Our purpose in this paper is to study when a planar differential system polynomial in one variable linearizes in the sense that it has an inverse integrating factor which can be constructed by means of the solutions of linear differential equations. We give several families of differential systems which illustrate how the integrability of the system passes through the solutions of a linear diff...

side weir is a flow control structure that used extensively in irrigation and drainage as well as sewer networks. throughout the present research, a study comprised of 162 experiments was done on sharp-crested semi-circular side weir. since the height in this side weir varies along its length, this feature enables it to more properly control flood than rectangular side weirs in various flood co...

An canonical process is stationary independent increment uncertain process whose increments are normal uncertain variables. Uncertain differential equation is a type of differential equation driven by the canonical process. This paper will give a concept of stability in p-th moment for uncertain differential equations. A sufficient and necessary condition for linear uncertain differential equat...

We derive properties of powers of a function satisfying a second-order linear differential equation. In particular we prove that the n-th power of the function satisfies an (n + 1)-th order differential equation and give a simple method for obtaining the differential equation. Also we determine the exponents of the differential equation and derive a bound for the degree of the polynomials, whic...

p−1 p p−1 . This equation is a special case of a general half-linear second order differential equation (r(t)Φ(x′))′ + c(t)Φ(x) = 0, (2) where Φ(x) := |x| sgn x, p > 1, and r, c are continuous functions, r(t) > 0 (in the studied equation (1) we have r(t) ≡ 1). Let us recall that similarly as in the linear case, which is a special case of (2) for p = 2 and equation (2) then reduces to the linear...

Uncertain differential equation with jumps, as a crucial tool to deal with a discontinuous uncertain system, is a type of differential equation driven by both canonical Liu process and uncertain renewal process. So far, a concept of stability in measure for an uncertain differential equation with jumps has been proposed. As a supplement, this paper proposes a concept of almost sure stability fo...

In this paper, the optimal power flow (OPF) problem is augmented to account for the costs associated with the loadfollowing stability of a power network. Load-following stability costs are expressed through the linear quadratic regulator (LQR). The power network is described by a set of nonlinear differential algebraic equations (DAEs). By linearizing the DAEs around a known equilibrium, a line...

The state of the art in the spectral theory of linear time-varying differential-algebraic equations (DAEs) is surveyed. To characterize the asymptotic behavior and the growth rate of solutions, basic spectral notions such as Lyapunovand Bohl exponents, and Sacker-Sell spectra are discussed. For DAEs in strangeness-free form, the results extend those for ordinary differential equations, but only...