×

Nonlocal problem for partial differential equations of fractional order. (Russian. English summary) Zbl 1413.35336

Summary: A nonlocal problem is investigated for the partial differential equation (diffusion equation of fractional order) in a finite domain. The boundary condition contains a linear combination of generalized operators of fractional integro-differentiation used on the solution in the characteristics and the solution and its derivative in the degenerating line. The uniqueness of the solution is proved by a modified Tricomi method. The existence of the solution is equivalently reduced to the question of the solvability of Fredholm integral equations of the second kind.

MSC:

35M12 Boundary value problems for PDEs of mixed type
PDFBibTeX XMLCite
Full Text: DOI MNR

References:

[1] [1] Samko St. G., Kilbas A. A., Marichev O. I., Fractional integrals and derivatives: theory and applications, Gordon and Breach, New York, NY, 1993, xxxvi+976 pp. · Zbl 0818.26003
[2] [2] Saigo M., “A remark on integral operators involving the Gauss hypergeometric function”, Math. Rep. Coll. Gen. Educ., Kyushu Univ., 11:2 (1978), 135–143 · Zbl 0399.45022
[3] [3] Repin O. A., Kraevye zadachi so smeshcheniem dlia uravnenii giperbolicheskogo i smeshannogo tipov [Boundary value problems with shift for equations of hyperbolic and mixed type], Saratov, 1992, 164 pp. (In Russian) · Zbl 0867.35002
[4] [4] Kochubei A. N., “Fractional-order diffusion”, Differ. Equ., 26:4 (1990), 485–492 · Zbl 0729.35064
[5] [5] Bitsadze A. N., Nekotorye klassy uravnenii v chastnykh proizvodnykh [Some Classes of Partial Differential Equations], Nauka, Moscow, 1981, 448 pp. (In Russian) · Zbl 0511.35001
[6] [6] Gekkieva S. Kh., “An analog of the Tricomi problem for a mixed type equation with a partial fractional derivative”, Izvestiya Kabardino-Balkarskaya Nauchnoogo Tsentra RAN, 2001, no. 2(7), 78–80 (In Russian)
[7] [7] Kilbas A. A., Repin O. A., “Solvability of a boundary value problem for a mixed-type equation with a partial Riemann–Liouville fractional derivative”, Differ. Equ., 46:10 (2010), 1457–1464 · Zbl 1213.35318 · doi:10.1134/S0012266110100095
[8] [8] Repin O. A., Kumykova S. K., “A nonlocal problem with fractional derivatives for the mixed type equation”, Russian Math. (Iz. VUZ), 58:8 (2014), 65–70 · Zbl 1302.35411 · doi:10.3103/S1066369X14080088
[9] [9] Repin O. A., Kumykova S. K., “On a boundary value problem with shift for an equation of mixed type in an unbounded domain”, Differ. Equ., 48:8 (2012), 1127–1136 · Zbl 1260.35080 · doi:10.1134/S0012266112080083
[10] [10] Repin O. A., Kumykova S. K., “A nonlocal problem for a mixed-type equation whose order degenerates along the line of change of type”, Russian Math. (Iz. VUZ), 57:8 (2013), 49–56 · Zbl 1286.35173 · doi:10.3103/S1066369X13080069
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.