On solutions of fractional Riccati differential equations

Advances in Difference Equations, Feb 2017

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.

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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 [, ]: cDα+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 © The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. 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) Jα+cDα+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)


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Mehmet Sakar, Ali Akgül, Dumitru Baleanu. On solutions of fractional Riccati differential equations, Advances in Difference Equations, 2017, pp. 39, 2017, DOI: 10.1186/s13662-017-1091-8