Fractional \(W\)-differentiation and its application to some singular boundary value problems (Q2713893)

Let \(B_x= \frac{d^2}{dx^2}+ \frac{\gamma}{x}\frac{d}{dx}\) denote the Bessel operator. The corresponding Fourier-Bessel transform is written as follows: NEWLINE\[NEWLINE F_B[\varphi](\xi)=\widehat{\varphi}(\xi)=\int_0^\infty j_\gamma(x\xi)\varphi(x) x^{\gamma} dx, NEWLINE\]NEWLINE with \(j_\gamma(t)= 2^{(\gamma-1)/2}\Gamma((\gamma+1)/2)J_{(\gamma-1)/2}(t)/t^{(\gamma-1)/2}\) \((J_{\gamma}\) is the Bessel function). Its inverse has a similar form. The author defines the fractional powers of the operator \(B_x\) by the equality \((B_x^{\alpha}\varphi)(x)=F_B^{-1}[|\xi|^{\alpha}\widehat{\varphi}(\xi)](x)\) and demonstrates that the usual semigroup properties are fulfilled, i.e., \(B_x^{\alpha}B_x^{\beta}=B_x^{\alpha+\beta}\). Moreover, some other properties of these fractional powers are established. In particular, the author describes explicit representations for these powers. The results are applied to the study of some simplest boundary value problems for the equation \((B_x u)(x,y)=D_y^2 u(x,y)\) (\(x,y>0\)).NEWLINENEWLINEFor the entire collection see [Zbl 0956.00039].