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As interest in the use and launching of spacecraft for communications, earth observation, scientific experiment and navigation purposes increases and manned missions to the Moon and Mars intensify, there is need for the design of efficient and high-fidelity relative motion dynamics to reduce spacecraft collisions and increase return on investment. The main aim of this work is to develop new approximate solution of spacecraft relative motion in elliptical orbit via power series method. Advantage of this method is that it does not involve evaluating complex integral I as employed for developing approximate solutions of linearized Tschauner–Hempel equations. Cauchy product, used for the discrete convolution of power series, is employed for the development of power series solutions of the approximated nonlinear spacecraft relative motion. Application of Cauchy criterion shows that the new solutions are convergent making them useful for spacecraft formation flying, proximity and rendezvous mission analysis.

Henstock--Kurzweil integral, a nonabsolute integral, is a natural extension of the Riemann integral that was studied independently by Ralph Henstock and Jaroslav Kurzweil. This paper will introduce the Henstock--Kurzweil--Stieltjes integral of $\mathcal{C}[a,b]$-valued functions defined on a closed interval $[f,g]\subseteq\mathcal{C}[a,b]$, where $\mathcal{C}[a,b]$ is the space of all continuous real-valued functions defined on $[a,b]\subseteq\mathbb{R}$. Some simple properties of this integral will be formulated including the Cauchy criterion and an existence theorem will be provided.

With the application of quaternion in technology, quaternion-valued neural networks (QVNNs) have attracted many scholars' attention in recent years. For the existing results, dynamical behavior is an important studying side. In this paper, we mainly research the existence, uniqueness and exponential stability criteria of solutions for the QVNNs with discrete time-varying delays and distributed delays by means of generalized 2-norm. In order to avoid the noncommutativity of quaternion multiplication, the QVDNN system is firstly decomposed into four real-number systems by Hamilton rules. Then, we obtain the sufficient criteria for the existence, uniqueness and exponential stability of solutions by special Lyapunov-type functional, Cauchy convergence principle and monotone function. Furthermore, several corollaries are derived from the main results. Finally, we give one numerical example and its simulated figures to illustrate the effectiveness of the obtained conclusion.

The admissible consensus tracking problem of nonlinear singular multiagent systems (SMASs) with time-varying delay, uncertainties, and external disturbances under jointly connected topologies is investigated in this article. First, the sliding-mode control (SMC) is applied to effectively reduce the adverse effects of uncertainties and nonlinearities of systems. Then, by the combination of admissible analysis, the Cauchy convergence criterion, and SMC, the sufficient conditions for the admissible consensus tracking and disturbance rejection of SMASs under jointly connected topologies are provided. Furthermore, a distributed SMC law is designed such that the sliding-mode dynamics trajectories reach the sliding surface in finite time. Finally, the simulation results are utilized to indicate the effectiveness of the presented methods.

In this work, using the concept of natural density, we introduce the $\left( p,q\right) $-analogue of the Stancu-beta operators of rough $\lambda$% -statistically $\rho$-Cauchy convergence on triple sequence spaces. We define the set of Bernstein Stancu beta opeators of rough statistical limit points of a triple sequence spaces and obtain to $\lambda-$statistical convergence criteria associated with this set. Also, we examine the relations between the set of Bernstein-Stancu beta operators of rough $\lambda$-statistically $\rho$-Cauchy convergence of triple sequences.

In this paper we introduce the λ -quasi Cauchy sequence of fuzzy numbers. We obtain the relation between strongly λ-quasi Cauchy convergence and statistically λ -quasi Cauchy convergence for fuzzy numbers.

In this paper, we investigate the global μ-stability problem of quaternion-valued neural networks (QVNNs) with unbounded and asynchronous time-varying delays. The decomposition of QVNNs is proposed first based on the Hamilton rules to avoid the non-communicative multiplication feature of the quaternion. Instead of establishing the Lyapunov-Krasovskii functional in the form of the Hermitian matrix, we combine the {ξ, ∞}-norm and the Cauchy convergence principle to derive the sufficient conditions that guarantee the existence, uniqueness, and μ-stability of the equilibrium point. The obtained criteria here are milder, owing to the consideration of inhibitory and excitatory between neurons. Finally, a specified example is presented to verify the correctness of the obtained results.

