MATHEMATICAL SCIENCES
Year:2013 | Volume:7 | Issue:-|Papers Count:6

Purpose: In this paper, we consider the maximum traveling salesman problem, a variation of the usual traveling salesman problem, in which the objective is to maximize the cost of a tour of the salesman. The main purpose of this paper is to develop a hybrid genetic algorithm (GA) for obtaining a heuristically optimal solution to the problem.Methods: First, a simple GA and then a hybrid GA have been proposed to solve the problem. As crossover operator plays a vital role in GAs, we modify the sequential constructive crossover operator for our simple GA to solve the problem. To improve the quality of the solution obtained by the crossover operator, restricted 2-opt search is applied. Then a hybrid GA is developed by incorporating a new local search algorithm to the simple GA in order to obtain a heuristic solution to the problem.Results: We compare the efficiency of our hybrid GA against an existing heuristic algorithm for symmetric traveling salesman problem library (TSPLIB) instances. Finally, we present solutions to the problem for asymmetric TSPLIB instances. Since, to the best of our knowledge, no literature presents solution for asymmetric instances, hence, we could not carry out any comparative study to show the efficiency of our hybrid GA for the asymmetric instances.Conclusions: The comparative study shows the effectiveness of our hybrid GA.

Eigenvalue problem for p-Laplacian with mixed boundary conditions is concerned on a bounded domain. The existence of nonnegative eigenvalues are obtained by using the Lusternik-Schnirelman principle. Boundedness of eigenfunctions is obtained by using the Moser iteration. The simplicity and isolation of the first eigenvalue are proved. The existence of the second eigenvalue is also illustrated.

This paper is concerned with a quasilinear elliptic system, which involves the Caffarelli-Kohn-Nirenberg inequality and multiple critical exponents. The existence and multiplicity results of positive solutions are obtained by variational methods.

For a group G, and a subset S of G such that 1G∉S, let X=Cay (G, S) be the corresponding Cayley graph. Then X is said to be normal edge transitive if NAut (X) (G) is transitive on edges. In this paper, we determine all connected directed Cayley graphs of finite abelian groups with valency at most 3 which are normal edge transitive but not normal.

The purpose of this paper is to characterize the conditions for the convergence of the implicit Mann iterative scheme with error term to the unique fixed point of f-hemicontractive mappings in a nonempty convex subset of an arbitrary Banach space.

In this paper, a numerical method is proposed to approximate the numeric solutions of nonlinear Fisher's reaction–diffusion equation with modified cubic B-spline collocation method. The method is based on collocation of modified cubic B-splines over finite elements, so we have continuity of the dependent variable and its first two derivatives throughout the solution range. We apply modified cubic B-splines for spatial variable and derivatives, which produce a system of first-order ordinary differential equations. We solve this system by using SSP-RK54 scheme. The proposed method needs less storage space that causes less accumulation of numerical errors. The numerical approximate solution to the nonlinear Fisher's reaction–diffusion equation has been computed without using any transformation and linearization process. Illustrative three test examples are included to establish the effectiveness and pertinence of the technique. Easy and economical implementation is the strength of this method.