Generalized Picard singular integrals

Anastassiou, George A.; Aral, Ali

doi:10.1016/j.camwa.2008.09.026

Cited by 17 publications

(15 citation statements)

References 11 publications

(11 reference statements)

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“…In this section, we continue the study of the generalized Picard operator P λ ,β [16] depending on nonisotropic β -distance, in the direction of weighted approximation process. For this purpose, we first define weighted n-dimensional L p space by involving weight depending on nonisotropic distance.…”

Section: Generalized Picard Operatorsmentioning

confidence: 99%

“…For the brief knowledge, it may be useful to refer to the work of Anastassiou and Aral [16] and references therein. As it is known, one of the central research directions of approximation theory is singular integral operators.…”

Section: Introductionmentioning

confidence: 99%

“…Yet, q-generalization of Picard and Gauss-Weierstrass singular integral operators has been stated, and some approximation properties in weighted space have been discussed; also complex variants of them have been studied [24,26,28]. Recently, another interesting improvement related to the multivariate q-Picard singular operators depending on nonisotropic norm, P λ ,β , has been subsequently stated in [16]. Here, the authors have investigated pointwise convergence of the family of P λ ,β ( f ) to f at the so called β -Lebesgue points depending on nonisotropic β -distance.…”

Section: Introductionmentioning

confidence: 99%

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Applications of q-Calculus in Operator Theory

Aral¹,

Gupta²,

Agarwal³

2013

363

206

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In order to approximate integrable functions on the interval [0, 1], Kantorovich gave modified Bernstein polynomials. Later in the year 1967 Durrmeyer [58] considered a more general integral modification of the classical Bernstein polynomials, which were studied first by Derriennic [47]. Also some other generalizations of the Bernstein polynomials are available in the literature. The other most popular generalization as considered by Goodman and Sharma [82], namely, genuine Bernstein-Durrmeyer operators. In this chapter we discuss the q analogues of various integral modifications of Bernstein polynomials. The results were discussed in recent papers [45, 62, 86, 89, 92, 94, 121], etc. q-Bernstein-Kantorovich OperatorsRecently, Dalmanoglu [45] proposed the q-Kantorovich-Bernstein operators asIn case q = 1, the operators (4.1) reduce to well-known Bernstein-Kantorovich operators Direct ResultsFor the operators (4.1), Dalmanoglu [45] obtained the following theorems:Theorem 4.1. If the sequence (q n ) satisfies the conditions lim n→∞ q n = 1 and lim n→∞Proof. First, we haveAlso by definition of q-integral q-Bernstein-Kantorovich Operators 115Again by definition of q-integralTo estimate K n,q (t 2 , x), we haveTherefore using [k + 1] q = q[k] q + 1 and using the similar methods as above, we haveReplacing q by a sequence {q n } such that lim n→∞ q n = 1, it is easily seen that K n,q (t i , x), i = 0, 1, 2 converges uniformly to t i . Thus the result follows by Korovkin's theorem. for all f ∈ C[0, a] and δ n = K n,q ((t − x) 2 , x).

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Section: Generalized Picard Operatorsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Applications of q-Calculus in Operator Theory

Aral¹,

Gupta²,

Agarwal³

2013

363

206

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“…While for a nonhomogeneous elastic material m=0, we have a NVIE. Many authors have interested in solving the nonlinear integral equations with different kernels using different methods, see [33][34][35][36][37][38].…”

Section:    mentioning

confidence: 99%

On the Solution of Nonlinear Viscoelastic Material

2017

IJIET

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-Here, we consider the problem of twisting of a cylindrical bar of elliptic cross-section. The bar is assumed to consist of a nonlinear viscoelastic material (NVeM). In addition, we consider the bar is subjected to nobody forces and free from forces acting on its lateral surface. Under certain conditions, the problem yields to a nonlinear Volterra integral equation (NVIE) of the second kind. The existence and uniqueness of solution of the integral equation is considered. Moreover, the best numerical method, Product Nystrom method (PNM) is used to obtain the numerical solution. Finally, numerical results are considered and the error estimate is computed

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“…In Ref. [3], Anastassiou and Aral introduced the multivariate of the q-Picard singular IE to obtain the rate of convergence and to define a new modulus of continuity. In Ref.…”

Section: Introductionmentioning

confidence: 99%

On Numerical Solution Approach for the Volterra-Fredholm Integral Equation

Ismail

Khamis

Abdou

et al. 2015

j comput theor nanosci

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In this paper, authors discussed the existence of a unique solution of Volterra-Fredholm integral equation of second kind in the space L p × C 0 T with a generalized kernel, in the domain of integration of several spaces, and t ∈ 0 T is the time is considered. A numerical technique used to obtain a system of Fredholm integral equations, Also authors considered the existence and the uniqueness of the system of equations. Moreover, the Toeplitz matrix method and Product Nystrom method are used to obtain numerically the solution of Fredholm integral equations with logarithmic kernel and Carleman function. Many special cases are considered and established from the work.

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Generalized Picard singular integrals

Cited by 17 publications

References 11 publications

Applications of q-Calculus in Operator Theory

Applications of q-Calculus in Operator Theory

On the Solution of Nonlinear Viscoelastic Material

On Numerical Solution Approach for the Volterra-Fredholm Integral Equation

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