Reproducing Kernel Method for Fractional Riccati Differential Equations

Abstract

This paper is devoted to a new numerical method for fractional Riccati differential equations. The method combines the reproducing kernel method and the quasilinearization technique. Its main advantage is that it can produce good approximations in a larger interval, rather than a local vicinity of the initial position. Numerical results are compared with some existing methods to show the accuracy and effectiveness of the present method.

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X. Y. Li 1 and B. Y. Wu 2 and R. T. Wang 1

Academic Editor:Youyu Wang

1, Department of Mathematics, Changshu Institute of Technology, Suzhou, Jiangsu 215500, China
2, Department of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang 150001, China

Received 29 December 2013; Accepted 7 April 2014; 27 April 2014

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

This paper deals with the numerical solution of the following fractional Riccati differential equation: [figure omitted; refer to PDF] where ^{u α} ( x ) denotes the Caputo fractional derivative of order α and [figure omitted; refer to PDF] Riccati differential equations arise in many fields [1]. The problem has attracted much attention and has been studied by many authors. However, deriving its analytical solution in an explicit form seems to be unlikely except for certain special situations. Recently, many numerical methods [2-9] have been proposed to solve integer order Riccati differential equations. However, the discussion on the numerical methods for fractional order Riccati differential equations is rare. Odibat and Momani [10] developed a modified homotopy perturbation method for fractional Riccati differential equations. Li [11] presented a numerical method for fractional differential equations based on Chebyshev wavelets. Hosseinnia et al. [12] applied an enhanced homotopy perturbation method for fractional Riccati differential equations. Yüzbasi [13] introduced a numerical method for fractional Riccati differential equations using the Bernstein polynomial. Khader [14] developed the fractional Chebyshev finite difference method for fractional Riccati differential equations. Yang and Baleanu [15], Yang et al. [16], and Baleanu et al. [17] proposed local fractional variation iteration for fractional heat conduction and wave equations.

Recently, based on the reproducing kernel theory, Cui, Geng, and Lin presented the reproducing kernel method (RKM) for linear and nonlinear operator equations [18-21]. The method has been developed and applied to many problems [22-26].

The aim of this paper is to present a new method for fractional Riccati differential equations, based on the RKM and the quasilinearization technique.

The rest of the paper is organized as follows. In the next section, the quasilinearization technique is applied to fractional Riccati differential equation. The RKM for reduced linear fractional differential equations is introduced in Section 3. The numerical examples are presented in Section 4. Section 5 ends this paper with a brief conclusion.

2. Quasilinearization of Riccati Differential Equation (1)

In this section, the quasilinearization technique is applied to reduce (1) to a series of linear fractional problems. Define f ( x , u ) = p ( x ) + r ( x ) ^{u 2} . By choosing an appropriate initial approximation _{u 0} ( x ) for the function u ( x ) in f ( x , u ) and expanding f ( x , u ) around _{u 0} ( x ) , it follows that [figure omitted; refer to PDF] Generally, one can write for k = 1,2 , ... ( k = iteration index) [figure omitted; refer to PDF] Therefore, the following iteration formula for (1) can be derived: [figure omitted; refer to PDF] where _{a k} ( x ) = - [ q ( x ) + ( ∂ f / ∂ u ) _{| u = u k - 1} ] = - [ q ( x ) + 2 r ( x ) _{u k - 1} ( x ) ] and _{f k} ( x ) = f ( x , _{u k - 1} ) - ( ∂ f / ∂ u ) _{| u = u k - 1} = p ( x ) - r ( x ) ^{u k - 1 2} ( x ) and _{u 0} ( x ) is the initial approximation.

Clearly, to solve (1), it suffices for us to solve the series of linear problem (5).

3. Method for Solving Linear Fractional Problem (5)

To illustrate how to solve (5) we consider the problem of solving [figure omitted; refer to PDF] where a ( x ) and f ( x ) are continuous.

By the definition of Caputo fractional derivative, (6) is equivalent to the following equation: [figure omitted; refer to PDF] If u ( t ) ∈ ^{C 1} [ 0 , T ] , then the improper integral ^{∫ 0 t} ... ( t - τ ^{) - α v [variant prime]} ( τ ) d τ exists and [figure omitted; refer to PDF] Define [figure omitted; refer to PDF] This gives us a continuous function on [ 0 , t ] and then integral ^{∫ 0 t} ... g ( τ , v ) d τ also exists and ^{∫ 0 t} ... ( ( ^{v [variant prime]} ( t ) - ^{v [variant prime]} ( τ ) ) / ( t - τ ^{) α} ) d τ = ^{∫ 0 t} ... g ( τ , v ) d τ .

