Nonlinear Differential Problem Research Articles

Long time behavior for a fractional Picard problem in a Hilbert space

Of concern is a nonlinear second order initial value differential problem involving a convolution of a singular kernel with the derivative of the state. The problem describes the dynamics of a single-degree-of-freedom fractional oscillator. It is a generalization of the standard harmonic oscillator. The model also generalizes some well-known fractionally damped second order differential equations such as the Bagley–Torvik equation. Moreover, it extends models using exponential non-viscous damping to the more challenging singular case. We prove an exponential stability result of the equilibrium using the multiplier technique. A new energy functional, different from the classical one and different from the one obtained by the diffusive representation, is introduced.

Finite Element Modeling of Eigenvibrations of a Loaded Bar

The nonlinear second-order differential eigenvalue problem describing eigenvibrations of a bar with elastically attached load is investigated. This problem has an increasing sequence of positive simple eigenvalues with limit point at infinity. The sequence of eigenvalues corresponds to a system of normalized eigenfunctions. The initial nonlinear eigenvalue problem is approximated by the quadrature finite element method on a uniform grid. The existence and accuracy of approximate solutions are studied. Investigations of the present paper can be generalized for the cases of more complicated and important problems on eigenvibrations of beams, plates and shells with elastically attached loads.

Velocity slip in mixed convective oblique transport of titanium oxide/water (nano-polymer) with temperature-dependent viscosity

Nano-polymers are the emerging development in cell coatings due to their enhanced durability and thermal efficiency. The viscosity of such fluids is considerably influenced by temperature. Motivated by fascinating applications of nano-polymers, the present article offers a mathematical study of steady, oblique transport of titanium water-based nano-polymer gels under mixed convection effects. To simulate real nano-polymer boundary interface dynamics, convective surface along with velocity slip are analysed at the wall. Viscosity is assumed to be temperature-dependent in our analysis. The conservation equations for mass, momentum (both normal and tangential) and energy are normalized using appropriate transformations leading to a multi degree nonlinear ordinary differential equations problem. Numerical solutions are attained using the Runge-Kutta-Fehlberg method with shooting quadrature in the symbolic software MATLAB. The influence of key evolving parameters, namely mixed convection parameter, velocity slip parameter, nanoparticles volume fraction on non-dimensional velocity components, temperature, normal and tangential skin friction coefficients and local heat flux is examined. Increasing the velocity slip parameter decreases the velocity components while it increases the temperature. The simulations performed reveal that the normal skin friction coefficient and the heat flux at the wall increase with enhancing the nanoparticles volume fraction.

A splitting method for overcoming the curse of dimensionality in Hamilton–Jacobi equations arising from nonlinear optimal control and differential games with applications to trajectory generation

Recent observations have been made that bridge splitting methods arising from optimization, to the Hopf and Lax formulas for Hamilton-Jacobi Equations with Hamiltonians $H(p)$. This has produced extremely fast algorithms in computing solutions of these PDEs. More recent observations were made in generalizing the Hopf and Lax formulas to state-and-time-dependent cases $H(x,p,t)$. In this article, we apply a new splitting method based on the Primal Dual Hybrid Gradient algorithm (a.k.a. Chambolle-Pock) to nonlinear optimal control and differential games problems, based on techniques from the derivation of the new Hopf and Lax formulas, which allow us to compute solutions at points $(x,t)$ directly, i.e. without the use of grids in space. This algorithm also allows us to create trajectories directly. Thus we are able to lift the curse of dimensionality a bit, and therefore compute solutions in much higher dimensions than before. And in our numerical experiments, we actually observe that our computations scale polynomially in time. Furthermore, this new algorithm is embarrassingly parallelizable.

Eigenvibrations of a simply supported beam with elastically attached load

The nonlinear differential eigenvalue problem describing eigenvibrations of a simply supported beam with elastically attached load is investigated. The existence of an increasing sequence of positive simple eigenvalues with limit point at infinity is established. To the sequence of eigenvalues, there corresponds a system of normalized eigenfunctions. To illustrate the obtained theoretical results, the initial problem is approximated by the finite difference method on a uniform grid. The accuracy of approximate solutions is studied. Investigations of the present paper can be generalized for the cases of more complicated and important problems on eigenvibrations of plates and shells with elastically attached loads.

Piecewise continuous approach to nonlinear differential equations approximation problem of computational structural mechanics

This paper presents a nonlinear differential equations system piecewise continuous approximation. The piecewise continuous approximation improves piecewise linear approximation through reducing the errors at the boundaries of different linear differential equations systems areas. The matrices of piecewise continuous differential and algebraic equations systems are estimated using nonlinear differential equations system time responses and random search method. The results of proposed approach application are presented.

