Numerical solution of fractionalorder Riccati differential equation by differential quadrature method based on Chebyshev polynomials
 Jianhua Hou^{1}Email author and
 Changqing Yang^{1}
https://doi.org/10.1186/s1366201714096
© The Author(s) 2017
Received: 1 June 2017
Accepted: 20 October 2017
Published: 17 November 2017
Abstract
We apply the Chebyshev polynomialbased differential quadrature method to the solution of a fractionalorder Riccati differential equation. The fractional derivative is described in the Caputo sense. We derive and utilize explicit expressions of weighting coefficients for approximation of fractional derivatives to reduce a Riccati differential equation to a system of algebraic equations. We present numerical examples to verify the efficiency and accuracy of the proposed method. The results reveal that the method is accurate and easy to implement.
Keywords
1 Introduction
The fractional differential equations have received considerable interest in recent years. In many applications, fractional derivatives and fractional integrals provide more accurate models of the systems than ordinary derivatives and integrals do. Many applications of fractional differential equations in the areas of solid mechanics and modeling of viscoelastic damping, electrochemical processes, dielectric polarization, colored noise, bioengineering, and various branches of science and engineering could be found, among others, in [1].
The existence and uniqueness of solutions of fractional differential equations have been investigated in [2, 3]. In general, most of the fractional differential equations have no exact solutions. Therefore, there has been significant interest in developing approximate methods for solving this kind of equations. Several methods have recently been proposed to solve these equations. These methods include the Adomian decomposition method [4], the homotopy analysis method [5], the AdamsBashforthMoulton method [6, 7], the Laplace transform method [8], the Bessel function method [9–11], and so on. However, few papers reported applications of the differential quadrature method to solve fractionalorder differential equations.
The differential quadrature method was introduced by Richard Bellman and his associates in the early 1970s, following the idea of integral quadrature [12]. The basic idea of the differential quadrature method is that any derivative at a mesh point can be approximated by a weighted linear sum of all the functional values along a mesh line. The key procedure in the differential quadrature method is the determination of weighting coefficients. Fung [13] introduced a modified differential quadrature method to incorporate initial conditions. He also discussed at length the stability of various grid patterns in the differential quadrature method.
2 Preliminaries and notation
In this section, we present some notation, definitions, and preliminary facts.
2.1 Basic definitions of fractional integration and differentiation
There are various definitions of fractional integration and derivatives. The widely used definition of a fractional integral is the RiemannLiouville definition, and that of a fractional derivative is the Caputo definition.
Definition 1
Definition 2
2.2 Chebyshev polynomials and their properties

The threeterm recurrence relationwith \(T_{0}(x)=1\) and \(T_{1}(x)=x\).$$ T_{k+1}(x)=2xT_{k}(x)T_{k1}(x) $$

The expression of \(T_{n}(x)\) in terms of x is given by [14]where$$ T_{n}(x)=\sum_{k=0}^{\lfloor n/2 \rfloor}c_{k}^{(n)}x^{n2k}, $$(5)and$$ c_{k}^{(n)}=(1)^{k}2^{n2k1} \frac{n}{nk}\dbinom{nk}{k} $$$$ c_{k}^{(2k)}=(1)^{k}\quad (k\geq 0). $$

Discrete orthogonality relation with the extrema of \(T_{n}(x)\) as nodes. Let \(n>0\), \(r,s\leq n\), and \(x_{i}=\cos(i\pi/n)\), \(i=0,1,\ldots,n\). Thenwhere \(K_{0}=K_{n}{}=n\) and \(K_{r}=\frac{1}{2}n\) for \(1\leq r\leq n1\). The double prime indicates that the terms with suffixes \(i=0\) and \(i=n\) are to be halved.$$ \sideset{} {^{\prime\prime}}\sum_{i=0}^{n} T_{r}(x_{i})T_{s}(x_{i})=K_{r} \delta_{rs}, $$(6)
3 Calculation of weighting coefficients of fractionalorder derivatives
4 Applications to fractional differential equation
5 Some useful lemmas
Lemma 1
([16])
6 Convergence analysis
Theorem 1
Proof
7 Illustrative examples
To illustrate the effectiveness of the proposed method, we carry out some test examples. The results obtained by this method reveal that it is very effective and convenient for fractional differential equations.
Example 1
Example 2
Comparison of the numerical solutions with the other methods for \(\pmb{\alpha=1/4}\) of Example 2
x  Present method N = 8  Method in [ 21 ]  Adams method in [ 21 ] 

