Abstract
The 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. The reduced equation is solved for the unknown in LT space. The solution of the original problem is retrieved by representing it as a Bromwich integral in the complex plane along a smooth curve. The Bromwich integral is approximated using the trapezoidal rule. Some numerical experiments are performed to validate our numerical scheme.
Highlights
IntroductionThe fractional differential equation (FDE) is an equation which contains derivatives of arbitrary order
In applied mathematics, the fractional differential equation (FDE) is an equation which contains derivatives of arbitrary order
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]
Summary
The fractional differential equation (FDE) is an equation which contains derivatives of arbitrary order. 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. E authors of [30] proposed a modified variational iteration method based on Adomain polynomials for the solution of RDE. We approximate the solution of fractional-order RDE with ABC derivative of the following form: Dβxχ(x) F(x, χ(x)), x ∈ [0, 1], β ∈ [0, 1], (1) χ(0) α, here, Dβx denotes Atangana–Baleanu fractional derivative of order β
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