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1

Bernstein operational matrix of differentiation and collocation approach for a class of three-point singular BVPs: error estimate and convergence analysis

Sriwastav Nikhil, Barnwal Amit K., Wazwaz Abdul-Majid, Singh Mehakpreet

Opuscula Mathematica

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2023

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Vol. 43, no. 4

575--601

EN

Singular boundary value problems (BVPs) have widespread applications in the field of engineering, chemical science, astrophysics and mathematical biology. Finding an approximate solution to a problem with both singularity and non-linearity is highly challenging. The goal of the current study is to establish a numerical approach for dealing with problems involving three-point boundary conditions. The Bernstein polynomials and collocation nodes of a domain are used for developing the proposed numerical approach. The straightforward mathematical formulation and easy to code, makes the proposed numerical method accessible and adaptable for the researchers working in the field of engineering and sciences. The priori error estimate and convergence analysis are carried out to affirm the viability of the proposed method. Various examples are considered and worked out in order to illustrate its applicability and effectiveness. The results demonstrate excellent accuracy and efficiency compared to the other existing methods.

2

Sensitivity and stability analysis for groundwater numerical modeling: a field study of finite element application in the arid region

Jafarzadeh Ahmad, Pourreza-Bilondi Mohsen, Akbarpour Abolfazl, Khashei-Siuki Abbas, Azizi Mohsen

Acta Geophysica

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2023

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Vol. 71, no. 2

1045--1062

EN

This study intends to investigate the impacts of scheme type, time step, and error threshold on the stability of numerical simulation in the groundwater modeling. Hence, a two-dimensional finite element (FE) was implemented to simulate groundwater flow in a synthetic test case and a real-world study (Birjand aquifer). To verify the proposed model in both cases, the obtained results were compared with analytical solutions and observed values. The stability of numerical results was analyzed through different schemes and time-step sizes. Besides, the effect of the error threshold was examined by considering different threshold values. The results confirmed that the FE model has a good capacity to simulate groundwater fluctuations even for the real problem with more complexities. Examination of implicit outputs indicated that groundwater simulations based on this scheme have good accuracy, stability, and proper convergence in all time intervals. However, in the explicit and Crank–Nicolson schemes the time interval should be less than or equal to 0.001 and 0.1 day, respectively. Also, results reveal that for making stability in all schemes the value of the error threshold should not be more than 0.0001 m. Moreover, it derived that the boundary conditions of the aquifer influence the stability of numerical outputs. Finally, it was comprehended that as time interval and error threshold increases, the oscillation rate propagated.

3

A Computational Technique for Solving Singularly Perturbed Delay Partial Differential Equations

Gürbüz Burcu

Foundations of Computing and Decision Sciences

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2021

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Vol. 46, No. 3

221--233

EN

In this work, a matrix method based on Laguerre series to solve singularly perturbed second order delay parabolic convection-diffusion and reaction-diffusion type problems involving boundary and initial conditions is introduced. The approximate solution of the problem is obtained by truncated Laguerre series. Moreover convergence analysis is introduced and stability is explained. Besides, a test case is given and the error analysis is considered by the different norms in order to show the applicability of the method.

4

Constrained Output Iterative Learning Control

Yovchev Kaloyan, Delchev Kamen, Krastev Evgeniy

Archives of Control Sciences

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2020

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Vol. 30, No. 1

157--176

EN

Iterative Learning Control (ILC) is a well-known method for control of systems performing repetitive jobs with high precision. This paper presents Constrained Output ILC (COILC) for non-linear state space constrained systems. In the existing literature there is no general solution for applying ILC to such systems. This novel method is based on the Bounded Error Algorithm (BEA) and resolves the transient growth error problem, which is a major obstacle in applying ILC to non-linear systems. Another advantage of COILC is that this method can be applied to constrained output systems. Unlike other ILC methods the COILC method employs an algorithm that stops the iteration before the occurrence of a violation in any of the state space constraints. This way COILC resolves both the hard constraints in the non-linear state space and the transient growth problem. The convergence of the proposed numerical procedure is proved in this paper. The performance of the method is evaluated through a computer simulation and the obtained results are compared to the BEA method for controlling non-linear systems. The numerical experiments demonstrate that COILC is more computationally effective and provides better overall performance. The robustness and convergence of the method make it suitable for solving constrained state space problems of non-linear systems in robotics.

5

Approximation solvability for a system of implicit nonlinear variational inclusions with H-monotone operators

Kim J. K., Bhat M. I.

Demonstratio Mathematica

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2018

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Vol. 51, nr 1

241--254

EN

In this paper, we introduce and study a new system of variational inclusions which is called a system of nonlinear implicit variational inclusion problems with A-monotone and H-monotone operators in semi-inner product spaces. We define the resolvent operator associated with A-monotone and H-monotone operators and prove its Lipschitz continuity. Using resolvent operator technique, we prove the existence and uniqueness of solution for this new system of variational inclusions. Moreover, we suggest an iterative algorithm for approximating the solution of this system and discuss the convergence analysis of the sequences generated by the iterative algorithm under some suitable conditions.

6

A numerical solution for a class of time fractional diffusion equations with delay

Pimenov V. G., Hendy A. S.

