Fractional BVPs with strong time singularities and the limit properties of their solutions

Svatoslav Staněk

Open Mathematics, Tome 12 (2014), p. 1638-1655 / Harvested from The Polish Digital Mathematics Library

Résumé

In the first part, we investigate the singular BVP ${\frac{d}{d t}}^{c} D^{α} u + {(a / t)}^{c} D^{α} u = ℋ u$ , u(0) = A, u(1) = B, c D α u(t)|t=0 = 0, where $ℋ$ is a continuous operator, α ∈ (0, 1) and a < 0. Here, c D denotes the Caputo fractional derivative. The existence result is proved by the Leray-Schauder nonlinear alternative. The second part establishes the relations between solutions of the sequence of problems ${\frac{d}{d t}}^{c} D^{α_{n}} u + {(a / t)}^{c} D^{α_{n}} u = f (t, u,^{c} D^{β_{n}} u)$ , u(0) = A, u(1) = B, ${^{c} D^{α_{n}} u (t)|}_{t = 0} = 0$ where a < 0, 0 < β n ≤ α n < 1, limn→∞ β n = 1, and solutions of u″+(a/t)u′ = f(t, u, u′) satisfying the boundary conditions u(0) = A, u(1) = B, u′(0) = 0.

Publié le : 2014-01-01

Zbl 1312.34025

EUDML-ID : urn:eudml:doc:269803

@article{bwmeta1.element.doi-10_2478_s11533-014-0435-9,
     author = {Svatoslav Stan\v ek},
     title = {Fractional BVPs with strong time singularities and the limit properties of their solutions},
     journal = {Open Mathematics},
     volume = {12},
     year = {2014},
     pages = {1638-1655},
     zbl = {1312.34025},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-014-0435-9}
}

Svatoslav Staněk. Fractional BVPs with strong time singularities and the limit properties of their solutions. Open Mathematics, Tome 12 (2014) pp. 1638-1655. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-014-0435-9/

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