Resolvents for weakly singular kernels and fractional differential equations

Abstract

The solution R(t,s) of the resolvent equation (1)R(t,s)=B(t,s)+∫stB(t,u)R(u,s)du, 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)=λ(t−s)q−1 of the scalar Abel integral equation of the second kind (2)x(t)=a(t)+λ∫0t1(t−s)1−qx(s)ds 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)=λs(t2−s2)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.