Unbounded Nonlinearities Research Articles

On the nonstandard maximum principle and its application for construction of monotone finite-difference schemes for multidimensional quasilinear parabolic equations

UDC 517.9 We consider the difference maximum principle with input data of variable sign and its application to the investigation of the monotonicity and convergence of finite-difference schemes (FDSs). Namely, we consider the Dirichlet initial-boundary value problem for multidimensional quasilinear parabolic equation with an unbounded nonlinearity. Unconditionally monotone linearized finite-difference schemes of the second-order of accuracy are constructed on uniform grids. A two-sided estimate for the grid solution, which is completely consistent with similar estimates for the exact solution, is obtained. These estimates are used to prove the convergence of FDSs in the grid L 2 -norm. We also present a study aimed at constructing second-order monotone difference schemes for the parabolic convection-diffusion equation with boundary conditions of the third kind and unlimited nonlinearity without using the initial differential equation on the domain boundaries. The goal is a combination of the assumption of existence and uniqueness of a smooth solution and the regularization principle. In this case, the boundary conditions are directly approximated on a two-point stencil of the second order.

Jumping Unbounded Nonlinearities and ALP Condition

Jumping Unbounded Nonlinearities and ALP Condition

Weak formulations of the nonlinear Poisson-Boltzmann equation in biomolecular electrostatics

We consider the nonlinear Poisson-Boltzmann equation in the context of electrostatic models for a biological macromolecule, embedded in a bounded domain containing a solution of an arbitrary number of ionic species which is not necessarily charge neutral. The resulting semilinear elliptic equation combines several difficulties: exponential growth and lack of sign preservation in the nonlinearity accounting for ion mobility, measure data arising from point charges inside the molecule, and discontinuous permittivities across the molecule boundary. Exploiting the modeling assumption that the point sources and the nonlinearity are active on disjoint parts of the domain, one can use a linear decomposition of the potential into regular and singular components. A variational argument can be used for the regular part, but the unbounded nonlinearity makes the corresponding functional not differentiable in Sobolev spaces. By proving boundedness of minimizers, these are related to standard H1 weak formulations for the regular component and in the framework of Boccardo and Gallouët for the full potential. Finally, a result of uniqueness of this type of weak solutions for more general semilinear problems with measure data validates the strategy, since the different decompositions and test spaces considered must then lead to the same solution.

On nonlinear discrete Sturm–Liouville boundary value problems with unbounded nonlinearities

This paper is devoted to the study of nonlinear Sturm–Liouville problems in the discrete setting. We consider problems for which nonlinearities in both the dynamics and boundary conditions may be unbounded. For such problems, we provide a criteria for the existence of solutions.

Long Time Dynamics for Generalized Korteweg–de Vries and Benjamin–Ono Equations

We provide an accurate description of the long time dynamics of the solutions of the generalized Korteweg-De Vries (gKdV) and Benjamin-Ono (gBO) equations on the one dimension torus, without external parameters, and that are issued from almost any (in probability and in density) small and smooth initial data. We stress out that these two equations have unbounded nonlinearities. In particular, we prove a long-time stability result in Sobolev norm: given a large constant r and a sufficiently small parameter $\epsilon$, for generic initial datum u(0) of size $\epsilon$, we control the Sobolev norm of the solution u(t) for times of order $\epsilon$^{--r}. These results are obtained by putting the system in rational normal form : we conjugate, up to some high order remainder terms, the vector fields of these equations to integrable ones on large open sets surrounding the origin in high Sobolev regularity.

Approximations to Problems of Optimal Control of Leading Coefficients of Elliptic Equations in Nondivergence Form with an Unbounded Nonlinearity in the Coefficients

Approximations to Problems of Optimal Control of Leading Coefficients of Elliptic Equations in Nondivergence Form with an Unbounded Nonlinearity in the Coefficients

Periodic Solutions for Systems with p-Relativistic Operator and Unbounded Discontinuous Nonlinearities

We are concerned with the existence of periodic solutions for potential differential inclusions involving the p-relativistic operator $$\begin{aligned} {\mathcal {R}}_pu:= \left( \frac{|u'|^{p-2}u'}{(1-|u'|^p)^{1-1/p}} \right) ' \end{aligned}$$ and an (possible) unbounded discontinuous gradient. The approach relies on critical point theory for locally Lipschitz perturbations of convex, lower semicontinuous functions. The solutions we obtain appear as either minimizers or saddle points of the corresponding energy functional. Some examples of applications illustrating the general results are provided.

