Mathematical Analysis and Numerical Simulation of a Nonlinear Thermoelastic System

Abstract

In this paper, we give a theoretical and numerical analysis of a model for small vertical vibrations of an elastic membrane coupled with a heat equation and subject to linear mixed boundary conditions. We establish the existence, uniqueness, and a uniform decay rate for global solutions to this nonlinear non-local thermoelastic coupled system with linear boundary conditions. Furthermore, we introduced a numerical method based on finite element discretization in a spatial variable and finite difference scheme in time which results in a nonlinear system to be solved by Newton’s method. Numerical experiments for one-dimensional and two-dimensional cases are presented in order to estimate the rate of convergence of the numerical solution that confirm our analysis and show the efficiency of the method.     

Solvability of Linear and Quasilinear Elliptic Boundary Value Problems via the A-Proper Mapping Theory

ABSTRACT

In this paper we use the theory of A-proper mappings and their uniform limits to obtain in a simple way general variational approximation-solvability and/or existence theorems for quasilinear elliptic BVProblems in divergence form of order 2m. In Section 1 we first show how our simple approach is used to obtain constructive solvability of linear elliptic equations under a somewhat different definition of strong ellipticity condition which allows us to dispense with the usage of the Garding inequality. In view of this, the same approach is also used in Section 2 to establish the constructive solvability of nonlinear BVProblems when the leading part satisfies a nonlinear version of the strong ellipticity condition used here. This result is then used to establish a new existence result (see Theorem 2.3) for a not necessarily coercive quasilinear elliptic BVProblem which properly includes earlier results of Vi?ik, Browder, Poho?ayev, Leray and Lions and others. In Section 3 some special cases of our abstract results are stated. These are applicable to ODE's and PDE's which are not in divergence form.     

The Alexandrov-Bakelman-Pucci Weak Maximum Principle for Fully Nonlinear Equations in Unbounded Domains

ABSTRACT

We obtain Alexandrov-Bakelman-Pucci type estimates for semicontinuous viscosity solutions of fully nonlinear elliptic inequalities in unbounded domains. Under suitable assumptions relating the geometry of the domain with structural conditions on the differential operator, we establish the validity of the weak maximum principle for solutions which are bounded from above. Two variants are also given, namely one for unbounded solutions in narrow domains and one for operators with possibly changing sign zero order coefficients in domains of small measure.     

Partly zero eigenvectors and matrices that satisfy Au = b

For given vectors u,b ∈ F ⁿ (where F is a field with at least 3 elements), we establish criteria for deciding whether a digraph allows in its pattern class a matrix A which satisfies Au = b. As corollaries to this we give necessary and sufficient conditions for a pattern class to allow a matrix which has an eigenvector with a particular zero/nonzero pattern. Moreover we establish whether or not that eigenvector can correspond to a zero or nonzero eigenvalue. We use these results to establish the analogous results for loopfree digraphs, and thus we obtain results additional to work already done by Maybee, Olesky, and Van Den Driessche in this area. We then use bipartite graphs to generalize our criteria for solutions to Au = b to the rectangular case.     

Blow-up phenomena for a semilinear heat equation with nonlinear boundary condition, I

We consider the blow-up of the solution to a semilinear heat equation with nonlinear boundary condition. We establish conditions on nonlinearities sufficient to guarantee that u(x, t) exists for all time t > 0 as well as conditions on data forcing the solution u(x, t) to blow up at some finite time t*. Moreover, an upper bound for t* is derived. Under somewhat more restrictive conditions, lower bounds for t* are also derived.     

Modelling and analysis of haemoglobin catalytic reaction kinetic system

ABSTRACT

In order to study whether haemoglobin (Hb) can replace peroxidase and has good catalytic properties. The key to exploring the characteristics of Hb peroxidase is to establish a suitable kinetic model, which is studied in this paper. First, according to the Hb catalytic reaction, a nonlinear system is established and improved. It is proved that the established system is in line with the practical significance. The stability of the original system is judged by analysing the stability of the simplified system. Then, considering the effect of time delay on Hb catalytic reaction, a nonlinear time-delay catalytic reaction system is obtained. For convenient application, the system is linearized using Taylor’s formula, and the dynamic characteristics of Hopf bifurcation are analysed. The response diagrams of three system are plotted by setting perturbation parameters, and their variations are observed to analyse the differences among them. The results show that the nonlinear time-delay system can better describe the characteristics of the catalytic reaction.     

Nonlinear Hyperbolic Equations in Infinite Homogeneous Waveguides

ABSTRACT

In this paper we prove global and almost global existence theorems for nonlinear wave equations with quadratic nonlinearities in infinite homogeneous waveguides. We can handle both the case of Dirichlet boundary conditions and Neumann boundary conditions. In the case of Neumann boundary conditions we need to assume a natural nonlinear Neumann condition on the quasilinear terms. The results that we obtain are sharp in terms of the assumptions on the dimensions for the global existence results and in terms of the lifespan for the almost global results. For nonlinear wave equations, in the case where the infinite part of the waveguide has spatial dimension three, the hypotheses in the theorem concern whether or not the Laplacian for the compact base of the waveguide has a zero mode or not.     