0.1  0.526735  0.458224  0.487151 
0.2  0.515880  0.532957  0.540879 
0.3  0.560879  0.563197  0.571773 
0.4  0.608031  0.588243  0.593261 
0.5  0.612075  0.606988  0.609616 
0.6  0.611336  0.620195  0.622749 
0.7  0.631897  0.630670  0.633677 
0.8  0.648453  0.640302  0.643005 
0.9  0.647655  0.649310  0.651121 
1.0  0.657946  0.656773  0.658290 
Comparison of the numerical solutions with the other methods for \(\pmb{\alpha=2/4}\) of Example 2
x  Present method N = 8  Method in [ 19 ] k = 6, m = 2  Method in [ 21 ] 

0.1  0.330148  0.330159  0.303073 
0.2  0.429343  0.436737  0.423370 
0.3  0.498133  0.504842  0.495336 
0.4  0.553348  0.553802  0.546941 
0.5  0.590546  0.591265  0.585991 
0.6  0.617818  0.621026  0.616680 
0.7  0.643355  0.645480  0.641789 
0.8  0.665723  0.666016  0.662967 
0.9  0.682298  0.683560  0.681065 
1.0  0.697835  0.696506 
Comparison of the numerical solutions with the other methods for \(\pmb{\alpha=3/4}\) of Example 2
x  Present method N = 8  Method in [ 19 ] k = 6, m = 2  Method in [ 21 ] 

0.1  0.185553  0.190108  0.178692 
0.2  0.305933  0.309886  0.301614 
0.3  0.400771  0.404552  0.397897 
0.4  0.478985  0.481638  0.476155 
0.5  0.543038  0.545178  0.540568 
0.6  0.595789  0.597790  0.594017 
0.7  0.640095  0.641801  0.638671 
0.8  0.677553  0.678835  0.676201 
0.9  0.709031  0.710182  0.707923 
1.0  0.735843  0.734913 
Absolute errors for \(\pmb{\alpha=1}\) using \(\pmb{N=4,12}\) for Example 2
x  N = 4  N = 12  Method in [ 20 ] k = 1, M = 5  Method in [ 19 ] k = 6, M = 2 

0.1  2.5206 × 10^{−4}  8.3141 × 10^{−11}  1.9995 × 10^{−5}  1.9000 × 10^{−5} 
0.2  2.4144 × 10^{−4}  9.1576 × 10^{−11}  4.1680 × 10^{−5}  5.9000 × 10^{−5} 
0.3  8.3597 × 10^{−5}  7.5812 × 10^{−11}  3.3875 × 10^{−6}  2.7000 × 10^{−5} 
0.4  7.1204 × 10^{−5}  1.1151 × 10^{−10}  3.6962 × 10^{−5}  3.4000 × 10^{−5} 
0.5  1.2313 × 10^{−4}  5.5890 × 10^{−11}  3.4157 × 10^{−5}  8.9000 × 10^{−5} 
0.6  5.8666 × 10^{−5}  7.8642 × 10^{−11}  7.4330 × 10^{−6}  9.0000 × 10^{−6} 
0.7  5.9751 × 10^{−5}  6.2746 × 10^{−11}  3.7223 × 10^{−5}  6.7000 × 10^{−5} 
0.8  1.3955 × 10^{−4}  5.3920 × 10^{−11}  1.5230 × 10^{−5}  4.3000 × 10^{−5} 
0.9  1.2508 × 10^{−4}  4.8389 × 10^{−11}  3.3870 × 10^{−5}  3.6000 × 10^{−5} 
8 Conclusion
A general formulation for the Chebyshev polynomialbased weighting coefficient matrix for approximation of fractional derivatives has been derived. The fractional derivatives are described in the Caputo sense. The matrix is used to get approximate numerical solutions of fractional Riccati differential equations. Our numerical results are compared with the solutions obtained by the wavelet and artificial neural network methods. The solution obtained using the present method shows that this approach can effectively solve the problem.
Declarations
Acknowledgements
The authors are very grateful to the referees for carefully reading the paper and for their comments and suggestions, which have improved the paper.
Authors’ contributions
Both authors contributed equally to the writing of this paper. Both authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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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