A Fourier generalized convolution transform and applications to integral equations (original) (raw)

The Fourier–Laplace Generalized Convolutions and Applications to Integral Equations

Vietnam Journal of Mathematics, 2013

In this paper we introduce two generalized convolutions for the Fourier cosine, Fourier sine and Laplace integral transforms. Convolution properties and their applications to solving integral equations and systems of integral equations are considered. Keywords Fourier sine transform • Fourier cosine transform • Laplace transform Mathematics Subject Classification (2000) 33C10 • 44A35 • 45E10 • 45J05 • 47A30 • 47B15 1 Introduction Convolutions for integral transforms are studied in the early years of the 20th century, such as convolutions for the Fourier transform (see [2, 9, 13]), the Laplace transform (see [1, 2, 8, 13, 16-19]), the Mellin transform (see [8, 13]), the Hilbert transform (see [2, 3]), the Fourier cosine and sine transforms (see [5, 7, 13, 14]), and so on. These convolutions have many important applications in image processing, partial differential equations, integral equations, inverse heat problems (see [2-4, 8, 11-13, 15-18]).

Generalized Convolutions for the Integral Transforms of Fourier Type and Applications

2009

In this paper we provide several new generalized convolutions for the Fourier-cosine and the Fourier-sine transforms and consider some applications. Namely, the linear space L 1 (R d ), equipped with each of the convolution multiplications constructed, becomes a normed ring, and the explicit solution in L 1 (R d ) of the integral equation with the mixed Toeplitz-Hankel kernel is obtained.

On the Polyconvolution with the Weight Function for the Fourier Cosine, Fourier Sine, and the Kontorovich-Lebedev Integral Transforms

Mathematical Problems in Engineering, 2010

The polyconvolution with the weight function γ of three functions f, g, and h for the integral transforms Fourier sine F s , Fourier cosine F c , and Kontorovich-Lebedev K iy , which is denoted by γ * f, g, h (x), has been constructed. This polyconvolution satisfies the following factorization property Fc γ * f, g, h y sin y F s f y • F c g y • K iy h y , for all y > 0. The relation of this polyconvolution to the Fourier convolution and the Fourier cosine convolution has been obtained. Also, the relations between the polyconvolution product and others convolution product have been established. In application, we consider a class of integral equations with Toeplitz plus Hankel kernel whose solution in closed form can be obtained with the help of the new polyconvolution. An application on solving systems of integral equations is also obtained.

Generalized convolution transforms and Toeplitz plus Hankel integral equations

FRACTIONAL CALCULUS AND …, 2008

We study integral transforms of the form g(x) = ∞ 0 K 1 (u)[f (|x + u − 1|) + f (|x − u − 1|) − f (x + u + 1) − f (|x − u + 1|)]du + ∞ 0 K 2 (u)[f (|x − u|) − f (x + u)]du from L p (R +) to L q (R +), (1 p 2, p −1 + q −1 = 1) with the help of a generalized convolution and prove Watson's and Plancherel's theorems. Using generalized convolutions a class of Toeplitz plus Hankel integral equations, and also a system of integro-differential equations are solved in closed form.

The solvability and explicit solutions of two integral equations via generalized convolutions

Journal of Mathematical Analysis and Applications, 2010

This paper presents the necessary and sufficient conditions for the solvability of two integral equations of convolution type; the first equation generalizes from integral equations with the Gaussian kernel, and the second one contains the Toeplitz plus Hankel kernels. Furthermore, the paper shows that the normed rings on L 1 (R d) are constructed by using the obtained convolutions, and an arbitrary Hermite function and appropriate linear combination of those functions are the weight-function of four generalized convolutions associating F and F. The open question about Hermitian weight-function of generalized convolution is posed at the end of the paper.

A Generalization of Integral Transform

European Journal of Pure and Applied Mathematics, 2018

In this paper, the generalization of integral transform (GIT) of the func-tion G{f (t)} is introduced for solving both differential and interodif-ferential equations. This transform generalizes the integral transformswhich use exponential functions as their kernels and the integral trans-form with polynomial function as a kernel. The generalized integraltransform converts the differential equation in us domain (the trans-formed variables) and reconvert the result by its inverse operator. Inparticular, if u = 1, then the generalized integral transform coincideswith the Laplace transform and this result can be written in anotherform as the polynomial integral transform.

A new generalized integral transform and applications

Cornell University - arXiv, 2022

In this work, we introduce a new generalized integral transform involving many potentially known or new transforms as special cases. Basic properties of the new integral transform, that investigated in this work, include the existence theorem, the scaling property, elimination property a Parseval-type identity, and inversion formula. The relationships of the new transform with well-known transforms are characterized by integral identities. The new transform is applied to solve certain initial boundary value problems. Some illustrative examples are given. The results established in this work extend and generalize recently published results.