A new modified definition of Caputo-fabrizio fractional-order derivative and their applications to the multi step homotopy analysis method (MHAM). (English) Zbl 1402.26005
Summary: In this paper, we present a new definition of fractional-order derivative with a smooth kernel based on the Caputo-Fabrizio fractional-order operator which takes into account some problems related with the conventional Caputo-Fabrizio factional-order derivative definition. The Modified-Caputo-Fabrizio fractional-order derivative here introduced presents some advantages when some approximated analytical methods are applied to solve non-linear fractional differential equations. We consider two approximated analytical methods to find analytical solutions for this novel operator; the homotopy analysis method (HAM) and the multi step homotopy analysis method (MHAM). The results obtained suggest that the introduction of the Modified-Caputo-Fabrizio fractional-order derivative can be applied in the future to many different scenarios in fractional dynamics.
MSC:
| 26A33 | Fractional derivatives and integrals |
| 34A08 | Fractional ordinary differential equations |
| 34A45 | Theoretical approximation of solutions to ordinary differential equations |
| 65L99 | Numerical methods for ordinary differential equations |
Keywords:
fractional calculus; homotopy analysis method; multistep homotopy analysis method; modified-Caputo-Fabrizio derivativeReferences:
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