On solutions of fractional Riccati differential equations
- Mehmet Giyas Sakar^{1},
- Ali Akgül^{2}Email authorView ORCID ID profile and
- Dumitru Baleanu^{3, 4}
https://doi.org/10.1186/s13662-017-1091-8
© The Author(s) 2017
Received: 11 August 2016
Accepted: 18 January 2017
Published: 3 February 2017
Abstract
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.
Keywords
iterative reproducing kernel Hilbert space method inner product fractional Riccati differential equation analytic approximation1 Introduction
The Riccati differential equation is named after the Italian nobleman Count Jacopo Francesco Riccati (1676-1754). The book of Reid [1] includes the main theories of Riccati equation, with implementations to random processes, optimal control, and diffusion problems [2].
Fractional Riccati differential equations arise in many fields, although discussions on the numerical methods for these equations are rare. Odibat and Momani [3] investigated a modified homotopy perturbation method for fractional Riccati differential equations. Khader [4] researched the fractional Chebyshev finite difference method for fractional Riccati differential equations. Li et al. [5] 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 [6, 7]. Those papers show that this method has many outstanding advantages [8]. Cui has presented the Hilbert function spaces. This useful framework has been utilized for obtaining approximate solutions to many nonlinear problems [9]. Convenient references for this method are [10–13].
This paper is arranged as follows. Reproducing kernel Hilbert space theory is given in Section 2. Implementation of the IRKHSM is shown in Section 3. Exact and approximate solutions of the problems are presented in Section 4. Some numerical examples are given in Section 5. A summary of the results of this investigation is given in Section 6.
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.
Definition 2.1
Definition 2.2
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 3.1
Definition 3.2
Theorem 3.1
Definition 3.3
4 Solutions to the fractional Riccati differential equations in RKHS
Theorem 4.1
If \(\{ {\eta_{i} } \}_{i = 1}^{\infty}\) is dense on \([ {0, T} ]\), then \(\{ {\psi_{i} ( \eta )} \}_{i = 1}^{\infty}\) is a complete system of \(W_{2}^{2} [ {0, T} ]\), and we have \(\psi_{i} ( \eta ) = {L_{t} R_{\eta}( t )} | _{t = \eta_{i} } \).
Proof
Theorem 4.2
Proof
Remark 4.1
We notice the following two cases in order to solve equations (1)-(2) by using RKHS.
Case 1: If equation (1) is linear, that is, \(p(\eta )=0\), then an approximate solution can be obtained directly from equation (12) .
Theorem 4.3
If \(u(\eta)\in W_{2}^{2}[0,T]\), then there exist \(E > 0\) and \(\Vert u(\eta)\Vert _{C[0,T]} = {\max} _{\eta\in[0,T]}\vert u(\eta)\vert \) such that \(\Vert u^{(i)}(\eta)\Vert _{C[0,T]}\le E\Vert u(\eta )\Vert _{W_{2}^{2}[0,T]}\), \(i =0,1\).
Proof
We obtain \(u^{(i)} ( \eta ) = \langle u ( t ),\partial_{\eta}^{i} R_{\eta}( t )\rangle_{W_{2}^{2}[0,T] } \) for any η, \(t \in [ {0, T} ]\) and \(i = 0, 1\). Then, we get \(\Vert {\partial_{\eta}^{i} R_{\eta}( t )} \Vert _{W_{2}^{2}[0,T] } \le E_{i} \), \(i = 0,1\), by \(R_{\eta}( t )\).
Thus, we get \(\Vert {u^{(i)} ( \eta )} \Vert _{C[0,T]} \le\max\{E_{0} , E_{1} \}\Vert {u ( \eta )} \Vert _{W_{2}^{2}[0,T] }\) for \(i=0, 1\). This completes the proof. □
Theorem 4.4
The approximate solution \(u_{n} ( \eta )\) and its first derivative \({u}'_{n} ( \eta )\) are uniformly convergent in \([0,T]\).
Proof
5 Numerical examples
To give a clear overview of this technique, we give the following informative examples. All of the computations have been applied by utilizing the Maple software package. The results attained by the method are compared with the exact solution of each example and are found to be in good agreement.
Example 5.1
Comparison of IRKHSM solution with other methods for Example 5.1 ( \(\pmb{\alpha=1}\) )
\(\boldsymbol{\eta_{i}}\) | Exact Sol. | IRKHSM | Method in [ 5 ] | Method in [ 16 ] | MHPM [ 3 ] |
---|---|---|---|---|---|
0.2 | 0.197375 | 0.197375 | 0.19738 | 0.197375 | 0.197375 |
0.4 | 0.379949 | 0.379949 | 0.379956 | 0.379948 | 0.379944 |
0.6 | 0.537049 | 0.537049 | 0.537061 | 0.537049 | 0.536857 |
0.8 | 0.664037 | 0.664037 | 0.664053 | 0.664036 | 0.661706 |
1.0 | 0.761594 | 0.761614 | 0.761618 | 0.761594 | 0.746032 |
Comparison of IRKHSM solution with other methods for Example 5.1 ( \(\pmb{\alpha=0.9}\) )
Comparison of IRKHSM solution with other methods for Example 5.1 ( \(\pmb{\alpha=0.75}\) )
Example 5.2
Comparison of absolute errors for some methods for Example 5.2 ( \(\pmb{\alpha=1}\) )
Comparison of IRKHSM solution with other methods for Example 5.2 ( \(\pmb{\alpha=0.9}\) )
Comparison of IRKHSM solution with other methods for Example 5.2 ( \(\pmb{\alpha=0.75}\) )
Example 5.3
when \(\alpha=1\). Using IRKHSM for equations (24)-(25) and taking \(T=1\), \(\eta_{i} = \frac{i}{N}\), \(i = 1, 2,\ldots,N\), the numerical solution \(u_{n}^{N} ( \eta )\) is computed. Comparison of the IRKHSM solutions \(u_{3}^{4}(\eta)\) for Example 5.3 with different values of α is given in Figure 1. The absolute error of IRKHSM solution \(u_{3}^{4}(\eta)\) for Example 5.3 with \(\alpha=1\) is shown in Figure 2.
6 Conclusion
IRKHSM were successfully implemented to get approximate solutions of the fractional Riccati differential equations. Numerical results were compared with the existing methods to prove the efficiency of the method. The IRKHSM is very powerful and accurate in obtaining approximate solutions for wide classes of the problem. The approximate solution attained by the IRKHSM is uniformly convergent. The series solution methodology can be implemented to much more complicated nonlinear equations. This method can be extended to solve the other fractional differential equations.
Declarations
Acknowledgements
The authors would like to thank Mehmet EFE from Siirt University for his kind contribution.
Open Access 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.
Authors’ Affiliations
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