A Collocation Method for Solving Fractional Riccati Differential Equation

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.

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 𝐷^𝛼_∗𝑦(𝑥)=𝐴(𝑥)+𝐵(𝑥)𝑦𝜆𝑥+𝛽+𝐶(𝑥)𝑦²(𝑥),𝑥>0,0<𝛼≤1 (1) subject to the initial conditions 𝑦(0)=𝜆, (2) 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 𝐷_𝑎^𝑘𝛼𝑦(𝑥)∈𝐶(𝑎,𝑏], 𝑦_𝑁(𝑥)=^𝑁_𝑖=0(𝑥−𝑎)^𝑖𝛼Γ()𝐷𝑖𝛼+1^𝑎𝑖𝛼𝑦(𝑥)_𝑥=𝑎, (3) 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 𝑥_𝑖=𝑖𝑁,𝑖=0,1,…,𝑁, (4) 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 function 𝑓(𝑥), 𝑥>0 is said to be in space 𝐶_𝜇, 𝜇∈𝑅∈ if there exists a real number 𝑝(>𝜇), such that 𝑓(𝑥)=𝑥^𝑝𝑓₁(𝑥), where 𝑓₁(𝑥)∈[0,∞), and it is said to be in the space 𝐶^𝑚_𝜇 iff 𝑓^(𝑚)∈𝐶_𝜇, 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 _𝑎𝐷^𝛼_𝑥𝑓(𝑥)=1Γ()𝑑−𝛼+𝑚+1𝑑𝑥^𝑚+1𝑚+1×∫^𝑥_𝑎(𝑥−𝜏)^𝑚−𝛼𝑓(𝜏)𝑑𝑑𝜏,0≤𝑚≤𝛼<𝑚+1_𝑚𝑓(𝑥)𝑑𝑡^𝑚,𝛼=𝑚+1∈𝑁. (5)

Definition 3. The Riemann-Liouville fractional integral of order 𝛼 is _𝑎𝐷_𝑥^−𝛼𝑓(𝑥)=1Γ(𝛼)^𝑥_𝑎(𝑥−𝜏)^𝛼−1𝑓(𝜏)𝑑𝜏,𝑝>0. (6)

Definition 4. The fractional derivative of 𝑓(𝑥) by means of Caputo sense is defined as 𝐷^𝛼𝑓(𝑥)=1Γ(𝑛−𝛼)^𝑥₀(𝑥−𝜏)^{𝑛−𝛼−1}𝑓^(𝑛)(𝜏)𝑑𝜏 (7) for 𝑛−1<𝛼≤𝑛, 𝑛∈𝑁, 𝑥>0, 𝑓∈𝐶^𝑛₋₁.
For the Caputo’s derivative we have 𝐷^𝛼𝐶=0, 𝐶 is a constant, 𝐷^𝛼𝑥^𝑛=⌈𝛼⌉Γ()0,𝑛∈𝑁,𝑛<𝑛+1Γ()𝑥𝑛+1−𝛼^𝑛−𝛼,𝑛∈𝑁,𝑛<⌊𝛼⌋. (8)

Theorem 5 (generalized Taylor formula). Suppose that 𝐷_𝑎^𝑘𝛼𝑓(𝑥)∈𝐶(𝑎,𝑏] for 𝑘=0,1,…,𝑛+1, where 0<𝛼≤1, then one has [23] 𝑓(𝑥)=^𝑛_𝑖=0(𝑥−𝑎)^𝑖𝛼Γ()𝐷𝑖𝛼+1^𝑎𝑖𝛼𝑓(𝑎)+𝐷_𝑎^(𝑛+1)𝑓(𝜉)Γ))()((𝑛+1𝛼+1𝑥−𝑎^(𝑛+1)𝛼 (9) with 𝑎≤𝜉≤𝑥, for all 𝑥∈(𝑎,𝑏], where 𝐷_𝑎^𝑛𝛼=𝐷^𝛼_𝑎⋅𝐷^𝛼_𝑎⋅𝐷^𝛼_𝑎⋯𝐷^𝛼_𝑎(𝑛𝑡𝑖𝑚𝑒𝑠). (10)

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 𝑦_𝑁(𝑥)𝑦_𝑁(𝑥)=𝐓(𝑥)𝐀=𝐗(𝑥)𝐌₀𝐀, (11) where 𝐗(𝑥)=1(𝑥−𝑐)^𝛼(𝑥−𝑐)^2𝛼⋯(𝑥−𝑐)^𝑁𝛼,𝐌₀=1Γ(1)0100⋯0Γ()1𝛼+10⋯000Γ()12𝛼+1⋯0⋮⋮⋮⋱⋮000⋯Γ(),𝐷𝑁𝛼+1𝐀=^∗0𝛼𝑦(𝑐)𝐷_∗^1𝛼𝑦(𝑐)𝐷_∗^2𝛼𝑦(𝑐)⋮𝐷_∗^𝑁𝛼𝑦(𝑐). (12) Then, the matrix representation of the function 𝐷^𝛼_∗𝑦_𝑁(𝑥) becomes 𝐷^𝛼_∗𝑦_𝑁(𝑥)=𝐷^𝛼_∗𝐗(𝑥)𝐌₀𝐀, (13) where we compute the 𝐷^𝛼_∗𝐗(𝑥), then𝐷^𝛼_∗𝐗(𝑥)=𝐷^𝛼_∗1𝐷^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, (14)

