On the Fourier cosine-Kontorovich-Lebedev generalized convolution transforms

Some results of Watson and Plancherel‐type integral transforms related to the Hartley, Fourier convolutions and applications

“…These above convolutions are also used to solve in a closed form some classes of integral equations of Fredholm (see Tuan and Hong 10,11 ) and Toeplitz plus Hankel's type with parabolic equation (see Tuan et al 12 ) and consider the boundedness solutions of these problems (see Tuan et al, 12,14 Hoang et al, 13 and van Anh & Thao 15 ). The limitation in Hong et al, 6 Tuan, 7 and Tuan et al 14 is that it cannot show the Plancherel‐type theorem and the boundedness of Watson integral transform operator. The reason here is that in the above results, the convolutions used are related to Kontorovich‐Lebedev integral transformation with the kernel as the Macdonald function.…”

“…In recent years, there have been some results of integral transformations for convolutions involving Fourier sine, Fourier cosine, and Kontorovich‐Lebedev transformations in case $$$ D $$$ is the differential operator of order 2 or $$$ 2n $$$ (refer to Britvina, 3 Al‐Musallam & Tuan, 4 and Tuan 5 ). From there, the authors use these results to solve in a closed form some classes of the differential equations (see Hong et al 6 and Tuan 7 ), and the convolution integral equations (see Tuan et al 8 and Yakubovich & Britvina 9 ). These above convolutions are also used to solve in a closed form some classes of integral equations of Fredholm (see Tuan and Hong 10,11 ) and Toeplitz plus Hankel's type with parabolic equation (see Tuan et al 12 ) and consider the boundedness solutions of these problems (see Tuan et al, 12,14 Hoang et al, 13 and van Anh & Thao 15 ).…”