrature rule on the for the integrals of the formWavelets in this article are those discovered by Daubechies [I. Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. 41 (1988) 909–996], where ϕ is the scaling function and ψ is the wavelet function. [Copyright &y& Elsevier]

Abstract: In this research, we introduce a new type of weighted quadrature rules asin which ; λ ∈ R; m ∈ N; ρ(x) is a positive function; f (m) (x) denotes the mth derivative of the function f(x) and is the error function. We determine the error function analytically and obtain the unknowns explicitly so that the above formula is exact for all polynomials of degree at most 2n + m −1. In particular, we emphasize on the sub-casewith the precision 2n (one degree higher than Gauss quadrature precision degree) and show that under some specific conditions the two foresaid formulas can be connected to the current weighted quadrature rules. The best application of the case m =1 in the second formula is when λ is a known root of the function f(x). For instance, and are two samples in which f(λ)=0. Finally, we present various analytic examples of above rules and introduce a more general form of the mentioned formulas as [Copyright &y& Elsevier]

Abstract: The purpose of this article is to study Adomian decomposition method as well as its stability and also to comparatively investigate it with other methods. [Copyright &y& Elsevier]

Abstract: In this paper, we bring three theorems that enable us to approximate the solution of Ferdholm integral equations of the second kind. Then we use Coifman wavelets or Coiflets as scaling functions for projection that satisfied the conditions of theorems for approximation. Also we use this projection to convert the integral equation to a Galerkin system, which is the most important of the expansion methods for solving linear integral equations. Finally, by using numerical examples we show that our estimation have a good degree of accuracy. [Copyright &y& Elsevier]

Abstract: In this paper, We use the wavelet bases of Hermite cubic splines to solve the second kind integral equationsA pair of wavelets are constructed on the basis of Hermite cubic splines. This wavelets are in C1 and supported on [0,2]. Moreover, one wavelet is symmetric, and the other is anti-symmetric. This spline wavelets are then adapted to the interval [0,1]. The computational results demonstrate the advantage of the wavelet basis. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to present a new method based on radial basis functions (RBFs) and iterative procedure for the solution of linear integral equations of Fredholm and Volterra types of the second kind. To obtain a primary approximate solution, the collocation method with a small number of nodes, based on the RBFs and their first-order derivatives, can be adapted. Moreover, we present an iterative procedure from the integral equation to improve the accuracy of solution. Numerical examples show that our approach has the potentiality to become an efficient method. [Copyright &y& Elsevier]

Abstract: In this paper, a new method using radial basis function (RBF) networks is presented for solving the linear second kind integral equations of Fredholm and Volterra types. This method employs a growing neural network as the approximate solution of the integral equations, whose the activation functions are RBFs. The parameters (weights, centers and widths) of the approximate solution are adjusted by using an unconstrained optimization problem. Numerical results show that our method has the potentiality to become an efficient approach for solving integral equations. [Copyright &y& Elsevier]

Abstract: The mechanical quadrature method described for two-dimensional nonlinear singular integral equation with Hilbert kernel in Banach space. The results are compared with the exact solution of the integral equation. [Copyright &y& Elsevier]

Abstract: In this paper we consider Fredholm–Volterra integral equation in one dimension with time dependent. The kernel of the Fredholm term is assumed to be singular while the kernel of Volterra term is continuous. The existence of the solution is proved and a numerical method is presented to get an approximate to such solution. [Copyright &y& Elsevier]

Numerical algorithms are presented for two classical free boundary problems for ideal flow past an obstacle: Helmhotz–Kirchhoff and reentrant jet flows. The Levi–Civita representation of the log-hodograph function is used in each case to derive non-linear integral equations for the boundary correspondence between the obstacle and the parameter domain. The integral equations are solved by a method of successive conjugation implemented with the fast Fourier transform. For the reentrant jet flow an additional non-linear system must be solved to update certain flow parameters at each iteration. Several examples are computed for polygonal and curvilinear obstacles. A convergence result is given for the Helmhotz–Kirchhoff flow iteration. [Copyright &y& Elsevier]

In this paper, we present a Taylor-series expansion method for a class of Volterra integral equations of second kind with smooth or weakly singular kernels. This method use Taylor-series approximation method for integral equation and transform the integral equation to an nth order, linear differential equation. Boundary conditions for differential equation produce in easy way. This method gives an approximate simple and closed form solution for integral equation. Some numerical example use to illustrate the accuracy of method. [Copyright &y& Elsevier]

In this work, the Fredholm integral equation of the second kind with logarithmic kernel is established from dynamic elastic problem. Such problem appears, for example, when a stamp of length 2a is impressed into the boundary of a strip y=h by a variable force P(t) with time t∈[0,T], T<∞. Also the strip, without fraction, lies on a rigid elastic layer of thickness H. The solution of the integral equation is obtained using Chebyshev polynomial. A numerical result is obtained for the problem. [Copyright &y& Elsevier]

