# A Collocation Method for Solving Fractional Riccati Differential Equation

Authors: Yalçın Öztürk; Ayşe Anapalı; Mustafa Gülsu; Mehmet Sezer
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

We have introduced a Taylor collocation method, which is based on collocation method for solving fractional Riccati differential equation with delay term. This method is based on first taking the truncated Taylor expansions of the solution function in the fractional Riccati differential equation and then substituting their matrix forms into the equation. Using collocation points, we have the system of nonlinear algebraic equation. Then, we solve the system of nonlinear algebraic equation using Maple 13, and we have the coefficients of the truncated Taylor sum. In addition, illustrative examples are presented to demonstrate the effectiveness of the proposed method. Comparing the methodology with some known techniques shows that the present approach is relatively easy and highly accurate.


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## Body

## 1. Introduction

The concept of fractional or noninteger order derivation and integration can be traced back to the genesis of integer order calculus itself [1, 2]. The recent investigations in science and engineering have demonstrated that the dynamics of many systems may be described more accurately by using differential equations of noninteger order. The fractional differential equations FDEs have shown to be adequate models for various physical phenomena in areas like damping laws, diffusion processes, and so forth. For example, the nonlinear oscillation of earthquake can be modeled with fractional derivatives [3] and the fluid-dynamic traffic model with fractional derivatives [4], psychology [5] and so forth, [6–9].In this paper, we present numerical and analytical solutions for the fractional Riccati differential equation with delay term(1)D*αy(x)=A(x)+B(x)y(λx+β)+C(x)y2(x),x>0,0<α≤1
subject to the initial conditions
(2)y(0)=λ,
where A(x), B(x), and C(x) are given functions, α is a parameter describing the order of the fractional derivative and λ, β are appropriate constants, and λx+β>0 for all x∈[0,1]. The general response expression contains a parameter describing the order of the fractional derivative that can be varied to obtain various responses. In the case of α=1, the fractional equation reduces to the classical Riccati differential equation. The importance of this equation usually arises in the optimal control problems [10]. The existing literature on fractional differential equations tends to focus on particular values for the order α. In modern applications (see e.g., [11]) much more general values of the order appear in the equations, and therefore one needs to consider numerical and analytical methods to solve differential equations of arbitrary order. This equation is solved the numerically in [12–14].We seek the approximate solution of (1) under the conditions (2) with the fractional Taylor series as Dakαy(x)∈C(a,b],
(3)yN(x)=∑i=0N(x-a)iαΓ(iα+1)(Daiαy(x))x=a,
where 0<α≤1. In recently, collocation method has become a very useful technique for solving equations [15–22]. This method transforms each part of the equation into matrix form and using the collocation points as
(4)xi=iN,i=0,1,…,N,
we get the nonlinear algebraic equation. Then this equation is solved, we obtained the coefficients, the approximate solutions for various N. All computations are performed on the computer algebraic system Maple 13 in this paper.


## 2. Basic Definitions

In this section, we first give some basic definitions and then present properties of fractional calculus [1, 2, 23].Definition 1.
A real functionf(x), x>0 is said to be in space Cμ, μ∈R∈ if there exists a real number p(>μ), such that f(x)=xpf1(x), where f1(x)∈[0,∞), and it is said to be in the space Cμm iff f(m)∈Cμ, m∈N.Definition 2.
The Riemann-Liouville fractional derivative of order α with respect to the variable t and with the starting point at x=a is
(5)Daxαf(x)={1Γ(-α+m+1)dm+1dxm+1×∫ax(x-τ)m-αf(τ)dτ,0≤m≤α<m+1dmf(x)dtm,α=m+1∈N.Definition 3.
The Riemann-Liouville fractional integral of orderα is
(6)Dax-αf(x)=1Γ(α)∫ax(x-τ)α-1f(τ)dτ,p>0.Definition 4.
The fractional derivative off(x) by means of Caputo sense is defined as
(7)Dαf(x)=1Γ(n-α)∫0x(x-τ)n-α-1f(n)(τ)dτ
for n-1<α≤n, n∈N, x>0, f∈C-1n.
For the Caputo’s derivative we haveDαC=0, C is a constant,
(8)Dαxn={0,n∈N,n<⌈α⌉Γ(n+1)Γ(n+1-α)xn-α,n∈N,n<⌊α⌋.Theorem 5 (generalized Taylor formula).
Suppose thatDakαf(x)∈C(a,b] for k=0,1,…,n+1, where 0<α≤1, then one has [23]
(9)f(x)=∑i=0n(x-a)iαΓ(iα+1)(Daiαf)(a)+(Da(n+1)f)(ξ)Γ((n+1)α+1)(x-a)(n+1)α
with a≤ξ≤x, for all x∈(a,b], where
(10)Danα=Daα·Daα·Daα⋯Daα(ntimes).