Stochastic differential games and inverse optimal control and stopper policies

ABSTRACTIn this paper, we consider a two-player stochastic differential game problem over an infinite time horizon where the players invoke controller and stopper strategies on a nonlinear stochastic differential game problem driven by Brownian motion. The optimal strategies for the two players are given explicitly by exploiting connections between stochastic Lyapunov stability theory and stochastic Hamilton–Jacobi–Isaacs theory. In particular, we show that asymptotic stability in probability of the differential game problem is guaranteed by means of a Lyapunov function which can clearly be seen to be the solution to the steady-state form of the stochastic Hamilton–Jacobi–Isaacs equation, and hence, guaranteeing both stochastic stability and optimality of the closed-loop control and stopper policies. In addition, we develop optimal feedback controller and stopper policies for affine nonlinear systems using an inverse optimality framework tailored to the stochastic differential game problem. These results are then used to provide extensions of the linear feedback controller and stopper policies obtained in the literature to nonlinear feedback controllers and stoppers that minimise and maximise general polynomial and multilinear performance criteria.

An exact solution to a Stefan problem with variable thermal conductivity and a Robin boundary condition

In this article it is proved the existence of similarity solutions for a one-phase Stefan problem with temperature-dependent thermal conductivity and a Robin condition at the fixed face. The temperature distribution is obtained through a generalized modified error function which is defined as the solution to a nonlinear ordinary differential problem of second order. It is proved that the latter has a unique non-negative bounded analytic solution when the parameter on which it depends assumes small positive values. Moreover, it is shown that the generalized modified error function is concave and increasing, and explicit approximations are proposed for it. Relation between the Stefan problem considered in this article with those with either constant thermal conductivity or a temperature boundary condition is also analysed.

A simple and effective nonlinear elastic one-dimensional model for the structural analysis of masonry arches

In this paper the static response of a masonry arch is studied by way of a one-dimensional nonlinear elastic model in which masonry is regarded as a material with bounded tensile and compressive strengths. By following an approach analogous to that followed in the theory of bending of elastic beams, the equilibrium problem for the arch leads to a free-boundary, nonlinear differential problem. An approximate solution to such problem can be pursued by means of an ad hoc iterative procedure, illustrated in detail herein. The results obtained in three case studies are compared with some numerical and experimental results available in the literature. In addition, the case of an actual arch undergoing spreading of the springings is considered, and the distribution and possible evolution of the cracking pattern discussed.

A numerical scheme for solutions of a class of nonlinear differential equations

In this paper, a collocation method based on Bessel functions of the first kind is presented to compute the approximate solutions of a class of high-order nonlinear differential equations under the initial and boundary conditions. First, the matrix forms of the Bessel functions of the first kind and their derivatives are constructed. Second, by using these matrix forms, collocation points and the matrix operations, a nonlinear differential equation problem is converted to a system of nonlinear algebraic equations. The solutions of this system give the coefficients of the assumed approximate solution. To demonstrate the validity and applicability of the technique, numerical examples are included and comparisons are made with existing results. The results show the efficiency and accuracy of the present work.

Existence and uniqueness of the modified error function

Abstract This article is devoted to the proof of the existence and uniqueness of the modified error function introduced in Cho and Sunderland (1974). This function is defined as the solution to a nonlinear second order differential problem depending on a real parameter. We prove here that this problem has a unique non-negative analytic solution when the parameter assumes small positive values.

A poroelastic mixture model of mechanobiological processes in biomass growth: theory and application to tissue engineering

In this article we propose a novel mathematical description of biomass growth that combines poroelastic theory of mixtures and cellular population models. The formulation, potentially applicable to general mechanobiological processes, is here used to study the engineered cultivation in bioreactors of articular chondrocytes, a process of Regenerative Medicine characterized by a complex interaction among spatial scales (from nanometers to centimeters), temporal scales (from seconds to weeks) and biophysical phenomena (fluid-controlled nutrient transport, delivery and consumption; mechanical deformation of a multiphase porous medium). The principal contribution of this research is the inclusion of the concept of cellular “force isotropy” as one of the main factors influencing cellular activity. In this description, the induced cytoskeletal tensional states trigger signalling transduction cascades regulating functional cell behavior. This mechanims is modeled by a parameter which estimates the influence of local force isotropy by the norm of the deviatoric part of the total stress tensor. According to the value of the estimator, isotropic mechanical conditions are assumed to be the promoting factor of extracellular matrix production whereas anisotropic conditions are assumed to promote cell proliferation. The resulting mathematical formulation is a coupled system of nonlinear partial differential equations comprising: conservation laws for mass and linear momentum of the growing biomass; advection–diffusion–reaction laws for nutrient (oxygen) transport, delivery and consumption; and kinetic laws for cellular population dynamics. To develop a reliable computational tool for the simulation of the engineered tissue growth process the nonlinear differential problem is numerically solved by: (1) temporal semidiscretization; (2) linearization via a fixed-point map; and (3) finite element spatial approximation. The biophysical accuracy of the mechanobiological model is assessed in the analysis of a simplified 1D geometrical setting. Simulation results show that: (1) isotropic/anisotropic conditions are strongly influenced by both maximum cell specific growth rate and mechanical boundary conditions enforced at the interface between the biomass construct and the interstitial fluid; (2) experimentally measured features of cultivated articular chondrocytes, such as the early proliferation phase and the delayed extracellular matrix production, are well described by the computed spatial and temporal evolutions of cellular populations.

