Using the parametric approach to solve the continuous-time linear fractional max–min problems

A numerical algorithm based on parametric approach is proposed in this paper to solve a class of continuous-time linear fractional max-min programming problems. We shall transform this original problem into a continuous-time non-fractional programming problem, which unfortunately happens to be a continuous-time nonlinear programming problem. In order to tackle this nonlinear problem, we propose the auxiliary problem that will be formulated as a parametric continuous-time linear programming problem. We also introduce a dual problem of this parametric continuous-time linear programming problem in which the weak duality theorem also holds true. We introduce the discrete approximation method to solve the primal and dual pair of parametric continuous-time linear programming problems by using the recurrence method. Finally, we provide two numerical examples to demonstrate the usefulness of this algorithm.     

The existence of an extremal solution to a nonlinear system with the right-handed Riemann–Liouville fractional derivative

A monotone iterative method is applied to show the existence of an extremal solution for a nonlinear system involving the right-handed Riemann–Liouville fractional derivative with nonlocal coupled integral boundary conditions. Two comparison results are established. As an application, an example is presented to demonstrate the efficacy of the main result.     

A reliable approach to the solution of Navier–Stokes equations

In this paper we will demonstrate an affective approach of solving Navier–Stokes equations by using a very reliable transformation method known as the Cole–Hopf transformation, which reduces the problem from nonlinear into a linear differential equation which, in turn, can be solved effectively.     

Analytical solution of the damped Helmholtz–Duffing equation

In this paper, we derive a class of analytical solution of the damped Helmholtz–Duffing oscillator that is based on a recently developed exact solution for the undamped case. Our solution procedure indicates that this solution holds for specific system parametric choice values.     

From the Klein–Gordon–Zakharov system to a singular nonlinear Schrödinger system

In this paper, we continue our investigation of the high-frequency and subsonic limits of the Klein–Gordon–Zakharov system. Formally, the limit system is the nonlinear Schrödinger equation. However, for some special case of the parameters going to the limits, some new models arise. The main object of this paper is the derivation of those new models, together with convergence of the solutions along the limits.     

Analysis of the nonlinear vibration of a two-mass–spring system with linear and nonlinear stiffness

An analytical approach is developed for areas of nonlinear science such as the nonlinear free vibration of a conservative, two-degree-of-freedom mass–spring system having linear and nonlinear stiffnesses. The main contribution of this research is twofold. First, it introduces the transformation of two nonlinear differential equations for a two-mass system using suitable intermediate variables into a single nonlinear differential equation and, more significantly, the treatment of a nonlinear differential system by linearization coupled with Newton’s method. Secondly, the major section is the solving of the governing nonlinear differential equation where the displacement of the two-mass system can be obtained directly from the linear second-order differential equation using a first-order variational approach. The aforementioned approach proposed by J.H. He, who actually developed the method, is exactly He’s variational method. This approach is an explicit method with high validity for resolving strong nonlinear oscillation system problems. Two examples of nonlinear two-degree-of-freedom mass–spring systems are analyzed, and verified with published results and exact solutions. The method can be easily extended to other nonlinear oscillations and so could be widely applicable in engineering and science.     

An open problem on the optimality of an asymptotic solution to Duffing’s nonlinear oscillation problem

Based on the renormalization group method, Kirkinis (2012) [8] obtained an asymptotic solution to Duffing’s nonlinear oscillation problem. Kirkinis then asked if the asymptotic solution is optimal. In this paper, an affirmative answer to the open problem is given by means of the homotopy analysis method.     

On the resolution of certain discrete univariate max–min problems

Graph coloring is a classical NP-hard combinatorial optimization problem with many practical applications. A broad range of heuristic methods exist for tackling the graph coloring problem: from fast greedy algorithms to more time-consuming metaheuristics. Although the latter produce better results in terms of minimizing the number of colors, the former are widely employed due to their simplicity. These heuristic methods are centralized since they operate by using complete knowledge of the graph. However, in real-world environmets where each component only interacts with a limited number of other components, the only option is to apply decentralized methods. This paper explores a novel and simple algorithm for decentralized graph coloring that uses a fixed number of colors and iteratively reduces the edge conflicts in the graph. We experimentally demonstrate that, for most of the tested instances, the new algorithm outperforms a recent and very competitive algorithm for decentralized graph coloring in terms of coloring quality. In our experiments, the fixed number of colors used by the new algorithm is controlled in a centralized manner.     

Optimal control of a nonlinear parabolic–elliptic system

In this paper, we shall study the problem of optimal control of the parabolic–elliptic system

_ut+_(f(t,x,u))x+g(t,x,u)+_Px−_(a(t,x)ux)x=_f0+B^∗ν$u_t+{\left(f(t,x,u)\right)}_x+g(t,x,u)+P_x-{\left(a(t,x)u_x\right)}_x=f_0+B^{\ast }\nu$

Nonuniform continuity of the solution map to the two component Camassa–Holm system

We prove that the solution map of the two-component Camassa–Holm system is not uniformly continuous as a map from a bounded subset of the Sobolev space H^s(T)×H^r(T)$H^s(\mathbb{T})\times H^r(\mathbb{T})$ to C([0,1],H^s(T)×H^r(T))$C([0,1],H^s(\mathbb{T})\times H^r(\mathbb{T}))$ when s?1$s?1$ and r?0$r?0$. We also demonstrate the nonuniform continuous property in the continuous function space C¹(T)×C¹(T)$C^1(\mathbb{T})\times C^1(\mathbb{T})$.     

