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

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