Archive | 2021
The heat semigroup and equation related to a Bessel-type operators and the canonical Fourier Bessel transform
Abstract
In this paper we study a translation operator associated with the\ncanonical Fourier Bessel transform\n$\\mathcal{F}_{\\nu}^{\\mathbf{m}}.$\nWe then use it to derive a convolution product and study some of its\nimportant properties. As a direct application, we introduce the heat\nsemigroup generated by the Bessel-type operators\n$$\\Delta_{\\nu}^{\\mathbf{m}^{-1}}=\\frac{d^{2}}{dx^{2}}+\\left(\n\\frac{2\\nu +1}{x}+2i\n\\frac{a}{b} x\\right)\n\\frac{d}{dx}-\\left(\n\\frac{a^{2}}{b^{2}}x^{2}-2i\\left(\n\\nu +1\\right)\n\\frac{a}{b}\\right) $$ and use it to\nsolve the initial value problem for the heat equation governed by\n$\\Delta_{\\nu}^{\\mathbf{m}^{-1}}.$