A numerical simulation for the blow-up of semi-linear diffusion equations

Many mathematical models have the property of developing singularities at a finite time; in particular, the solution u(x, t) of the semi-linear parabolic Equation (1) may blow up at a finite time T. In this paper, we consider the numerical solution with blow-up. We discretize the space variables with a spectral method and the discrete method used to advance in time is an exponential time differencing scheme. This numerical simulation confirms the theoretical results of Herrero and Velzquez [M.A. Herrero and J.J.L. Velzquez, Blow-up behavior of one-dimensional semilinear parabolic equations, Ann. Inst. Henri Poincare 10 (1993), pp. 131–189.] in the one-dimensional problem. Later, we use this method as an experimental approach to describe the various possible asymptotic behaviours with two-space variables.     

Global and blow-up solutions for nonlinear parabolic equations with Robin boundary conditions

In this paper we discuss the blow-up for classical solutions to the following class of parabolic equations with Robin boundary condition: $\{\begin{array}{cc}{\left(b\left(u\right)\right)}_t=??\left(g\left(u\right)?u\right)+f\left(u\right)\hfill & \text{in }\Omega \times (0,T)\text{,}\hfill \\ \frac{?u}{?n}+\gamma u=0\hfill & \text{on }?\Omega \times (0,T)\text{,}\hfill \\ u(x,0)=h\left(x\right)\ge 0\hfill & \text{in }\overline{\Omega }\text{,}\hfill \end{array}$ where $\Omega$ is a bounded domain of ${\mathbb{R}}^N(N\ge 2)$ with smooth boundary $?\Omega$. By constructing some appropriate auxiliary functions and using a first-order differential inequality technique, we derive conditions on the data which guarantee the blow-up or the global existence of the solution. For the blow-up solution, a lower bound on blow-up time is also obtained. Moreover, some examples are presented to illustrate the applications.     

A modified L-scheme to solve nonlinear diffusion problems

In this work, we propose a linearization technique for solving nonlinear elliptic partial differential equations that are obtained from the time-discretization of a wide variety of nonlinear parabolic problems. The scheme is inspired by the L-scheme, which gives unconditional convergence of the linear iterations. Here we take advantage of the fact that at a particular time step, the initial guess for the iterations can be taken as the solution of the previous time step. First it is shown for quasilinear equations that have linear diffusivity that the scheme always converges, irrespective of the time step size, the spatial discretization and the degeneracy of the associated functions. Moreover, it is shown that the convergence is linear with convergence rate proportional to the time step size. Next, for the general case it is shown that the scheme converges linearly if the time step size is smaller than a certain threshold which does not depend on the mesh size, and the convergence rate is proportional to the square root of the time step size. Finally numerical results are presented that show that the scheme is at least as fast as the modified Picard scheme, faster than the L-scheme and is more stable than the Newton or the Picard scheme.     

Oscillation of solutions for second-order nonlinear difference equations with nonlinear neutral term

Some Riccati type difference inequalities are given for the second-order nonlinear difference equations with nonlinear neutral term.

and using these inequalities, we obtain some oscillation criteria for the above equation.     

A posteriori error estimates of two-grid finite volume element methods for nonlinear elliptic problems

In this paper, we study the a posteriori error estimates of two-grid finite volume element method for second-order nonlinear elliptic equations. We derive the residual-based a posteriori error estimator and prove the computable upper and lower bounds on the error in $H^1$-norm. The a posteriori error estimator can be used to assess the accuracy of the two-grid finite volume element solutions in practical applications. Numerical examples are provided to illustrate the performance of the proposed estimator.     

Non-smooth simultaneous stabilization of nonlinear systems: An interpolation method

In this paper, we introduce a method which enables us to construct a continuous simultaneous stabilizer for pairs of systems in which cannot be simultaneously stabilized by means of C¹ feedback. We extend this method to higher-dimensional systems and show that any pair of asymptotically stabilizable nonlinear systems can be simultaneously stabilized (not asymptotically) by means of continuous feedback.     