where 𝐌₁=0Γ(𝛼+1)Γ(1)Γ()0⋯0002𝛼+1Γ()Γ()𝛼+1⋯0⋮⋮⋮⋱⋮000⋯𝑁𝛼+1Γ((𝑁−1)𝛼+1)000⋯0. (15) Then, so the matrix representation of fractional differential part as 𝐷^𝛼_∗𝑦_𝑁(𝑥)=𝐗(𝑥)𝐌₁𝐌₀𝐀. (16) Additionally, using (11) we can write 𝑦_𝑁𝜆𝑥+𝛽=𝐗𝜆𝑥+𝛽𝐌₀𝐀=𝐗(𝑥)𝐌₀𝐀, (17) where 𝐗(𝑥)=1=𝐗𝜆𝑥+𝛽𝜆𝑥+𝛽−𝑐^𝛼⋯𝜆𝑥+𝛽−𝑐^𝑁𝛼. (18) Moreover, since [21, 22] 𝐘^𝑚=𝐘^𝑚−1𝐘, (19) where 𝐘^𝑚−1(𝑥)=𝑦^𝑚−1(𝑥)𝑦^𝑚−1(𝑥)⋮𝑦^𝑚−1(𝑥),𝐘(𝑥)=𝑦(𝑥)(𝑥)(𝑥),0⋯00𝑦⋯0⋮⋮⋱⋮00⋯𝑦 (20) and using collocation points in (11) 𝐘=𝐓𝐀, (21) where 𝐓(𝑥)=𝐓(𝑥)(𝑥)(𝑥),0⋯00𝐓⋯0⋮⋮⋱⋮00⋯𝐓𝐀=𝐀0⋯00𝐀⋯0⋮⋮⋱⋮00⋯𝐀 (22) then we construct the following relation 𝑦𝑥_𝑖𝑦𝑥_𝑖=𝑦𝑥_𝑖𝑦𝑥_𝑖=𝐓𝐀𝐗𝑥_𝑖𝐌₀𝐀. (23) Hence, the fundamental matrix relation of (1) is 𝐗(𝑥)𝐌₁𝐌₀−𝐵(𝑥)𝐗(𝑥)𝐌₀−𝐶(𝑥)𝐓𝐀𝐗(𝑥)𝐌₀𝐀=𝐴(𝑥). (24) Finally, we obtained matrix representation of the condition in (2) 𝐔₀=𝐗(0)𝐌₀𝐀=𝑢₀𝑢₁𝑢₂⋯𝑢_𝑁=[𝜆]. (25)

4. Method of Solution

Using collocation points in (4), we can write (24) 𝐗𝑥_𝑖𝐌₁𝐌₀−𝐵𝑥_𝑖𝐗𝑥_𝑖𝐌₀−𝐶𝑥_𝑖𝐓𝐀𝐗𝑥_𝑖𝐌₀𝐀=𝐴𝑥_𝑖 (26) or briefly the fundamental matrix equation 𝐗𝐌₁𝐌₀−𝐁𝐗𝐌₀−𝐂𝐓𝐀𝐗𝐌₀𝐀=𝐅, (27) where 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(𝑥_𝑁−𝑐)^𝛼(𝑥_𝑁−𝑐)^2𝛼⋯(𝑥_𝑁−𝑐)^𝑁𝛼𝐴𝑥,𝐅=⁰𝐴𝑥₁𝐴𝑥₂⋮𝐴𝑥_𝑁,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(𝜆𝑥_𝑁+𝛽−𝑐)^𝛼(𝜆𝑥_𝑁+𝛽−𝑐)^2𝛼⋯(𝜆𝑥_𝑁+𝛽−𝑐)^𝑁𝛼,𝑇𝑥𝐓=⁰𝑥00⋯00𝑇₁𝑥0⋯000𝑇₂𝑥⋯0⋮⋮⋮⋱⋮000⋯𝑇_𝑁𝐵𝑥,𝐁=⁰𝑥00⋯00𝐵₁𝑥0⋯000𝐵₂𝑥⋯0⋮⋮⋮⋱⋮000⋯𝐵_𝑁,𝐶𝑥𝐂=⁰𝑥00⋯00𝐶₁𝑥0⋯000𝐶₂𝑥⋯0⋮⋮⋮⋱⋮000⋯𝐶_𝑁. (28)

