In this paper, we consider a class of multiobjective E-convex programming problems with inequality constraints, where the objective and constraint functions are E-convex functions which were firstly introduced by Youness (J. Optim. Theory Appl. 102:439-450, 1999). Fritz-John and Kuhn-Tucker necessary and sufficient optimality theorems for the multiobjective E-convex programming are established under the weakened assumption of the theorems in Megahed et al. (J. Inequal. Appl. 2013:246, 2013 and Youness (Chaos Solitons Fractals 12:1737-1745. A mixed duality for the primal problem is formulated and weak and strong duality theorems between primal and dual problems are explored. Illustrative examples are given to explain the obtained results. MSC: 90C29; 90C30; 69K05

Abstract. A quadratic B-spline finite element method for the spatial variable combined with a Newton method for the time variable is proposed to approximate a solution of Benjamin-Bona-Mahony-Burgers (BBMB) equation. Two examples were considered to show the efficiency of the proposed scheme. The numerical solutions obtained for various viscosity were compared with the exact solutions. The numerical results show that the scheme is efficient and feasible.

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ABSTRACT. A numerical scheme is proposed to control the BBMB (Benjamin-Bona-MahonyBurgers) equation, and the scheme consists of three steps. Firstly, BBMB equation is converted to a finite set of nonlinear ordinary differential equations by the quadratic B-spline finite element method in spatial. Secondly, the controller is designed based on the linear quadratic regulator (LQR) theory; Finally, the system of the closed loop compensator obtained on the basis of the previous two steps is solved by the backward Euler method. The controlled numerical solutions are obtained for various values of parameters and different initial conditions. Numerical simulations show that the scheme is efficient and feasible.

A reduced-order model for distributed feedback control of the Benjamin-Bona-Mahony-Burgers (BBMB) equation is discussed. To retain more information in our model, we first calculate the functional gain in the full-order case, and then invoke the proper orthogonal decomposition (POD) method to design a low-order controller and thereby reduce the order of the model. Numerical experiments demonstrate that a solution of the reduced-order model performs well in comparison with a solution for the full-order description.