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 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.

中文翻译：

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从Caputo型分数阶导数的意义上Riccati分数阶微分方程的数值逼近

Riccati微分方程是一个著名的非线性微分方程，在工程和科学领域具有不同的应用，例如鲁棒镇定，随机实现理论，网络综合和最优控制以及金融数学。在这项研究中，我们旨在近似分数阶Riccati方程的解与Atangana–Baleanu衍生物（ABC）。我们的数值方案基于拉普拉斯变换（LT）和正交规则。我们将LT应用于给定的分数阶微分方程，从而将其简化为代数方程。对于LT空间中的未知数，可求解简化方程。通过将其表示为沿着平滑曲线的复平面中的Bromwich积分，可以检索原始问题的解。Bromwich积分使用梯形法则进行近似。进行了一些数值实验以验证我们的数值方案。