Numerical Approximation of Riccati Fractional Differential Equation in the Sense of Caputo-Type Fractional Derivative

)e Riccati differential equation is a well-known nonlinear differential equation and has different applications in engineering and science domains, such as robust stabilization, stochastic realization theory, network synthesis, and optimal control, and in financial mathematics. In this study, we aim to approximate the solution of a fractional Riccati equation of order 0< β< 1 with Atangana–Baleanu derivative (ABC). Our numerical scheme is based on Laplace transform (LT) and quadrature rule. We apply LT to the given fractional differential equation, which reduces it to an algebraic equation. )e reduced equation is solved for the unknown in LT space. )e solution of the original problem is retrieved by representing it as a Bromwich integral in the complex plane along a smooth curve. )e Bromwich integral is approximated using the trapezoidal rule. Some numerical experiments are performed to validate our numerical scheme.


Introduction
In applied mathematics, the fractional differential equation (FDE) is an equation which contains derivatives of arbitrary order. Fractional derivatives first appeared in 1695 [1]. Recently, the research community took more interest in fractional calculus because of its applications in engineering and other sciences [2][3][4]. Many physical systems show fractional order behavior that may vary with time or space. Fractional derivatives have many kinds. e three important and most commonly used fractional derivatives are Gr € unwald-Letnikov derivative, Riemann-Liouville fractional derivative, and Caputo derivative [5]. However, these classical fractional derivatives have a singular kernel, and hence, they may face difficulties in describing the nonlocality of real-world dynamics. In order to handle the nonlocal systems in a better way, recently, new fractional derivatives with nonsingular kernels are defined such as the Caputo-Fabrizio (CF) derivative and Atangana-Baleanu (ABC) derivative [6,7]. Fractional derivatives with nonsingular kernels become more valuable due to the fact that numerous phenomena cannot properly be modeled by fractional derivatives with singular kernels [8].
In this work, we aim to approximate a Riccati differential equation (RDE) with ABC derivative. RDEs have many applications such as random processes, optimal control, and diffusion process [9]. e RDE of fractional order has been studied by many authors; for example, in [10], the authors developed the Adomain decomposition method for the solution of RDE of fractional order. In [11], some analytic techniques are presented for the solution of RDE. e authors [12] obtained the solution of RDE using the differential transform method. In [13], the authors have developed a Laplace-transform Adomain decomposition method for the solution of RDE. Other works on the analytic solution of RDE can be found in [14,15] and the references therein.
Most of the time, the exact/analytical solution of FDEs cannot be found, so numerical approximations must be utilized [16][17][18][19]. Numerous numerical methods have been developed for numerical approximation of FDEs such as the Chebyshev collocation method [20], variation iteration method [16], reproducing kernel Hilbert space method [18,19], and homotopy perturbation method ( [21] and references therein). e numerical solutions of fractionalorder RDE have been studied by a large number of researchers. For example, the authors [20] studied fractional-order RDE with ABC derivatives, and they established the existence and uniqueness results using Banach fixedpoint theorem. A reproducing kernel Hilbert space method [19] for approximating RDEs and Bernoulli differential equations of fractional order with ABC derivative has been presented. In [18], the authors developed an iterative reproducing kernel Hilbert space method for numerical approximation of fractional RDE. e authors [22] studied the numerical solution of fractional-order RDE using the modified homotopy perturbation method. Numerical solution of fractional-order RDEs using Bernstein polynomials is considered [23]. A fractional Chebyshev finite difference method [24] for numerical investigation of RDE of fractional order is proposed. Jafari and Tajadodi [25] proposed a variational iteration method for solving the RDE of fractional order. e authors in [26] proposed a hybrid method based on the quasilinearization technique and reproducing kernel method for RDE of fractional order. In [27], the solution of fractional-order RDE is approximated using fractional polynomial approximations. e authors in [28] developed a method based on finite difference and Padévariational iteration method for solving the RDE of noninteger order. A numerical method based on the path following methods and the Tau Legendre is presented [29] for the solution of fractional-order RDE. e authors of [30] proposed a modified variational iteration method based on Adomain polynomials for the solution of RDE. Khashan et al. [31] have utilized the Haar wavelets for the approximation of fractional-order RDE. e authors of [32] utilized the finite difference scheme for the approximation of RDE. Other works on the numerical approximation of the solution of fractional-order RDE can be found in [33][34][35][36][37][38][39][40][41][42] and references therein. In this work, we approximate the solution of fractional-order RDE with ABC derivative of the following form: here, D β x denotes Atangana-Baleanu fractional derivative of order β.

