Approximations of the Fractional Integral and Numerical Solutions of Fractional Integral Equations

Yuri Dimitrov

doi:10.1007/s42967-021-00132-7

Communications on Applied Mathematics and Computation ›› 2021, Vol. 3 ›› Issue (3) : 545 -569. DOI: 10.1007/s42967-021-00132-7

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Approximations of the Fractional Integral and Numerical Solutions of Fractional Integral Equations

Yuri Dimitrov ¹^,²^,a

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Abstract

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 orders $\alpha ,1+\alpha ,2+\alpha ,3+\alpha $ and $4+\alpha $. The approximations are applied for computation of the numerical solutions of the ordinary fractional relaxation and the fractional oscillation equations expressed as fractional integral equations.

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Yuri Dimitrov. Approximations of the Fractional Integral and Numerical Solutions of Fractional Integral Equations. Communications on Applied Mathematics and Computation, 2021, 3(3): 545-569 DOI:10.1007/s42967-021-00132-7