Scalar Nonlinear Equation Research Articles

Optimal distributed control of linear parabolic equations by spectral decomposition

AbstractWe construct an algorithm for solving a constrained optimal control problem for a first‐order evolutionary system governed by a positive self‐adjoint operator. The problem consists in identifying distributed control that minimizes a given cost functional, which comprises a cost of the control and a trajectory regulation term, while steering the final state close to a given target. The approach explores the dual problem and it generalizes the Hilbert Uniqueness Method (HUM). The practical implementation of the algorithm is based on a spectral decomposition of the operator determining the dynamics of the system. Once this decomposition is available – which can be done offline and saved for future use – the optimal control problem is solved almost instantaneously. It is practically reduced to a scalar nonlinear equation for the optimal Lagrange multiplier. The efficiency of the algorithm is demonstrated through numerical examples corresponding to different types of control operators and penalization terms.

General class of optimal Sobolev inequalities and nonlinear scalar field equations

We find a class of optimal Sobolev inequalities(∫RN|∇u|2dx)NN−2≥CN,G∫RNG(u)dx,u∈D1,2(RN),N≥3, where the nonlinear function G:R→R of class C1 satisfies general growth assumptions in the spirit of the fundamental works of Berestycki and Lions. We admit, however, a wider class of problems involving zero, positive and infinite mass cases as well as G need not be even. We show that any minimizer is radial up to a translation. Moreover, up to a dilation, it is a least energy solution of the nonlinear scalar field equation−Δu=g(u)in RN,with g=G′. In particular, if G(u)=u2log⁡|u|, then the sharp constant is CN,G:=2⁎(N2)2N−2e2(N−1)N−2(π)NN−2 and uλ(x)=eN−12−λ22|x|2 with λ>0 constitutes the whole family of minimizers up to translations. The optimal inequality provides a new proof of the classical logarithmic Sobolev inequality based on a Pohozaev manifold approach. Moreover, if N≥4, then there is at least one nonradial solution and if, in addition, N≠5, then there are infinitely many nonradial solutions of the nonlinear scalar field equation. The energy functional associated with the problem may be infinite on D1,2(RN) and is not Fréchet differentiable in its domain. We present a variational approach to this problem based on a new variant of Lions' lemma in D1,2(RN).

Convergence and Stability of a Parametric Class of Iterative Schemes for Solving Nonlinear Systems

A new parametric class of iterative schemes for solving nonlinear systems is designed. The third- or fourth-order convergence, depending on the values of the parameter being proven. The analysis of the dynamical behavior of this class in the context of scalar nonlinear equations is presented. This study gives us important information about the stability and reliability of the members of the family. The numerical results obtained by applying different elements of the family for solving the Hammerstein integral equation and the Fisher’s equation confirm the theoretical results.

Some new generalizations of $ F- $contraction type mappings that weaken certain conditions on Caputo fractional type differential equations

<abstract><p>In this paper, firstly, we introduce some new generalizations of $ F- $contraction, $ F- $Suzuki contraction, and $ F- $expanding mappings. Secondly, we prove the existence and uniqueness of the fixed points for these mappings. Finally, as an application of our main result, we investigate the existence of a unique solution of an integral boundary value problem for scalar nonlinear Caputo fractional differential equations with a fractional order (1, 2).</p></abstract>

Highly Efficient Derivative Free Numerical Method for Solving Nonlinear Scalar Equations by Multiple Precision Arithmetic

Highly Efficient Derivative Free Numerical Method for Solving Nonlinear Scalar Equations by Multiple Precision Arithmetic

The Active Flux Scheme for Nonlinear Problems

The Active Flux scheme is a finite volume scheme with additional point values distributed along the cell boundary. It is third order accurate and does not require a Riemann solver. Instead, given a reconstruction, the initial value problem at the location of the point value is solved. The intercell flux is then obtained from the evolved values along the cell boundary by quadrature. Whereas for linear problems an exact evolution operator is available, for nonlinear problems one needs to resort to approximate evolution operators. This paper presents such approximate operators for nonlinear hyperbolic systems in one dimension and nonlinear scalar equations in multiple spatial dimensions. They are obtained by estimating the wave speeds to sufficient order of accuracy. Additionally, an entropy fix is introduced and a new limiting strategy is proposed. The abilities of the scheme are assessed on a variety of smooth and discontinuous setups.

Дихотомия. Дихотомия? Дихотомия! Основные положения, проблемы терминологии и инспекционный анализ метода дихотомии

When creating modern systems of automatic control of various processes and objects operating in real time, very often one has to face the problem of solving various kinds of nonlinear scalar equations. Currently, there are a number of computational methods and algorithms for its solution, one of which is the dichotomy method. This method has a number of advantages in comparison with other known methods for solving nonlinear equations, but at present it has not found wide practical use. The main reason for its low popularity is the low rate of convergence of the sequence of approximate solutions and a large amount of computation required to obtain sufficiently accurate solutions. The purpose of the study is to consider in detail distinctive features of the dichotomy method and show the preference of its use in comparison with other known methods. We propose a modified version of the dichotomy method that allows one to obtain more rapidly converging sequences of approximate solutions to nonlinear scalar equations and requires significantly fewer computations required to obtain solutions with the desired accuracy. By solving a number of specific nonlinear equations, it is possible to illustrate the higher convergence rate of the sequence of approximate solutions calculated using the modified dichotomy method and, thereby, to substantiate the advantage of the new method for its use in creating various automatic control and regulation systems. Based on the results obtained a modification of the method for segment bisection is proposed. It has all the main advantages of the modified method. The research results can be used in the development of modern automatic control systems for various technological processes and objects.