Hence, the fundamental matrix equation (27) corresponding to (1) can be written in the form 𝐖𝐀=𝐅or[𝐖;𝐅],𝐖=𝑤_𝑖,𝑗,𝑖,𝑗=0,1,…,𝑁, (29) where 𝐖=𝐗𝐌₁𝐌₀−𝐁𝐗𝐌₀−𝐂𝐓𝐀𝐗𝐌₀. (30) 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, 𝐅=𝑤𝐖;⁰⁰𝑤₀₁⋯𝑤_0𝑁;𝐴𝑥₀𝑤₁₀𝑤₁₁⋯𝑤_1𝑁;𝐴𝑥₁𝑤⋮⋮⋱⋮⋮⋮_𝑁−20𝑤_𝑁−21⋯𝑤_𝑁−2𝑁⋮𝐴𝑥_𝑁−2𝑤_𝑁−10𝑤_𝑁−11⋯𝑤_𝑁−1𝑁;𝐴𝑥_𝑁−1𝑢₀𝑢₁⋯𝑢_𝑁;𝜆. (31) 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 𝑦_𝑁(𝑥) and its derivatives are substituted in (1), the resulting equation must be satisfied approximately; that is, for 𝑥=𝑥_𝑞∈[0,1],𝑞=0,1,2,…𝐸_𝑁𝑥_𝑞=𝐷^𝛼_∗𝑦𝑥_𝑞−𝐴𝑥_𝑞−𝐵𝑥_𝑞𝑦𝜆𝑥_𝑞𝑥+𝛽−𝐶_𝑞𝑦²𝑥_𝑞≅0. (32)

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 between 𝑁th-order approximate values 𝑦_𝑁 and the corresponding exact values 𝑦_ex as 𝑁_𝑒=|𝑦_𝑁−𝑦_ex| are determined and all computations performed computer algebraic system with mathematical programing in Maple 13.

Example 1. Consider the following fractional Riccati equation: 𝐷^𝛼_∗𝑦(𝑥)=𝑦²(𝑥)−𝑥²𝑦()+Γ(3)𝑥+1Γ()𝑥5/2^3/2−2𝑥³−𝑥²,𝑥>0,0<𝛼≤1 (33) with initial conditions 𝑦(0)=0. (34) Then, 𝐴(𝑥)=Γ(3)𝑥^3/2/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 𝑦_𝑁 by Taylor series, for 𝑐=0, 𝑁=4𝑦₄(𝑥)=⁴_𝑘=0𝑥^𝑘𝛼Γ()𝐷𝑘𝛼+1^∗𝑘𝛼𝑦(𝑥)_𝑥=0 (35) with collocation points being 𝑥₀=0,𝑥₁=14,𝑥₂=24,𝑥₃=34,𝑥₄=1. (36) Fundamental matrix relation of this problem is 𝐗𝐌₁𝐌₀−𝐁𝐗𝐌₀−𝐂𝐓𝐀𝐗𝐌₀𝐀=𝐅, (37) where 11𝐗=10000√414√41161116√212√24141√3√4343√34√49,1611111,𝐁=100000100000100000100000111𝐗=12222√41+14√+14116+11116+1√21+12√+1241+141√+13√43+143√+134√49+1,,𝐌16+112222𝐂=0000000.0625000000.2500000000.5625000001₀=0210000√𝜋4000001000003√𝜋0100002,𝐌₁=20100000√𝜋4000001000003√𝜋.00000,𝐅=0.0000000.1880630.5319230.9772051.504505 (38) Also, we have the matrix representation of conditions, 𝑦(0)=1000000𝐀=[0] (39) and so we solve (37) and obtain the coefficients of the Taylor series𝐀=0−0.478771𝑒−40.189414𝑒−3−0.307615𝑒−32.000189. (40)
Hence, for 𝑁=4, the approximate solution of Example 1 is given as 𝑦₅=0.957543√𝑥√𝜋−+0.189414𝑒−3𝑥0.410153𝑥^3/2√𝜋+1.000094𝑥². (41) Comparison of numerical results with the exact solution is shown in Table 1 for various 𝑁.

Table 1 

Numerical result for Example 1.

Example 2. Let us consider the following fractional Riccati equation [13] 𝐷^𝛼_∗𝑦(𝑥)=𝑦²(𝑥)+1,0<𝛼≤1 (42) subject to the initial condition 𝑦(0)=1. (43) Then, 𝐴(𝑥)=1, 𝐵(𝑥)=0, 𝐶(𝑥)=1. Fundamental matrix relation of this problem is 𝐗𝐌₁𝐌₀−𝐂𝐓𝐀𝐗𝐌₀𝐀=𝐅. (44) Also, we have the matrix representation of conditions, 𝑦(0)=1000000𝐀=[0]. (45) The exact solution, when 𝛼=1, is 𝑦(𝑥)=𝑒^2𝑥−1𝑒^2𝑥.+1 (46) We approximately solve the fractional Ricatti equation for 𝑁=12 and obtained the approximate solution for 𝛼=1, 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. (47) 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 𝑦_PM is so closed to the exact solution.

Table 2 

The numerical results with comparison HP and PM (𝑁=12).

Figure 1 

Comparison of the HPM and PMN for 𝛼=0.5.

Figure 2 

Comparison of the HPM and PM for 𝛼=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.