Numerical inversion of Abel’s equation with unbounded right-hand side. (English. Russian original) Zbl 0842.65093
J. Math. Sci., New York 74, No. 6, 1342-1347 (1995); translation from Issled. Prikl. Mat. 20, 74-83 (1992).
On the closed interval \([a,b]\) of the real line we consider Abel’s integral equation
\[
\int^x_a {\varphi(t)dt\over (x- t)^\alpha}= {g(x)\over (x- a)^{\gamma_a}(b- x)^{\gamma_b}},\quad 0< \alpha< 1, \max(\gamma_a, \gamma_b)< \alpha, \min (\gamma_a, \gamma_b)> 0,\tag{1}
\]
where \(g(x)\) is a given real-valued function satisfying a Lipschitz condition: \(|g(x)- g(t)|\leq A|x- t|^\mu\), \(x, t\in [a, b]\), and
\[
\mu> 1- \alpha.\tag{2}
\]
We obtain an inversion formula for equation (1) that does not involve differentiation. We show that under a certain strengthening of the requirement (2) the resulting formula assumes a form convenient for numerical implementation.
MSC:
65R20 | Numerical methods for integral equations |
45E10 | Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) |
Keywords:
numerical inversion; unbounded right-hand side; Abel’s integral equation; inversion formulaReferences:
[1] | A. V. Maister, ”On approximation by polygonal lines in solving Abel’s equation,”Issled. po Prikl. Mat., No. 18 (1990). · Zbl 0842.65092 |
[2] | A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev,Integrals and Series, Gordon and Breach, New York (1986). · Zbl 0606.33001 |
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