A variable step size numerical method based on fractional type quadratures for linear integro-differential equations

In this work a discretization in time of variable step size of the convolution equation u(t)=∫0t(t-s)α-1u(s)ds, based on a fractional type quadrature is studied. The convergence is directly proved thanks to a suitable representation of the error by means of Peano kernels. Practical illustrations showing the efficiency of our numerical scheme are provided.