2019
Том 71
№ 2

All Issues

Virchenko N. A.

Articles: 6
Article (English)

On the generalized convolution for $F_c$, $F_c$, and $K - L$ integral transforms

Thao N. X., Virchenko N. A.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2012. - 64, № 1. - pp. 81-91

We study new generalized convolutions $f \overset{\gamma}{*} g$ with weight function $\gamma(y) = y$ for the Fourier cosine, Fourier sine, and Kontorovich-Lebedev integral transforms in weighted function spaces with two parameters $L(\mathbb{R}_{+}, x^{\alpha} e^{-\beta x} dx)$. These generalized convolutions satisfy the factorization equalities $$F_{\left\{\frac SC\right\}} (f \overset{\gamma}{*} g)_{\left\{\frac 12\right\}}(y) = y (F_{\left\{\frac SC\right\}} f)(y) (K_{sy}g) \quad \forall y > 0$$ We establish a relationship between these generalized convolutions and known convolutions, and also relations that associate them with other convolution operators. As an example, we use these new generalized convolutions for the solution of a class of integral equations with Toeplitz-plus-Hankel kernels and a class of systems of two integral equations with Toeplitz-plus-Hankel kernels.

Article (English)

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

Thao N. X., Virchenko N. A.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2010. - 62, № 10. - pp. 1388–1399

The polyconvolution $∗_1(f,g,h)(x)$ of three functions $f, g$ and $h$ is constructed for the Fourier cosine $(F_c)$, Fourier sine $(F_s)$, and Kontorovich–Lebedev $(K_{iy})$ integral transforms whose factorization equality has the form $$F_c(∗_1(f,g,h))(y)=(F_sf)(y).(F_sg)(y).(K_{iy}h)\;\;∀y>0.$$ The relationships between this polyconvolution, the Fourier convolution, and the Fourier cosine convolution are established. In addition, we also establish the relationships between the product of the new polyconvolution and the products of the other known types of convolutions. As an application, we consider a class of integral equations with Toeplitz and Hankel kernels whose solutions can be obtained with the help of the new polyconvolution in the closed form. We also present the applications to the solution of systems of integral equations.

Article (Ukrainian)

On some integral equations with the generalized Legendre function

Sichkar' Yu, V., Virchenko N. A.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 1999. - 51, № 2. - pp. 263–267

We solve and investigate an integral equation with the generalized associated Legendre functionP k m,n (z) by using the fractional integro-differential calculus.

Article (Ukrainian)

Method of N-ary integral equations

Virchenko N. A.

Full text (.pdf)

Ukr. Mat. Zh. - 1991. - 43, № 11. - pp. 1494–1504

Article (Ukrainian)

Applications of Legendre's generalized associated functions

Virchenko N. A.

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Ukr. Mat. Zh. - 1987. - 39, № 2. - pp. 534–538

Article (Ukrainian)

Certain hybrid dual integral equations

Virchenko N. A.

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Ukr. Mat. Zh. - 1984. - 36, № 2. - pp. 139 - 142