On solutions of fractional Riccati differential equations
Sakar et al. Advances in Difference Equations
On solutions of fractional Riccati differential equations
Mehmet Giyas Sakar 2
Ali Akgül 0
Dumitru Baleanu 1 3
0 Department of Mathematics, Art and Science Faculty, Siirt University , Siirt, 56100 , Turkey
1 Department of Mathematics and Computer Sciences, Art and Science Faculty, Çankaya University , Ankara, 06300 , Turkey
2 Department of Mathematics, Faculty of Sciences, Yuzuncu Yil University , Van, 65080 , Turkey
3 Department of Mathematics, Institute of Space Sciences , Bucharest , Romania
We apply an iterative reproducing kernel Hilbert space method to get the solutions of fractional Riccati differential equations. The analysis implemented in this work forms a crucial step in the process of development of fractional calculus. The fractional derivative is described in the Caputo sense. Outcomes are demonstrated graphically and in tabulated forms to see the power of the method. Numerical experiments are illustrated to prove the ability of the method. Numerical results are compared with some existing methods.
iterative reproducing kernel Hilbert space method; inner product; fractional Riccati differential equation; analytic approximation
1 Introduction
In this work, we present an iterative reproducing kernel Hilbert spaces method (IRKHSM)
for investigating the fractional Riccati differential equation of the following form [, ]:
cDα+u(η) = p(η)u(η) + q(η)u(η) + r(η),
with the initial condition
u() = ,
The Riccati differential equation is named after the Italian nobleman Count Jacopo
Francesco Riccati (-). The book of Reid [] includes the main theories of
Riccati equation, with implementations to random processes, optimal control, and diffusion
problems [].
Fractional Riccati differential equations arise in many fields, although discussions on
the numerical methods for these equations are rare. Odibat and Momani [] investigated
a modified homotopy perturbation method for fractional Riccati differential equations.
Khader [] researched the fractional Chebyshev finite difference method for fractional
Riccati differential equations. Li et al. [] have solved this problem by quasi-linearization
technique.
There has been much attention in the use of reproducing kernels for the solutions to
many problems in the recent years [, ]. Those papers show that this method has many
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outstanding advantages []. Cui has presented the Hilbert function spaces. This useful
framework has been utilized for obtaining approximate solutions to many nonlinear
problems []. Convenient references for this method are [–].
This paper is arranged as follows. Reproducing kernel Hilbert space theory is given in
Section . Implementation of the IRKHSM is shown in Section . Exact and approximate
solutions of the problems are presented in Section . Some numerical examples are given
in Section . A summary of the results of this investigation is given in Section .
2 Preliminaries
The fractional derivative has good memory influences compared with the ordinary
calculus. Fractional differential equations are attained in model problems in fluid flow,
viscoelasticity, finance, engineering, and other areas of implementations.
x ∂n y(r)
(n – α) ∂xn (x – r)n–α dr,
We need the following properties:
(i) Jα+cDα+y(x) = y(x) –
3 Reproducing kernel functions
We describe the notion of reproducing kernel Hilbert spaces, show some particular
instances of these spaces, which will play an important role in this work, and define some
well-known properties of these spaces in this section.
Definition . The inner product space W[, T ] is presented as []
f ∈ L[, T ], f () = ,
where L[, T ] = {f | T f (t) dt < ∞},
f (t), g(t) W[,T] = f ()g() + f (T )g(T ) +
f (t)g (t) dt
f W =
f , f W , f , g ∈ W[, T ],
are the inner product and norm in W[, T ].
Theorem . W[, T ] is an RKHS. There exist Rx(t) ∈ W[, T ] for any f (t) ∈ W[, T ]
and each fixed x ∈ [, T ], t ∈ [, T ], such that f (t), Rx(t) W = f (x). The reproducing kernel
Rx(t) can be written as []
Definition . W[, T ] is given as []
f (t), g(t) W[,T] = f ()g() +
f (t)g (t) dt,
f W =
f , f W , f , g ∈ W[, T ],
are the inner product and norm in W[, T ].
W[, T ] is am RKHS, and its reproducing kernel function is obtained as
4 Solutions to the fractional Riccati differential equations in RKHS
The solution of ()-() has been obtained in the RKHS W[, T ]. To get through with the
problem, we investigate equation () as
i= k=
i= k=
βik (...truncated)