Solution In Continuous Space Research Articles

A Study on Solutions for a Class of Higher-Order System of Singular Boundary Value Problem

In this article, we propose a fourth-order non-self-adjoint system of singular boundary value problems (SBVPs), which arise in the theory of epitaxial growth by considering hte equation 1rβrβ1rβ(rβΘ′)′′′=12rβK11μ′Θ′2+2μΘ′Θ″+K12μ′φ′2+2μφ′φ″+λ1G1(r),1rβrβ1rβ(rβφ′)′′′=12rβK21μ′Θ′2+2μΘ′Θ″+K22μ′φ′2+2μφ′φ″+λ2G2(r), where λ1≥0 and λ2≥0 are two parameters, μ=pr2β−2,p∈R+, G1,G2∈L1[0,1] such that M1*≥G1(r)≥M1>0,M2*≥G2(r)≥M2>0 and K12>0, K11≥0, and K21>0, K22≥0 are constants that are connected by the relation (K12+K22)≥(K11+K21) and β>1. To study the governing equation, we consider three different types of homogeneous boundary conditions. We use the transformation t=r1+β1+β to deduce the second-order singular boundary value problem. Also, for β=p=G1(r)=G2(r)=1, it admits dual solutions. We show the existence of at least one solution in continuous space. We derive a sign of solutions. Furthermore, we compute the approximate bound of the parameters to point out the region of nonexistence. We also conclude bounds are symmetric with respect to two different transformations.

A Hybrid Particle-Swarm Tabu Search Algorithm for Solving Job Shop Scheduling Problems

This paper proposes a method for the job shop scheduling problem (JSSP) based on the hybrid metaheuristic method. This method makes use of the merits of an improved particle swarm optimization (PSO) and a tabu search (TS) algorithm. In this work, based on scanning a valuable region thoroughly, a balance strategy is introduced into the PSO for enhancing its exploration ability. Then, the improved PSO could provide diverse and elite initial solutions to the TS for making a better search in the global space. We also present a new local search strategy for obtaining better results in JSSP. A real-integer encode and decode scheme for associating a solution in continuous space to a discrete schedule solution is designed for the improved PSO and the tabu algorithm to directly apply their solutions for intensifying the search of better solutions. Experimental comparisons with several traditional metaheuristic methods demonstrate the effectiveness of the proposed PSO–TS algorithm.