In this paper, we consider the Cauchy problems for a uniformly parabolic equation and a Hamilton–Jacobi equation. Our aim is to show the optimality of an upper estimate of the L ∞ -norm of the gradient of solution to each Cauchy problem. In order to achieve this aim, we investigate a lower estimate of this L ∞ -norm.

The aim of this paper is to prove various ill-posedness and well-posedness results on the Cauchy problem associated to a class of fractional Kadomtsev-Petviashvili (KP) equations including the KP version of the Benjamin-Ono and Intermediate Long Wave equations., 42 pages

In this work, we present piecewise constant Galerkin method for a class of Cauchy singular integral equations of the second kind with constant coefficients in L 2 ( [ 0 , 1 ] , C ) , using a sequence of orthogonal finite rank projections. We prove the existence and uniqueness theorems for the Cauchy integral equation and the approximate equation, respectively. We perform the error analysis for which we give new and improved estimates for the rates of convergence. Numerical example illustrates the theoretical results.

This work is concerned with the discussion of the numerical approximation for random Cauchy transport model in one dimension. The random (forward time, backward space) finite difference scheme is used to find the stochastic solution. The impression of the consistency and the random von-Neumann stability technique under the mean square sense are studied. Using some examples, we can support our main objective of this model statistically.

We study the Cauchy problem for a system of complex linear differential equations in scales of spaces of functions of exponential type with an integral metric. Conditions under which this problem is well posed are obtained. These sufficient conditions are shown to be also necessary for the well-posedness of the Cauchy problem in the case of systems of ordinary differential equations with a parameter.

The Cauchy problem for the Helmholtz equation is considered. It was demonstrated in a previous paper by the authors that the alternating algorithm suggested by V.A. Kozlov and V.G. Maz’ya does not ...

In this paper, a novel scheme is proposed to solve the first kind Cauchy integral equation over a finite interval. For this purpose, the regularization method is considered. Then, the collocation method with Fibonacci base function is applied to solve the obtained second kind singular integral equation. Also, the error estimate of the proposed scheme is discussed. Finally, some sample Cauchy integral equations stem from the theory of airfoils in fluid mechanics are presented and solved to illustrate the importance and applicability of the given algorithm. The tables in the examples show the efficiency of the method.

We obtain closed-form solutions of singular integral equations with Cauchy kernels in the case of an infinite integration domain. Further, we find natural conditions for the uniqueness of solutions of these equations for the case in which the solution contains arbitrary functions or arbitrary constants.

The present paper deals with a solution of the Riemann–Hilbert problem with piecewise continuous coefficients in the class of Cauchy type integrals with densities in grand Lebesgue spaces. Necessary and sufficient solvability condition is established. In solvability case the solutions in explicit form are constructed.

In this paper, we study the Cauchy problem of the generalized Fokas–Olver–Resenau–Qiao equation. Firstly, by means of transport equation and Littlewood–Paley theory, we obtain the local well-posedn...

In this paper, we investigate a Cauchy problem associated with Helmholtz-type equation in an infinite strip. This is a classical severely ill-posed problem, i.e., the solution (if it exists) does not depend continuously on the data (or Cauchy data), a small perturbation in the data can cause a dramatically large error in the solution for 0

We study new sequence spaces associated to sequences in normed spaces and the band matrix F̂ defined by the Fibonacci sequence. We give some characterizations of continuous linear operators and weakly unconditionally Cauchy series by means of completeness of the new sequence spaces. Also, we characterize the barreledness of a normed space via weakly∗ unconditionally Cauchy series in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$X^{*}$\end{document}X∗.

In this paper, we establish the generalized Cauchy theorems on the para-sphere and the generalized Cauchy integral formulae on the strong para-sphere in Clifford analysis. As applications, the generalized Cauchy theorems and the generalized Cauchy integral formulae on the closed smooth surface and the cylindroid with crooked tips are respectively obtained. And these directly result in the Painleve theorem and the generalization of the Sochocki–Plemelj formula for the difference of boundary values in Clifford analysis. Then, by using these results the Riemann jump boundary value problems and Dirichlet boundary value problems for regular functions in Clifford analysis are discussed. Some singular integral equations are also solved and the inversion formula for Cauchy principal value is obtained by the results based on these boundary value problems solved.