A genetic algorithm for the simultaneous delivery and pickup problems with time window

This paper concerns a Simultaneous Delivery and Pickup Problem with Time Windows (SDPPTW). A mixed binary integer programming model was developed for the problem and was validated. Due to its NP nature, a co-evolution genetic algorithm with variants of the cheapest insertion method was proposed to speed up the solution procedure. Since there were no existing benchmarks, this study generated some test problems which revised from the well-known Solomon’s benchmark for Vehicle Routing Problem with Time Windows (VRPTW). From the comparison with the results of Cplex software and the basic genetic algorithm, the proposed algorithm showed that it can provide better solutions within a comparatively shorter period of time.     

Time second-order finite difference/finite element algorithm for nonlinear time-fractional diffusion problem with fourth-order derivative term

In this article, we study and analyze a Galerkin mixed finite element (MFE) method combined with time second-order discrete scheme for solving nonlinear time fractional diffusion equation with fourth-order derivative term. We firstly introduce an auxiliary variable $\sigma =△u$, reduce the fourth-order problem into a coupled system with two equations, discretize the obtained coupled system at time $t_{k?\frac{\alpha }{2}}$ by a second-order difference scheme with second-order approximation for fractional derivative, then formulate mixed weak formulation and fully discrete MFE scheme. Further, we give the detailed proof for stability of scheme, the existence and uniqueness of MFE solution, and a priori error estimates. Finally, by some numerical computations, we test the theoretical results, which illustrate that we can obtain the numerical results for two variables, moreover, we arrive at second-order time convergence orders, which are higher than the ones yielded by the $L1$-approximation.     

Solution of nonlinear evolution problems by parallelized collocation-interpolation methods

This paper deals with the solution of initial-boundary value problems for nonlinear evolution equations. The solution technique is based on collocation-interpolation methods which are improved in order to reduce computational errors at fixed discretizations of the independent variables. The method consists in parallelizing the approximation of the space derivatives so that the same approximation is reached by a lower number of collocation points, and hence, by a lower computation time. The analysis includes a theoretical and computational estimate both of the approximation error and of the computational time.     

On finite dimensional realization theory of discrete time nonlinear systems

We characterize finite dimensional realizability of discrete time nonlinear systems which have a Volterra series development with separable structure of the Volterra kernels.     

Finite-horizon dynamic optimization of nonlinear systems in real time

A novel scheme for constructing and tracking the solution trajectories to regular, finite-horizon, deterministic optimal control problems with nonlinear dynamics is devised. The optimal control is obtained from the states and costates of Hamiltonian ODEs, integrated online. In the one-dimensional case the initial costate is found by successively solving two first-order, quasi-linear, partial differential equations, whose independent variables are the time-horizon duration T and the final penalty coefficient S. These PDEs should in general be integrated off-line, the solution rendering not only the missing initial condition sought in the particular (T,S)-situation, but additional information on the boundary values of the whole two-parameter family of control problems, which can be used for designing the definitive objective functional. Optimal trajectories for the model are then generated in real time and used as references to be followed by the physical system.  Numerical improvements are suggested for accurate integration of naturally unstable Hamiltonian dynamics, and strategies are proposed for tracking their results, in finite time or asymptotically, when perturbations in the state of the system appear. The whole procedure is tested in models arising in aero-navigation optimization.     

The method of particular solutions for solving nonlinear Poisson problems

Based on the recent development in the method of particular solutions, we re-exam three approaches using different basis functions for solving nonlinear Poisson problems. We further propose to simplify the solution procedure by removing the insolvency condition when the radial basis functions are augmented with high order polynomial basis functions. We also specify the deficiency of some of these methods and provide necessary remedy. The traditional Picard method is introduced to compare with the recent proposed methods using MATLAB optimization toolbox solver for solving nonlinear Poisson equations. Ranking on these three approaches are given based on the results of numerical experiment.     

