Resolvents for weakly singular kernels and fractional differential equations

The solution R(t, s) of the resolvent equation R(t, s) = B(t, s) + integral(t)(s) B(t, u) R(u, s) du, (1) where B(t, s) denotes a given weakly singular matrix, is obtained by means of fixed point mappings. The result is a series that begins with some singular terms after which the remainder of the terms defines a continuous function. In particular, the resolvent is calculated for the kernel B(t, s) = lambda(t - s)(q-1) of the scalar Abel integral equation of the second kind x(t) = a(t) + lambda integral(t)(0) 1/(t - s)(1-q) x(s) ds (2) where 0 < q < 1. It is then used to derive closed-form formulas for the resolvents corresponding to q = 1/2 and 1/3. Furthermore, a closed-form formula for the resolvent for B(t, s) = lambda s(t(2) - s(2))(q-1) is derived thereby demonstrating that the results of this paper apply not only to kernels of convolution type but also to those of non-convolution type as well. Finally, the resolvent for the kernel of (2) is used to find a general solution of a linear fractional differential equation of Caputo type. (C) 2012 Elsevier Ltd. All rights reserved.