In this paper, a fractional order nonlinear dynamical model of interpersonal relationships has been introduced. The stability of equilibrium points is studied. Numerical simulations are also presented to verify the obtained results.
In this paper, we study the quenching behavior of the solution of a semilinear heat equation with a singular boundary outflux. We prove a finite-time quenching for the solution. Further, we show that quenching occurs on the boundary under certain conditions and we show that the time derivative blows up at a quenching point. Finally, we get a quenching rate and a lower bound for the quenching time.Theorem 4. If u 0 satisfies (2), (3), (4) and (5), then there exists a positive constant C 1 such thatfor t sufficiently close to T .since u x < 0, J(x, t) cannot attain a positive interior maximum. On the other hand, J(x, 0) ≤ 0 by (4) and J(0, t) = 0, J(1, t) = 0, for t ∈ (0, T ). By the maximum principle, we obtain that J(Integrating for t from t to T we getwhere C 1 = (p + 1) 1/(p+1) . Remark 3. We can calculate a lower bound for the quenching time. From Theorem 4, a lower bound is (1 − u 0 (0)) p+1 /(p + 1) for quenching time T . If we choose, as in Remark 1, u 0 (x) = 0.9 − 2 3 x 4.5 , then we have T = 10 −11 for p = 9.
In recent years, unmanned aerial vehicles-UAVs represent one of the most demanding technologies in aeronautics, and they have tremendous appeal because of their operability with considerable autonomy (by using minimal human intervention). UAVs have to operate in complex environments with different constraints such as obstacles, threatening zones, UAV kinematics, etc. In this technology, path planning plays a crucial role for high autonomy operations, although absolute autonomy is still an open question. In this paper, we tried to discuss, how a feasible path planning for a UAV can be done in the 3-dimensional environment by avoiding threats such as a radar network which contains several radars with different detection ranges. The proposed methodology is implemented with using genetic algorithms, and a parallel approach is used for reducing path planning calculations. The environment is represented as 3 dimensional structure by using World Wind, which is an open-source and accurate 3D environment browser. The developed methodology can provide fast and safe routes for autonomous single UAVs or operatorassisted flight.
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