### Abstract

In this article, we consider some classes of nonlinear fractional differential equations with singularity take the form t^{α}∂^{α}u(t,z)/∂t^{α}=F(t,z,u,∂u/∂z) 0<α<1, where t ∈ J:= [0,1] and z ∈ U:= {z ∈ C: z < 1}. Our purpose is to establish a result similar to the k-summability known in the case of singular ordinary differential equations. It's shown that, under some conditions, all formal solutions are Borel summable or k-summable with respect to z ∈ U in all directions except at most a countable number.

Original language | English |
---|---|

Pages (from-to) | 339-344 |

Number of pages | 6 |

Journal | Proceedings of the Pakistan Academy of Sciences |

Volume | 50 |

Issue number | 4 |

Publication status | Published - Dec 2013 |

Externally published | Yes |

### Keywords

- Borel summable
- Fractional calculus
- Fractional differential equation
- Holomorphic solution
- Nonlinear
- Riemann-liouville operators
- Singular fractional differential equation
- Unit disk

### ASJC Scopus subject areas

- Social Sciences(all)

### Cite this

*Proceedings of the Pakistan Academy of Sciences*,

*50*(4), 339-344.

**Borel summability for fractional differential equation in the unit disk.** / Ibrahim, Rabha W.; Darus, Maslina.

Research output: Contribution to journal › Article

*Proceedings of the Pakistan Academy of Sciences*, vol. 50, no. 4, pp. 339-344.

}

TY - JOUR

T1 - Borel summability for fractional differential equation in the unit disk

AU - Ibrahim, Rabha W.

AU - Darus, Maslina

PY - 2013/12

Y1 - 2013/12

N2 - In this article, we consider some classes of nonlinear fractional differential equations with singularity take the form tα∂αu(t,z)/∂tα=F(t,z,u,∂u/∂z) 0<α<1, where t ∈ J:= [0,1] and z ∈ U:= {z ∈ C: z < 1}. Our purpose is to establish a result similar to the k-summability known in the case of singular ordinary differential equations. It's shown that, under some conditions, all formal solutions are Borel summable or k-summable with respect to z ∈ U in all directions except at most a countable number.

AB - In this article, we consider some classes of nonlinear fractional differential equations with singularity take the form tα∂αu(t,z)/∂tα=F(t,z,u,∂u/∂z) 0<α<1, where t ∈ J:= [0,1] and z ∈ U:= {z ∈ C: z < 1}. Our purpose is to establish a result similar to the k-summability known in the case of singular ordinary differential equations. It's shown that, under some conditions, all formal solutions are Borel summable or k-summable with respect to z ∈ U in all directions except at most a countable number.

KW - Borel summable

KW - Fractional calculus

KW - Fractional differential equation

KW - Holomorphic solution

KW - Nonlinear

KW - Riemann-liouville operators

KW - Singular fractional differential equation

KW - Unit disk

UR - http://www.scopus.com/inward/record.url?scp=84890942568&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84890942568&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:84890942568

VL - 50

SP - 339

EP - 344

JO - Proceedings of the Pakistan Academy of Sciences

JF - Proceedings of the Pakistan Academy of Sciences

SN - 0377-2969

IS - 4

ER -