Damped Klein-Gordon equation with variable diffusion coefficient

<p style='text-indent:20px;'>We consider a damped Klein-Gordon equation with a variable diffusion coefficient. This problem is challenging because of the equation's unbounded nonlinearity. First, we study the nonlinearity's continuity properties. Then the existence and the uniqueness of the solutions is established. The main result is the continuity of the solution map on the set of admissible parameters. Its application to the parameter identification problem is considered.</p>

CREEP OF A THICKWALLED QUASILINEAR VISCOELASTIC TUBE UNDER A CONSTANT EXTERNAL AND INTERNAL PRESSURE

We consider the creep problem for a quasilinear viscoelastic model of a thickwalled tube, loaded with constant internal and external pressure; the material is supposed to be incompressible. An exact solution to this problem was received by one of the authors in previous papers, assuming the state of a tube to be plain deformation; hereby we study properties of this solution for arbitrary material functions of quasilinear viscoelasticity constitutive relation. A criterion of stress stationarity is derived; the stress field of a thickwalled tube under a constant pressure evolves in time in the case of unbounded creep function and arbitrary nonlinearity function, except some particular types. The monotonicity of stress field components is studied: the radial stress monotonicity depends only on internal and external pressure values (for internal pressure, greater than an external one, it is negative and increases in radii). For other stress components, there are derived sufficient conditions of monotonicity. For an exponential nonlinearity function and unbounded creep function, a creep curve is determined to be concave up at the initial moment, and concave down during prolonged observation; the creep curve of a bipower nonlinearity function model may change its convexity. The stressstrain state of a model with a bounded creep function is proved to be bounded.

Fast-convergent fault detection and isolation in a class of nonlinear uncertain systems

The present work proposes a fast-convergent fault detection and isolation (FDI) scheme for linear systems affected by model uncertainties, such as unknown inputs or unbounded nonlinearities. The finite-time convergence is attained by transforming the I/O signals through Volterra operators with suitably designed kernel functions. A novel feature of the proposed approach is the exploitation of a system decomposition that allows removing the effect of intractable uncertainties while recasting the system dynamics in a form applicable for Volterra operators to achieve non-asymptotic estimation. Remarkably, the proposed approach can reconstruct the state variables of the system in an arbitrarily short time and the fault can be diagnosed efficiently by imposing detection and isolation thresholds on transformed signals. The detectability and isolability of the fault are also characterized. The proposed FDI scheme is applied in simulation to a web process system to diagnose the presence of actuator faults. Simulation results confirm the effectiveness of the proposed scheme in two scenarios with nonlinear uncertainties. Furthermore, comparisons are made between the proposed method and a Sliding Mode Control (SMC) method in terms of estimation performance and computational complexity.

Finite-difference method for the Gamma equation on non-uniform grids

We propose a new monotone finite-difference scheme for the second-order local approximation on a nonuniform grid that approximates the Dirichlet initial boundary value problem (IBVP) for the quasi-linear convection-diffusion equation with unbounded nonlinearity, namely, for the Gamma equation obtained by transformation of the nonlinear Black-Scholes equation into a quasilinear parabolic equation. Using the difference maximum principle, a two-sided estimate and an a priori estimate in the C-norm are obtained for the solution of the difference schemes that approximate this equation.

On Convergence of Difference Approximations to Problems of Optimal Control in the Coefficients of Elliptic Equations with Mixed Derivatives and Unbounded Non-Linearity

This paper deals with optimization problems for elliptic PDEs with mixed derivatives and nonlinearities of unlimited growth in the coefficients (i.e. the conditions imposed on the equation coefficients are assumed to hold only in a neighborhood of the exact solution to the original problem). The controls are contained in the leading part of the state equation. Convergence and approximation analysis for the optimization problems is investigated. The convergence of the approximations with regard to the state, functional and control is established, the approximations are regularized.

Maxwell's equations with arbitrary self-action nonlinearity in a waveguiding theory: Guided modes and asymptotic of eigenvalues

The paper treats a nonlinear eigenvalue problem that describes propagation of a transverse-magnetic wave in a plane dielectric waveguide having perfectly conducted walls at both sides. The dielectric's permittivity is characterised by an arbitrary monotonically increasing self-action nonlinearity. The full set of guided modes is described by eigenvalues of the corresponding (nonlinear) Maxwell operator with appropriate boundary conditions. We give a comprehensive analysis of this problem and develop an original approach to study its solvability and properties of solutions. Several results about existence of the eigenvalues are proved, their distribution and asymptotic are found; zeros of the eigenfunctions and their location are also determined; criterion of periodicity for the eigenfunctions is found, comparison theorem is derived, etc. It is shown that unbounded nonlinearities lead to appearance of nonlinearised solutions. This results in the existence of a novel guided regime that cannot be described within the framework of perturbation of linear guided wave regimes.

