Spectral-collocation method for fractional Fredholm integro-differential equations (Q2870483)

The authors construct an approximate scheme to solve the fractional order integro-differential equation of the form NEWLINE\[NEWLINE D^{\gamma}y(t)=y(t) + \int_{0}^{t} k_{1}(t,\tau)y(\tau)d\tau +\int_{0}^{T}k_{2}(t,\zeta)y(\zeta)d\zeta +f(t),NEWLINE\]NEWLINE NEWLINE\[NEWLINE y(0)=y_{0}, \qquad 0<\gamma <1,\qquad t\in[0,T], NEWLINE\]NEWLINE in the space \( L^{2}_{\omega}(-1,1)\).NEWLINENEWLINEIn this equation the integrals are approximated using the Gauss quadrature formula with the collocation points as the roots of the Jacobi polynomials. The convergence of the method is proved and three numerical examples illustrate the theoretical predictions.