Definition 1.
e Mittag-Leffler (ML) function with one parameter is defined as [52,53] Definition 2. e two-parameter Mittag-Leffler (ML) function is defined as [53,54] Definition 3. e ABC fractional derivative of order 0 < β < 1 of χ * * ∈ S 1 (a, b) with base point a at x ∈ (a, b) is defined as [7] D β where S 1 is a first-order Sobolev space equipped with L 2 -norm over the region Ω ⊂ R, which is defined as and the term M(β) is given as Definition 5.
e Laplace transform of a piecewise continuous function χ(x) is defined as Definition 6. e LT of ML function with one parameter is defined as [4] Definition 7. e LT of two-parameter ML function is defined as [4] Definition 8. If 0 < β ≤ 1, then the LT of the ABC derivative is defined by [7] L D β

Laplace Transform Method for Fractional Riccati Equation with ABC Derivative
In this section, we give a detailed description of our proposed numerical scheme for approximating the inverse Laplace transform. First, we will apply the Laplace transform to the given fractional problem, which will transform it to an algebraic equation. After solving the reduced equation, the solution of the original problem can be obtained by representing it as a contour integral in the left half of the complex plane. e trapezoidal rule is then utilized to approximate the contour integral. Applying the Laplace transform to equation (1), we obtain which can be written in simplified form as where In our method, first we represent the solution χ(x) of original problem (1) as a contour integral: where, for Re s ≥ ω, ω is appropriately large and Γ is an initially appropriately chosen line Γ 0 perpendicular to the real axis in the complex plane, with Im s ⟶ ± ∞. Integral (15) is just the inverse transform of χ(s), with the condition that it must be analytic to the right of Γ 0 . To make sure the contour of integration remains in the domain of analyticity of χ(x), we select Γ as a deformed contour in the set Σ ϑ ϕ � s ≠ 0: |arg s| < ϕ ∪ 0 { }, which behaves as a pair of asymptotes in the left half plane, with Re s ⟶ − ∞ when Im s ⟶ ± ∞, which force e sx to decay towards both ends of Γ. In our work, we choose Γ as where By writing s � x + ιy, we notice that (16) is the left branch of the following hyperbola: e asymptotes for (18) are (17) confirms that Γ lies in the sector Σ ϑ ϕ � ϑ + Σ ϕ ⊂ Σ ϕ and grows into the left half plane. From (16) and (15), we obtain e trapezoidal rule is used for the approximation of equation (19) with step k as follows: where λ j � jk, s j � s(λ j ), and s j ′ � s ′ (λ j ).

Error Analysis
In the process of obtaining the solution of problem equations (1)-(13), the fractional integrodifferential equation is first transformed to an algebraic equation using Laplace transform, and this causes no error. e transformed equation is then solved for the unknown in the Laplace space. Finally, the solution is obtained using inverse Laplace transform via integral representation (19). e integral is then approximated using quadrature rule. In the process of approximating integral (19), convergence is achieved at different rates depending on the path Γ. In approximating integral (19), the convergence order relies on the step k of the quadrature rule and the time domain [t 0 , T]. e proof for the order of quadrature error is given in the next theorem.

Results and Discussion
Most of the time analytical methods cannot be applied to handle a real-world problem. So we need numerical methods to approximate the solutions of the problems. In this section, we consider fractional-order Riccati equations to validate our method.

Problem 1.
Here, we consider the fractional Riccati equation as follows: with . In this experiment, the optimal parameters utilized are θ � 0. 10 Table 1. e absolute errors for various quadrature nodes N and fractional order β are depicted in Table 2.

Journal of Mathematics 5
Problem 3. Here, we consider the fractional Riccati equation as follows: with where, e problem has exact solution χ(x) � x (β+1) + 1. In this experiment, the same set of optimal parameters is utilized. e values of approximate solution for different fractional orders β and x ∈ [0, 1] are depicted in Table 4. e absolute errors for various values of N and fractional order β are shown in Table 5. Figure 5 shows the comparison between the absolute error and error_est, and a good agreement between them is observed. e plots of numerical and exact solutions for different fractional orders are displayed in Figures 6-9. It can be seen that the proposed method has produced good results and the exact and numerical solutions are in good agreement. is shows that this method can with χ(0) � 1.     Table 6. Figure 10(a) displays the comparison between the absolute error and error estimate. Figure 10(b) shows the error function for β � 1.

Problem 5.
Here, we consider the fractional Riccati equation as follows: with χ(0) � 0. e problem has exact solution χ(x) � − x 2 − βx. In this experiment, the same set of optimal parameters is utilized. e values of approximate solution for different fractional orders β and x ∈ [0, 1] are displayed in Table 7. e comparison between absolute error and error estimate is shown in Figure 11(a), and Figure 11(b) shows the error function for fractional order β � 0.95.

Conclusion
In this work, we developed a numerical scheme based on LT and inverse LT for approximation of the solution of fractional Riccati equations with ABC derivative. e inverse LT is approximated using the quadrature rule. e proposed method approximated the fractional Riccati equation with ABC derivative accurately and efficiently. From the results, it can be seen that this method is an excellent alternative for approximation of such type of equations.

Data Availability
Data are included within this paper.

Conflicts of Interest
e authors declare that they have no conflicts of interest.

Authors' Contributions
Xin Liu improved literature review and gave numerical applications of results and present examples. Kamran wrote the paper, and Yukun Yao gave error analysis and plotted error functions.