Mauro Antonio Rincon

Effect of moving boundaries on the vibrating elastic string

A new derivation of a wave equation for small vibrations of elastic strings fastened at ends varying with time is presented. The model takes into account the change of length during the vibration and the nonlinear behavior of elastic strings in general. This model is a generalization of the Kirchhoff equation which contains a nonlinear term involving the displacement gradient. Numerical simulations of the model are based on finite difference approximations. Differences between linear and nonlinear aspects and the assumptions of numerical and theoretical analysis are briefly discussed and comparisons are made for linear and nonlinear elastic strings as well as the Kirchhoff model and the linear model without the term containing the displacement gradient.

Beam equation with weak-internal damping in domain with moving boundary

The small-amplitude motion of an elastic beam with internal damping is investigated in a One-dimensional domain with moving boundary. Existence, uniqueness, asymptotic behavior, and numerical analysis of solutions are shown to the mixed problem associated with the beam equation with fully clamped boundary conditions.

Numerical method, existence and uniqueness for thermoelasticity system with moving boundary

In this work, we are interested in obtaining existence, uniqueness of the solution and an approximate numerical solution for the model of linear thermoelasticity with moving boundary. We apply finite element method with finite difference for evolution in time to obtain an approximate numerical solution. Some numerical experiments were presented to show the moving boundarys effects for problems in linear thermoelasticity.

Successive linear approximation for finite elasticity

A method of successive Lagrangian formulation of linear approximation for solving boundary value problems of large deformation in finite elasticity is proposed. Instead of solving the nonlinear problem, by assuming time steps small enough and the reference configuration updated at every step, we can linearize the constitutive equation and reduce it to linear boundary value problems to be solved successively with incremental boundary data. Moreover, nearly incompressible elastic body is considered as an approximation to account for the condition of incompressibility. For the proposed method, numerical computations of pure shear of a square block for Mooney-Rivlin material are considered and the results are compared with the exact solutions. Mathematical subject classification: Primary: 65C20; Secondary: 74B20.

Numerical analysis and simulation for a nonlinear wave equation

In this work we study a nonlinear wave equation, depending on different norms of the initial conditions, has bounded solution for all t 0 or 0 < t < T 0 for some T 0 0 . We also prove that the solution may blow-up at T 0 . Proofs of some the analytical results listed are sketched or given. For approximate numerical solutions we use the finite element method in the spatial variable and the finite difference method in time. The nonlinear system for each time step is solved by Newtons modified method. We present numerical analysis for error estimates and numerical simulations to illustrate the convergence of the theoretical results. We present too, the singularity points ( x ? , t ? ) , where the blow-up occurs for different ? values in a numerical simulation.

Analysis and Numerical Simulation of Viscous Burgers Equation

This work is concerned with the viscous Burgers equation inside a time dependent domain. We establish theoretical and numerical analysis such as existence and uniqueness of solutions, asymptotic behavior to the energy, and numerical discretization by Finite difference and Finite element methods.

Analysis and numerical solution of Benjamin-Bona-Mahony equation with moving boundary

In this work we present the existence, the uniqueness and numerical solutions for a mathematical model associated with equations of Benjamin-Bona-Mahony type in a domain with moving boundary. We apply the Galerkin method, multiplier techniques, energy estimates and compactness results to obtain the existence and uniqueness. For numerical solutions, we shall employ the finite element method together with the Crank-Nicolson method. Some numerical experiments are presented to show the moving boundary for the problem.

Numerical studies of the damped Korteweg-de Vries system

We present a numerical analysis of the Korteweg-de Vries (KdV) system in a bounded interval under the effect of a localized damping mechanism. For the sake of completeness, we include the proofs of existence and uniqueness of the weak solution by means of the Faedo-Galerkin method. Error estimates of finite element approximations, for both semi-discrete and fully discrete schemes in the energy norm are provided and numerical experiments are performed.

Variational inequalities applied to option market problem

Abstract Since the classic work of Black and Scholes [F. Black, M. Scholes, The pricing of options and corporate liabilities, J. Polit. Econ. 81 (1973) 354–637] many techniques to calculate the value of European option have been developed. When we are interested in assessing the American option, these techniques must change to adapt to the early exercise possibility. To solve the American options problem, we obtain an inequality variational system and use numerical methods over it. This work aims to get the put American option price using the finite elements method and finite differences method. Numerical results are presented.

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.