## 3. Fundamental Relations

In this section, we consider the fractional Ricatti differential equations (1). We use the Taylor matrix method [15–22] to find the truncated Taylor series expansions of each term in expression at x=c and their matrix representations for solving αth order linear fractional part and nonlinear part. We first consider the solution y(x) of (1) defined by a truncated Taylor series (3). Then, we have the matrix form of the solution yN(x)(11)[yN(x)]=T(x)A=X(x)M0A,
where
(12)X(x)=[1(x-c)α(x-c)2α⋯(x-c)Nα],M0=[1Γ(1)00⋯001Γ(α+1)0⋯0001Γ(2α+1)⋯0⋮⋮⋮⋱⋮000⋯1Γ(Nα+1)],A=[D*0αy(c)D*1αy(c)D*2αy(c)⋮D*Nαy(c)].
Then, the matrix representation of the function D*αyN(x) becomes
(13)D*αyN(x)=D*αX(x)M0A,
where we compute the D*αX(x), then(14)D*αX(x)=[D*α1D*α(x-c)αD*α(x-c)2α⋯D*α(x-c)Nα]=[0Γ(α+1)Γ(1)Γ(2α+1)Γ(α+1)(x-c)α⋯Γ(Nα+1)Γ((N-1)α+1)(x-c)(N-1)α]=X(x)M1,where(15)M1=[0Γ(α+1)Γ(1)0⋯000Γ(2α+1)Γ(α+1)⋯0⋮⋮⋮⋱⋮000⋯Γ(Nα+1)Γ((N-1)α+1)000⋯0].
Then, so the matrix representation of fractional differential part as
(16)D*αyN(x)=X(x)M1M0A.
Additionally, using (11) we can write
(17)[yN(λx+β)]=X(λx+β)M0A=X-(x)M0A,
where
(18)X-(x)=X(λx+β)=[1(λx+β-c)α⋯(λx+β-c)Nα].
Moreover, since [21, 22]
(19)Ym=Ym-1Y-,
where
(20)Ym-1(x)=[ym-1(x)ym-1(x)⋮ym-1(x)],Y-(x)=[y(x)0⋯00y(x)⋯0⋮⋮⋱⋮00⋯y(x)],
and using collocation points in (11)
(21)Y-=T-A-,
where
(22)T-(x)=[T(x)0⋯00T(x)⋯0⋮⋮⋱⋮00⋯T(x)],A-=[A0⋯00A⋯0⋮⋮⋱⋮00⋯A]
then we construct the following relation
(23)y(xi)y(xi)=y(xi)y(xi)=(T-A-)X(xi)M0A.
Hence, the fundamental matrix relation of (1) is
(24)(X(x)M1M0-B(x)X-(x)M0-C(x)(T-A-)X(x)M0)A=A(x).
Finally, we obtained matrix representation of the condition in (2)
(25)U0=X(0)M0A=[u0u1u2⋯uN]=[λ].


