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Nesir Huseyin - One of the best experts on this subject based on the ideXlab platform.
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Approximation of the Sections of the Set of Trajectories of the Control System Described by a Nonlinear Volterra Integral Equation
Mathematical Modelling and Analysis, 2015Co-Authors: Anar Huseyin, Nesir Huseyin, Khalik G. GuseinovAbstract:Approximation of the sections of the set of trajectories of the control system described by a nonlinear Volterra Integral Equation is studied. The admissible control functions are chosen from the closed ball of the space Lp, p > 1, with radius µ and centered at the origin. The set of admissible control functions is replaced by the set of control functions, which includes a finite number of control functions and generates a finite number of trajectories. It is proved that the sections of the set of trajectories can be approximated by the sections of the set of trajectories, generated by a finite number of control functions.
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approximate construction of the set of trajectories of the control system described by a Volterra Integral Equation
Mathematische Nachrichten, 2015Co-Authors: Nesir Huseyin, Khalik G. Guseinov, Vladimir N UshakovAbstract:In this paper the set of trajectories of the control system is investigated. It is assumed that the behavior of the control system is described by a Volterra Integral Equation which is nonlinear with respect to the state vector and is affine with respect to the control vector, and the control functions have an Integral constraint. An approximation of the set of trajectories by the set which consists of a finite number of trajectories is given. The Hausdorff distance between the set of trajectories of the system and the set, consisting of a finite number of trajectories is evaluated.
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Dependence on the parameters of the set of trajectories of the control system described by a nonlinear Volterra Integral Equation
Applications of Mathematics, 2014Co-Authors: Anar Huseyin, Nesir HuseyinAbstract:In this paper the control system with limited control resources is studied, where the behavior of the system is described by a nonlinear Volterra Integral Equation. The admissible control functions are chosen from the closed ball centered at the origin with radius µ in Lp (p > 1). It is proved that the set of trajectories generated by all admissible control functions is Lipschitz continuous with respect to µ for each fixed p, and is continuous with respect to p for each fixed µ. An upper estimate for the diameter of the set of trajectories is given.
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precompactness of the set of trajectories of the controllable system described by a nonlinear Volterra Integral Equation
Mathematical Modelling and Analysis, 2012Co-Authors: Anar Huseyin, Nesir HuseyinAbstract:Abstract In this paper the controllable system whose behaviour is described by a nonlinear Volterra Integral Equation, is studied. The set of admissible control functions is the closed ball of the space L p (p > 1) with radius µ 0 and centered at the origin. It is shown that the set of trajectories of the system is a bounded and precompact subset of the space of continuous functions.
Teresa Diogo - One of the best experts on this subject based on the ideXlab platform.
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a hybrid collocation method for a nonlinear Volterra Integral Equation with weakly singular kernel
Journal of Computational and Applied Mathematics, 2010Co-Authors: Magda Rebelo, Teresa DiogoAbstract:This work is concerned with the numerical solution of a nonlinear weakly singular Volterra Integral Equation. Owing to the singular behavior of the solution near the origin, the global convergence order of product integration and collocation methods is not optimal. In order to recover the optimal orders a hybrid collocation method is used which combines a non-polynomial approximation on the first subinterval followed by piecewise polynomial collocation on a graded mesh. Some numerical examples are presented which illustrate the theoretical results and the performance of the method. A comparison is made with the standard graded collocation method.
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Extrapolation Methods for a Nonlinear Weakly Singular Volterra Integral Equation
2010Co-Authors: Teresa Diogo, Pedro M. Lima, Magda RebeloAbstract:In this work we consider a nonlinear Volterra Integral Equation with weakly singular kernel. An asymptotic error expansion for the explicit Euler’s method is obtained and this allows the use of certain extrapolation procedures. The performance of the extrapolation method is illustrated by several numerical examples.
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numerical methods for a Volterra Integral Equation with non smooth solutions
Journal of Computational and Applied Mathematics, 2006Co-Authors: Teresa Diogo, Pedro M. Lima, Neville J Ford, Svilen S ValtchevAbstract:We consider the numerical treatment of a singular Volterra Integral Equation with an infinite set of solutions, one of which is smooth and all others have infinite gradient at the origin. This Equation has been the subject of previous works, where we have dealt with the approximation of the smooth solution. Here we present numerical methods which enable us to obtain approximations to any of the infinite class of solutions. Some numerical examples are given which illustrate the performance of the methods employed.
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numerical solution of a nonlinear abel type Volterra Integral Equation
Communications on Pure and Applied Analysis, 2006Co-Authors: Teresa Diogo, Pedro M. Lima, Magda RebeloAbstract:We are concerned with the analytical and numerical analysis of a nonlinear weakly singular Volterra Integral Equation. Owing to the singularity of the solution at the origin, the global convergence order of Euler's method is less than one. The smoothness properties of the solution are investigated and, by a detailed error analysis, we prove that first order of convergence can be achieved away from the origin. Some numerical results are included confirming the theoretical estimates.
