Approximate Analytic Solution for Fractional Differential Equations with a Generalized Fractional Derivative of Caputo-Type

Abstract

This paper introduces the analytic series solution of the differential equation with fractional Caputo-type derivative including two parameters using the homotopy analysis method (HAM). The main properties of the fractional derivative with two parameters are illustrated. The standard HAM converges for a short domain, so we modify the method to overcome this issue by dividing the domain into finite subintervals and applying the method to each one. The initial conditions in each subinterval can be obtained from the previous one In this way, a continuous piecewise function that converges to the exact solution can be constructed. The effect of each fractional parameter on the solution behaviors is presented in figures and tables. Several examples are presented to verify the validity of the algorithm. A comparison with the exact solution in the case of integer derivative and with the Adaptive predictor corrected algorithm in the case of fractional one demonstrates the efficiency of the method.

Keywords

• Fractional calculus
• Homotopy analysis method
• Riccati equation

References

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