International Journal of Applied Mathematics and Computer Science

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2017

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Vol. 27, no. 3

477--488

EN

This paper describes a numerical scheme for a class of fractional diffusion equations with fixed time delay. The study focuses on the uniqueness, convergence and stability of the resulting numerical solution by means of the discrete energy method. The derivation of a linearized difference scheme with convergence order O(τ 2−α + h4) in L ∞-norm is the main purpose of this study. Numerical experiments are carried out to support the obtained theoretical results.

7

Computing the Stackelberg/Nash equilibria using the extraproximal method: Convergence analysis and implementation details for Markov chains games

Trejo K. K., Clempner J. B., Poznyak A. S.

International Journal of Applied Mathematics and Computer Science

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2015

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Vol. 25, no. 2

337--351

EN

In this paper we present the extraproximal method for computing the Stackelberg/Nash equilibria in a class of ergodic controlled finite Markov chains games. We exemplify the original game formulation in terms of coupled nonlinear programming problems implementing the Lagrange principle. In addition, Tikhonov's regularization method is employed to ensure the convergence of the cost-functions to a Stackelberg/Nash equilibrium point. Then, we transform the problem into a system of equations in the proximal format. We present a two-step iterated procedure for solving the extraproximal method: (a) the first step (the extra-proximal step) consists of a “prediction” which calculates the preliminary position approximation to the equilibrium point, and (b) the second step is designed to find a “basic adjustment” of the previous prediction. The procedure is called the “extraproximal method” because of the use of an extrapolation. Each equation in this system is an optimization problem for which the necessary and efficient condition for a minimum is solved using a quadratic programming method. This solution approach provides a drastically quicker rate of convergence to the equilibrium point. We present the analysis of the convergence as well the rate of convergence of the method, which is one of the main results of this paper. Additionally, the extraproximal method is developed in terms of Markov chains for Stackelberg games. Our goal is to analyze completely a three-player Stackelberg game consisting of a leader and two followers. We provide all the details needed to implement the extraproximal method in an efficient and numerically stable way. For instance, a numerical technique is presented for computing the first step parameter (λ) of the extraproximal method. The usefulness of the approach is successfully demonstrated by a numerical example related to a pricing oligopoly model for airlines companies.

8

Iterative learning control with sampled-data feedback for robot manipulators

Delchev K., Boiadjiev G., Kawasaki H., Mouri T.

Archives of Control Sciences

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2014

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Vol. 24, no. 3

299--319

EN

This paper deals with the improvement of the stability of sampled-data (SD) feedback control for nonlinear multiple-input multiple-output time varying systems, such as robotic manipulators, by incorporating an off-line model based nonlinear iterative learning controller. The proposed scheme of nonlinear iterative learning control (NILC) with SD feedback is applicable to a large class of robots because the sampled-data feedback is required for model based feedback controllers, especially for robotic manipulators with complicated dynamics (6 or 7 DOF, or more), while the feedforward control from the off-line iterative learning controller should be assumed as a continuous one. The robustness and convergence of the proposed NILC law with SD feedback is proven, and the derived sufficient condition for convergence is the same as the condition for a NILC with a continuous feedback control input. With respect to the presented NILC algorithm applied to a virtual PUMA 560 robot, simulation results are presented in order to verify convergence and applicability of the proposed learning controller with SD feedback controller attached.

9

A simple scheme for semi-recursive identification of Hammerstein system nonlinearity by Haar wavelets

Śliwiński P., Hasiewicz Z., Wachel P.

International Journal of Applied Mathematics and Computer Science

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2013

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Vol. 23, no. 3

507--520

EN

A simple semi-recursive routine for nonlinearity recovery in Hammerstein systems is proposed. The identification scheme is based on the Haar wavelet kernel and possesses a simple and compact form. The convergence of the algorithm is established and the asymptotic rate of convergence (independent of the input density smoothness) is shown for piecewise-Lipschitz nonlinearities. The numerical stability of the algorithm is verified. Simulation experiments for a small and moderate number of input-output data are presented and discussed to illustrate the applicability of the routine.

10

On upper and lower strong quasi-uniform convergence of multivalued maps

Dominik I., Sochaczewska A.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

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2012

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Vol. 17

25-35

EN

The concept of strong convergence of functions and multifunctions was introduced by I. Kupka, V. Toma and A. Sochaczewska. In this paper we consider new deﬁnitions of convergence for the nets of multifunctions – upper and lower strong quasi-uniform convergence.

11

On-line wavelet estimation of Hammerstein system nonlinearity

Śliwiński P.

International Journal of Applied Mathematics and Computer Science

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2010

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Vol. 20, no 3

513-523

EN

A new algorithm for nonparametric wavelet estimation of Hammerstein system nonlinearity is proposed. The algorithm works in the on-line regime (viz., past measurements are not available) and offers a convenient uniform routine for nonlinearity estimation at an arbitrary point and at any moment of the identification process. The pointwise convergence of the estimate to locally bounded nonlinearities and the rate of this convergence are both established.

12

A new nonlinear Lagrangian method for nonconvex semidefinite programming

Li Y., Zhang L.

Journal of Applied Analysis

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2009

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Vol. 15, nr 2

149-172

EN

We study convergence properties of a new nonlinear Lagrangian method for nonconvex semidefinite programming. The convergence analysis shows that this method converges locally when the penalty parameter is less than a threshold and the error bound of solution is proportional to the penalty parameter under the constraint nondegeneracy condition, the strict complementarity condition and the strong" second order sufficient conditions. The major tools used in the analysis include the second implicit function theorem and differentials of Lowner operators.