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numerical solution of a nonuniquely solvable Volterra Integral Equation using extrapolation methods
Journal of Computational and Applied Mathematics, 2002Co-Authors: Pedro M. Lima, Teresa DiogoAbstract:In this work the numerical solution of a Volterra Integral Equation with a certain weakly singular kernel, depending on a real parameter µ, is considered. Although for certain values of µ this Equation possesses an infinite set of solutions, we have been able to prove that Euler's method converges to a particular solution. It is also shown that the error allows an asymptotic expansion in fractional powers of the stepsize, so that general extrapolation algorithms, like the E-algorithm, can be applied to improve the numerical results. This is illustrated by means of some examples.
Anar Huseyin - One of the best experts on this subject based on the ideXlab platform.
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Approximation of the Sections of the Set of Trajectories of the Control System Described by a Nonlinear Volterra Integral Equation
Mathematical Modelling and Analysis, 2015Co-Authors: Anar Huseyin, Nesir Huseyin, Khalik G. GuseinovAbstract:Approximation of the sections of the set of trajectories of the control system described by a nonlinear Volterra Integral Equation is studied. The admissible control functions are chosen from the closed ball of the space Lp, p > 1, with radius µ and centered at the origin. The set of admissible control functions is replaced by the set of control functions, which includes a finite number of control functions and generates a finite number of trajectories. It is proved that the sections of the set of trajectories can be approximated by the sections of the set of trajectories, generated by a finite number of control functions.
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Dependence on the parameters of the set of trajectories of the control system described by a nonlinear Volterra Integral Equation
Applications of Mathematics, 2014Co-Authors: Anar Huseyin, Nesir HuseyinAbstract:In this paper the control system with limited control resources is studied, where the behavior of the system is described by a nonlinear Volterra Integral Equation. The admissible control functions are chosen from the closed ball centered at the origin with radius µ in Lp (p > 1). It is proved that the set of trajectories generated by all admissible control functions is Lipschitz continuous with respect to µ for each fixed p, and is continuous with respect to p for each fixed µ. An upper estimate for the diameter of the set of trajectories is given.
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precompactness of the set of trajectories of the controllable system described by a nonlinear Volterra Integral Equation
Mathematical Modelling and Analysis, 2012Co-Authors: Anar Huseyin, Nesir HuseyinAbstract:Abstract In this paper the controllable system whose behaviour is described by a nonlinear Volterra Integral Equation, is studied. The set of admissible control functions is the closed ball of the space L p (p > 1) with radius µ 0 and centered at the origin. It is shown that the set of trajectories of the system is a bounded and precompact subset of the space of continuous functions.
M Serrano C Perez - One of the best experts on this subject based on the ideXlab platform.
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nonlinear Volterra Integral Equation of the second kind and biorthogonal systems
Abstract and Applied Analysis, 2010Co-Authors: M I Berenguer, D Gamez, A I Garraldaguillem, M Serrano C PerezAbstract:We obtain an approximation of the solution of the nonlinear Volterra Integral Equation of the second kind, by means of a new method for its numerical resolution. The main tools used to establish it are the properties of a biorthogonal system in a Banach space and the Banach fixed point theorem.
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analytical techniques for a numerical solution of the linear Volterra Integral Equation of the second kind
Abstract and Applied Analysis, 2009Co-Authors: M I Berenguer, D Gamez, A I Garraldaguillem, Ruiz M Galan, M Serrano C PerezAbstract:In this work we use analytical tools—Schauder bases and Geometric Series theorem—in order to develop a new method for the numerical resolution of the linear Volterra Integral Equation of the second kind.
Khalik G. Guseinov - One of the best experts on this subject based on the ideXlab platform.
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Approximation of the Sections of the Set of Trajectories of the Control System Described by a Nonlinear Volterra Integral Equation
Mathematical Modelling and Analysis, 2015Co-Authors: Anar Huseyin, Nesir Huseyin, Khalik G. GuseinovAbstract:Approximation of the sections of the set of trajectories of the control system described by a nonlinear Volterra Integral Equation is studied. The admissible control functions are chosen from the closed ball of the space Lp, p > 1, with radius µ and centered at the origin. The set of admissible control functions is replaced by the set of control functions, which includes a finite number of control functions and generates a finite number of trajectories. It is proved that the sections of the set of trajectories can be approximated by the sections of the set of trajectories, generated by a finite number of control functions.
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approximate construction of the set of trajectories of the control system described by a Volterra Integral Equation
Mathematische Nachrichten, 2015Co-Authors: Nesir Huseyin, Khalik G. Guseinov, Vladimir N UshakovAbstract:In this paper the set of trajectories of the control system is investigated. It is assumed that the behavior of the control system is described by a Volterra Integral Equation which is nonlinear with respect to the state vector and is affine with respect to the control vector, and the control functions have an Integral constraint. An approximation of the set of trajectories by the set which consists of a finite number of trajectories is given. The Hausdorff distance between the set of trajectories of the system and the set, consisting of a finite number of trajectories is evaluated.
