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Theoretical and Computational Aspects of Fractional Hybrid Differential Equations |
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PP: 119-133 |
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doi:10.18576/pfda/090109
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Author(s) |
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Habibulla Akhadkulov,
Ali Fareed Jameel,
Teh Yuan Ying,
Sokhobiddin Akhatkulov,
Abdel-Karrem Alomari,
Dulfikar Jawad Hashim,
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Abstract |
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| This paper is devoted to study the existence and uniqueness of a solution for the following fractional hybrid differential equations defined by Riemann-Liouville differential operator order of 0 < α < 1
http://dx.doi.org/10.18576/pfda/090109
Dα x(t)−f(t,x(t))=ft,x(t), a.et∈J, t0+ 1 2
(1.0) whereDα istheRiemann-Liouvillefractionalderivativeorderof0<α<1,J=[t ,t +a],forsomet ∈R,a>0, f (·,x)∈Cα(J,R)
x(t0) = x0 ∈ R,
t0+ α 00 0 1
for all x ∈ R and f2 ∈ L (J × R, R). We prove the existence and uniqueness of a solution of the equation (1.0) by using a coupled fixed point theorem. This result extends the existence theorems of [1,2,3,4]. Moreover, we investigate Picard iterations of an operator T defined on a space of continuous functions under two different weak construction conditions. It is shown that Picard iterations of T converge to the unique fixed point if the weak contraction function is a tangent hyperbolic function. If the weak contraction is a fractional linear function then Picard iterations of T converge to the unique fixed point with an algebraic rate. Finally, we investigate approximate solutions of fractional hybrid differential equations via the homotopy analysis method. |
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