Therefore, (6) can be converted into [figure omitted; refer to PDF]

Applying Hermite's quadrature formula to ^{∫ 0 t} ... ( ( ^{v [variant prime]} ( t ) - ^{v [variant prime]} ( τ ) ) / ^{( t - τ ) α} ) d τ , one obtains [figure omitted; refer to PDF] where g ¯ ( x , u ( x ) ) = g ( ( t / 2 ) ( 1 + x ) , u ( ( t / 2 ) ( 1 + x ) ) ) , M is the number of nodes, and _{x k} = cos ... ( ( 2 k - 1 ) / 2 M ) , k = 1 , ... , M .

Then (6) can be further equivalently approximated to [figure omitted; refer to PDF]

To apply the RKM to (12), it is necessary to construct the following reproducing kernel Hilbert space ^{W 3} [ 0 , T ] .

Definition 1.

^{W 3} [ 0 , T ] = { u ( x ) |" ^{u [variant prime][variant prime]} ( x ) is an absolutely continuous real value function, ^{u ( 3 )} ( x ) ∈ ^{L 2} [ 0 , T ] , u ( 0 ) = 0 } . The inner product and norm in ^{W 3} [ 0 , T ] are given, respectively, by [figure omitted; refer to PDF]

Theorem 2.

^{W 3} [ 0 , T ] is a reproducing kernel space and its reproducing kernel is [figure omitted; refer to PDF] where _{k 1} ( x , y ) = ^{y 2} ( 7 x ^{y 4} - ^{y 5} + 35 ^{x 3} y ( 4 + y ) - 21 ^{x 2} ( - 60 + ^{y 3} ) ) / 5040 .

Definition 3.

^{W 1} [ 0 , T ] = { u ( x ) |" u ( x ) is an absolutely continuous real value function, ^{u [variant prime]} ( x ) ∈ ^{L 2} [ 0 , T ] } . The inner product and norm in ^{W 1} [ 0 , T ] are given, respectively, by [figure omitted; refer to PDF]

Theorem 4.

^{W 1} [ 0 , T ] is a reproducing kernel space and its reproducing kernel is [figure omitted; refer to PDF]

Put [figure omitted; refer to PDF] Clearly, L : ^{W 3} [ 0 , T ] [arrow right] ^{W 1} [ 0 , T ] is a bounded linear operator. Put _{[straight phi] i} ( x ) = k ¯ ( x , _{x i} ) and _{ψ i} ( x ) = ^{L *} _{[straight phi] i} ( x ) , where ^{L *} is the adjoint operator of L . The orthonormal system { _{ψ ¯ i} ( x ) ^{} i = 1 ∞} of ^{W 3} [ 0 , T ] can be derived from Gram-Schmidt orthogonalization process of { _{ψ i} ( x ) ^{} i = 1 ∞} , [figure omitted; refer to PDF]

Theorem 5.

If { _{x i} ^{} i = 1 ∞} is dense on [ 0 , T ] , then { _{ψ i} ( x ) ^{} i = 1 ∞} is the complete system of ^{W 3} [ 0 , T ] .

Theorem 6.

If { _{x i} ^{} i = 1 ∞} is dense on [ 0 , T ] and the solution of (12) is unique, then the solution of (12) is [figure omitted; refer to PDF]

Now, an approximate solution _{V N} ( x ) of (6) can be obtained by the N -term intercept of the exact solution v ( x ) and [figure omitted; refer to PDF]

Similarly, the approximate solutions _{u k} ( x ) can be obtained: [figure omitted; refer to PDF] where _{A j} = ^{∑ l = 1 j} _{β j l f k} ( _{x l} ) .

4. Numerical Examples

Example 1.

Consider the following fractional Riccati differential equation [10-13]: [figure omitted; refer to PDF] The exact solution for α = 1 can be easily determined to be [figure omitted; refer to PDF] Applying the proposed method, taking T = 1 , k = 3 , M = 30 , N = 50 , the numerical results compared with other methods are listed in Tables 1 and 2. Taking T = 4 , k = 3 , M = 30 , N = 50 , the numerical results on [ 0,4 ] are listed in Table 3. From Table 3, it is easily found that the present approximations are effective for a larger interval, rather than a local vicinity of the initial position.

