Higher-Order Numerical Solutions of the Fractional Relaxation-Oscillation Equation using Fractional Integration

In the present paper we derive the asymptotic expansion formula for the trapezoidal approximation of the fractional integral. We use the expansion formula to obtain approximations for the fractional integral of order $\alpha ,1+\alpha ,2+\alpha ,3+\alpha$ and $4+\alpha$. The approximations are applied for computing the numerical solutions of the fractional relaxation-oscillation equation.