Single species population dynamics in seasonal environment with short reproduction period

We present a periodic nonlinear scalar delay differential equation model for a population with short reproduction period. By transforming the equation to a discrete dynamical system, we reduce the infinite dimensional problem to one dimension. We determine the basic reproduction number not merely as the spectral radius of an operator, but as an explicit formula and show that is serves as a threshold parameter for the stability of the trivial equilibrium and for permanence.

Return mapping algorithm for elasto-plastic deformation of thin walled pipes

Return mapping algorithms are popular procedures for evaluating elasto-plastic material behaviour. This paper presents a dedicated return mapping algorithm for thin walled pipes and vessels with Von Mises plasticity and isotropic hardening, for which it can be assumed that the hoop stress is constant. This applies for example to pipe line sections during offshore laying operations. In the algorithm a nonlinear scalar equation is solved iteratively. A general 3D radial return mapping algorithm with an iteration loop to fulfil the assumptions on hoop and radial stress is used to verify results obtained with the dedicated algorithm. Results of both algorithms are identical, while the new algorithm is more efficient.

D-Stability of the Initial Value Problem for Symmetric Nonlinear Functional Differential Equations

This paper presents a method of establishing the D-stability terms of the symmetric solution of scalar symmetric linear and nonlinear functional differential equations. We determine the general conditions of the unique solvability of the initial value problem for symmetric functional differential equations. Here, we show the conditions of the symmetric property of the unique solution of symmetric functional differential equations. Furthermore, in this paper, an illustration of a particular symmetric equation is presented. In this example, all theoretical investigations referred to earlier are demonstrated. In addition, we graphically demonstrate two possible linear functions with the required symmetry properties.

Nonradial solutions of nonlinear scalar field equations

We prove new results concerning the nonlinear scalar field equation with a nonlinearity g satisfying the general assumptions due to Berestycki and Lions. In particular, we find at least one nonradial solution for any N ⩾ 4 minimizing the energy functional on the Pohozaev constraint in a subspace of consisting of nonradial functions. If in addition N ≠ 5, then there are infinitely many nonradial solutions. These solutions are sign-changing. The results give a positive answer to a question posed by Berestycki and Lions in [, ]. Moreover, we build a critical point theory on a topological manifold, which enables us to solve the above equation as well as to treat new elliptic problems.

On the maximal parameter range of global stability for a nonlocal thermostat model

The global asymptotic stability of the unique steady state of a nonlinear scalar parabolic equation with a nonlocal boundary condition is studied. The equation describes the evolution of the temperature profile that is subject to a feedback control loop. It can be viewed as a model of a rudimentary thermostat, where a parameter controls the intensity of the heat flow in response to the magnitude of the deviation from the reference temperature at a boundary point. The system is known to undergo a Hopf bifurcation when the parameter exceeds a critical value. Results on the characterization of the maximal parameter range where the reference steady state is globally asymptotically stable are obtained by analyzing a closely related nonlinear Volterra integral equation. Its kernel is derived from the trace of a fundamental solution of a linear heat equation. A version of the Popov criterion is adapted and applied to the Volterra integral equation to obtain a sufficient condition for the asymptotic decay of its solutions.

Using structural tensors for inelastic material modeling in the finite strain regime – A novel approach to anisotropic damage

The idea of the paper is to apply the well established structural tensors approach to complex inelastic material modeling in the finite strain regime. Using this concept, a very general and new anisotropic damage model is derived which is the major novel contribution of the paper. In this framework, a second order damage tensor plays the role of a structural tensor. The damage model is coupled to finite elasto-plasticity such that its first purpose is to model damage-induced anisotropy in ductile materials, in particular metals. Further structural tensors could be introduced to represent initial anisotropy of materials such as soft tissues, composites or thermoplastics. The main results of the derivation are the suitable choice of the Helmholtz free energy function, the development of yield and damage potentials as well as the formulation of physically reasonable evolution equations for the plastic deformation, the damage tensor as well as plastic and damage hardening. It is important to mention that all tensorial internal variables are symmetric and referred to the undeformed configuration. To the knowledge of the authors, the interpretation of a structural tensor of the undeformed configuration as damage tensor is new. Two-point tensors such as the plastic part of the deformation gradient show up in the theoretical derivation but are never computed. The plastic spin remains undetermined which has the advantage that a statement about its evolution is not needed. Further, the evolution equation for the plastic deformation includes only six non-linear scalar equations instead of nine. The numerical implementation of the model is straightforward. The number of material parameters remains moderate. A strategy to identify them by means of standard experiments is discussed. Nevertheless, the validation of the model by means of own experiments or experimental results from the literature is left for further work.