Table 1: Comparison of the numerical solutions with the other methods for α = 0.75 .

x	Ours	[10]	[11]
0.20	0.469516	0.428892	0.584307
0.40	0.933596	0.891404	1.024974
0.50	1.14488	1.132763	1.198621
0.60	1.33098	1.370240	1.349150
0.80	1.62153	1.794879	1.599235
1.00	1.81865	2.087384	1.801763

Table 2: Comparison of the numerical solutions with the other methods for α = 0.90 .

x	Ours	[13]	IABMM [12]	MHPM [10]
0.20	0.312985	0.314869	--	--
0.40	0.695357	0.697544	--	--
0.50	0.901484	0.903695	0.8621	0.9010
0.60	1.10576	1.107866	--	--
0.80	1.47606	1.477707	--	--
1.00	1.76417	1.764520	1.7356	1.8720

Table 3: Numerical results for α = 0.99 , 1 on [ 0,4 ] .

x	Ours ( α = 0.99 )	Ours ( α = 1 )	Exact solution ( α = 1 )
0.5	0.769552	0.756446	0.756014
1.0	1.69828	1.68978	1.689500
1.5	2.19159	2.19599	2.195630
2.0	2.35063	2.35838	2.357770
2.5	2.39373	2.40046	2.40028
3.0	2.40539	2.41058	2.41081
3.5	2.40899	2.41302	2.41339
4.0	2.41047	2.41374	2.41401

Example 2.

Consider the following fractional Riccati differential equation [10-14]: [figure omitted; refer to PDF] The exact solution for α = 1 can be easily determined to be [figure omitted; refer to PDF] According to the present method, taking T = 1 , k = 3 , M = 50 , N = 50 , the numerical results compared with other methods are given in Tables 4, 5, and 6. Taking T = 4 , k = 5 , M = 50 , N = 80 , the numerical results on [ 0,4 ] are shown in Figures 1 and 2. From these figures we can conclude that the obtained numerical solutions are in excellent agreement with the exact solution for a larger interval.

Table 4: Comparison of the numerical solutions with the other methods for α = 0.75 .

x	Ours	[13]	IABMM [12]	MHPM [10]
0.20	0.307359	0.3099755	0.3117	0.3138
0.40	0.480346	0.4816318	0.4855	0.4929
0.60	0.597542	0.5977827	0.6045	0.5974
0.80	0.679657	0.6788495	0.6880	0.6604
1.00	0.738213	0.7368368	0.7478	0.7183

Table 5: Comparison of the numerical solutions with the other methods for α = 0.90 .

x	Ours	[13]	IABMM [12]	MHPM [10]
0.20	0.237652	0.2387891	0.2393	0.2391
0.40	0.421766	0.4225830	0.4234	0.4229
0.60	0.565673	0.5661715	0.5679	0.5653
0.80	0.674464	0.6746270	0.6774	0.6740
1.00	0.754632	0.7545890	0.7584	0.7569

Table 6: Comparison of the numerical solutions with the other methods for α = 1.0 .

x	Exact	Ours	[13]	MHPM [10]
0.20	0.197375	0.19738	0.197375	0.197375
0.40	0.379949	0.379956	0.379948	0.379944
0.60	0.53705	0.537061	0.537049	0.536857
0.80	0.664037	0.664053	0.664036	0.661706
1.00	0.761594	0.761618	0.761594	0.746032

The behavior of approximate solution with different values of α ((a) α = 0.99 ; (b) α = 0.75,0.99 ; (c) α = 0.5,0.75,0.99 ).

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

(c) [figure omitted; refer to PDF]

Comparison of approximate solutions with the exact solutions for α = 1 ((a) exact solution; (b) absolute errors).

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

5. Conclusion

In this paper, combining the RKM, the numerical integral, and quasilinearization techniques, a new numerical method is proposed for fractional Riccati differential equations. The main advantage of this method is that it can provide accurate numerical approximations on a larger interval. Numerical results compared with the existing methods show that the present method is a powerful method for solving fractional Riccati differential equations.

Acknowledgment

This work was supported by the National Natural Science Foundation of China (Grant nos. 11326237, 11271100, and 11126222).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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