Exponential decay for elastic systems with structural damping and infinite delay

ABSTRACT

In this paper, we consider nonlinear evolution equations of second order in Banach spaces involving unbounded delay, which can model an elastic system with structural damping involving infinite delays. By using fixed point for condensing maps, we prove the existence and exponential decay of mild solutions. The obtained results can be applied to the nonlinear vibration equation of elastic beams with structural damping and infinite delay.     

Asymptotic Analysis for a Class of Infinite Systems of First-Order PDE: Nonlinear Parabolic PDE in the Singular Limit

Abstract

We study the asymptotic behavior of solutions of the Cauchy problem for a functional partial differential equation with a small parameter as the parameter tends to zero. We establish a convergence theorem in which the limit problem is identified with the Cauchy problem for a nonlinear parabolic partial differential equation. We also present comparison and existence results for the Cauchy problem for the functional partial differential equation and the limit problem.     

DIRICHLET PROBLEM OF ELLIPTIC EQUATIONS WITH STRONG RESONANCES

ABSTRACT

This paper is concerned with strong resonant problems for nonlinear elliptic Dirichlet BVPs. Classifying the decay rates of the nonlinearity at infinity and using a Morse theoretical approach developed in our earlier work, we are able to establish several multiplicity results of positive and sign-changing solutions.     

General decay of solutions to a viscoelastic wave equation with linear damping,nonlinear damping and source term

ABSTRACT

This paper is concerned with the decay property of a nonlinear viscoelastic wave equation with linear damping, nonlinear damping and source term. Under weaker assumption on the relaxation function, we establish a general decay result, which extends the result obtained in Messaoudi [Exponential decay of solutions of a nonlinearly damped wave equation. Nodea-Nonlinear Differ Equat Appl. 2005;12:391–399].     

On Perturbed Discrete Boundary Value Problems

In this paper, we study nonlinear discrete boundary value problems of the form x ( t +1)= A ( t ) x ( t )+ h ( t )+ k f ( t , x ( t ), k ) subject to Bx (0)+ Dx ( J )= u + k g ( x (0), x ( J ), k ) where k is a "small" parameter. Our main concern is the case of resonance, that is, the situation where the associated linear homogeneous boundary value problem x ( t +1)= A ( t ) x ( t ), Bx (0)+ Dx ( J )=0 admits nontrivial solutions. We establish conditions for the solvability of the nonlinear boundary value problem when k is "small". We also establish qualitative properties of these solutions.     

Extremal solutions to a system of n nonlinear differential equations and regularly varying functions

The strongly increasing and strongly decreasing solutions to a system of n nonlinear first order equations are here studied, under the assumption that both the coefficients and the nonlinearities are regularly varying functions. We establish conditions under which such solutions exist and are (all) regularly varying functions, we derive their index of regular variation and establish asymptotic representations. Several applications of the main results are given, involving n‐th order nonlinear differential equations, equations with a generalized ?‐Laplacian, and nonlinear partial differential systems.     

A Nonlinear Diffusion Problem with Localized Large Diffusion

Abstract

This paper is concerned with a nonlinear parabolic problem, with nonlinear boundary conditions, for which the diffusion coefficient becomes very large in a sub-region of the physical domain.     

Optimality conditions for vector optimization problems with non-cone constraints in image space

ABSTRACT

In this paper, we employ the image space analysis method to investigate a vector optimization problem with non-cone constraints. First, we use the linear and nonlinear separation techniques to establish Lagrange-type sufficient and necessary optimality conditions of the given problem under convexity assumptions and generalized Slater condition. Moreover, we give some characterizations of generalized Lagrange saddle points in image space without any convexity assumptions. Finally, we derive the vectorial penalization for the vector optimization problem with non-cone constraints by a general way.     

Apery Sets of Numerical Semigroups #

Let S be a numerical semigroup. We examine a particular subset of the Apery set of S and establish a correspondence between this subset and the holes of S . This correspondence allows us to establish conditions for S to be almost symmetric.     

Local asymptotic normality for multivariate nonlinear AR processes

In this paper, we establish local asymptotic normality (LAN) for the log-likelihood ratio of a class of multivariate nonlinear autoregressive processes generated by independent identically distributed (i.i.d.) random vectors. This result generalizes LAN for multivariate linear processes to multivariate nonlinear autoregressive processes.     

Existence of positive solutions for Sturm-Liouville boundary value problems on the half-line

In this paper, we establish sufficient conditions to guarantee the existence of at least one positive solution, a unique positive solution, and multiple positive solutions for the Sturm-Liouville boundary value problem on the half-line. By using an effective operator, the fixed point theorems in cone, especially Krasnoselskii fixed point theorem, can be applied to such systems and then existence criteria are established. The interesting point of the results is that the nonlinear term f can be sign-changing.     

An application of <Emphasis Type="Italic">H</Emphasis>-differentiability to nonnegative and unrestricted generalized complementarity problems

This paper deals with nonnegative nonsmooth generalized complementarity problem, denoted by GCP(f,g). Starting with H-differentiable functions f and g, we describe H-differentials of some GCP functions and their merit functions. We show how, under appropriate conditions on H-differentials of f and g, minimizing a merit function corresponding to f and g leads to a solution of the generalized complementarity problem. Moreover, we generalize the concepts of monotonicity, P ₀-property and their variants for functions and use them to establish some conditions to get a solution for generalized complementarity problem. Our results are generalizations of such results for nonlinear complementarity problem when the underlying functions are C ¹, semismooth, and locally Lipschitzian.