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
1 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
References
- Reid, WT: Riccati Differential Equations. Math. Sci. Eng., vol. 86. Academic Press, New York (1972) MATHGoogle Scholar
- Khader, MM: Numerical treatment for solving fractional Riccati differential equation. J. Egypt. Math. Soc. 21, 32-37 (2013) MathSciNetView ArticleMATHGoogle Scholar
- Odibat, Z, Momani, S: Modified homotopy perturbation method: application to quadratic Riccati differential equation of fractional order. Chaos Solitons Fractals 36(1), 167-174 (2008) MathSciNetView ArticleMATHGoogle Scholar
- Khader, MM: Numerical treatment for solving fractional Riccati differential equation. J. Egypt. Math. Soc. 21(1), 32-37 (2013) MathSciNetView ArticleMATHGoogle Scholar
- Li, XY, Wu, BY, Wang, RT: Reproducing kernel method for fractional Riccati differential equations. Abstr. Appl. Anal. 2014, Article ID 970967 (2014) MathSciNetGoogle Scholar
- Takeuchi, T, Yamamoto, M: Tikhonov regularization by a reproducing kernel Hilbert space for the Cauchy problem for an elliptic equation. SIAM J. Sci. Comput. 31, 112-142 (2008) MathSciNetView ArticleMATHGoogle Scholar
- Hon, YC, Takeuchi, T: Discretized Tikhonov regularization by reproducing kernel Hilbert space for backward heat conduction problem. Adv. Comput. Math. 34, 167-183 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Wang, W, Yamamoto, M, Han, B: Numerical method in reproducing kernel space for an inverse source problem for the fractional diffusion equation. Inverse Probl. 29, 095009 (2013) MathSciNetView ArticleMATHGoogle Scholar
- Ghasemi, M, Fardi, M, Ghaziani, RK: Numerical solution of nonlinear delay differential equations of fractional order in reproducing kernel Hilbert space. Appl. Math. Comput. 268, 815-831 (2015) MathSciNetGoogle Scholar
- Inc, M, Akgül, A: The reproducing kernel Hilbert space method for solving Troesch’s problem. J. Assoc. Arab Univ. Basic Appl. Sci. 14, 19-27 (2013) Google Scholar
- Inc, M, Akgül, A: Approximate solutions for MHD squeezing fluid flow by a novel method. Bound. Value Probl. 2014, 18 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Inc, M, Akgül, A, Geng, F: Reproducing kernel Hilbert space method for solving Bratu’s problem. Bull. Malays. Math. Soc. 38, 271-287 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Inc, M, Akgül, A, Kilicman, A: Numerical solutions of the second order one-dimensional telegraph equation based on reproducing kernel Hilbert space method. Abstr. Appl. Anal. 2013, 13 (2013) MathSciNetGoogle Scholar
- Podlubny, I: Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Mathematics in Science and Engineering, vol. 198, Academic Press, San Diego (1999). ISBN:0-12-558840-2 MATHGoogle Scholar
- Sakar, MG: Iterative reproducing kernel Hilbert spaces method for Riccati differential equations. J. Comput. Appl. Math. 309, 163-174 (2017) MathSciNetView ArticleMATHGoogle Scholar
- Yüzbaşı, Ş: Numerical solutions of fractional Riccati type differential equations by means of the Bernstein polynomials. Appl. Math. Comput. 219, 6328-6343 (2013) MathSciNetMATHGoogle Scholar
- Hosseinnia, SH, Ranjbar, A, Momani, S: Using an enhanced homotopy perturbation method in fractional differential equations via deforming the linear part. Comput. Math. Appl. 56, 3138-3149 (2008) MathSciNetView ArticleMATHGoogle Scholar
- Batiha, B, Noorani, MSM, Hashim, I: Application of variational iteration method to a general Riccati equation. Int. Math. Forum 2, 2759-2770 (2007) MathSciNetView ArticleMATHGoogle Scholar
- Mabood, F, Ismail, AI, Hashim, I: Application of optimal homotopy asymptotic method for the approximate solution of Riccati equation. Sains Malays. 42, 863-867 (2013) MATHGoogle Scholar
- Li, YL: Solving a nonlinear fractional differential equation using Chebyshev wavelets. Commun. Nonlinear Sci. Numer. Simul. 15, 2284-2292 (2010) MathSciNetView ArticleMATHGoogle Scholar