### Abstract

The integro-differential equations d^{2n}/dx^{2n} ∫_{-1} ^{1} (a[(x - t)^{2}] ln |x - t| + b[(x - t)^{2}])φ(t)dt = f(x) of the convolution on an interval with infinitely differentiable functions a(s) and b(s) decreasing at infinity are considered. The Fourier symbol is assumed to be sectorial, that is, it has positive projection on some direction in the complex plane. The existence and uniqueness of solutions in the classes of functions representable in the form φ(t) = (1 - t^{2})^{δn} φ(t), δ_{n} = n - 1 + ε, ε < 0, φ ∈ C^{1} [-1, 1] are proved. Properties concerning the smoothness of solutions are described.

Original language | English |
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Pages (from-to) | 1161-1165 |

Number of pages | 5 |

Journal | Journal of Mathematical Sciences |

Volume | 79 |

Issue number | 4 |

DOIs | |

Publication status | Published - 1 Jan 1996 |

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### Scopus subject areas

- Statistics and Probability
- Mathematics(all)
- Applied Mathematics

### Cite this

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**Integro-differential equations of the convolution on a finite interval with kernel having a logarithmic singularity.** / Andronov, I. V.

Research output

TY - JOUR

T1 - Integro-differential equations of the convolution on a finite interval with kernel having a logarithmic singularity

AU - Andronov, I. V.

PY - 1996/1/1

Y1 - 1996/1/1

N2 - The integro-differential equations d2n/dx2n ∫-1 1 (a[(x - t)2] ln |x - t| + b[(x - t)2])φ(t)dt = f(x) of the convolution on an interval with infinitely differentiable functions a(s) and b(s) decreasing at infinity are considered. The Fourier symbol is assumed to be sectorial, that is, it has positive projection on some direction in the complex plane. The existence and uniqueness of solutions in the classes of functions representable in the form φ(t) = (1 - t2)δn φ(t), δn = n - 1 + ε, ε < 0, φ ∈ C1 [-1, 1] are proved. Properties concerning the smoothness of solutions are described.

AB - The integro-differential equations d2n/dx2n ∫-1 1 (a[(x - t)2] ln |x - t| + b[(x - t)2])φ(t)dt = f(x) of the convolution on an interval with infinitely differentiable functions a(s) and b(s) decreasing at infinity are considered. The Fourier symbol is assumed to be sectorial, that is, it has positive projection on some direction in the complex plane. The existence and uniqueness of solutions in the classes of functions representable in the form φ(t) = (1 - t2)δn φ(t), δn = n - 1 + ε, ε < 0, φ ∈ C1 [-1, 1] are proved. Properties concerning the smoothness of solutions are described.

UR - http://www.scopus.com/inward/record.url?scp=53349118858&partnerID=8YFLogxK

U2 - 10.1007/BF02362880

DO - 10.1007/BF02362880

M3 - Article

AN - SCOPUS:53349118858

VL - 79

SP - 1161

EP - 1165

JO - Journal of Mathematical Sciences

JF - Journal of Mathematical Sciences

SN - 1072-3374

IS - 4

ER -