Symmetry Analysis, Exact Solutions and Conservation Laws of a Benjamin–Bona–Mahony–Burgers Equation in 2+1-Dimensions

The Benjamin–Bona–Mahony equation describes the unidirectional propagation of small-amplitude long waves on the surface of water in a channel. In this paper, we consider a family of generalized Benjamin–Bona–Mahony–Burgers equations depending on three arbitrary constants and an arbitrary function G(u). We study this family from the standpoint of the theory of symmetry reductions of partial differential equations. Firstly, we obtain the Lie point symmetries admitted by the considered family. Moreover, taking into account the admitted point symmetries, we perform symmetry reductions. In particular, for G′(u)≠0, we construct an optimal system of one-dimensional subalgebras for each maximal Lie algebra and deduce the corresponding (1+1)-dimensional nonlinear third-order partial differential equations. Then, we apply Kudryashov’s method to look for exact solutions of the nonlinear differential equation. We also determine line soliton solutions of the family of equations in a particular case. Lastly, through the multipliers method, we have constructed low-order conservation laws admitted by the family of equations.

A note on the exact boundary controllability for an imperfect transmission problem

In this note, we consider a hyperbolic system of equations in a domain made up of two components. We prescribe a homogeneous Dirichlet condition on the exterior boundary and a jump of the displacement proportional to the conormal derivatives on the interface. This last condition is the mathematical interpretation of an imperfect interface. We apply a control on the external boundary and, by means of the Hilbert Uniqueness Method, introduced by J. L. Lions, we study the related boundary exact controllability problem. The key point is to derive an observability inequality by using the so called Lagrange multipliers method, and then to construct the exact control through the solution of an adjoint problem. Eventually, we prove a lower bound for the control time which depends on the geometry of the domain, on the coefficients matrix and on the proportionality between the jump of the solution and the conormal derivatives on the interface.

Optimal Solution of the Fractional Order Breast Cancer Competition Model

This paper discusses the fractional order breast cancer competition model (F-BCCM), which considers population dynamics among cancer stem, tumor and healthy cells, as well as the effects of excess estrogen and the body’s natural immune response on the cell populations. Generalized shifted Legendre polynomials and their operational matrices are presented in the scope of a general procedure for the solution of the F-BCCM. The application of the Lagrange multipliers method transforms the F-BCCM into a system of algebraic equations. Additionally, the convergence analysis of the method and two illustrative numerical examples complement the study.Mathematics Subject Classification: 97M60; 41A58; 92C42.

A discrete formulation of Kirchhoff rods in large-motion dynamics

A nonlinear model for the dynamics of a Kirchhoff rod in the three-dimensional space is developed in the framework of a discrete elastic theory. The formulation avoids the use of Euler angles for the orientation of the rod cross-sections to provide a computationally singularity-free parameterization of rotations along the motion trajectories. The material directions related to the principal axes of the cross-sections are specified using auxiliary points that must satisfy constraints enforced by the Lagrange multipliers method. A generalization of this approach is presented to take into account Poisson’s effect in an orthotropic rod. Numerical simulations are performed to test the presented formulation.

Dynamic Answer of a Human Ankle Joint Implant

This paper addresses to a research of a dynamic answer obtained through numerical simulations of a human ankle joint implant with finite element method. The research background consists of an inverse dynamic analysis based on Newton-Euler formalism completed with Lagrange’s multipliers method. Thus, a parameterized virtual model of a human ankle joint was elaborated and simulated together with the implant, in dynamic conditions similar with real ones in reality. A results numerical processing was obtained with the aid of MSC Nastran and important results were obtained for orthopedic implants design.