Duan, Jun-Sheng Time- and space-fractional partial differential equations. (English) Zbl 1076.26006 J. Math. Phys. 46, No. 1, 013504, 8 p. (2005). Summary: The fundamental solution for time- and space-fractional partial differential operator \(D_t^\lambda+a^2(-\Delta)^{\gamma/2}\) (\(\lambda\), \(\gamma > 0\)) is given in terms of the Fox’s \(H\)-function. Here the time-fractional derivative in the sense of generalized functions (distributions) \(D_t^\lambda\) is defined by the convolution \(D_t^\lambda f(t)=-\Phi_\lambda (t)^*f(t)\), where \(\Phi_\lambda(t)=t_+^{\lambda-1}/\Gamma(\lambda)\) and \(f(t) \equiv 0\) as \(t<0\), and the fractional \(n\)-dimensional Laplace operator \((-\Delta)^{\gamma/2}\) is defined by its Fourier transform with respect to spatial variable \(\mathcal {F}[(-\Delta)^{\gamma/2} g(x)] = |\omega|^{\gamma}\mathcal {F}[g(x)]\). The solutions for initial value problems for time- and space-fractional partial differential equation in the sense of Caputo and Riemann-Liouville time-fractional derivatives, respectively, are obtained by the fundamental solution. Cited in 21 Documents MSC: 26A33 Fractional derivatives and integrals 35S10 Initial value problems for PDEs with pseudodifferential operators 44A30 Multiple integral transforms 45K05 Integro-partial differential equations PDF BibTeX XML Cite \textit{J.-S. Duan}, J. Math. Phys. 46, No. 1, 013504, 8 p. (2005; Zbl 1076.26006) Full Text: DOI OpenURL References: [1] Braaksma B. L. J., Compos. Math. 15 pp 239– (1964) [2] DOI: 10.1006/jmaa.1998.6078 · Zbl 0932.58038 [3] Chazarain J., Introduction to the Theory of Linear Partial Differential Equations (1982) · Zbl 0487.35002 [4] DOI: 10.1103/PhysRevE.53.4191 [5] DOI: 10.1007/978-1-4757-5512-1 [6] Fox C., Trans. Am. Math. Soc. 98 pp 395– (1961) [7] Fujita Y., Osaka Math. J. 27 pp 309– (1990) [8] Gel’fand I. M., Generalized Functions (1964) [9] DOI: 10.1007/BF01058445 · Zbl 0945.82559 [10] DOI: 10.1016/S0377-0427(00)00288-0 · Zbl 0973.35012 [11] DOI: 10.1007/BFb0084665 · Zbl 0717.45002 [12] DOI: 10.1016/S0378-4371(99)00469-0 [13] DOI: 10.1021/jp9936289 [14] DOI: 10.1016/0893-9659(96)00089-4 · Zbl 0879.35036 [15] DOI: 10.1016/0960-0779(95)00125-5 · Zbl 1080.26505 [16] Mathai A. M., The H-function with Applications in Statistics and Other Disciplines (1978) · Zbl 0382.33001 [17] DOI: 10.1016/0378-4371(94)90064-7 [18] Podlubny I., Fractional Differential Equations (1999) · Zbl 0924.34008 [19] DOI: 10.1016/S0375-9601(00)00518-1 · Zbl 1050.82545 [20] DOI: 10.1063/1.528578 · Zbl 0692.45004 [21] Srivastava H. M., The H-functions of One and Two Variables with Applications (1982) · Zbl 0506.33007 [22] DOI: 10.1103/PhysRevE.55.99 [23] DOI: 10.1063/1.527251 · Zbl 0632.35031 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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.