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A note on generalized Abel equations with constant coefficients. (English) Zbl 1487.45007

Summary: The generalized Abel equation (GAE) with constant coefficients, \(\gamma_1 \, {}_aD_x^{-\sigma} u + \gamma_{2} \, {}_xD_b^{-\sigma} u = f\), \(0 < \sigma < 1\), has renewed interest since it has appeared as a component in double-sided fractional diffusion equations. Motivated by direct applications in modeling, we seek the necessary and sufficient conditions for interchanging the order of differentiation and fractional integration in GAEs. Put another way, assume functions \(u, v, f\) lie in Hölder spaces that admit integrable singularities at the endpoints and \[ \gamma_1 \, {}_aD_x^{-\sigma} u + \gamma_2 \, {}_xD_b^{-\sigma} u = f\quad \text{ and }\quad \gamma_1 \, {}_aD_x^{-\sigma} v + \gamma_2 \, {}_xD_b^{-\sigma} v = Df, \] then we prove that \(Du = v\) if and only if \(u(a) = u(b) = 0\).

MSC:

45J05 Integro-ordinary differential equations
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
26A33 Fractional derivatives and integrals
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