Iranian Journal of Numerical Analysis and Optimization
Volume:6 Issue: 1, Winter and Spring 2016

ýConsidering a damped wave system defined on a two-dimensional domainý, ýwith a dissipative term localized in an unknown subset with an unknown damping parameterý, ýwe address the shape design ill-posed problem which consists of optimizing the shape of the unknown subset in order to minimize the energy of the system at a given timeý. ýBy using a new approach based on the embedding processý, ýfirstý, ýthe system is formulated in variational form; thený, ýby transferring the problem into polar coordinates and defining two positive Radon measuresý, ýwe represent the problem in a space of measuresý. ýIn this wayý, ýthe shape design problem is changed into an infinite linear one whose solution is guaranteedý. ýIn this stageý, ýby applying two subsequent approximation stepsý, ýthe optimal solution (optimal controlý, ýoptimal regioný, ýoptimal damping parameter and optimal energy) is identified by a three-phase optimization search techniqueý. ýNumerical simulations are also given in order to compare this new method with another oneý.

In this paper, we propose an efficient implementation of the Chebyshev Galerkin method for rst order Volterra and Fredholm integro-differential equations of the second kind. Some numerical examples are presented to show the accuracy of the method.

The Kudryashov method to look for the exact solutions of the nonlinear differential equations is presented. The Kudryashov method is applied to search for the exact solutions of the Liouville equation and the Sinh-Poisson equation. The equations of magnetohydrostatic equilibria for a plasma in a gravitational field are investigated analytically. An investigation of a family of isothermal magnetostatic atmospheres with one ignorable coordinate cor-responding to a uniform gravitational field in a plane geometry is carried out.
The distributed current in the model J is directed along the x-axis where x is the horizontal ignorable coordinate. These equations transform to a single nonlinear elliptic equation for the magnetic vector potential u. This equation depends on an arbitrary function of u that must be specified.

ýýIn this paperý, ýwe introduce fractional-order for a model of tritrophic food chain Lotka-Volterraý. ýMoreoverý, ýwe discuss the stability analysis of fractional systemý. ýThe nonstandard finite difference (NSFD) scheme is implementedý ýto study the dynamic behaviors in the fractional-order Lotka-Volterra systemý. ýNumerical results show that theý ýNSFD approach is easy to implement and accurate when applied to fractional -order Lotka-Volterra systemý.

ýThe wide variety of available interactive methods brings the need for creating generalý ýinteractive algorithms enabling the decision maker (DM) to apply freely several convenient methods which best fit his/her preferencesý. ýTo this endý, ýin this paperý, ýwe propose a general scalarizing problem for multiobjective programming problemsý.
ýThe relation between optimal solutions of the introduced scalarizing problem and (weakly) efficient as well as properly efficient solutions of the main multiobjective optimization problem (MOP) is discussedý. ýIt is shown that some of the scalarizing problems used in different interactive methods can be obtained from proposed formulation by selecting suitable transformationsý. ýBased on the suggested scalarizing problemý, ýwe propose a general interactive algorithm (GIA) that enables the DM to specify his/her preferences in six different ways with capability to change his/her preferences any time during the iterations of the algorithmý.
ýFinallyý, ýa numerical example demonstrating the applicability of the algorithm is providedý.

This paper presents a trust-region procedure for solving systems of nonlinear equations. The proposed approach takes advantages of an effective adaptive trust-region radius and a nonmonotone strategy by combining both of them appropriately. It is believed that selecting an appropriate adaptive radius based on a suitable nonmonotone strategy can improve the efficiency and robustness of the trust-region framework as well as can decrease the computational cost of the algorithm by decreasing the number of subproblems that must be solved. The global convergence to first order stationary points as well as the local q-quadratic convergence of the proposed approach are proved. Numerical experiments show that the new algorithm is promising and attractive for solving nonlinear systems.

ýIn this paperý, ýwe use the definition of fuzzy normed spaces given by Bag and Samanta and provide four types of fuzzy versions of contractioný. ýWe show that these mappings necessarily have unique fixed points in fuzzy normed linear spacesý. ýWe will show that the presented theorems are indeed fuzzy extensions of their classical counterpartsý.