Numerical solution of an inverse problem for a two-dimensional mathematical model of sorption dynamics

We consider a two-dimensional mathematical model of sorption that allows for inner-diffusion kinetics as well as longitudinal and transverse diffusion. The inverse problem of determining the sorption isotherm from an experimental dynamic output curve is investigated for this model and stable solution methods are proposed for the inverse and the direct problem. The efficiency of the solution methods is explored in computer experiments.     

On the stability loss for an Euler beam resting on a tensionless Pasternak foundation

In the present work, the tensionless contact problem of an Euler–Bernoulli beam of finite length resting on a two-parameter Pasternak-type foundation is investigated. Owing to the tensionless character of the contact, the beam may lift-off the foundation and the point where contact ceases and detachment begins, named contact locus, needs be assessed. In this situation, a one-dimensional free boundary problem is dealt with. An extra condition, in the form of a homogeneous second-order equation in the displacement and its derivatives, is demanded to set the contact locus and it gives the problem its nonlinear feature. Conversely, the loading and the beam length may be such that the beam rests entirely supported on the foundation, which situation is governed by a classical linear boundary value problem. In this work, contact evolution is discussed for a continuously varying loading condition, starting from a symmetric layout and at a given beam length, until overturning is eventually reached. In particular, stability is numerically assessed through the energy criterion, which is shown to stand for the free boundary situation as well. At overturning, a descending pathway in the system energy appears and stability loss is confirmed.     

An iterative method for the shape reconstruction of the inverse Euler problem

This article is concerned with the shape reconstruction for the inviscid fluid governed by the Euler equations. By formulating the domain derivative of the Euler equations and applying a regularized Gauss‐Newton iterative algorithm, the numerical examples are given for recovering the shape. The results show that our theory is useful for practical purpose and the proposed algorithm is feasible. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 587–596, 2012     

Quasi-acoustic scheme for the Euler equations of gas dynamics

We suggest an original scheme and an algorithm for the numerical solution of the Euler equations of gas dynamics. The construction of the scheme is based on the mass, momentum, and energy conservation laws. The flux computation is carried out by summation of elementary fluxes formed by small-amplitude running waves that satisfy the linearized equations of gas dynamics. The scheme contains no artificial regularizers, has second-order accuracy on smooth solutions, and is quasimonotone in a neighborhood of the discontinuities. Examples of one- and two-dimensional computations are given.     

An inverse problem for an immobilized enzyme model

A method for estimating unknown kinetic parameters in a mathematical model for catalysis by an immobilized enzyme is studied. The model consists of a semilinear parabolic partial differential equation modeling the reaction‐diffusion process coupled with an ordinary differential equation for the rate transport. The well posedness of the model is proven; a PDE‐constrained optimization approach is applied to the stated inverse problem; and finally, some numerical simulations are presented.     

Strain gradient beam model for dynamics of microscale pipes conveying fluid

Based on the strain gradient theory, we present a microstructure-dependent Bernoulli–Euler model to analyze the vibration and stability of microscale pipes conveying fluid. The equation of motion and boundary conditions are derived using Hamilton’s principle. The proposed strain gradient beam model contains three material length scale parameters to capture the size effect. This new model may be reduced to the modified couple stress beam model when two of these three material length scale parameters vanish and may be reduced to the classical beam model in the absence of all the material length scale parameters. From the numerical calculations for micropipes with both ends positively supported, it is found that the natural frequency and the critical flow velocity are size-dependent. The results show that the microscale pipe displays remarkable size effect when its outside diameter becomes comparable to the material length scale parameter, while the size effect is almost diminishing as the diameter is far greater than the material length scale parameter. Moreover, the size effect predicted by the current strain gradient beam model is stronger than that predicted by the modified couple stress beam model, since two other material length scale parameters have been accounted for in the former.     

Numerical solution of an inverse problem of nonisothermal sorption dynamics

We consider the inverse problem for a mathematical model of sorption dynamics that incorporates diffusion, intradiffusion kinetics, and the heat balance. Two numerical methods are proposed. Their efficiency is investigated in a computer experiment. Translated from Prikladnaya Matematika i Informatika, No. 29, pp. 56–63, 2008.     

Numerical method for solving an inverse problem for a population model

The inverse problem of determining the growth rate coefficient of biological objects from additional information on their time-dependent density is considered. Two nonlinear integral equations are derived for the unknown coefficient, which is determined on part of its domain from one equation and on the remaining part from the other equation. The nonlinear integral equations are solved by iterative methods. The convergence conditions for the iterative methods are formulated, and results of numerical experiments are presented.     

