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

Svatoslav Staněk

Open Mathematics (2014)

  • Volume: 12, Issue: 11, page 1638-1655
  • ISSN: 2391-5455

Abstract

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In the first part, we investigate the singular BVP 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 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.

How to cite

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Svatoslav Staněk. "Fractional BVPs with strong time singularities and the limit properties of their solutions." Open Mathematics 12.11 (2014): 1638-1655. <http://eudml.org/doc/269803>.

@article{SvatoslavStaněk2014,
abstract = {In the first part, we investigate the singular BVP \[\tfrac\{d\}\{\{dt\}\}^c D^\alpha u + (a/t)^c D^\alpha u = \mathcal \{H\}u\] , u(0) = A, u(1) = B, c D α u(t)|t=0 = 0, where \[\mathcal \{H\}\] 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 \[\tfrac\{d\}\{\{dt\}\}^c D^\{\alpha \_n \} u + (a/t)^c D^\{\alpha \_n \} u = f(t,u,^c D^\{\beta \_n \} u)\] , u(0) = A, u(1) = B, \[\left. \{^c D^\{\alpha \_n \} u(t)\} \right|\_\{t = 0\} = 0\] 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.},
author = {Svatoslav Staněk},
journal = {Open Mathematics},
keywords = {Singular fractional differential equation; Time singularity; Caputo fractional derivative; Dirichlet conditions; Leray-Schauder alternative; Limit properties of solutions; singular fractional differential equation; time singularity; limit properties of solutions},
language = {eng},
number = {11},
pages = {1638-1655},
title = {Fractional BVPs with strong time singularities and the limit properties of their solutions},
url = {http://eudml.org/doc/269803},
volume = {12},
year = {2014},
}

TY - JOUR
AU - Svatoslav Staněk
TI - Fractional BVPs with strong time singularities and the limit properties of their solutions
JO - Open Mathematics
PY - 2014
VL - 12
IS - 11
SP - 1638
EP - 1655
AB - In the first part, we investigate the singular BVP \[\tfrac{d}{{dt}}^c D^\alpha u + (a/t)^c D^\alpha u = \mathcal {H}u\] , u(0) = A, u(1) = B, c D α u(t)|t=0 = 0, where \[\mathcal {H}\] 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 \[\tfrac{d}{{dt}}^c D^{\alpha _n } u + (a/t)^c D^{\alpha _n } u = f(t,u,^c D^{\beta _n } u)\] , u(0) = A, u(1) = B, \[\left. {^c D^{\alpha _n } u(t)} \right|_{t = 0} = 0\] 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.
LA - eng
KW - Singular fractional differential equation; Time singularity; Caputo fractional derivative; Dirichlet conditions; Leray-Schauder alternative; Limit properties of solutions; singular fractional differential equation; time singularity; limit properties of solutions
UR - http://eudml.org/doc/269803
ER -

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