×

Existence conditions for a classical solution of the Cauchy problem for the diffusion-wave equation with a partial Caputo derivative. (English. Russian original) Zbl 1196.35069

Dokl. Math. 75, No. 3, 407-410 (2007); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 414, No. 4, 451-454 (2007).
From the text: We consider the Cauchy problem
\[ (^c D^\alpha_{0+,t}u)(x,t)=\lambda^2\Delta_xu(x,t),\quad x\in\mathbb R^m,\;t>0,\;\alpha>0,\;\lambda>0,\tag{1} \]
\[ \frac{\partial^ku}{\partial t^k}(x,0+)=f_k(x),\quad k=0,1,\dots,n-1,\;n=-[-\alpha],\;x\in\mathbb R^m.\tag{2} \]
We examine the existence conditions for a classical solution to problem (1), (2) when \(0 <\alpha< 2\). Specifically, our study is based on the scheme proposed in [A. N. Kochubei, Differ. Equations 26, No. 4, 485–492 (1990); translation from Differ. Uravn. 26, No. 4, 660–670 (1990; Zbl 0729.35064)] for analyzing the Cauchy problem with a regularized fractional derivative of order \(0<\alpha<1\).

MSC:

35G10 Initial value problems for linear higher-order PDEs
35S10 Initial value problems for PDEs with pseudodifferential operators
26A33 Fractional derivatives and integrals

Citations:

Zbl 0729.35064
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] I. Podlubny, Fractional Differential Equations (Academic, San Diego, 1999). · Zbl 0924.34008
[2] A. N. Kochubei, Differ. Uravn. 26, 660–670 (1990).
[3] A. A. Voroshilov and A. A. Kilbas, Differ. Equations 42, 638–649 (2006) [Differ. Uravn. 42, 599–609 (2006)]. · Zbl 1123.35302 · doi:10.1134/S0012266106050041
[4] A. A. Kilbas and M. Saigo, H-Transforms (Chapman &amp; Hall/CRC, Boca Raton, FL, 2004). · Zbl 1056.44001
[5] V. A. Zorich, Mathematical Analysis (MTsNMO, Moscow, 2002), Vol. 2 [in Russian]. · Zbl 1024.00504
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.