## 4. Method of Solution

Using collocation points in (4), we can write (24)
(26)(X(xi)M1M0-B(xi)X-(xi)M0-C(xi)(T-A-)X(xi)M0)A=A(xi)
or briefly the fundamental matrix equation
(27)(XM1M0-BX-M0-C(T-A-)XM0)A=F,
where (28)X=[1(x0-c)α(x0-c)2α⋯(x0-c)Nα1(x1-c)α(x1-c)2α⋯(x1-c)Nα1(x2-c)α(x2-c)2α⋯(x2-c)Nα⋮⋮⋮⋱⋮1(xN-c)α(xN-c)2α⋯(xN-c)Nα],F=[A(x0)A(x1)A(x2)⋮A(xN)],X-=[1(λx0+β-c)α(λx0+β-c)2α⋯(λx0+β-c)Nα1(λx1+β-c)α(λx1+β-c)2α⋯(λx1+β-c)Nα1(λx2+β-c)α(λx2+β-c)2α⋯(λx2+β-c)Nα⋮⋮⋮⋱⋮1(λxN+β-c)α(λxN+β-c)2α⋯(λxN+β-c)Nα],T-=[T(x0)00⋯00T(x1)0⋯000T(x2)⋯0⋮⋮⋮⋱⋮000⋯T(xN)],B=[B(x0)00⋯00B(x1)0⋯000B(x2)⋯0⋮⋮⋮⋱⋮000⋯B(xN)],C=[C(x0)00⋯00C(x1)0⋯000C(x2)⋯0⋮⋮⋮⋱⋮000⋯C(xN)].Hence, the fundamental matrix equation (27) corresponding to (1) can be written in the form
(29)WA=For[W;F],W=[wi,j],i,j=0,1,…,N,
where
(30)W=XM1M0-BX-M0-C(T-A-)XM0.
To obtain the solution of (1) under conditions (2), by replacing the row matrices (25) by the last 2 rows of the matrix (26), we have the new augmented matrix,
(31)[W~;F~]=[w00w01⋯w0N;A(x0)w10w11⋯w1N;A(x1)⋮⋮⋱⋮⋮⋮wN-20wN-21⋯wN-2N⋮A(xN-2)wN-10wN-11⋯wN-1N;A(xN-1)u0u1⋯uN;λ].
So, we obtained a system of (N+1) nonlinear algebraic equations with unknown Taylor coefficients.We can easily check the accuracy of the method. Since the truncated fractional Taylor series (3) is an approximate solution of (1), when the solution yN(x) and its derivatives are substituted in (1), the resulting equation must be satisfied approximately; that is, for x=xq∈[0,1],q=0,1,2,…(32)EN(xq)=|D*αy(xq)-A(xq)-B(xq)y(λxq+β)-C(xq)y2(xq)|≅0.


