On some fractional integrals and their applications. (English) Zbl 0617.35006
Behandelt werden Operatoren der Form
\[
I_ p(\eta,\alpha)f(x):=2^{\alpha -1} p^{1-\alpha} x^{-(\alpha +\eta)}\int^{x}_{0}u^ n[\frac{x-u}{x}]^{(\alpha -1)/2}\cdot J_{\alpha -1}\{p\sqrt{(x^ 2-xu)}\}f(u)du
\]
mit \(\alpha >0\), \(p\geq 0\), \(J_{\alpha -1}..\). Besselfunktion erster Art. Es wird gezeigt, daß die Beziehung
\[
I_ p(\eta,\alpha)M^{(x)}_{2(\alpha +\eta)}f(x)=[M^{(x)}_{2(\alpha +\eta)}+(px)^ 2]I_ p(\eta,\alpha)f(x),\quad x>0,
\]
gilt, mit \(M_{\gamma}^{(x)}:=x^{- (\gamma -1)}Dx^{\gamma +1} D\), \(D:=d/dx.\)
Mittels dieser Beziehung werden Lösungsdarstellungen für die verallgemeinerte biaxialsymmetrische Helmholtzgleichung (GBSHE) \[ \partial^ 2 v/\partial x^ 2+\partial^ 2 v/\partial y^ 2+(2\alpha /x)\partial v/\partial x+(2\beta /y)\partial v/\partial y+k^ 2 v=0,\quad k\geq 0 \] angegeben. In ähnlicher Weise werden Gleichungen der Form \[ \sum^{n}_{i=1}\partial^ 2 w/\partial x^ 2_ i+\partial^ 2 w/\partial \rho^ 2+(s/\rho)\partial w/\partial \rho +k^ 2 w=0,\quad k\geq 0,\quad s>-1, \]
\[ \partial^ 2 v/\partial x^ 2+\partial^ 2 v/\partial y^ 2+(2\alpha /y)\partial v/\partial y+k^ 2 v=0,\quad k\geq 0, \] behandelt und eine Inversionsformel der Kontorovich-Lebedev-Transformation hergeleitet.
Mittels dieser Beziehung werden Lösungsdarstellungen für die verallgemeinerte biaxialsymmetrische Helmholtzgleichung (GBSHE) \[ \partial^ 2 v/\partial x^ 2+\partial^ 2 v/\partial y^ 2+(2\alpha /x)\partial v/\partial x+(2\beta /y)\partial v/\partial y+k^ 2 v=0,\quad k\geq 0 \] angegeben. In ähnlicher Weise werden Gleichungen der Form \[ \sum^{n}_{i=1}\partial^ 2 w/\partial x^ 2_ i+\partial^ 2 w/\partial \rho^ 2+(s/\rho)\partial w/\partial \rho +k^ 2 w=0,\quad k\geq 0,\quad s>-1, \]
\[ \partial^ 2 v/\partial x^ 2+\partial^ 2 v/\partial y^ 2+(2\alpha /y)\partial v/\partial y+k^ 2 v=0,\quad k\geq 0, \] behandelt und eine Inversionsformel der Kontorovich-Lebedev-Transformation hergeleitet.
Reviewer: R.Heersink
MSC:
35A25 | Other special methods applied to PDEs |
35C05 | Solutions to PDEs in closed form |
35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
26A33 | Fractional derivatives and integrals |
Keywords:
operators of fractional integration; inversion formula; representation of solutions; generalized biaxial-symmetric Helmholtz equation; Kontorovich- Lebedev-TransformationReferences:
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