The asymptotic properties of estimates of the parameters of nonlinear time series

Asymptotic properties of nonlinear time series parameter estimators constructed on trajectories of stochastic systems under stationary and transient conditions are studied with the use of the least-squares method. The investigation method is based on the study of asymptotic properties of extremal sets of random functions. Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 62–72, March–April, 2000     

Instability in supercritical nonlinear wave equations: Theoretical results and symplectic integration

Nonlinear wave evolutions involve a dynamical balance between linear dispersive spreading of the waves and nonlinear self-interaction of the waves. In sub-critical settings, the dispersive spreading is stronger and therefore solutions are expected to exist globally in time. We show that in the supercritical case, the nonlinear self-interaction of the waves is much stronger. This leads to some sort of instability of the waves. The proofs are based on the construction of high frequency approximate solutions. Preliminary numerical simulations that support these theoretical results are also reported.     

On the asymptotic stability of coupled delay differential and continuous time difference equations

A two-step Liapunov-Krasovskii methodology for checking the asymptotic stability of nonlinear coupled delay differential and continuous time difference equations is proposed here. The feasibility of such methodology is shown by means of Liapunov-Krasovskii functionals with nonconstant kernels in the integrals, for instance discretized Liapunov-Krasovskii ones. An illustrative example taken from the literature, showing the effectiveness of the proposed method, is reported.     

A new feature-preserving nonlinear anisotropic diffusion for denoising images containing blobs and ridges

Blobs and ridges underlie many important features in biological, biometric and remote sensing images. These images are likely to be corrupted by noise, such as live cells in fluorescent biological images, ridges and valleys in fingerprints and moving targets in synthetic aperture radar and infrared images. In this paper we present a diffusion method for denoising low-signal-to-ratio images containing blob and ridge features. A commonly used denoising method makes use of edge information in an image to achieve a good balance between noise removal and feature preserving. However, if edges are partly lost to a certain extent or contaminated severely by noise, such an approach may not be able to preserve these features, leading to loss of important information. To overcome this problem, we propose a novel second-order nonlocal derivative as a robust blob and ridge detector and incorporate it into a diffusion process to form a novel feature-preserving nonlinear anisotropic diffusion model. Experiments show that the new diffusion filter outperforms many popular filters for preserving blobs and ridges, reducing noise and minimizing artifacts.     

Periodic solutions of a class of nonlinear difference equations via critical point method

In this paper, we obtain some new sufficient conditions for the existence of nontrivial m-periodic solutions of the following nonlinear difference equation

by using the critical point method, where f: Z × R → R is continuous in the second variable, m ≥ 2 is a given positive integer, p_n+m = p_n for any n  Z and f(t + m, z) = f(t, z) for any (t, z)  Z × R, (−1)^δ = −1 and δ > 0.     

An asymptotic property of nonlinear estimators arising as solutions to a certain class of convex programming problems

In this paper we examine estimators that generalize classical conditional expectation. Via a convergence principle we analyze their role in statistical inference of stochastic processes and the degradation they undergo when our observations of the process in question are distorted and occur at only finitely many times.     

The use of Kernel set and sample memberships in the identification of nonlinear time series

The problem of system modeling and identification has attracted considerable attention in the nonlinear time series analysis mostly because of a large number of applications in diverse fields like financial management, biomedical system, transportation, ecology, electric power systems, hydrology, and aeronautics. Many papers have been presented on the study of time series clustering and identification. Nonetheless, we would like to point out that in dealing with clustering time series, we should also take the vague case as they belong to two or more classes simultaneously into account. Because many patterns of grouping in time series really are ambiguous, those phenomena should not be assigned to certain specific classes inflexibly. In this paper, we propose a procedure that can effectively cluster nonlinear time series into several patterns based on kernel set. This algorithm also combines with the concept of a fuzzy set. The membership degree of each datum corresponding to the cluster centers is calculated and is used for performance index grouping. We also suggest a principle for extending the fuzzy set by kernel set and further interpret events in a sensible light through these sets. Finally, the procedure is demonstrated by set off RRI data and its performance is shown to compare favorably with other procedures published in the literature.We are grateful to the referees for their careful reading and helpful comments.