On the polyconvolution for the Fourier cosine, Fourier sine, and Kontorovich–Lebedev integral transforms

Thảo, Nguyễn Xuân; Virchenko, Nina

doi:10.1007/s11253-011-0453-8

Cited by 4 publications

(4 citation statements)

References 4 publications

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“…Using the Definition 1.1, the polyconvolutions generated by various linear operators can be constructed. In particular, this method can be used to construct the polyconvolution involving the Hankel integral transform [7,8] and Fourier-Kontorovich-Lebedev integral transforms [29], which leads to the study of the operational properties of polyconvolution and can be applied to solving some classes of integral equations, and systems of integral equations. In this paper, extending the notions in [18], we introduce a new polyconvolution operator involving the Fourier cosine( )-Laplace() integral transforms and apply it to study the solvability in closed-form of the equations.…”

Section: Introductionmentioning

confidence: 99%

New polyconvolution product for Fourier-cosine and Laplace integral operators and their applications

Tuan

2023

Preprint

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The goal of this paper is to introduce the notion of polyconvolution for Fourier-cosine, Laplace integral operators, and its applications. The structure of this polyconvolution operator and associated integral transforms is investigated in detail. The Watson-type theorem is given, to establish necessary and sufficient conditions for this operator to be unitary on L 2 ( R ) , and to get its inverse represented in the conjugate symmetric form. The correlation between the existence of polyconvolution with some weighted spaces is shown, and Young’s type theorem, as well as the norm-inequalities in weighted space, are also obtained. As applications of the Fourier cosine–Laplace polyconvolution, the solvability in closed-form of some classes for integral equations of Toeplitz plus Hankel type and integro-differential equations of Barbashin type is also considered. Several examples are provided for illustrating the obtained results to ensure their validity and applicability.

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Section: Introductionmentioning

confidence: 99%

New polyconvolution product for Fourier-cosine and Laplace integral operators and their applications

Tuan

2023

Preprint

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“…Using Definition 1.1, the polyconvolutions generated by various linear operators can be constructed. In particular, this method can be used to construct the polyconvolution involving the Hankel integral transform [6, 7] and Fourier–Kontorovich–Lebedev integral transforms [8], which leads to the study of the operational properties of polyconvolution and applied to solving some classes of integral equations, and systems of integral equations. In this paper, extending the notions in [5], we introduce a new structure of polyconvolution operator involving the Fourier cosine

\left({F}_c\right)

‐Laplace

\left(\mathcal{L}\right)

integral transforms and apply it to study the solvability in closed‐form of classes for integral equations.…”

Section: Introductionmentioning

confidence: 99%

New polyconvolution product for Fourier‐cosine and Laplace integral operators and their applications

Tuan

2023

Math Methods in App Sciences

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The goal of this paper is to introduce the notion of polyconvolution for Fourier‐cosine, Laplace integral operators, and its applications. The structure of this polyconvolution operator and associated integral transforms are investigated in detail. The Watson‐type theorem is given, to establish necessary and sufficient conditions for this operator to be isometric isomorphism (unitary) on , and to get its inverse represented in the conjugate symmetric form. The correlation between the existence of polyconvolution with some weighted spaces is shown, and Young's type theorem, as well as the norm‐inequalities in weighted space, is also obtained. As applications, we investigate the solvability of a class of Toeplitz plus Hankel type integral equations and linear Barbashin's equations with the help of factorization identities of such polyconvolution. Several examples are provided to illustrate the obtained results to ensure their validity and applicability.

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“…[2,11,12] proposed solutions in closed-form expression for some particular cases of TPH kernels by the generalized convolution. In [13,14], a new polyconvolution was introduced and applied to solve a system of integral equations with TPH kernels. In [10], integral mean value theorem was used to numerically solve the N-th order Fredholm integro-differential equations.…”

Section: Introductionmentioning

confidence: 99%

An Application of the Weighted Mean Value Method to Fredholm integral equations with Toeplitz plus Hankel kernels

Altürk¹,

Şahin²

2017

Journal of Interpolation and Approximation in Scientific Comput

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In the present work, we study Fredholm integral equations of the second kind with Toeplitz, Hankel, and Toeplitz plus Hankel kernels. The weighted mean-value method is employed and extended to obtain numerical solutions for one-and multidimensional Fredholm integral equations of the second kind. By a successful application of this method, we first convert the integral equation into a system of algebraic equations, then solve this system and obtain promising results. Finally, numerical examples are provided to show the simplicity and applicability of the method.

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On the polyconvolution for the Fourier cosine, Fourier sine, and Kontorovich–Lebedev integral transforms

Cited by 4 publications

References 4 publications

New polyconvolution product for Fourier-cosine and Laplace integral operators and their applications

New polyconvolution product for Fourier-cosine and Laplace integral operators and their applications

New polyconvolution product for Fourier‐cosine and Laplace integral operators and their applications

An Application of the Weighted Mean Value Method to Fredholm integral equations with Toeplitz plus Hankel kernels

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