On the Hölder regularity of the weak solution to a drift–diffusion system with pressure

In this paper we address the regularity issue of weak solution for the following linear drift–diffusion system with pressure

$$\begin{aligned} \partial _t u + b\cdot \nabla u -\Delta u + \nabla p = 0,\quad \mathrm {div}\,u=0,\quad u|_{t=0}(x)=u_0(x), \end{aligned}$$

where \(x\in \mathbb {R}^n\) and b is a given divergence-free vector field. Under some assumptions of the drift field b in the critical sense, and for the initial data \(u_0\in (L^2(\mathbb {R}^n))^n\), we prove that there exists a weak solution u(t) to this system such that u(t) for any time \(t>0\) is \(\alpha \)-Hölder continuous with \(\alpha \in (0,1)\). The proof of the Hölder regularity result utilizes a maximum-principle type method to improve the regularity of weak solution step by step.

     

A nonlinear analytic approach to the splitting theorem of Jacobs–de Leeuw–Glicksberg

In this paper, we study nonlinear analytic methods for linear contractive semigroups in Banach spaces and apply them to the splitting theorem of Jacobs–de Leeuw–Glicksberg. Using these results, we obtain the extension of Lin’s proposition for a group of linear operators to a semigroup.     

On the concentration of semi-classical states for a nonlinear Dirac–Klein–Gordon system

In the present paper, we study the semi-classical approximation of a Yukawa-coupled massive Dirac–Klein–Gordon system with some general nonlinear self-coupling. We prove that for a constrained coupling constant there exists a family of ground states of the semi-classical problem, for all ?   small, and show that the family concentrates around the maxima of the nonlinear potential as ?→0$?\to 0$. Our method is variational and relies upon a delicate cutting off technique. It allows us to overcome the lack of convexity of the nonlinearities.     

The nonlinear future stability of the FLRW family of solutions to the Euler–Einstein system with a positive cosmological constant

In this article, we study small perturbations of the family of Friedmann–Lemaître–Robertson–Walker cosmological background solutions to the 1 + 3 dimensional Euler–Einstein system with a positive cosmological constant. These background solutions describe an initially uniform quiet fluid of positive energy density evolving in a spacetime undergoing accelerated expansion. Our nonlinear analysis shows that under the equation of state ${p = c^2_s \rho}$ , ${0 < c^2_s < 1/3}$ , the background solutions are globally future-stable. In particular, we prove that the perturbed spacetime solutions, which have the topological structure ${[0,\infty) \times \mathbb{T}^3}$ , are future-causally geodesically complete. These results are extensions of previous results derived by the author in a collaboration with I. Rodnianski, in which the fluid was assumed to be irrotational. Our novel analysis of a fluid with non-zero vorticity is based on the use of suitably defined energy currents.     

Analytical solution of a spatially variable coefficient advection–diffusion equation in up to three dimensions

Analytical solutions are provided for the two- and three-dimensional advection–diffusion equation with spatially variable velocity and diffusion coefficients. We assume that the velocity component is proportional to the distance and that the diffusion coefficient is proportional to the square of the corresponding velocity component. There is a simple transformation which reduces the spatially variable equation to a constant coefficient problem for which there are available a large number of known analytical solutions for general initial and boundary conditions. These solutions are also solutions to the spatially variable advection–diffusion equation. The special form of the spatial coefficients has practical relevance and for divergent free flow represent corner or straining flow. Unlike many other analytical solutions, we use the transformation to obtain solutions of the spatially variable coefficient advection–diffusion equation in two and three dimensions. The analytical solutions, which are simple to evaluate, can be used to validate numerical models for solving the advection–diffusion equation with spatially variable coefficients. For numerical schemes which cannot handle flow stagnation points, we provide analytical solution to the spatially variable coefficient advection–diffusion equation for two-dimensional corner flow which contains an impermeable flow boundary. The impermeable flow boundary coincides with a streamline along which the fluid velocity is finite but the concentration vanishes. This example is useful for validating numerical schemes designed to predict transport around a curved boundary.     

Grasping the phantom – a new approach to nonlinear spectral theory

A new definition of a spectrum for nonlinear operators is suggested, which is called phantom. The phantom naturally divides into two subsets which in the linear case correspond to the point spectrum, and to the union of the continuous and residual spectrum. There is also a natural nonlinear analogue to the approximate point spectrum. The phantom may be considered as a variant of the spectrum of M. Furi, M. Martelli, and A. Vignoli (Ann. Mat. Pura Appl. 118 (1978), 229–294), which reflects the behavior of the operator on bounded sets and which is based on a new definition of an eigenvalue that was suggested earlier by the authors (Nonlinear Anal. 40 (2000), 565–576).?To define the phantom, the class of stably 0-epi maps is introduced and studied. This is a subclass of 0-epi maps satisfying a Rouché type theorem. Received: November 15, 1999?Published online: October 2, 2001     

On the topology of the set of singularities of a solution to the Hamilton–Jacobi equation

We address the topology of the set of singularities of a solution to a Hamilton–Jacobi equation. For this, we will apply the idea of the first two authors (Cannarsa and Cheng, Generalized characteristics and Lax–Oleinik operators: global result, preprint, arXiv:1605.07581, 2016) to use the positive Lax–Oleinik semi-group to propagate singularities.