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Zbl 0823.45002
Kilbas, Anatoly A.; Saigo, Megumi
On solution of integral equation of Abel-Volterra type. (English)
[J] Differ. Integral Equ. 8, No.5, 993-1011 (1995). ISSN 0893-4983

The authors study asymptotic expansions of the solution of the equation $$\varphi (x) = a(x) \int\sb 0\sp x (x - t)\sp{\alpha - 1} \varphi (t) dt + f(x),$$ as $x \to 0 +$, where $0 < \alpha < 1$. It is assumed that $a$ and $f$ have expansions of the form $a(x) \sim \sum\sb{k = - 1}\sp \infty a\sb k x\sp{\alpha k}$, $f(x) \sim \sum\sb{k = - 1}\sp \infty f\sb k x\sp{\alpha k}$, and it is shown that then the solution has a similar expansion $\varphi (x) \sim \sum\sb{k = - 1}\sp \infty \varphi\sb k x\sp{\alpha k}$ when $x \to 0 +$. Recursive formulas for calculating the coefficients $\varphi\sb k$ are given. The case where $a(x) = ax\sp{\alpha m}$, $m = - 1,0, \dots$, and where one can find an explicit expression for the solution is studied, too.
[G.Gripenberg (Helsinki)]

MSC 2000:: *45E10 Integral equations of the convolution type
45D05 Volterra integral equations
41A60 Asymptotic problems in approximation

Keywords: integral equation of Abel-Volterra kind; asymptotic expansions

Cited in: Zbl 1246.39005 Zbl 0935.33012

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