An improved adaptation of homotopy analysis method

An improved adaptation of the well-known homotopy analysis method (HAM) is proposed to approximate the solutions of strongly nonlinear differential problems in terms of a rapidly convergent series. The proposed method involves simpler integrals and less computations than the standard HAM. The method is illustrated using different numerical examples. The comparative analysis confirms the applicability and efficiency of the proposed technique.

Piecewise-linear and bilinear approaches to nonlinear differential equations approximation problem of computational structural mechanics

This paper presents a bilinear approach to nonlinear differential equations system approximation problem. Sometimes the nonlinear differential equations right-hand sides linearization is extremely difficult or even impossible. Then piecewise-linear approximation of nonlinear differential equations can be used. The bilinear differential equations allow to improve piecewise-linear differential equations behavior and reduce errors on the border of different linear differential equations systems areas. The matrices of bilinear differential equations system are estimated through nonlinear programming. The results of proposed approach application are presented.

Darcy-Forchheimer flow due to a curved stretching surface with Cattaneo-Christov double diffusion: A numerical study

This communication addresses the Darcy-Forchheimer flow by an unsteady curved stretching surface. Flow in porous medium is characterized by Darcy-Forchheimer relation. Cattaneo-Christov double diffusion expressions are implemented in modelling. Suitable transformations correspond to a strong nonlinear ordinary differential problems. The problems are numerically solved. Outcomes of emerging flow variables on velocity, temperature and concentration fields are plotted. Skin friction coefficient and local Nusselt and Sherwood numbers are computed and examined. Our findings reveal that the skin friction coefficient is an increasing function of Forchheimer number.

Nondifferentiable augmented Lagrangian, ε-proximal penalty methods and applications

The purpose of this work is to prove results concerning the duality theory and to give detailed study on the augmented Lagrangian algorithms and $\varepsilon$-proximal penalty method which are considered, today, as the most strong algorithms to solve nonlinear differentiable and nondifferentiable problems of optimization. We give an algorithm of primal-dual type, where we show that sequences $\left\{\lambda^k\right\}_k$ and $\left\{x^k\right\}_k$ generated by this algorithm converge globally, with at least the Slater condition, to $\bar{\lambda}$ and $\bar{x}$. Numerical simulations are given.

Improving the Accuracy of Chebyshev Tau Method for Nonlinear Differential Problems

The spectral properties convergence of the Tau method allow to obtain good approximate solutions for linear differential problems advantageously. However, for nonlinear differential problems the method may produce ill-conditioned matrices issued from the approximations obtained in the iterations from the linearization process. In this work we introduce a procedure to approximate nonlinear terms in the differential equations and a new way to build the corresponding algebraic problem improving the stability of the overall algorithm. Introducing the linearization coefficients of orthogonal polynomials in the Tau method within the iterative process, we can go further in the degree to approximate the solution of the differential problems, avoiding the consequences of ill-conditioning.

A coincidence point theorem for sequentially continuous mappings

The aim of this paper is to present a coincidence point theorem for sequentially weakly continuous maps. Moreover, as a consequence, a critical point theorem for functionals possibly containing a nonsmooth part is obtained. Finally, as an application, existence results for nonlinear differential problems depending also on the derivative of the solution are established.

Asymptotic and chaotic solutions of a singularly perturbed Nagumo-type equation33Supported by the project Equazioni differenziali ordinarie sulla retta reale of GNAMPA-I.N.d.A.M., Italy. The second author is supported by the P.R.I.N. Project ‘Variational and perturbative aspects of nonlinear differential problems’

We deal with the singularly perturbed Nagumo-type equationϵ2u′′+u(1−u)(u−a(s))=0, ?>where is a real parameter and is a piecewise constant function satisfying 0 < a(s) < 1 for all s. For small ε, we prove the existence of chaotic, homoclinic and heteroclinic solutions. We use a dynamical systems approach, based on the Stretching Along Paths technique and on the Conley–Waewski’s method.

Application of the Schwarz alternating method for simulating the contact interaction of a system of bodies

A numerical algorithm for simulating the multicontact thermoelastic interaction of a system of many bodies is developed. The algorithm is based on the iterative Schwarz alternating method modified for the class of problems under consideration. The nonlinear differential problem in question is discretized using the finite element method. Numerical results are presented, including the thermomechanical interaction of 350 bodies.