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

Research output

Abstract

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.

Original languageEnglish
Pages (from-to)1161-1165
Number of pages5
JournalJournal of Mathematical Sciences
Volume79
Issue number4
DOIs
Publication statusPublished - 1 Jan 1996

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Integrodifferential equations
Convolution
Integro-differential Equation
Logarithmic
Singularity
kernel
Interval
Existence and Uniqueness of Solutions
Argand diagram
Differentiable
Smoothness
Infinity
Projection
Form
Class

Scopus subject areas

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

Cite this

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abstract = "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.",
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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.

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