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1

Numerical study of time delay singularly perturbed parabolic differential equations involving both small positive and negative space shifts

Sahu Subal Ranjan, Mohapatra Jugal

Journal of Applied Analysis

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2022

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Vol. 28, nr 1

121--134

EN

A time dependent singularly perturbed differential-difference equation is considered. The problem involves time delay and general small space shift terms. Taylor series approximation is used to expand the space shift term. A robust numerical scheme based on the backward Euler scheme for the time and classical upwind scheme for space is proposed. The convergence analysis is carried out. It is observed that the proposed scheme converges almost first order up to a logarithm term and optimal first order in space on the Shishkin and Bakhvalov-Shishkin mesh, respectively. Numerical results confirm the efficiency of the proposed scheme, which are in agreement with the theoretical bounds.

2

Korovkin-type approximation by operators in Riesz spaces via power series method

Chil Elmiloud, Assili Marwa

Demonstratio Mathematica

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2019

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Vol. 52, nr 1

490--495

EN

In this paper we prove an Ozguç, Yurdakadim and Taş version of the Korovkin-type approximation by operators in the sense of the power series method. That is, we try to extend the Korovkin approximation theorems, obtained by Ozguç and Taş in 2016, and Taş and Yurdakadim in 2017, for concrete classes of Banach spaces to the class of Riesz spaces. Some applications are presented.

3

On the almost uniform convergence

Drozdowski R., Jędrzejewski J., Sochaczewska A.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

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2013

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Vol. 18

11--17

EN

Uniform convergence for continuous real functions sequences preserves continuity of the limit of such sequences. There are weaker types of convergence which have similar properties. We consider such types of convergence for functions from one topological space into another one.

4

On the quasi-uniform convergence

Drozdowski R., Jędrzejewski J., Sochaczewska A.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

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2011

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Vol. 16

19--22

EN

Arzelá [1] considered the weaker form of uniform convergence which is as good as uniform convergence of sequences of functions in respect to continuity of the limit of a sequence of continuous functions. Some generalization of such convergence can be found in [5]. Similar kinds of convergence of function sequences were considered in [3] and [4]. In our article we generalize those kinds of convergence for functions defined in a topological space with values in a topological space. In the article we use terminology which is explained in Engelking's monograph “General Topology” [2]. Among others, we use the notion of a star with respect to an open over. If X is a topological space and α is a cover of this space, then the star St(x, α) of a point x ϵ X with respect to the cover α is defined as the union of all the sets from α which contain the point x, i.e. [wzór].

5

Riemann integrability and quasi-uniform convergence

Ger R., Jędrzejewski J.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

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2010

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Vol. 15

31--34

EN

We consider quasi-uniform convergence of sequences of functions in a context of Riemann integrability of its limit. Some generalizations are discussed as well.

6

On Whitney convergence

Kowalczyk S.

Journal of Applied Analysis

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2009

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Vol. 15, nr 1

139-148

EN

In this paper we describe the W-convergence in C(X, Y) where X is a normal topological space, (Y, d) is a metric space and C(X, Y) is the space of all continuous functions from X to Y. The main theorem of this paper states that the sequence (fn)n∈N of functions from C(X, Y) is W-convergent to a function f ∈ C(X, Y) if and only if it is uniformly convergent to f and there exists a countably compact subset K ⊂ X such that if U is a neighborhood of K then there exists n0 ∈ N for which [wzór]

7

Time-parametric control : uniform convergence of the optimal value functions of discretized problems

Gugat M.

Control and Cybernetics

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1999

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Vol. 28, no 1

7-33

EN

The problem of time-optimal control of linear hyperbolic systems is equivalent to the computation of the root of the optimal value function of a time-parametric program, whose feasible set is described by a countable system of moment equations. To compute this root, discretized problems with a finite number of equality constraints can be used. In this paper, we show that on a certain time-interval, the optimal value functions of the discretized problems converge uniformly to the optimal value function of the original problem. We also give sufficient conditions fot Lipschitz and Hoelder continuity of the optimal value function of the original problem.