## 5. Examples

In order to illustrate the effectiveness of the method proposed in this paper, several numerical examples are carried out in this section. In the following computations, for convenience, absolute errors betweenNth-order approximate values yN and the corresponding exact values yex as Ne=|yN-yex| are determined and all computations performed computer algebraic system with mathematical programing in Maple 13.Example 1.
Consider the following fractional Riccati equation:(33)D*αy(x)=y2(x)-x2y(x+1)+Γ(3)Γ(5/2)x3/2-2x3-x2,x>0,0<α≤1
with initial conditions
(34)y(0)=0.
Then, A(x)=Γ(3)x3/2/Γ(5/2)-2x3-x2, B(x)=-x2, C(x)=1. We assume that α=1/2, 0≤x≤1, and we seek the approximate solutions yN by Taylor series, for c=0, N=4(35)y4(x)=∑k=04xkαΓ(kα+1)(D*kαy(x))x=0
with collocation points being
(36)x0=0,x1=14,x2=24,x3=34,x4=1.
Fundamental matrix relation of this problem is
(37)(XM1M0-BX-M0-C(T-A-)XM0)A=F,
where
(38)X=[100001141441611611212241413434334491611111],B=[1000001000001000001000001],X-=[12222114+114+1416+1116+1112+112+124+114+1134+134+13344+1916+112222],C=[0000000.0625000000.2500000000.5625000001],M0=[1000002π0000010000043π0000012],M1=[01000002π0000010000043π00000],F=[0.0000000.1880630.5319230.9772051.504505].
Also, we have the matrix representation of conditions,
(39)y(0)=[1000000]A=[0]
and so we solve (37) and obtain the coefficients of the Taylor series(40)A=[0-0.478771e-40.189414e-3-0.307615e-32.000189].
Hence, forN=4, the approximate solution of Example 1 is given as
(41)y5=0.957543xπ+0.189414e-3x-0.410153x3/2π+1.000094x2.
Comparison of numerical results with the exact solution is shown in Table 1 for various N.Table 1
Numerical result for Example1.
x
Exactsolution
Present Method
N
=
4
N
e
=
4
N
=
5
N
e
=
5
N
=
6
N
e
=
6
0.0
0.000
0.000000
0.000
E
-
0
0.000000
0.000
E
-
0
0.00000
0.000
E
-
0
0.2
0.040
0.039985
0.150
E
-
4
0.040000
0.553
E
-
6
0.040000
0.740
E
-
8
0.4
0.160
0.159993
0.700
E
-
5
0.160000
0.352
E
-
6
0.160000
0.530
E
-
8
0.6
0.360
0.359996
0.446
E
-
5
0.360000
0.330
E
-
6
0.360000
0.497
E
-
8
0.8
0.640
0.63996
0.446
E
-
5
0.640000
0.337
E
-
6
0.640000
0.528
E
-
8
1.0
1.000
0.999998
0.219
E
-
5
1.000000
0.464
E
-
6
1.000000
0.673
E
-
8Example 2.
Let us consider the following fractional Riccati equation [13]
(42)D*αy(x)=y2(x)+1,0<α≤1
subject to the initial condition
(43)y(0)=1.
Then, A(x)=1, B(x)=0, C(x)=1. Fundamental matrix relation of this problem is
(44)(XM1M0-C(T-A-)XM0)A=F.
Also, we have the matrix representation of conditions,
(45)y(0)=[1000000]A=[0].
The exact solution, when α=1, is
(46)y(x)=e2x-1e2x+1.
We approximately solve the fractional Ricatti equation for N=12 and obtained the approximate solution for α=1,
(47)y12=x-0.559171e-6x2-0.333319x3-0.155476e-3x4+0.134388x5-0.467422e-2x6-0.398472e-1x7-0.295050e-1x8+0.640148e-1x9-0.390402e-1x10+0.109027x11-0.116904e-2x12.
Table 2 shows the approximate solutions for Example 2 obtained by the present method and compare with homotopy perturbation method for different α. From the numerical results in Table 2, present method is in high agreement with the exact solution than homotopy perturbation method [13]. Moreover, using the numerical result in Table 2, Figures 1 and 2 are plotted. Figure 2 shows that numerical solution yPM is so closed to the exact solution.Table 2
The numerical results with comparison HP and PM(N=12).
x
y
exact
y
HPM
y
PM
α
=
0.5
α
=
1
α
=
0.5
α
=
1
0.0
0.000000
0.000000
0.000000
0.000000
0.000000
0.1
0.096668
0.273875
0.099668
0.329929
0.096668
0.2
0.197375
0.454125
0.197375
0.436741
0.197375
0.3
0.291313
0.573932
0.291312
0.504823
0.291313
0.4
0.379949
0.644422
0.379944
0.553733
0.379949
0.5
0.462117
0.674137
0.462078
0.591156
0.462117
0.6
0.537050
0.671987
0.536857
0.620983
0.537050
0.7
0.604368
0.648003
0.603631
0.645461
0.604368
0.8
0.664037
0.613306
0.661706
0.665998
0.664037
0.9
0.716298
0.579641
0.709919
0.683545
0.716298
1.0
0.761594
0.558557
0.746032
0.698852
0.761594Figure 1
Comparison of the HPM and PMN forα=0.5.Figure 2
Comparison of the HPM and PM forα=1.


## 6. Conclusion

In real world systems, delays can be recognised everywhere and there has been widespread interest in the study of delay differential equations for many years. Although it seems natural to model certain processes and systems in engineering and other sciences with this kind of equation, only in the last few years has the attention of the scientific community been devoted to them.In this study, we present a Taylor collocation method for the numerical solutions of fractional Riccati differential equation with delay term. This method transforms fractional Riccati differential equation with delay term into matrix equations. The desired approximate solutions can be determined by solving the resulting system, which can be effectively computed using symbolic computing codes on Maple 13. Examples show that Taylor collocation method has been successfully applied to find the approximate solutions of the fractional Riccati differential equation. Graphics and tables show that this method is extremely effective and practical for this sort of approximate solutions.



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