The leaderless consensus problem in a class of high-order multiagent systems with Lipschitz nonlinearities is studied in this paper. Despite existing leaderless consensus protocols devoted to high-order multiagent systems with Lipschitz nonlinearities, the proposed protocol in this paper guarantees achieving consensus in networks of agents with jointly connected topologies. To achieve this goal, first a consensus protocol for high-order multiagent systems with general linear models is proposed. By introducing a common Lyapunov candidate for the set of switching topologies and based on the Cauchy convergence criterion, achieving consensus under the proposed protocol is studied. Then, sufficient conditions are derived under which the results are extended to the case of Lipschitz nonlinearities in agents models. Numerical examples validate the proposed consensus protocol.

The Cauchy problem is studied for a multidimensional difference equation in a class of functions defined at the integer points of a rational cone. We give an easy-to-check condition on the coefficients of the characteristic polynomial of the equation sufficient for solvability of the problem. A multidimensional analog of the condition ensuring stability of the Cauchy problem is stated on using the notion of amoeba of an algebraic hypersurface.

We give examples of Lebesgue spaces on which Cauchy and Cauchy–Leray–Fantappie operators are not bounded.

This paper is concerned with the Cauchy problem connected with the Helmholtz equation. We investigate the denseness of Fourier-Bessel functions and propose a numerical method for approximating the solution to the problem. The convergence and stability are analyzed with truncated singular value decomposition method. Numerical experiments are also presented to show the effectiveness of our method.

ACL; International audience; This work is a contribution to non-destructive testing in the context of uncertainties. It consists in identifying boundary conditions on an inaccessible part of a solid body boundary, from the knowledge of over-specified data given on an accessible part of this boundary. This problem is well known as Cauchy problem. The material properties are considered uncertain. Polynomial chaos expansion is used in order to solve this Cauchy problem in the random context. A stochastic constrained optimization problem is formulated and solved. Numerical experiments are presented

We establish the local well-posedness for a generalized Novikov equation in nonhomogeneous Besov spaces. Besides, we obtain a blow-up criteria and provide a sufficient condition for strong solutions to blow up in finite time.

The paper analyzes the theoretical aspects of constructing a family of single-stage multi-step methods for solving the Cauchy problem with prehistory for ordinary differential equations. The authors consider general issues related to discretization, approximation, convergence, and stability. The problem of improving the accuracy of numerical solutions is analyzed in detail. The results presented in the paper are also applicable for the numerical solution of partial differential equations.

It is proved that the iterative sequence constructed by the Basic Trust-Region Algorithm (see Conn et al. in Trust-region methods, MPS-SIAM series on optimization, Philadelphia, 2000), which uses the Cauchy point method, is locally stable and linearly convergent in a neighborhood of a nonsingular local minimizer.

Shirvani and Soltani suggested a method for characterizing and simulating one-sided truncated Cauchy random variables. For simulation, that involves solving a nonlinear transformation in ea...

This paper is concerned with the Cauchy problem of the modified Hunter-Saxton equation, which was proposed by by J. Hunter and R. Saxton [SIAM J. Appl. Math. 51(1991) 1498-1521]. Using the approximate solution method, the local well-posedness of the model equation is obtained in Sobolev spaces $H^{s}$ with $s > 3/2$, in the sense of Hadamard, and its data-to-solution map is continuous but not uniformly continuous. However, if a weaker $H^{r}$-topology is used then it is shown that the solution map becomes Holder continuous in $H^{s}$.

In this paper, we examine the Cauchy problem of the Laplace equation. Motivated by the incompleteness of the single-layer potential function method, we investigate the double-layer potential function method. Through the use of a layer approach to the solution, we devise a numerical method for approximating the solution of the Cauchy problem, which are well known to be highly ill-posed in nature. The ill-posedness is dealt with Tikhonov regularization, whilst the optimal regularization parameter is chosen by Morozov discrepancy principle. Convergence and stability estimates of the proposed method are then given. Finally, some examples are given for the efficiency of the proposed method. Especially, when the single-layer potential function method does not give accurate results for some problems, it is shown that the proposed method is effective and stable.

