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.

In this paper, we consider the existence of at least three positive solutions for the 2nth order differential equations with integral boundary conditionswhere (-1) n f > 0 is continuous, and k i (t) ∈ L 1 [0, 1] (i = 0, 1, . . . , n -1) are nonnegative. The associated Green's function for the higher order differential equations with integral boundary conditions is first given, and growth conditions are imposed on f which yield the existence of multiple positive solutions by using the Leggett-Williams fixed point theorem.

Objective: To study the therapeutic effect of thalidomide on ankylosing spondylitis (AS) by meta-analysis. Methods: Personal digital library, Cochrane library, and China Biology Medicine disc (CMBdisc), as well as relevant pharmaceutical and medical journals, were collected and reviewed. After the analysis of characteristics of the selected document and the evaluation of the risk of bias, the therapeutic effect of thalidomide on ankylosing spondylitis (AS) and its influence on related indexes were analyzed by literature data. Results: The meta-analysis results of 8 pieces of literature showed that the total effective rate of thalidomide in the treatment of ankylosing spondylitis (AS) was significantly improved, compared with conventional treatment or sulfasalazine (SASP) treatment (P＜0.05). Furthermore, the time of morning stiffness, BASDAI score, C-reactive protein (CRP) level, and other related symptoms and indexes were significantly optimized (P＜0.05). Conclusion: By rational utilization of thalidomide in the treatment of ankylosing spondylitis (AS), related symptoms and indexes of patients can be effectively improved, the total effective rate of the treatment was significantly improved and the safety of the treatment can be guaranteed.