TY  - JOUR
T1  - Approximate Analytic Solution for Fractional Differential Equations with a Generalized Fractional Derivative of Caputo-Type
AU  - Alomarı, Abedel-karrem
AU  - Alsleeby, Anwar
PY  - 2023
DA  - December
DO  - 10.55549/epstem.1403008
JF  - The Eurasia Proceedings of Science Technology Engineering and Mathematics
JO  - EPSTEM
PB  - ISRES Publishing
WT  - DergiPark
SN  - 2602-3199
SP  - 50
EP  - 58
VL  - 25
LA  - en
AB  - 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.
KW  - Fractional calculus
KW  - Homotopy analysis method
KW  - Riccati equation
CR  - Almeida, R., Malinowska, A. B., & Odzijewicz, T. (2016). Fractional differential equations with dependence on the Caputo–Katugampola derivative. Journal of Computational and Nonlinear Dynamics,11(6), 061017
CR  - Cang, J., Tan, Y., Xu, H., & Liao, S. J. (2009). Series solutions of non-linear Riccati differential equations with fractional order. Chaos, Solitons & Fractals, 40(1), 1-9.
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Kilbas, A. A., Srivastava, H. M., & Trujillo, J. J. (2006). Theory and applications of fractional differential equations (Volume 204). Elsevier.‏
UR  - https://doi.org/10.55549/epstem.1403008
L1  - https://dergipark.org.tr/en/download/article-file/3591318
ER  -