Massera's theorems for various types of equations with discontinuous solutions

We present new Massera-type theorems for various types of equations with periodic right-hand sides. We deal with generalized ordinary differential equations, measure differential equations, impulsive equations (all of which might have discontinuous solutions), as well as dynamic equations on time scales. For scalar nonlinear equations, we find sufficient conditions guaranteeing that each bounded solution is asymptotic to a periodic solution. For linear systems, we show that the existence of a bounded solution implies the existence of a periodic solution. We include some examples to illustrate our results.

A conservative semi-Lagrangian finite difference WENO scheme based on exponential integrator for one-dimensional scalar nonlinear hyperbolic equations

<p style='text-indent:20px;'>In this paper, we propose a conservative semi-Lagrangian finite difference (SLFD) weighted essentially non-oscillatory (WENO) scheme, based on Runge-Kutta exponential integrator (RKEI) method, to solve one-dimensional scalar nonlinear hyperbolic equations. Conservative semi-Lagrangian schemes, under the finite difference framework, usually are designed only for linear or quasilinear conservative hyperbolic equations. Here we combine a conservative SLFD scheme developed in [<xref ref-type="bibr" rid="b21">21</xref>], with a high order RKEI method [<xref ref-type="bibr" rid="b7">7</xref>], to design conservative SLFD schemes, which can be applied to nonlinear hyperbolic equations. Our new approach will enjoy several good properties as the scheme for the linear or quasilinear case, such as, conservation, high order and large time steps. The new ingredient is that it can be applied to nonlinear hyperbolic equations, e.g., the Burgers' equation. Numerical tests will be performed to illustrate the effectiveness of our proposed schemes.</p>

A mass supercritical problem revisited

In any dimension $N\geq1$ and for given mass $m>0$, we revisit the nonlinear scalar field equation with an $L^2$ constraint: $$ -\Delta u=f(u)-\mu u, \quad u \in H^1(\mathbb{R}^N) \quad \text{with} \quad \|u\|^2_{L^2(\mathbb{R}^N)}=m. $$ where $\mu\in\mathbb{R}$ will arise as a Lagrange multiplier. Assuming only that the nonlinearity $f$ is continuous and satisfies weak mass supercritical conditions, we show the existence of ground states and reveal the basic behavior of the ground state energy $E_m$ as $m>0$ varies. In particular, to overcome the compactness issue when looking for ground states, we develop robust arguments which we believe will allow treating other $L^2$ constrained problems in general mass supercritical settings. Under the same assumptions, we also obtain infinitely many radial solutions for any $N\geq2$ and establish the existence and multiplicity of nonradial sign-changing solutions when $N\geq4$. Finally we propose two open problems.

Iterative Algorithm for Solving Scalar Fractional Differential Equations with Riemann–Liouville Derivative and Supremum

The initial value problem for a special type of scalar nonlinear fractional differential equation with a Riemann–Liouville fractional derivative is studied. The main characteristic of the equation is the presence of the supremum of the unknown function over a previous time interval. This type of equation is difficult to be solved explicitly and we need approximate methods for its solving. In this paper, initially, mild lower and mild upper solutions are defined. Then, based on these definitions and the application of the monotone-iterative technique, we present an algorithm for constructing two types of successive approximations. Both sequences are monotonically convergent from above and from below, respectively, to the mild solutions of the given problem. The suggested iterative scheme is applied to particular problems to illustrate its application.

Stability of solutions and the problem of Aizerman for sixth-order differential equations

This article is devoted to the investigation of stability of equilibrium of ordinary differential equations using the method of semi-definite Lyapunov’s functions. Types of scalar nonlinear sixth-order differential equations for which regular constant auxiliary functions are used are emphasized. Sufficient conditions of global asymptotic stability and instability of the zero solution have been obtained and it has been established that the Aizerman problem has a positive solution concerning the roots of the corresponding linear differential equation. Studies highlight the advantages of using semi-definite functions compared to definitely positive Lyapunov’s functions.

Generalized Fixed Point Results with Application to Nonlinear Fractional Differential Equations

The main objective of this paper is to introduce the ( α , β )-type ϑ -contraction, ( α , β )-type rational ϑ -contraction, and cyclic ( α - ϑ ) contraction. Based on these definitions we prove fixed point theorems in the complete metric spaces. These results extend and improve some known results in the literature. As an application of the proved fixed point Theorems, we study the existence of solutions of an integral boundary value problem for scalar nonlinear Caputo fractional differential equations with a fractional order in (1,2).

Multiple normalized standing-wave solutions to the scalar non-linear Klein-Gordon equation with two competing powers

In this work we prove the existence of standing-wave solutions to the scalar non-linear Klein-Gordon equation in dimension one and the stability of the ground-state, the set which contains all the minima of the energy constrained to the manifold of the states sharing a fixed charge. For non-linearities which are combinations of two competing powers we prove that standing-waves in the ground-state are orbitally stable. We also show the existence of a degenerate minimum and the existence of two positive and radially symmetric minima having the same charge.