The classification for dynamics of an STD model

In this paper, a 3(n+1) dimensional sexually transmitted disease model is studied. The model involves two competing strains 1 and 2, where females are divided into n different groups based on their susceptibility to two distinct pathogenic strains. The first and second reproduction numbers , which have their biological meaning, are defined for strain , respectively. With the non-degeneracy assumption (H), a complete classification for dynamics of this model in terms of is presented, which is based on the theory of type-K monotone dynamical systems.     

A method for solving an inverse biharmonic problem

This paper deals with the problem of determining of an unknown coefficient in an inverse boundary value problem. Using a nonconstant overspecified data, it has been shown that the solution to this inverse problem exists and is unique.     

A finite-difference algorithm for an inverse Sturm-Liouville problem

We study a method for approximating a potential q(x) in y(0)=y()=0 from finite spectral data. When the potential is symmetric,the data are the first M Dirichlet eigenvalues. In the generalcase, the first M terminal velocities are also specified. Acentred finite-difference scheme reduces the inverse Sturm-Liouvilleproblem to a matrix inverse eigenvalue problem. Our approachis motivated by the work of Paine, de Hoog and Anderssen, whoinvestigated the discrepancy between continuous and matrix eigenvaluesunder finite differences. Our modified Newton scheme is basedon choosing the number of interior mesh points in the discretizationto be 2M. The modified Newton scheme is shown to be convergentfor both the case of a symmetric and general potential. Somenumerical experiments are given. Supported in part by Institute for Scientific Computation,Texas A&M University.     

On an inverse problem of sorbtion dynamics with nonlinear flow velocity

In the present paper we study the inverse problem of determining a sorbtion isotherm from the output dynamic curve for the mathematical model that describes the sorbtion process in the case of a nonlinear flow velocity. We prove a uniqueness theorem for the solution of this inverse problem and propose a stable method of solving the inverse problem based on the use of a priori information on the solution. We construct the gradient of the discrepancy. We study the stability and convergence of the difference scheme that approximates the direct problem. We give the results of a computational experiment. Two figures. Bibliography: 12 titles. Translated fromMetody Matematicheskogo Modelirovaniya, 1998, pp. 142–150.     

A stochastic model for the direct and inverse problem of adhesive materials

This work deals with the generation of artificial data based on experimental data for adhesive materials and the application of this data to the inverse and the direct problem. In reality there are only a very limited number of experimental data available. Therefore, the prediction of material behaviour is difficult and a statistical analysis with a stochastic proved thesis is nearly impossible. In order to increase the number of tests a method of stochastic simulation based on time series analysis is applied. With artificial data an arbitrary number of data is available and the process of the parameter identification can be statistically analysed. Additionally, one example is shown, which adapts the analysed material parameter to the direct problem. The stochastic finite element method is used to take into account the distribution and deviation of the fracture strain. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Longtime dynamics for a novel piezoelectric beam model with creep and thermo-viscoelastic effects

In this work, a novel fully-dynamic piezoelectric beam model is considered. Electromagnetic and thermal effects are taken into consideration by Maxwell’s equations and the Coleman–Gurtin law (instead of the Fourier’s law), respectively. Our model accounts also for thermal and electromagnetic (creep) past histories, which are in line with the time response of PVDF at the applied stress in the longitudinal direction. Under suitable assumptions, the existence and uniqueness of solutions are proved by the semigroup theory. The main purpose of this paper is to establish the longtime dynamics of the model. Therefore, the quasi-stability property of the model and the existence of smooth global attractors with finite fractal dimension are obtained. The existence of exponential attractors for the associated dynamical system is also proved.     

Uniqueness of solution of some inverse problems of sorption dynamics for a model with mixed-diffusion kinetics

Translated from Pryamye i Obratnye Zadachi Matematicheskoi Fiziki, pp. 30–34, 1991.     

Finite time blow-up for a dyadic model of the Euler equations

We introduce a dyadic model for the Euler equations and the Navier-Stokes equations with hyper-dissipation in three dimensions. For the dyadic Euler equations we prove finite time blow-up. In the context of the dyadic Navier-Stokes equations with hyper-dissipation we prove finite time blow-up in the case when the dissipation degree is sufficiently small.

     

A regularizing algorithm for the solution of the inverse problem for an inhomogeneous string

We consider uniqueness of the solution of the inverse problem of determining the coefficient of the one-dimensional wave equation on the real halfline. Necessary conditions of existence of a unique solution of this inverse problem are obtained. A Tikhonov regularizing algorithm is constructed for approximate solution of the inverse problem. The algorithm has an efficient numerical implementation.Translated from Vychislitel'naya Matematika i Matematicheskoe Obespechenie EVM, pp. 55–66, 1985.