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Inversion formula for a Volterra equation of the first kind and its application. (English. Russian original) Zbl 0894.45005
Russ. Math. 40, No. 9, 14-17 (1996); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1996, No. 9(412), 16-20 (1996).
The authors show that, under certain assumptions, the solution of the equation \[ \int_x^\infty \varphi(t)_0F_1(1;-{\mu^2\over 4}(x^\alpha-t^\alpha)^2) dt = g(x), \] where \(\mu^2 = \lambda^2/\alpha^2\) is given by \[ \varphi(x) = x^{\alpha-1}\int_x^\infty J_0(| \mu| (x^\alpha-s^\alpha) (\alpha^2\mu^2 g(s)s^{\alpha-1} + (1-\alpha)g'(s)s^{-\alpha}+ g''(s)s^{1-\alpha}) ds. \] This result is used for solving the degenerate hyperbolic equation \[ (xy)^{1-\alpha}u_{xy} + {\lambda^2\over 4} u = 0. . \]

MSC:
45D05 Volterra integral equations
35L80 Degenerate hyperbolic equations
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