We establish the existence and uniqueness conditions for the weighted Cauchy type problem and the Cauchy problem for the linear Barbashin integro-differential equation with fractional partial derivative of order 0 < α ≤ 1 in the sense of Riemann–Liouville and Caputo. Bibliography: 8 titles.

The paper is devoted to investigating a Cauchy problem for homogeneous elliptic PDEs in the abstract Hilbert space given by u ′′ ⁢ ( t ) - A ⁢ u ⁢ ( t ) = 0 {u^{\prime\prime}(t)-Au(t)=0} , 0 < t < T {0 , u ⁢ ( 0 ) = φ {u(0)=\varphi} , u ′ ⁢ ( 0 ) = 0 {u^{\prime}(0)=0} , where A is a positive self-adjoint and unbounded linear operator. The problem is severely ill-posed in the sense of Hadamard [23]. We shall give a new regularization method for this problem when the operator A is replaced by A α = A ⁢ ( I + α ⁢ A ) - 1 {A_{\alpha}=A(I+\alpha A)^{-1}} and u ⁢ ( 0 ) = φ {u(0)=\varphi} is replaced by a nonlocal condition. We show the convergence of this method and we construct a family of regularizing operators for the considered problem. Convergence estimates are established under a priori regularity assumptions on the problem data. Some numerical results are given to show the effectiveness of the proposed method.

The method of radial basis function (RBF) is popularly used in the solution of partial differential equations (PDEs). We propose a multiple-scale MQ-RBF method to solve the linear elliptic PDEs and the corresponding inverse Cauchy problems in simply- and doubly-connected domains, where the multiple scales are automatically determined a priori by the collocation points and source points, which play a role of post-conditioner of linear system to determine the unknown expansion coefficients. In the solution of inverse Cauchy problems the multiple-scale MQ-RBF is quite accurate and stable against large noise level up to 10–30%. Even for a case with only a quarter of boundary being imposed over-specified data, the multiple-scale MQ-RBF can still recover 75% unknown data very well.

We pose the problem of determining the exact value of the operator norm of the Cauchy transform. A lower bound of the norm is presented.

We compute the generalized Cauchy index of some semi-almost periodic functions, which are important in the study of the Fredholm index of singular integral operators with shifts and slowly oscillating data.

In this paper, we consider a Cauchy problem for the semi-linear elliptic equation and use a modified boundary Tikhonov-type regularization method to overcome its ill-posedness. The existence, uniqueness and stability of the regularization solution are proven. Under an a-priori bound assumption for the exact solution, a convergence estimate of Holder type for this method is obtained. Finally, an iterative scheme is proposed to calculate the regularization solution, numerical results show that this method works well.

We derive an algorithm of optimal complexity which determines whether a given matrix is a Cauchy matrix, and which exactly recovers the Cauchy points defining a Cauchy matrix from the matrix entries. Moreover, we study how to approximate a given matrix by a Cauchy matrix with a particular focus on the recovery of Cauchy points from noisy data. We derive an approximation algorithm of optimal complexity for this task, and prove approximation bounds. Numerical examples illustrate our theoretical results. (C) 2015 Elsevier Inc. All rights reserved.

We obtain a representation of the solution to the Cauchy problem with a polynomial difference operator. We establish the solvability condition and specify it in the twodimensional case. Bibliography: 9 titles.

We first point out that the Cauchy convergence topology which was introduced by M.H. Clapp and R.C. Shiflett is not a topology. We redefine the notion of the Cauchy convergence topology for function space. We give examples to show that under the new Cauchy convergence topology some results which appear in the article “A necessary and sufficient condition for the equivalence of the topologies of uniform and compact convergence” which is written by M.H. Clapp and R.C. Shiflett are not true. Finally, we improve part of the conclusions which appear in the article “Cauchy convergence topologies on the space of continuous functions” which is written by Zuquan Li and point out that the corresponding conclusions in it are true under the new Cauchy convergence topology by a small correction.