In this paper we introduce a new method, based on Adomian method, for solving nonlinear integral equations. [Copyright &y& Elsevier]

Building on a new theory of parametric risk models initiated in Hu¨rlimann [Bla¨tter der Deutschen Gesellschaft fu¨r Versicherungsmathematik XXIII(3) (1998) 251], it is shown how mean scaled individual risk models can be constructed. The approximate computation of their distributions and related quantities can be done in the author's (1990) mathematical framework of generalized pseudo compound Poisson distributions, which are characterized by integral equations of Volterra type. The results by Dhaene and De Pril [Insurance: Mathematics and Economics 14 (1994) 181] are extended to this more general class of models. The method is applied to the construction of a mean scaled operational risk model. [Copyright &y& Elsevier]

By virtue of the introduction of a dependant variable and the separation of variables technique, the piezothermoelastic axisymmetric plane strain dynamic problem of a special non-homogeneous pyroelectric hollow cylinder is transformed to a Volterra integral equation of the second kind about a function with respect to time, which can be solved successfully by means of the interpolation method. Then the solutions of displacements, stresses, electric displacements and electric potential are obtained. The present method is suitable for a non-homogeneous pyroelectric hollow cylinder with an arbitrary thickness subjected to arbitrary axisymmetric thermal loads. Numerical results are presented graphically. [Copyright &y& Elsevier]

One is interested in the scattering of TE and TM electromagnetic plane waves on rough planes. The conventional way of tackling this problem requires the solution of an integral equation of the first kind. Using a different approach leads for the total field incident plus scattered to an integral equation of the second kind. This formulation, developed when the total field satisfies the Dirichlet or Neumann boundary condition is applied to TE, TM wave scattering from perfectly conducting rough planes. The Rayleigh–Ganz iterative technique gives the solution of the corresponding integral equations in the form of successive approximations and we concentrate on the first iterate to get in particular the differential cross-section. An illustrative application to grooved and self-affine rough planes is presented but at this stage no numerical computation is performed. [Copyright &y& Elsevier]

The resonator problem for a positive branch confocal unstable resonator reduces to a Fredholm homogeneous integral equation of the second kind, whose numerical solution here is based on a sequence of algebraic eigenvalue problems. We compare two algorithms for the solution of an optical resonator problem. These are obtained by (i) successive degenerate kernel approximation by Taylor polynomials of the Fredholm kernel and (ii) Nystro¨m’s method with Simpson’s rule as the subordinate numerical integration method. The numerical results arising from these routines compare well with other published results, and have the added advantage of simplicity and easy adaptability to other resonator problems. [Copyright &y& Elsevier]

In this article, we consider linear and non-linear systems of integral equations of the first kind. The problem of existence and uniqueness are considered. The Adomian decomposition method is being used to solve these linear and non-linear systems of Volterra integral equations of the first kind. Some examples are prepared to show the efficiency and simplicity of the method. [Copyright &y& Elsevier]

In this work, the solution of the integral equation of the mixed type in the domain Ω={(x,y,z)∈Ω:−∞ is obtained in the space L2(Ω)&ast;C(0,T), where t is the time, 0&les;t&les;T<∞, in a series form. Also, the solution of Wiener–Hopf integral equation of the second kind with Macdonald kernel is also obtained, and the convergence of the solution is discussed. Finally a numerical example is given. [Copyright &y& Elsevier]

In this paper we study existence theorems for continuous and discrete problems of forms y(t)=h(t)+∫t0K(t,s)f(s,y(s)) ds, t∈[0,∞) and y(i)=h(i)+∑j=0∞G(i,j)f(i,y(j)), i∈{0,1,2,…}, on the unbounded interval, where “∫” denote the Pettis integral and f is weakly–weakly sequentially continuous. [Copyright &y& Elsevier]

We give new criteria for nonuniform dichotomy of nonautonomous systems on the whole line in terms of admissibility relative to an integral equation. In our approach the input space I (R , X) is an intersection of spaces that can be successively minimized and the output space C (R , X) can be one of some well-known spaces of continuous functions. Using computational arguments, we show that the admissibility of (C (R , X) , I (R , X)) leads to a nonuniform exponential dichotomy. We expose a complete analysis of the connections between admissibility and nonuniform dichotomy on the whole line and we also discuss several interesting consequences. Moreover, we obtain the explicit expression of the growth rates for dichotomy in terms of the initial exponential growth and the norm of the input-output operator. Finally, we present a direct application of the main result in the case of evolution families which admit uniform exponential growth. [ABSTRACT FROM AUTHOR]