*Mathematical Methods in the Applied Sciences*The heat semigroup and equation related to a Bessel-type operators and
the canonical Fourier Bessel transform

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

28 May 2021

30 May 2021

30 May 2021

08 Jun 2021