Steady-State Problem of an S-K-T Competition Model with Spatially Degenerate Coefficients

In this paper, we study the steady-state problem of an S-K-T competition model with a spatially degenerate intraspecific competition coefficient. First, the global bifurcation continuum of positive steady-state solutions from its semitrivial steady-state solution is given, which depends on the spatial heterogeneity and cross-diffusion. Second, two limiting systems are derived as the cross-diffusion coefficient tends to infinity. Moreover, we demonstrate the existence of positive steady-state solutions near the two limiting systems, and show which one of the limiting systems characterizes the positive steady-state solution.

On a Semilinear Parabolic Problem with Four-Point Boundary Conditions

This paper studies a semilinear parabolic equation in 1D along with nonlocal boundary conditions. The value at each boundary point is associated with the value at an interior point of the domain, which is known as a four-point boundary condition. First, the solvability of a steady-state problem is addressed and a constructive algorithm for finding a solution is proposed. Combining this schema with the semi-discretization in time, a constructive algorithm for approximation of a solution to a transient problem is developed. The well-posedness of the problem is shown using the semigroup theory in C-spaces. Numerical experiments support the theoretical algorithms.

Quadratic Constrained Periodic Optimization for Bandlimited Linear Systems Via the Fourier-Based Method

Motivated by engineering applications, we address bounded steady-state optimal control of linear dynamical systems undergoing steady-state bandlimited periodic oscillations. The optimization can be cast as a minimization problem by expressing the state and the input as finite Fourier series expansions, and using the expansions coefficients as parameters to be optimized. With this parametrization, we address linear quadratic problems involving periodic bandlimited dynamics by using quadratic minimization with parametric time-dependent constraints. We hence investigate the implications of a discretization of linear continuous time constraints and propose an algorithm that provides a feasible suboptimal solution whose cost is arbitrarily close to the optimal cost for the original constrained steady-state problem. Finally, we discuss practical case studies that can be effectively tackled with the proposed framework, including optimal control of DC/AC power converters, and optimal energy harvesting from pulsating mechanical energy sources.

The periodic response of a fractional oscillator with a spring-pot and an inerter-pot

The fractional oscillation system with two Weyl-type fractional derivative terms $_{ - \infty }D_t^\beta x$ (0 < β < 1) and $_{ - \infty }D_t^\alpha x$ (1 < α < 2), which portray a “spring-pot” and an “inerter-pot” and contribute to viscoelasticity and viscous inertia, respectively, was considered. At first, it was proved that the fractional system with constant coefficients under harmonic excitation is equivalent to a second-order differential system with frequency-dependent coefficients by applying the Fourier transform. The effect of the fractional orders β (0 < β < 1) and α (1 < α < 2) on inertia, stiffness and damping was investigated. Then, the harmonic response of the fractional oscillation system and the corresponding amplitude–frequency and phase–frequency characteristics were deduced. Finally, the steady-state response to a general periodic incentive was obtained by utilizing the Fourier series and the principle of superposition, and the numerical examples were exhibited to verify the method. The results show that the Weyl fractional operator is extremely applicable for researching the steady-state problem, and the fractional derivative is capable of describing viscoelasticity and portraying a “spring-pot”, and also describing viscous inertia and serving as an “inerter-pot”.

The symmetry analysis of electrostatic micro-electromechanical system (MEMS)

The model of electrostatic Micro-Electromechanical System (MEMS) is considered without/with an external pressure. The model represents nonlinear elliptic or parabolic problem due to the steady or non-steady state problem, respectively. To obtain exact solutions in an explicit form, the symmetry analysis is considered. With symmetries, the transformations are obtained and by means of these transformations, solvable equations are hold. The obtained results have a major role in the literature so that the considered equation is seen in a large-scale applications.

On a Class of Reaction-Diffusion Equations with Aggregation

In this paper, global-in-time existence and blow-up results are shown for a reaction-diffusion equation appearing in the theory of aggregation phenomena (including chemotaxis). Properties of the corresponding steady-state problem are also presented. Moreover, the stability around constant equilibria and the non-existence of nonconstant solutions are studied in certain cases.

Power system steady state calculations using artificial neural networks

The power systems steady-state problem are described by a system of nonlinear equations, and for their solution are widely used iterative techniques such as the Newton-Raphson and others. Recently, techniques based on the use of genetic algorithms, the theory of fuzzy sets, artificial neural networks have been applied to solve this problem. In this article feedforward neural networks are used for calculating the steady-state regimes. The modeling results were obtained with the results of calculations using the Newton-Raphson method.

Convergence of a homotopy finite element method for computing steady states of Burgers’ equation

In this paper, the convergence of a homotopy method (1.1) for solving the steady state problem of Burgers’ equation is considered. When ν is fixed, we prove that the solution of (1.1) converges to the unique steady state solution as ε → 0, which is independent of the initial conditions. Numerical examples are presented to confirm this conclusion by using the continuous finite element method. In contrast, when ν = ε →, numerically we show that steady state solutions obtained by (1.1) indeed depend on initial conditions.

Simulation of Conjugate Heat Transfer in Thermal Processes with Open Source CFD

A verification and validation study was performed using the open source computational fluid dynamics software package OpenFOAM version 6-dev for conjugate heat transfer problems. The test cases had a growing complexity starting from a simple steady state problem over unsteady heat transfer to more realistic engineering applications. First, a fin effectiveness study was performed. Then, the external convection at pipes and internal pipe heat transfer were investigated. The validity of the techniques was shown for each test case by comparing the simulation results with experimental and analytic data available in the literature. Finally, a simplified shell-and-tube heat exchanger was simulated to demonstrate how these methods can be applied to plant scale engineering problems.

Subsea Cable Laying Problem

Subsea cable laying process is a difficult task for an engineer due to many uncertain situations which occur during the operation. It is very often that the cable being laid out is not perfectly fit on the route being planned, which results in the formation of slack. In order to control wastages during installation, the slack needs to be minimized and the movement of a ship/vessel needs to be synchronized with the cable being laid out. The current problem was addressed using a mathematical model by considering a number of defining parameters such as the external forces, the cable properties and geometry. Due to the complexity, the model is developed for a steady-state problem assuming velocity of the vessel is constant, seabed is flat and the effect of wind and wave is insignificant. Non-dimensional system is used to scale the engineering parameters and grouped them into only two main parameters which are the hydrodynamic drag of the fluid and the bending stiffness of the cable. There are two solutions generated in this article; numerical and asymptotic solutions. The result of these solutions suggests that the percentage of slack can be reduced by the increase of the prescribed cable tension, and also the increase in either the drag coefficient of the sea water or the bending stiffness of the cable, similarly will result in lower slack percentage.