Monotone Finite-Difference Schemes of Second-Order Accuracy for Quasilinear Parabolic Equations with Mixed Derivatives

We consider the initial-boundary value problem for quasilinear parabolic equation with mixed derivatives and an unbounded nonlinearity. We construct unconditionally monotone and conservative finite-difference schemes of the second-order accuracy for arbitrary sign alternating coefficients of the equation. For the finite-difference solution, we obtain a two-sided estimate completely consistent with similar estimates for the solution of the differential problem, and also obtain an important a priori estimate in the uniform C-norm. These estimates are used to prove the convergence of finite-difference schemes in the grid L2-norm. All theoretical results are obtained under the assumption that some conditions imposed only on the input data of the differential problem are satisfied.

Random exponential attractor for second order non-autonomous stochastic lattice dynamical systems with multiplicative white noise in weighted spaces

This paper is concerned with the random exponential attractor for second order non-autonomous stochastic lattice system with multiplicative white noise and unbounded nonlinearity. Firstly, we transfer the stochastic lattice system into a random lattice system without noise term whose solutions generate a continuous cocycle on a weighted space of infinite sequences. Then we present some sufficient conditions for the existence of a random exponential attractor for a continuous cocycle in the product weighted space of sequences, which improved the existing conditions. Finally, we prove the existence of a random exponential attractor for the considered system in weighted space of sequences.

On convergence of difference schemes for Dirichlet IBVP for two-dimensional quasilinear parabolic equations

For Dirichlet initial boundary value problem (IBVP) for two-dimensional quasilinear parabolic equations, a monotone linearised difference scheme is constructed. The uniform parabolicity condition 0 < k1 ≤ kα(u) ≤ k2, α = 1, 2 is assumed to be fulfilled for the sign alternating solution only in the domain of exact solution values (unbounded nonlinearity). On the basis of the proved new corollaries of the maximum principle not only two-sided estimates for the approximate solution y but its belonging to the domain of exact solution values are established. We assume that the solution is continuous and its first derivatives have discontinuities of the first kind in the neighbourhood of the finite number of discontinuity lines. No smoothness of the time derivative is assumed. Convergence of approximate solution to generalised solution of differential problem in the grid norm L2 is proved.

Distributed Consensus of Multi‐Agent Systems with Input Faults and Time‐Varying Delays

AbstractThis work studies the consensus problem of multi‐agent systems with input faults and time‐varying delays. The assumed faults in the system are loss of effectiveness of actuator and nonlinear additive term mixed with nominal input. For system faults that are nonlinear, constraints of constant norm‐bounded, sector nonlinearity, and unbounded nonlinearity with known basis functions are considered. Employing a Lyapunov–Krasovskii functional method, delay dependent consensus criteria are established to show the exponential behavior of the system. Finally, one simulation example is solved to demonstrate the advantage of the obtained results.

On the consistent two-side estimates for the solutions of quasilinear convection–diffusion equations and their approximations on non-uniform grids

A new second-order in space linearized difference scheme on non-uniform grid that approximates the Dirichlet problem for multidimensional quasilinear convection–diffusion equation with unbounded nonlinearity is constructed. Proposed algorithm is a novel nonlinear generalization of difference schemes for linear problems developed earlier by the authors. Nontrivial two-side pointwise estimates of the solution of the scheme fully consistent with the corresponding estimates for the differential problem are established. Such estimates permit to prove the nonnegativity of the exact solution, important in physical problems, and also to find sufficient conditions on the input data when the nonlinear problem is parabolic. As a result a priori estimates of the approximate solution in the grid norm C that depend on the initial and boundary conditions and on the right-hand side only are proved.

Consistent convergence rate estimates in the grid W 2,0 2 (ω) norm for difference schemes approximating nonlinear elliptic equations with mixed derivatives and solutions from W 2,0 m (Ω), 3 &lt; m ≤ 4

The Dirichlet boundary value problem for nonlinear elliptic equations with mixed derivatives and unbounded nonlinearity is considered. A difference scheme for solving this class of problems and an implementing iterative process are constructed and investigated. The convergence of the iterative process is rigorously analyzed. This process is used to prove the existence and uniqueness of a solution to the nonlinear difference scheme approximating the original differential problem. Consistent with the smoothness of the desired solution, convergence rate estimates in the discrete norm of W 2,0 2 (ω) for difference schemes approximating the nonlinear equation with unbounded nonlinearity are established.

Consistent two-sided estimates for the solutions of quasilinear parabolic equations and their approximations

For a linearized finite-difference scheme approximating the Dirichlet problem for a multidimensional quasilinear parabolic equation with unbounded nonlinearity, we establish pointwise two-sided solution estimates consistent with similar estimates for the differential problem. These estimates are used to prove the convergence of finite-difference schemes in the grid L 2 norm.