This study defines a Hadamard fractional sum by use of the time-scale theory. Then a [Formula: see text]-fractional difference is given and fundamental theorems are proved. Initial value problems of fractional difference equations are presented and their equivalent fractional sum equations are provided. The discrete Mittag-Leffler function solutions of linear fractional difference equations are obtained. It can be concluded that the new discrete fractional calculus of Hadamard type is well defined.

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Parameter estimation of uncertain differential equations becomes popular very recently. This paper suggests a new method based on fractional uncertain differential equations for the first time, which hold more parameter freedom degrees. The Adams numerical method and Adam algorithm are adopted for the optimization problems. The estimation results are compared to show a better forecast. Finally, the predictor–corrector method is adopted to solve the fractional uncertain differential equations. Numerical solutions are demonstrated with varied α-paths.

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