Shooting swarm algorithm for solving two-point boundary value problems

Boundary value problems (BVPs) are solved using the more detailed swarm algorithm (SA) based on particle swarm optimization (PSO) and firefly algorithm (FA). In the field of optimization techniques, both PSO and FA have good features to solve many problems in applied mathematics. Due to the sensitivity of the use of the controversial shooting method for solving BVPs, which can not able to reach the exact solution oftentimes. A shooting Swarm algorithm (SSA) is proposed based on PSO and FA. Several BVPs including stiff BVPs were principally used to investigate the SSA. The numerical experiments and analyses revealed that the algorithm was able to overcome the shooing method drawbacks. On another hand, the proposed method that is based on FA significantly reduces the number of iterations required for solving BVPs, because of its flexible properties in the exploration and exploitation phases, and it is in good agreement with the exact solution of BVPs. The SSA was investigated to solve stiff BVPs and Its efficacy has been proven with the accurate solutions.